Subgroups — C₆₇×#34273 #30713524

Order 17152.

3578 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#534×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2359	C₂	1	8	yes	maximal, complemented
4	C₆₇×#2358	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2325	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2315	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	#534×C₆₇	C₄	1	16	yes	complemented
13	C₂×#123×C₆₇	C₂²	1	4	yes	—
14	C₂×#121×C₆₇	C₂²	1	4	yes	—
15	C₂×#120×C₆₇	C₂²	1	4	yes	—
16	#518×C₆₇	—	2	16	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
19	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
20	C₂×#117×C₆₇	C₂²	1	4	yes	complemented
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
27	#390×C₆₇	C₂²	1	4	yes	complemented
28	#389×C₆₇	C₂²	1	4	yes	complemented
29	#388×C₆₇	C₂²	1	8	yes	complemented
30	#388×C₆₇	C₂²	1	8	yes	complemented
31	#387×C₆₇	C₂²	1	8	yes	complemented
32	#386×C₆₇	C₂²	1	8	yes	—
33	#386×C₆₇	C₂²	1	4	yes	complemented
34	#386×C₆₇	C₂²	1	4	yes	complemented
35	#386×C₆₇	C₂²	1	8	yes	complemented
36	#385×C₆₇	C₂²	1	8	yes	complemented
37	#385×C₆₇	C₂²	1	8	yes	complemented
38	#380×C₆₇	C₂²	1	4	yes	complemented
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
40	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
41	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
42	C₂×#95×C₆₇	C₂²	1	2	yes	complemented
43	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
44	C₂⁵×C₆₇	C₂³	1	2	yes	—
45	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	8	no	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
49	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	—	2	4	no	—
53	C₂³×C₄×C₆₇	—	4	4	no	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	#123×C₆₇	C₂×C₄	1	32	yes	complemented
63	#121×C₆₇	C₂×C₄	1	32	yes	complemented
64	#120×C₆₇	C₂×C₄	1	32	yes	complemented
65	C₄×D₄×C₆₇	—	2	32	no	—
66	C₄×D₄×C₆₇	—	2	32	no	—
67	C₄×D₄×C₆₇	—	4	8	no	—
68	#117×C₆₇	—	2	16	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
76	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
77	C₂×C₂²⋊C₄×C₆₇	D₄	1	8	yes	complemented
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄²×C₆₇	—	4	4	no	—
87	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
89	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
90	#95×C₆₇	C₂³	1	8	yes	—
91	#95×C₆₇	C₂³	1	8	yes	—
92	#95×C₆₇	C₂³	1	8	yes	—
93	#95×C₆₇	C₂³	1	8	yes	—
94	C₂⁴×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	2	4	no	—
96	C₂⁴×C₆₇	—	4	2	no	—
97	C₂⁴×C₆₇	—	4	2	no	—
98	C₂⁴×C₆₇	—	4	4	no	—
99	C₂⁴×C₆₇	—	4	4	no	—
100	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
101	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
103	C₂⁴×C₆₇	#41	1	4	yes	—
104	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
105	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
106	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂×D₄×C₆₇	—	2	64	no	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	2	8	no	—
111	C₂×D₄×C₆₇	—	2	8	no	—
112	C₂×D₄×C₆₇	—	4	4	no	—
113	C₂×D₄×C₆₇	—	4	4	no	—
114	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
115	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
116	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	complemented
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	2	8	no	—
125	C₂²×C₄×C₆₇	—	2	16	no	—
126	C₂²×C₄×C₆₇	—	4	2	no	—
127	C₂²×C₄×C₆₇	—	4	2	no	—
128	C₂²×C₄×C₆₇	—	4	4	no	—
129	C₂²×C₄×C₆₇	—	4	2	no	—
130	C₂²×C₄×C₆₇	—	4	2	no	—
131	C₂²×C₄×C₆₇	—	4	2	no	—
132	C₂²×C₄×C₆₇	—	4	4	no	—
133	C₂²×C₄×C₆₇	—	4	8	no	—
134	C₂²×C₄×C₆₇	—	4	8	no	—
135	C₂²×C₄×C₆₇	—	4	4	no	—
136	C₂²×C₄×C₆₇	—	4	16	no	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
138	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
146	C₄⋊C₄×C₆₇	—	2	16	no	—
147	C₄⋊C₄×C₆₇	—	4	4	no	—
148	C₄⋊C₄×C₆₇	—	4	4	no	—
149	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	complemented
150	C₂²⋊C₄×C₆₇	—	2	8	no	—
151	C₂²⋊C₄×C₆₇	—	2	8	no	—
152	C₂²⋊C₄×C₆₇	—	2	16	no	—
153	C₂²⋊C₄×C₆₇	—	2	16	no	—
154	C₂²⋊C₄×C₆₇	—	4	8	no	—
155	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	complemented
157	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
158	C₄²×C₆₇	—	2	16	no	—
159	C₄²×C₆₇	—	2	8	no	—
160	C₄²×C₆₇	—	4	4	no	—
161	C₂³×C₆₇	—	2	2	no	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	4	no	—
167	C₂³×C₆₇	—	2	8	no	—
168	C₂³×C₆₇	—	2	8	no	—
169	C₂³×C₆₇	—	2	16	no	—
170	C₂³×C₆₇	—	4	4	no	—
171	C₂³×C₆₇	—	4	8	no	—
172	C₂³×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	—	4	4	no	—
174	C₂³×C₆₇	—	4	16	no	—
175	C₂³×C₆₇	—	8	4	no	—
176	C₂³×C₆₇	—	8	4	no	—
177	C₂³×C₆₇	—	8	8	no	—
178	C₂³×C₆₇	C₄×D₄	1	16	yes	complemented
179	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
180	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
181	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
183	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
184	C₂³×C₆₇	C₂×#41	1	4	yes	—
185	C₂³×C₆₇	C₂×#41	1	2	yes	—
186	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
187	D₄×C₆₇	—	2	32	no	—
188	D₄×C₆₇	—	2	32	no	—
189	D₄×C₆₇	—	4	16	no	—
190	C₂×C₄×C₆₇	—	2	8	no	—
191	C₂×C₄×C₆₇	—	2	16	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	4	8	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	16	no	—
208	C₂×C₄×C₆₇	—	4	4	no	—
209	C₂×C₄×C₆₇	—	8	16	no	—
210	C₂×C₄×C₆₇	—	8	8	no	—
211	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
212	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
214	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	8	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	2	8	no	—
223	C₂²×C₆₇	—	2	8	no	—
224	C₂²×C₆₇	—	2	16	no	—
225	C₂²×C₆₇	—	4	16	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	—	4	4	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	8	16	no	—
230	C₂²×C₆₇	C₂×C₄×D₄	1	8	yes	—
231	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
232	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
233	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
234	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
236	C₂²×C₆₇	#585	1	2	yes	—
237	C₄×C₆₇	—	2	8	no	—
238	C₄×C₆₇	—	2	8	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	2	8	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	8	no	—
243	#34273	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	2	8	no	—
245	C₂×C₆₇	—	2	4	no	—
246	C₂×C₆₇	—	4	8	no	—
247	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
248	C₂×C₆₇	C₄×H(5,𝔽₂)	1	2	yes	—
249	C₂×C₆₇	#3490	1	1	yes	characteristic
250	C₂×C₆₇	#3491	1	1	yes	characteristic
251	C₂×C₆₇	#3496	1	2	yes	—
252	C₂×#534	C₂×C₆₇	1	1	yes	characteristic
253	C₂×#518	C₂×C₆₇	1	2	yes	complemented
254	#2359	C₂×C₆₇	1	8	yes	complemented
255	#2358	C₂×C₆₇	1	8	yes	complemented
256	#2325	C₂×C₆₇	1	4	yes	complemented
257	#2315	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₄×#121	C₂×C₆₇	1	4	yes	complemented
259	C₂×#386	C₂×C₆₇	1	2	yes	complemented
260	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₆₇	#34273	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
263	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
264	#534	C₄×C₆₇	1	16	yes	complemented
265	C₂×#123	C₂²×C₆₇	1	4	yes	—
266	C₂×#121	C₂²×C₆₇	1	4	yes	—
267	C₂×#120	C₂²×C₆₇	1	4	yes	—
268	#518	—	2	16	no	—
269	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
270	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
271	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
272	C₂×#117	C₂²×C₆₇	1	4	yes	complemented
273	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
274	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
275	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
277	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
278	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
279	#390	C₂²×C₆₇	1	4	yes	complemented
280	#389	C₂²×C₆₇	1	4	yes	complemented
281	#388	C₂²×C₆₇	1	8	yes	complemented
282	#388	C₂²×C₆₇	1	8	yes	complemented
283	#387	C₂²×C₆₇	1	8	yes	complemented
284	#386	C₂²×C₆₇	1	8	yes	—
285	#386	C₂²×C₆₇	1	4	yes	complemented
286	#386	C₂²×C₆₇	1	8	yes	complemented
287	#386	C₂²×C₆₇	1	4	yes	complemented
288	#385	C₂²×C₆₇	1	8	yes	complemented
289	#385	C₂²×C₆₇	1	8	yes	complemented
290	#380	C₂²×C₆₇	1	4	yes	complemented
291	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
292	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
293	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
294	C₂×#95	C₂²×C₆₇	1	2	yes	complemented
295	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
296	C₂⁵	C₂³×C₆₇	1	2	yes	—
297	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×D₄	—	2	8	no	—
299	C₂²×D₄	—	2	8	no	—
300	C₂²×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
301	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
302	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
303	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
304	C₂³×C₄	—	2	4	no	—
305	C₂³×C₄	—	4	4	no	—
306	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
308	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
309	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
310	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
311	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
314	#123	C₂×C₄×C₆₇	1	32	yes	complemented
315	#121	C₂×C₄×C₆₇	1	32	yes	complemented
316	#120	C₂×C₄×C₆₇	1	32	yes	complemented
317	C₄×D₄	—	2	32	no	—
318	C₄×D₄	—	2	32	no	—
319	C₄×D₄	—	4	8	no	—
320	#117	—	2	16	no	—
321	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
322	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
323	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
325	C₂×C₂²⋊C₄	—	2	8	no	—
326	C₂×C₂²⋊C₄	—	2	16	no	—
327	C₂×C₂²⋊C₄	—	4	8	no	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
329	C₂×C₂²⋊C₄	D₄×C₆₇	1	8	yes	complemented
330	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₄²	—	4	4	no	—
339	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
340	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
341	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
342	#95	C₂³×C₆₇	1	8	yes	—
343	#95	C₂³×C₆₇	1	8	yes	—
344	#95	C₂³×C₆₇	1	8	yes	—
345	#95	C₂³×C₆₇	1	8	yes	—
346	C₂⁴	—	2	4	no	—
347	C₂⁴	—	2	4	no	—
348	C₂⁴	—	4	2	no	—
349	C₂⁴	—	4	2	no	—
350	C₂⁴	—	4	4	no	—
351	C₂⁴	—	4	4	no	—
352	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
353	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
354	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
355	C₂⁴	#41×C₆₇	1	4	yes	—
356	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
357	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
358	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
359	C₂×D₄	—	2	64	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
367	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
368	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	complemented
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	2	16	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	4	no	—
381	C₂²×C₄	—	4	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	16	no	—
389	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
390	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
398	C₄⋊C₄	—	2	16	no	—
399	C₄⋊C₄	—	4	4	no	—
400	C₄⋊C₄	—	4	4	no	—
401	C₄⋊C₄	C₂²×C₄×C₆₇	1	16	yes	complemented
402	C₂²⋊C₄	—	2	8	no	—
403	C₂²⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	16	no	—
405	C₂²⋊C₄	—	2	16	no	—
406	C₂²⋊C₄	—	4	8	no	—
407	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
408	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	complemented
409	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
410	C₄²	—	2	16	no	—
411	C₄²	—	2	8	no	—
412	C₄²	—	4	4	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	8	no	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	16	no	—
422	C₂³	—	4	4	no	—
423	C₂³	—	4	8	no	—
424	C₂³	—	4	8	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	16	no	—
427	C₂³	—	8	4	no	—
428	C₂³	—	8	4	no	—
429	C₂³	—	8	8	no	—
430	C₂³	C₄×D₄×C₆₇	1	16	yes	complemented
431	C₂³	C₂³×C₄×C₆₇	1	8	yes	—
432	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
433	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
434	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
435	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
436	C₂³	C₂×#41×C₆₇	1	4	yes	—
437	C₂³	C₂×#41×C₆₇	1	2	yes	—
438	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
439	D₄	—	2	32	no	—
440	D₄	—	2	32	no	—
441	D₄	—	4	16	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	16	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	4	8	no	—
459	C₂×C₄	—	4	16	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	8	16	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
464	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
465	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
466	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	8	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	2	8	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	2	16	no	—
477	C₂²	—	4	16	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	8	no	—
481	C₂²	—	8	16	no	—
482	C₂²	C₂×C₄×D₄×C₆₇	1	8	yes	—
483	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
484	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
485	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
486	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
487	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
488	C₂²	#585×C₆₇	1	2	yes	—
489	C₄	—	2	8	no	—
490	C₄	—	2	8	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	8	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	8	no	—
495	C₂	—	2	8	no	—
496	C₂	—	2	4	no	—
497	C₂	—	4	8	no	—
498	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
499	C₂	C₄×H(5,𝔽₂)×C₆₇	1	2	yes	—
500	C₂	C₆₇×#3490	1	1	yes	characteristic
501	C₂	C₆₇×#3491	1	1	yes	characteristic
502	C₂	C₆₇×#3496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#534×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. C₂×#518×C₆₇	C₂	semidirect
3. C₆₇×#2359	C₂	semidirect
4. C₆₇×#2358	C₂	semidirect
5. C₆₇×#2325	C₂	semidirect
6. C₆₇×#2315	C₂	semidirect
7. C₄×#121×C₆₇	C₂	semidirect
8. C₂×#386×C₆₇	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
12. #534×C₆₇	C₄	semidirect
18. C₂×C₄×D₄×C₆₇	C₂²	semidirect
19. C₂×C₄×D₄×C₆₇	C₂²	semidirect
20. C₂×#117×C₆₇	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
25. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
26. C₂²×C₄²×C₆₇	C₂²	semidirect
27. #390×C₆₇	C₂²	semidirect
28. #389×C₆₇	C₂²	semidirect
29. #388×C₆₇	C₂²	semidirect
30. #388×C₆₇	C₂²	semidirect
31. #387×C₆₇	C₂²	semidirect
33. #386×C₆₇	C₂²	semidirect
34. #386×C₆₇	C₂²	semidirect
35. #386×C₆₇	C₂²	semidirect
36. #385×C₆₇	C₂²	semidirect
37. #385×C₆₇	C₂²	semidirect
38. #380×C₆₇	C₂²	semidirect
39. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
40. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
41. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
42. C₂×#95×C₆₇	C₂²	semidirect
43. C₂×#95×C₆₇	C₂²	semidirect
48. C₂²×D₄×C₆₇	C₂×C₄	semidirect
62. #123×C₆₇	C₂×C₄	semidirect
63. #121×C₆₇	C₂×C₄	semidirect
64. #120×C₆₇	C₂×C₄	semidirect
76. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
77. C₂×C₂²⋊C₄×C₆₇	D₄	semidirect
116. C₂×D₄×C₆₇	C₂²×C₄	semidirect
149. C₄⋊C₄×C₆₇	C₂²×C₄	semidirect
156. C₂²⋊C₄×C₆₇	C₂²×C₄	semidirect
178. C₂³×C₆₇	C₄×D₄	semidirect
243. #34273	C₆₇	direct (N × H)
253. C₂×#518	C₂×C₆₇	semidirect
254. #2359	C₂×C₆₇	semidirect
255. #2358	C₂×C₆₇	semidirect
256. #2325	C₂×C₆₇	semidirect
257. #2315	C₂×C₆₇	semidirect
258. C₄×#121	C₂×C₆₇	semidirect
259. C₂×#386	C₂×C₆₇	semidirect
260. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
261. C₆₇	#34273	direct (N × H)
264. #534	C₄×C₆₇	semidirect
270. C₂×C₄×D₄	C₂²×C₆₇	semidirect
271. C₂×C₄×D₄	C₂²×C₆₇	semidirect
272. C₂×#117	C₂²×C₆₇	semidirect
276. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
277. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
278. C₂²×C₄²	C₂²×C₆₇	semidirect
279. #390	C₂²×C₆₇	semidirect
280. #389	C₂²×C₆₇	semidirect
281. #388	C₂²×C₆₇	semidirect
282. #388	C₂²×C₆₇	semidirect
283. #387	C₂²×C₆₇	semidirect
285. #386	C₂²×C₆₇	semidirect
286. #386	C₂²×C₆₇	semidirect
287. #386	C₂²×C₆₇	semidirect
288. #385	C₂²×C₆₇	semidirect
289. #385	C₂²×C₆₇	semidirect
290. #380	C₂²×C₆₇	semidirect
291. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
292. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
293. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
294. C₂×#95	C₂²×C₆₇	semidirect
295. C₂×#95	C₂²×C₆₇	semidirect
300. C₂²×D₄	C₂×C₄×C₆₇	semidirect
314. #123	C₂×C₄×C₆₇	semidirect
315. #121	C₂×C₄×C₆₇	semidirect
316. #120	C₂×C₄×C₆₇	semidirect
328. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
329. C₂×C₂²⋊C₄	D₄×C₆₇	semidirect
368. C₂×D₄	C₂²×C₄×C₆₇	semidirect
401. C₄⋊C₄	C₂²×C₄×C₆₇	semidirect
408. C₂²⋊C₄	C₂²×C₄×C₆₇	semidirect
430. C₂³	C₄×D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#534×C₆₇	2
C₂×#518×C₆₇	2
C₆₇×#2359	2
C₆₇×#2358	2
C₆₇×#2325	2
C₆₇×#2315	2
C₄×#121×C₆₇	2
C₂×#386×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
#34273	67

Subgroups — C67×#34273 #30713524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34273 #30713524