Subgroups — C₆₇×#34423 #30713674

Order 17152.

2390 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2376	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2370	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2366	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2359	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2354	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2349	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2325	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2324	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2324	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
14	#540×C₆₇	C₄	1	16	yes	complemented
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#123×C₆₇	C₂²	1	4	yes	—
18	C₂×#121×C₆₇	C₂²	1	2	yes	—
19	C₂×#121×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₂²	1	4	yes	—
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#390×C₆₇	C₂²	1	2	yes	—
30	#390×C₆₇	C₂²	1	4	yes	—
31	#390×C₆₇	C₂²	1	2	yes	—
32	#390×C₆₇	C₂²	1	4	yes	—
33	#389×C₆₇	C₂²	1	2	yes	—
34	#389×C₆₇	C₂²	1	2	yes	—
35	#389×C₆₇	C₂²	1	2	yes	—
36	#389×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	C₂²	1	4	yes	—
38	#388×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	4	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#388×C₆₇	C₂²	1	4	yes	—
43	#387×C₆₇	C₂²	1	4	yes	—
44	#387×C₆₇	C₂²	1	4	yes	—
45	#387×C₆₇	C₂²	1	4	yes	—
46	#386×C₆₇	C₂²	1	4	yes	—
47	#386×C₆₇	C₂²	1	4	yes	—
48	#386×C₆₇	C₂²	1	4	yes	—
49	#386×C₆₇	C₂²	1	4	yes	—
50	#385×C₆₇	C₂²	1	2	yes	—
51	#385×C₆₇	C₂²	1	2	yes	—
52	#385×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	4	yes	—
55	#384×C₆₇	C₂²	1	1	yes	characteristic
56	#382×C₆₇	C₂²	1	1	yes	characteristic
57	#380×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	4	yes	—
59	#380×C₆₇	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
67	C₂×#95×C₆₇	C₂²	1	4	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
69	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
70	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
76	#127×C₆₇	C₂×C₄	1	8	yes	complemented
77	#125×C₆₇	C₂×C₄	1	8	yes	complemented
78	#123×C₆₇	C₂×C₄	1	32	yes	complemented
79	#121×C₆₇	C₂×C₄	1	32	yes	complemented
80	#121×C₆₇	C₂×C₄	1	16	yes	complemented
81	C₄×D₄×C₆₇	—	4	4	no	—
82	C₄×D₄×C₆₇	—	4	4	no	—
83	C₄×D₄×C₆₇	—	4	8	no	—
84	C₄×D₄×C₆₇	—	4	8	no	—
85	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
86	#117×C₆₇	—	4	2	no	—
87	#117×C₆₇	—	4	8	no	—
88	#117×C₆₇	C₂×C₄	1	8	yes	complemented
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄²×C₆₇	—	4	2	no	—
115	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
121	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
124	#95×C₆₇	C₂³	1	2	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	4	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	2	yes	—
130	#95×C₆₇	C₂³	1	2	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	4	yes	—
133	#95×C₆₇	C₂³	1	2	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
136	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	4	4	no	—
140	C₂×D₄×C₆₇	—	4	4	no	—
141	C₂×D₄×C₆₇	—	4	4	no	—
142	C₂×D₄×C₆₇	—	4	4	no	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
144	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
145	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	4	2	no	—
151	C₂²×C₄×C₆₇	—	4	4	no	—
152	C₂²×C₄×C₆₇	—	4	4	no	—
153	C₂²×C₄×C₆₇	—	4	2	no	—
154	C₂²×C₄×C₆₇	—	4	4	no	—
155	C₂²×C₄×C₆₇	—	4	4	no	—
156	C₂²×C₄×C₆₇	—	4	4	no	—
157	C₂²×C₄×C₆₇	—	4	4	no	—
158	C₂²×C₄×C₆₇	—	4	4	no	—
159	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂²×C₄	1	16	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	4	no	—
175	C₄⋊C₄×C₆₇	—	2	4	no	—
176	C₄⋊C₄×C₆₇	—	4	2	no	—
177	C₄⋊C₄×C₆₇	—	4	4	no	—
178	C₄⋊C₄×C₆₇	—	4	2	no	—
179	C₄⋊C₄×C₆₇	—	4	2	no	—
180	C₄⋊C₄×C₆₇	—	4	4	no	—
181	C₄⋊C₄×C₆₇	—	4	4	no	—
182	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
183	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
184	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
185	C₂²⋊C₄×C₆₇	—	2	8	no	—
186	C₂²⋊C₄×C₆₇	—	4	1	no	—
187	C₂²⋊C₄×C₆₇	—	4	4	no	—
188	C₂²⋊C₄×C₆₇	—	4	1	no	—
189	C₂²⋊C₄×C₆₇	—	4	4	no	—
190	C₂²⋊C₄×C₆₇	—	4	4	no	—
191	C₂²⋊C₄×C₆₇	—	4	4	no	—
192	C₂²⋊C₄×C₆₇	—	4	4	no	—
193	C₂²⋊C₄×C₆₇	—	4	4	no	—
194	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
195	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
196	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
197	C₄²×C₆₇	—	2	4	no	—
198	C₄²×C₆₇	—	2	8	no	—
199	C₄²×C₆₇	—	2	4	no	—
200	C₄²×C₆₇	—	4	1	no	—
201	C₄²×C₆₇	—	4	1	no	—
202	C₄²×C₆₇	—	4	4	no	—
203	C₄²×C₆₇	—	4	4	no	—
204	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
205	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
206	C₂³×C₆₇	—	2	4	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	4	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	1	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
217	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
218	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
219	D₄×C₆₇	—	2	16	no	—
220	D₄×C₆₇	—	4	16	no	—
221	D₄×C₆₇	—	4	16	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	8	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	4	16	no	—
249	C₂×C₄×C₆₇	—	4	8	no	—
250	C₂×C₄×C₆₇	—	8	8	no	—
251	C₂×C₄×C₆₇	—	8	8	no	—
252	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
253	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
254	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
256	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
258	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
259	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	8	no	—
262	C₂²×C₆₇	—	2	8	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	8	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	4	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	4	no	—
272	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
275	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
276	C₂²×C₆₇	#585	1	2	yes	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	4	no	—
279	C₄×C₆₇	—	2	8	no	—
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	4	2	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	—	4	2	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	#520	1	4	yes	—
288	#34423	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
289	C₂×C₆₇	—	4	2	no	—
290	C₂×C₆₇	—	4	8	no	—
291	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
292	C₂×C₆₇	C₄×H(5,𝔽₂)	1	2	yes	—
293	C₂×C₆₇	#3542	1	2	yes	—
294	C₂×C₆₇	#3548	1	1	yes	characteristic
295	C₂×C₆₇	#3557	1	1	yes	characteristic
296	C₂×#540	C₂×C₆₇	1	1	yes	characteristic
297	#2376	C₂×C₆₇	1	4	yes	complemented
298	#2370	C₂×C₆₇	1	2	yes	complemented
299	#2366	C₂×C₆₇	1	2	yes	complemented
300	#2359	C₂×C₆₇	1	4	yes	complemented
301	#2354	C₂×C₆₇	1	2	yes	complemented
302	#2349	C₂×C₆₇	1	2	yes	complemented
303	#2325	C₂×C₆₇	1	4	yes	complemented
304	#2324	C₂×C₆₇	1	2	yes	complemented
305	#2324	C₂×C₆₇	1	4	yes	complemented
306	#2314	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₄×#121	C₂×C₆₇	1	2	yes	complemented
308	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₆₇	#34423	1	1	yes	characteristic, direct product factor, Sylow
310	#540	C₄×C₆₇	1	16	yes	complemented
311	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×#123	C₂²×C₆₇	1	4	yes	—
314	C₂×#121	C₂²×C₆₇	1	4	yes	—
315	C₂×#121	C₂²×C₆₇	1	2	yes	—
316	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
317	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
320	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
321	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#117	C₂²×C₆₇	1	4	yes	—
323	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
324	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
325	#390	C₂²×C₆₇	1	4	yes	—
326	#390	C₂²×C₆₇	1	2	yes	—
327	#390	C₂²×C₆₇	1	4	yes	—
328	#390	C₂²×C₆₇	1	2	yes	—
329	#389	C₂²×C₆₇	1	2	yes	—
330	#389	C₂²×C₆₇	1	4	yes	—
331	#389	C₂²×C₆₇	1	2	yes	—
332	#389	C₂²×C₆₇	1	2	yes	—
333	#389	C₂²×C₆₇	1	2	yes	—
334	#388	C₂²×C₆₇	1	4	yes	—
335	#388	C₂²×C₆₇	1	4	yes	—
336	#388	C₂²×C₆₇	1	4	yes	—
337	#388	C₂²×C₆₇	1	4	yes	—
338	#388	C₂²×C₆₇	1	4	yes	—
339	#387	C₂²×C₆₇	1	4	yes	—
340	#387	C₂²×C₆₇	1	4	yes	—
341	#387	C₂²×C₆₇	1	4	yes	—
342	#386	C₂²×C₆₇	1	4	yes	—
343	#386	C₂²×C₆₇	1	4	yes	—
344	#386	C₂²×C₆₇	1	4	yes	—
345	#386	C₂²×C₆₇	1	4	yes	—
346	#385	C₂²×C₆₇	1	2	yes	—
347	#385	C₂²×C₆₇	1	4	yes	—
348	#385	C₂²×C₆₇	1	4	yes	—
349	#385	C₂²×C₆₇	1	2	yes	—
350	#385	C₂²×C₆₇	1	4	yes	—
351	#384	C₂²×C₆₇	1	1	yes	characteristic
352	#382	C₂²×C₆₇	1	1	yes	characteristic
353	#380	C₂²×C₆₇	1	2	yes	—
354	#380	C₂²×C₆₇	1	4	yes	—
355	#380	C₂²×C₆₇	1	4	yes	—
356	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
357	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
359	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
360	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
361	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
363	C₂×#95	C₂²×C₆₇	1	4	yes	—
364	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
365	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
366	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
367	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
369	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
370	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
371	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
372	#127	C₂×C₄×C₆₇	1	8	yes	complemented
373	#125	C₂×C₄×C₆₇	1	8	yes	complemented
374	#123	C₂×C₄×C₆₇	1	32	yes	complemented
375	#121	C₂×C₄×C₆₇	1	32	yes	complemented
376	#121	C₂×C₄×C₆₇	1	16	yes	complemented
377	C₄×D₄	—	4	4	no	—
378	C₄×D₄	—	4	4	no	—
379	C₄×D₄	—	4	8	no	—
380	C₄×D₄	—	4	8	no	—
381	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
382	#117	—	4	2	no	—
383	#117	—	4	8	no	—
384	#117	C₂×C₄×C₆₇	1	8	yes	complemented
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
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	4	yes	—
410	C₂×C₄²	—	4	2	no	—
411	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
420	#95	C₂³×C₆₇	1	2	yes	—
421	#95	C₂³×C₆₇	1	4	yes	—
422	#95	C₂³×C₆₇	1	4	yes	—
423	#95	C₂³×C₆₇	1	4	yes	—
424	#95	C₂³×C₆₇	1	4	yes	—
425	#95	C₂³×C₆₇	1	2	yes	—
426	#95	C₂³×C₆₇	1	2	yes	—
427	#95	C₂³×C₆₇	1	4	yes	—
428	#95	C₂³×C₆₇	1	4	yes	—
429	#95	C₂³×C₆₇	1	2	yes	—
430	#95	C₂³×C₆₇	1	4	yes	—
431	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
432	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	—	4	4	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
440	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
441	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	8	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	4	no	—
451	C₂²×C₄	—	4	4	no	—
452	C₂²×C₄	—	4	4	no	—
453	C₂²×C₄	—	4	4	no	—
454	C₂²×C₄	—	4	4	no	—
455	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
456	C₂²×C₄	C₂²×C₄×C₆₇	1	16	yes	—
457	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
466	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
467	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	4	no	—
474	C₄⋊C₄	—	4	4	no	—
475	C₄⋊C₄	—	4	2	no	—
476	C₄⋊C₄	—	4	4	no	—
477	C₄⋊C₄	—	4	2	no	—
478	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
479	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
480	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	4	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	4	no	—
486	C₂²⋊C₄	—	4	4	no	—
487	C₂²⋊C₄	—	4	4	no	—
488	C₂²⋊C₄	—	4	4	no	—
489	C₂²⋊C₄	—	4	4	no	—
490	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
491	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
492	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
493	C₄²	—	2	4	no	—
494	C₄²	—	2	8	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	4	no	—
499	C₄²	—	4	4	no	—
500	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
501	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
502	C₂³	—	2	4	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	4	no	—
512	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
513	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
514	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
515	D₄	—	2	16	no	—
516	D₄	—	4	16	no	—
517	D₄	—	4	16	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	4	16	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	8	8	no	—
547	C₂×C₄	—	8	8	no	—
548	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
549	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
550	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
551	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
552	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
553	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
554	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
555	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
556	C₂²	—	2	2	no	—
557	C₂²	—	2	8	no	—
558	C₂²	—	2	8	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	8	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	—	4	4	no	—
568	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
569	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
570	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
571	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
572	C₂²	#585×C₆₇	1	2	yes	—
573	C₄	—	2	4	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	4	no	—
576	C₄	—	2	8	no	—
577	C₄	—	2	4	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	#520×C₆₇	1	4	yes	—
584	C₂	—	4	2	no	—
585	C₂	—	4	8	no	—
586	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
587	C₂	C₄×H(5,𝔽₂)×C₆₇	1	2	yes	—
588	C₂	C₆₇×#3542	1	2	yes	—
589	C₂	C₆₇×#3548	1	1	yes	characteristic
590	C₂	C₆₇×#3557	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#540×C₆₇	C₂
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×#2376	C₂	semidirect
3. C₆₇×#2370	C₂	semidirect
4. C₆₇×#2366	C₂	semidirect
5. C₆₇×#2359	C₂	semidirect
6. C₆₇×#2354	C₂	semidirect
7. C₆₇×#2349	C₂	semidirect
8. C₆₇×#2325	C₂	semidirect
9. C₆₇×#2324	C₂	semidirect
10. C₆₇×#2324	C₂	semidirect
11. C₆₇×#2314	C₂	semidirect
12. C₄×#121×C₆₇	C₂	semidirect
13. C₆₇×#2296	C₂	semidirect
14. #540×C₆₇	C₄	semidirect
76. #127×C₆₇	C₂×C₄	semidirect
77. #125×C₆₇	C₂×C₄	semidirect
78. #123×C₆₇	C₂×C₄	semidirect
79. #121×C₆₇	C₂×C₄	semidirect
80. #121×C₆₇	C₂×C₄	semidirect
85. C₄×D₄×C₆₇	C₂×C₄	semidirect
88. #117×C₆₇	C₂×C₄	semidirect
288. #34423	C₆₇	direct (N × H)
297. #2376	C₂×C₆₇	semidirect
298. #2370	C₂×C₆₇	semidirect
299. #2366	C₂×C₆₇	semidirect
300. #2359	C₂×C₆₇	semidirect
301. #2354	C₂×C₆₇	semidirect
302. #2349	C₂×C₆₇	semidirect
303. #2325	C₂×C₆₇	semidirect
304. #2324	C₂×C₆₇	semidirect
305. #2324	C₂×C₆₇	semidirect
306. #2314	C₂×C₆₇	semidirect
307. C₄×#121	C₂×C₆₇	semidirect
308. #2296	C₂×C₆₇	semidirect
309. C₆₇	#34423	direct (N × H)
310. #540	C₄×C₆₇	semidirect
372. #127	C₂×C₄×C₆₇	semidirect
373. #125	C₂×C₄×C₆₇	semidirect
374. #123	C₂×C₄×C₆₇	semidirect
375. #121	C₂×C₄×C₆₇	semidirect
376. #121	C₂×C₄×C₆₇	semidirect
381. C₄×D₄	C₂×C₄×C₆₇	semidirect
384. #117	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#540×C₆₇	2
C₆₇×#2376	2
C₆₇×#2370	2
C₆₇×#2366	2
C₆₇×#2359	2
C₆₇×#2354	2
C₆₇×#2349	2
C₆₇×#2325	2
C₆₇×#2324	2
C₆₇×#2314	2
C₄×#121×C₆₇	2
C₆₇×#2296	2
#34423	67

Subgroups — C67×#34423 #30713674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34423 #30713674