Subgroups — C₆₇×#34194 #30713445

Order 17152.

3014 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2352	C₂	1	4	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2345	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2342	C₂	1	4	yes	maximal, complemented
7	D₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
8	C₆₇×#2319	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2318	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2280	C₂	1	2	yes	maximal, complemented
13	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2275	C₂	1	2	yes	maximal, complemented
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#390×C₆₇	C₂²	1	4	yes	—
33	#389×C₆₇	C₂²	1	4	yes	complemented
34	#388×C₆₇	C₂²	1	8	yes	complemented
35	#388×C₆₇	C₂²	1	8	yes	complemented
36	#387×C₆₇	C₂²	1	8	yes	complemented
37	#386×C₆₇	C₂²	1	8	yes	—
38	#385×C₆₇	C₂²	1	8	yes	complemented
39	#384×C₆₇	C₂²	1	2	yes	complemented
40	#383×C₆₇	C₂²	1	2	yes	complemented
41	#382×C₆₇	C₂²	1	2	yes	complemented
42	#382×C₆₇	C₂²	1	2	yes	complemented
43	#381×C₆₇	C₂²	1	2	yes	—
44	#381×C₆₇	C₂²	1	4	yes	complemented
45	#380×C₆₇	C₂²	1	1	yes	characteristic
46	#380×C₆₇	C₂²	1	2	yes	—
47	#380×C₆₇	C₂²	1	4	yes	—
48	#380×C₆₇	C₂²	1	1	yes	characteristic
49	#380×C₆₇	C₂²	1	2	yes	—
50	#380×C₆₇	C₂²	1	4	yes	complemented
51	#380×C₆₇	C₂²	1	4	yes	complemented
52	#380×C₆₇	C₂²	1	4	yes	complemented
53	#379×C₆₇	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
55	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
57	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
58	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
60	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
61	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
62	#376×C₆₇	C₂²	1	2	yes	complemented
63	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
64	C₂⁵×C₆₇	C₂³	1	2	yes	—
65	C₂²×D₄×C₆₇	—	2	8	no	—
66	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	—
67	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	—	4	8	no	—
71	C₂³×C₄×C₆₇	C₂×C₄	1	16	yes	complemented
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₄×D₄×C₆₇	—	2	16	no	—
82	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
83	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
84	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
85	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
86	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
87	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
88	#117×C₆₇	—	4	4	no	—
89	#117×C₆₇	—	4	8	no	—
90	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
91	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄²×C₆₇	C₂×C₄	1	8	yes	complemented
114	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	#95×C₆₇	C₂³	1	2	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	8	yes	—
124	#95×C₆₇	C₂³	1	2	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	8	yes	—
127	#95×C₆₇	C₂³	1	4	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	C₂⁴×C₆₇	—	2	8	no	—
131	C₂⁴×C₆₇	—	2	4	no	—
132	C₂⁴×C₆₇	—	2	4	no	—
133	C₂⁴×C₆₇	—	4	4	no	—
134	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
135	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
136	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
137	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	2	32	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₆₇	#41	1	8	yes	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	16	no	—
152	C₂²×C₄×C₆₇	—	4	32	no	—
153	C₂²×C₄×C₆₇	—	4	4	no	—
154	C₂²×C₄×C₆₇	—	4	4	no	—
155	C₂²×C₄×C₆₇	—	4	4	no	—
156	C₂²×C₄×C₆₇	—	4	8	no	—
157	C₂²×C₄×C₆₇	—	4	8	no	—
158	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
162	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
163	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
164	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
165	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₆₇	—	2	8	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	8	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	4	8	no	—
182	C₄⋊C₄×C₆₇	—	4	4	no	—
183	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
184	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
185	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
186	C₂²⋊C₄×C₆₇	—	2	16	no	—
187	C₂²⋊C₄×C₆₇	—	2	4	no	—
188	C₂²⋊C₄×C₆₇	—	2	8	no	—
189	C₂²⋊C₄×C₆₇	—	2	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	8	no	—
191	C₂²⋊C₄×C₆₇	—	4	8	no	—
192	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
193	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
194	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
195	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
196	C₄²×C₆₇	—	2	8	no	—
197	C₄²×C₆₇	—	4	4	no	—
198	C₄²×C₆₇	—	4	8	no	—
199	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
200	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
201	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
202	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	4	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	2	2	no	—
207	C₂³×C₆₇	—	2	2	no	—
208	C₂³×C₆₇	—	2	16	no	—
209	C₂³×C₆₇	—	2	16	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	4	8	no	—
215	C₂³×C₆₇	—	4	16	no	—
216	C₂³×C₆₇	—	4	16	no	—
217	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
218	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
219	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
220	C₂³×C₆₇	C₂×#41	1	8	yes	—
221	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
222	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
223	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
224	D₄×C₆₇	—	2	16	no	—
225	D₄×C₆₇	—	2	8	no	—
226	D₄×C₆₇	#117	1	16	yes	complemented
227	C₂×C₄×C₆₇	—	2	8	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	8	no	—
238	C₂×C₄×C₆₇	—	2	16	no	—
239	C₂×C₄×C₆₇	—	2	16	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	8	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	8	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	4	8	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	—	4	8	no	—
252	C₂×C₄×C₆₇	—	8	16	no	—
253	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
258	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
259	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
262	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	2	8	no	—
268	C₂²×C₆₇	—	2	4	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	8	no	—
271	C₂²×C₆₇	—	2	4	no	—
272	C₂²×C₆₇	—	2	4	no	—
273	C₂²×C₆₇	—	2	4	no	—
274	C₂²×C₆₇	—	4	32	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	C₂×#117	1	8	yes	—
278	C₂²×C₆₇	#520	1	4	yes	—
279	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
280	C₂²×C₆₇	C₂²×#41	1	2	yes	—
281	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
282	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	4	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	4	8	no	—
291	C₄×C₆₇	—	4	4	no	—
292	C₄×C₆₇	—	4	4	no	—
293	C₄×C₆₇	C₂×#117	1	4	yes	—
294	C₄×C₆₇	#518	1	4	yes	—
295	#34194	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
296	C₂×C₆₇	—	2	4	no	—
297	C₂×C₆₇	—	2	8	no	—
298	C₂×C₆₇	—	4	4	no	—
299	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
300	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
301	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
302	C₂×C₆₇	#3519	1	2	yes	—
303	C₂×C₆₇	#3520	1	2	yes	—
304	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
305	#2354	C₂×C₆₇	1	2	yes	complemented
306	#2352	C₂×C₆₇	1	4	yes	complemented
307	D₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
308	#2345	C₂×C₆₇	1	2	yes	complemented
309	#2342	C₂×C₆₇	1	4	yes	complemented
310	D₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
311	#2319	C₂×C₆₇	1	2	yes	complemented
312	#2318	C₂×C₆₇	1	4	yes	complemented
313	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#2280	C₂×C₆₇	1	2	yes	complemented
316	C₂×#380	C₂×C₆₇	1	2	yes	complemented
317	#2275	C₂×C₆₇	1	2	yes	complemented
318	C₆₇	#34194	1	1	yes	characteristic, direct product factor, Sylow
319	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
320	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
321	C₂×C₄×D₄	C₄×C₆₇	1	16	yes	complemented
322	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
323	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
326	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
328	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#117	C₂²×C₆₇	1	2	yes	—
330	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
331	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
332	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
333	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
334	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
335	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
336	#390	C₂²×C₆₇	1	4	yes	—
337	#389	C₂²×C₆₇	1	4	yes	complemented
338	#388	C₂²×C₆₇	1	8	yes	complemented
339	#388	C₂²×C₆₇	1	8	yes	complemented
340	#387	C₂²×C₆₇	1	8	yes	complemented
341	#386	C₂²×C₆₇	1	8	yes	—
342	#385	C₂²×C₆₇	1	8	yes	complemented
343	#384	C₂²×C₆₇	1	2	yes	complemented
344	#383	C₂²×C₆₇	1	2	yes	complemented
345	#382	C₂²×C₆₇	1	2	yes	complemented
346	#382	C₂²×C₆₇	1	2	yes	complemented
347	#381	C₂²×C₆₇	1	2	yes	—
348	#381	C₂²×C₆₇	1	4	yes	complemented
349	#380	C₂²×C₆₇	1	2	yes	—
350	#380	C₂²×C₆₇	1	4	yes	—
351	#380	C₂²×C₆₇	1	1	yes	characteristic
352	#380	C₂²×C₆₇	1	2	yes	—
353	#380	C₂²×C₆₇	1	1	yes	characteristic
354	#380	C₂²×C₆₇	1	4	yes	complemented
355	#380	C₂²×C₆₇	1	4	yes	complemented
356	#380	C₂²×C₆₇	1	4	yes	complemented
357	#379	C₂²×C₆₇	1	4	yes	—
358	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
359	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
360	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
361	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
362	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
364	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
365	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
366	#376	C₂²×C₆₇	1	2	yes	complemented
367	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
368	C₂⁵	C₂³×C₆₇	1	2	yes	—
369	C₂²×D₄	—	2	8	no	—
370	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	—
371	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
372	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂³×C₄	—	4	8	no	—
375	C₂³×C₄	C₂×C₄×C₆₇	1	16	yes	complemented
376	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
379	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
380	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
381	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
382	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
385	C₄×D₄	—	2	16	no	—
386	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
387	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
388	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
389	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
390	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
391	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
392	#117	—	4	4	no	—
393	#117	—	4	8	no	—
394	C₂×C₄⋊C₄	—	4	8	no	—
395	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
398	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
400	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
401	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	4	8	no	—
406	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
407	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄²	C₂×C₄×C₆₇	1	8	yes	complemented
418	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
419	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
420	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
421	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
425	#95	C₂³×C₆₇	1	2	yes	—
426	#95	C₂³×C₆₇	1	4	yes	—
427	#95	C₂³×C₆₇	1	8	yes	—
428	#95	C₂³×C₆₇	1	2	yes	—
429	#95	C₂³×C₆₇	1	4	yes	—
430	#95	C₂³×C₆₇	1	8	yes	—
431	#95	C₂³×C₆₇	1	4	yes	—
432	#95	C₂³×C₆₇	1	4	yes	—
433	#95	C₂³×C₆₇	1	4	yes	—
434	C₂⁴	—	2	8	no	—
435	C₂⁴	—	2	4	no	—
436	C₂⁴	—	2	4	no	—
437	C₂⁴	—	4	4	no	—
438	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
439	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
440	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
441	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	32	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
446	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
447	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
448	C₂×D₄	#41×C₆₇	1	8	yes	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	16	no	—
456	C₂²×C₄	—	4	32	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	4	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	4	8	no	—
461	C₂²×C₄	—	4	8	no	—
462	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
463	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
464	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
465	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
466	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
467	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
468	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
469	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
470	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
473	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	C₄⋊C₄	—	2	8	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	8	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	4	8	no	—
486	C₄⋊C₄	—	4	4	no	—
487	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
488	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
489	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
490	C₂²⋊C₄	—	2	16	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	8	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	4	8	no	—
495	C₂²⋊C₄	—	4	8	no	—
496	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
497	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
498	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
499	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
500	C₄²	—	2	8	no	—
501	C₄²	—	4	4	no	—
502	C₄²	—	4	8	no	—
503	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
504	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
505	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
506	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	2	16	no	—
513	C₂³	—	2	16	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	4	8	no	—
519	C₂³	—	4	16	no	—
520	C₂³	—	4	16	no	—
521	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
522	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
523	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
524	C₂³	C₂×#41×C₆₇	1	8	yes	—
525	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
526	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
527	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
528	D₄	—	2	16	no	—
529	D₄	—	2	8	no	—
530	D₄	#117×C₆₇	1	16	yes	complemented
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	8	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	16	no	—
543	C₂×C₄	—	2	16	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	8	16	no	—
557	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
558	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
559	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
560	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
561	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
562	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
563	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
564	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
566	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	4	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	8	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	32	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	C₂×#117×C₆₇	1	8	yes	—
582	C₂²	#520×C₆₇	1	4	yes	—
583	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
584	C₂²	C₂²×#41×C₆₇	1	2	yes	—
585	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
586	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
587	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	4	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	4	no	—
595	C₄	—	4	8	no	—
596	C₄	—	4	4	no	—
597	C₄	C₂×#117×C₆₇	1	4	yes	—
598	C₄	#518×C₆₇	1	4	yes	—
599	C₂	—	2	4	no	—
600	C₂	—	2	8	no	—
601	C₂	—	4	4	no	—
602	C₂	C₂²×#117×C₆₇	1	1	yes	characteristic
603	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
604	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
605	C₂	C₆₇×#3519	1	2	yes	—
606	C₂	C₆₇×#3520	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2354	C₂	semidirect
3. C₆₇×#2352	C₂	semidirect
4. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
5. C₆₇×#2345	C₂	semidirect
6. C₆₇×#2342	C₂	semidirect
8. C₆₇×#2319	C₂	semidirect
9. C₆₇×#2318	C₂	semidirect
10. C₆₇×#2285	C₂	semidirect
11. C₆₇×#2283	C₂	semidirect
12. C₆₇×#2280	C₂	semidirect
13. C₂×#380×C₆₇	C₂	semidirect
14. C₆₇×#2275	C₂	semidirect
17. C₂×C₄×D₄×C₆₇	C₄	semidirect
26. C₂×#117×C₆₇	C₂²	semidirect
28. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
33. #389×C₆₇	C₂²	semidirect
34. #388×C₆₇	C₂²	semidirect
35. #388×C₆₇	C₂²	semidirect
36. #387×C₆₇	C₂²	semidirect
38. #385×C₆₇	C₂²	semidirect
39. #384×C₆₇	C₂²	semidirect
40. #383×C₆₇	C₂²	semidirect
41. #382×C₆₇	C₂²	semidirect
42. #382×C₆₇	C₂²	semidirect
44. #381×C₆₇	C₂²	semidirect
50. #380×C₆₇	C₂²	semidirect
51. #380×C₆₇	C₂²	semidirect
52. #380×C₆₇	C₂²	semidirect
54. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
55. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
56. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
57. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
60. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
61. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
62. #376×C₆₇	C₂²	semidirect
63. C₂×#95×C₆₇	C₂²	semidirect
71. C₂³×C₄×C₆₇	C₂×C₄	semidirect
84. C₄×D₄×C₆₇	C₂×C₄	semidirect
85. C₄×D₄×C₆₇	C₂×C₄	semidirect
86. C₄×D₄×C₆₇	C₂×C₄	semidirect
87. C₄×D₄×C₆₇	C₂×C₄	semidirect
91. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
102. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
113. C₂×C₄²×C₆₇	C₂×C₄	semidirect
226. D₄×C₆₇	#117	semidirect
295. #34194	C₆₇	direct (N × H)
305. #2354	C₂×C₆₇	semidirect
306. #2352	C₂×C₆₇	semidirect
307. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
308. #2345	C₂×C₆₇	semidirect
309. #2342	C₂×C₆₇	semidirect
311. #2319	C₂×C₆₇	semidirect
312. #2318	C₂×C₆₇	semidirect
313. #2285	C₂×C₆₇	semidirect
314. #2283	C₂×C₆₇	semidirect
315. #2280	C₂×C₆₇	semidirect
316. C₂×#380	C₂×C₆₇	semidirect
317. #2275	C₂×C₆₇	semidirect
318. C₆₇	#34194	direct (N × H)
321. C₂×C₄×D₄	C₄×C₆₇	semidirect
330. C₂×#117	C₂²×C₆₇	semidirect
332. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
337. #389	C₂²×C₆₇	semidirect
338. #388	C₂²×C₆₇	semidirect
339. #388	C₂²×C₆₇	semidirect
340. #387	C₂²×C₆₇	semidirect
342. #385	C₂²×C₆₇	semidirect
343. #384	C₂²×C₆₇	semidirect
344. #383	C₂²×C₆₇	semidirect
345. #382	C₂²×C₆₇	semidirect
346. #382	C₂²×C₆₇	semidirect
348. #381	C₂²×C₆₇	semidirect
354. #380	C₂²×C₆₇	semidirect
355. #380	C₂²×C₆₇	semidirect
356. #380	C₂²×C₆₇	semidirect
358. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
359. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
360. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
361. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
364. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
365. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
366. #376	C₂²×C₆₇	semidirect
367. C₂×#95	C₂²×C₆₇	semidirect
375. C₂³×C₄	C₂×C₄×C₆₇	semidirect
388. C₄×D₄	C₂×C₄×C₆₇	semidirect
389. C₄×D₄	C₂×C₄×C₆₇	semidirect
390. C₄×D₄	C₂×C₄×C₆₇	semidirect
391. C₄×D₄	C₂×C₄×C₆₇	semidirect
395. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
406. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
417. C₂×C₄²	C₂×C₄×C₆₇	semidirect
530. D₄	#117×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2354	2
C₆₇×#2352	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2345	2
C₆₇×#2342	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#2319	2
C₆₇×#2318	2
C₆₇×#2285	2
C₆₇×#2283	2
C₆₇×#2280	2
C₂×#380×C₆₇	2
C₆₇×#2275	2
#34194	67

Subgroups — C67×#34194 #30713445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34194 #30713445