Subgroups — C₆₇×#34956 #30714207

Order 17152.

3394 conjugacy classes of subgroups.

The subgroup lattice has 508 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	2	yes	maximal, complemented
2	C₆₇×#2600	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2579	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2534	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2405	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2403	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#2392	C₂	1	4	yes	maximal, complemented
9	C₄×#123×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₄×#120×C₆₇	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	complemented
15	C₂×#123×C₆₇	C₂²	1	4	yes	—
16	C₂×#120×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₆₇	—	2	16	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
21	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
22	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
27	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
28	#398×C₆₇	C₂²	1	4	yes	complemented
29	#397×C₆₇	C₂²	1	4	yes	complemented
30	#396×C₆₇	C₂²	1	8	yes	complemented
31	#394×C₆₇	C₂²	1	4	yes	complemented
32	#394×C₆₇	C₂²	1	8	yes	complemented
33	#392×C₆₇	C₂²	1	4	yes	—
34	#388×C₆₇	C₂²	1	8	yes	complemented
35	#387×C₆₇	C₂²	1	8	yes	complemented
36	#386×C₆₇	C₂²	1	4	yes	—
37	#386×C₆₇	C₂²	1	4	yes	—
38	#386×C₆₇	C₂²	1	8	yes	complemented
39	#385×C₆₇	C₂²	1	8	yes	complemented
40	#385×C₆₇	C₂²	1	4	yes	complemented
41	#385×C₆₇	C₂²	1	4	yes	complemented
42	#381×C₆₇	C₂²	1	4	yes	complemented
43	#381×C₆₇	C₂²	1	2	yes	complemented
44	#380×C₆₇	C₂²	1	4	yes	complemented
45	#379×C₆₇	C₂²	1	1	yes	characteristic
46	#379×C₆₇	C₂²	1	1	yes	characteristic
47	#379×C₆₇	C₂²	1	4	yes	complemented
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
50	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
51	C₂⁵×C₆₇	C₂³	1	2	yes	—
52	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	—	2	8	no	—
54	C₂²×D₄×C₆₇	—	2	16	no	—
55	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	—	2	8	no	—
59	C₂³×C₄×C₆₇	—	4	4	no	—
60	C₂³×C₄×C₆₇	—	4	8	no	—
61	C₂³×C₄×C₆₇	D₄	1	8	yes	complemented
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
71	#123×C₆₇	—	2	16	no	—
72	#120×C₆₇	—	2	16	no	—
73	C₄×D₄×C₆₇	—	2	16	no	—
74	C₄×D₄×C₆₇	—	2	16	no	—
75	C₄×D₄×C₆₇	—	2	16	no	—
76	C₄×D₄×C₆₇	—	2	16	no	—
77	C₄×D₄×C₆₇	—	4	8	no	—
78	#117×C₆₇	—	4	4	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₄²×C₆₇	—	2	8	no	—
99	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	#95×C₆₇	C₂³	1	8	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	#95×C₆₇	C₂³	1	4	yes	—
106	#95×C₆₇	C₂³	1	4	yes	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	8	no	—
109	C₂⁴×C₆₇	—	2	8	no	—
110	C₂⁴×C₆₇	—	2	8	no	—
111	C₂⁴×C₆₇	—	4	2	no	—
112	C₂⁴×C₆₇	—	4	4	no	—
113	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
114	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
116	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
117	C₂×D₄×C₆₇	—	2	64	no	—
118	C₂×D₄×C₆₇	—	2	8	no	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	2	8	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	2	16	no	—
123	C₂×D₄×C₆₇	—	2	8	no	—
124	C₂×D₄×C₆₇	—	4	4	no	—
125	C₂×D₄×C₆₇	#41	1	16	yes	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	8	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	8	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	8	no	—
134	C₂²×C₄×C₆₇	—	2	8	no	—
135	C₂²×C₄×C₆₇	—	2	16	no	—
136	C₂²×C₄×C₆₇	—	4	16	no	—
137	C₂²×C₄×C₆₇	—	4	4	no	—
138	C₂²×C₄×C₆₇	—	4	2	no	—
139	C₂²×C₄×C₆₇	—	4	8	no	—
140	C₂²×C₄×C₆₇	—	4	4	no	—
141	C₂²×C₄×C₆₇	—	4	32	no	—
142	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	16	no	—
157	C₄⋊C₄×C₆₇	—	4	4	no	—
158	C₄⋊C₄×C₆₇	—	4	4	no	—
159	C₂²⋊C₄×C₆₇	—	2	8	no	—
160	C₂²⋊C₄×C₆₇	—	2	8	no	—
161	C₂²⋊C₄×C₆₇	—	2	8	no	—
162	C₂²⋊C₄×C₆₇	—	2	8	no	—
163	C₂²⋊C₄×C₆₇	—	2	16	no	—
164	C₂²⋊C₄×C₆₇	—	2	8	no	—
165	C₂²⋊C₄×C₆₇	—	2	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	16	no	—
167	C₂²⋊C₄×C₆₇	—	4	2	no	—
168	C₂²⋊C₄×C₆₇	—	4	2	no	—
169	C₂²⋊C₄×C₆₇	—	4	8	no	—
170	C₄²×C₆₇	—	2	8	no	—
171	C₄²×C₆₇	—	2	8	no	—
172	C₄²×C₆₇	—	2	8	no	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	16	no	—
176	C₂³×C₆₇	—	2	16	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	8	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	8	no	—
183	C₂³×C₆₇	—	4	4	no	—
184	C₂³×C₆₇	—	4	16	no	—
185	C₂³×C₆₇	—	4	32	no	—
186	C₂³×C₆₇	—	4	8	no	—
187	C₂³×C₆₇	—	4	8	no	—
188	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
189	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
190	C₂³×C₆₇	C₂×#41	1	8	yes	—
191	C₂³×C₆₇	C₂×#41	1	8	yes	—
192	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
193	D₄×C₆₇	—	2	32	no	—
194	D₄×C₆₇	—	2	32	no	—
195	D₄×C₆₇	—	4	16	no	—
196	C₂×C₄×C₆₇	—	2	16	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	16	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	8	no	—
206	C₂×C₄×C₆₇	—	2	8	no	—
207	C₂×C₄×C₆₇	—	2	8	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	8	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	8	no	—
213	C₂×C₄×C₆₇	—	4	16	no	—
214	C₂×C₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	—	4	8	no	—
216	C₂×C₄×C₆₇	—	4	16	no	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
221	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
222	C₂²×C₆₇	—	2	4	no	—
223	C₂²×C₆₇	—	2	4	no	—
224	C₂²×C₆₇	—	2	16	no	—
225	C₂²×C₆₇	—	2	4	no	—
226	C₂²×C₆₇	—	2	8	no	—
227	C₂²×C₆₇	—	2	2	no	—
228	C₂²×C₆₇	—	2	16	no	—
229	C₂²×C₆₇	—	4	16	no	—
230	C₂²×C₆₇	—	4	32	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	8	no	—
234	C₂²×C₆₇	—	4	4	no	—
235	C₂²×C₆₇	#525	1	8	yes	—
236	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
237	C₂²×C₆₇	C₂²×#41	1	4	yes	—
238	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
239	C₄×C₆₇	—	2	8	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	2	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	—	4	4	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₄×C₆₇	#534	1	4	yes	—
247	#34956	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₆₇	—	2	8	no	—
249	C₂×C₆₇	—	4	2	no	—
250	C₂×C₆₇	—	4	8	no	—
251	C₂×C₆₇	C₂×#525	1	2	yes	—
252	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
253	C₂×C₆₇	#3481	1	4	yes	—
254	C₂²×C₄×D₄	C₂×C₆₇	1	2	yes	complemented
255	#2600	C₂×C₆₇	1	4	yes	complemented
256	#2579	C₂×C₆₇	1	4	yes	complemented
257	#2534	C₂×C₆₇	1	4	yes	complemented
258	#2483	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#2405	C₂×C₆₇	1	2	yes	complemented
260	#2403	C₂×C₆₇	1	1	yes	characteristic
261	#2392	C₂×C₆₇	1	4	yes	complemented
262	C₄×#123	C₂×C₆₇	1	4	yes	complemented
263	C₄×#120	C₂×C₆₇	1	4	yes	complemented
264	#2280	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₆₇	#34956	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
267	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
268	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	complemented
269	C₂×#123	C₂²×C₆₇	1	4	yes	—
270	C₂×#120	C₂²×C₆₇	1	4	yes	—
271	C₂×C₄×D₄	—	2	16	no	—
272	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
273	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
274	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
275	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
276	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
277	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
278	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
279	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
280	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
281	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
282	#398	C₂²×C₆₇	1	4	yes	complemented
283	#397	C₂²×C₆₇	1	4	yes	complemented
284	#396	C₂²×C₆₇	1	8	yes	complemented
285	#394	C₂²×C₆₇	1	8	yes	complemented
286	#394	C₂²×C₆₇	1	4	yes	complemented
287	#392	C₂²×C₆₇	1	4	yes	—
288	#388	C₂²×C₆₇	1	8	yes	complemented
289	#387	C₂²×C₆₇	1	8	yes	complemented
290	#386	C₂²×C₆₇	1	4	yes	—
291	#386	C₂²×C₆₇	1	4	yes	—
292	#386	C₂²×C₆₇	1	8	yes	complemented
293	#385	C₂²×C₆₇	1	8	yes	complemented
294	#385	C₂²×C₆₇	1	4	yes	complemented
295	#385	C₂²×C₆₇	1	4	yes	complemented
296	#381	C₂²×C₆₇	1	2	yes	complemented
297	#381	C₂²×C₆₇	1	4	yes	complemented
298	#380	C₂²×C₆₇	1	4	yes	complemented
299	#379	C₂²×C₆₇	1	1	yes	characteristic
300	#379	C₂²×C₆₇	1	1	yes	characteristic
301	#379	C₂²×C₆₇	1	4	yes	complemented
302	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
303	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
304	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
305	C₂⁵	C₂³×C₆₇	1	2	yes	—
306	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
307	C₂²×D₄	—	2	8	no	—
308	C₂²×D₄	—	2	16	no	—
309	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
310	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
311	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
312	C₂³×C₄	—	2	8	no	—
313	C₂³×C₄	—	4	4	no	—
314	C₂³×C₄	—	4	8	no	—
315	C₂³×C₄	D₄×C₆₇	1	8	yes	complemented
316	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
317	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
319	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
321	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
322	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
325	#123	—	2	16	no	—
326	#120	—	2	16	no	—
327	C₄×D₄	—	2	16	no	—
328	C₄×D₄	—	2	16	no	—
329	C₄×D₄	—	2	16	no	—
330	C₄×D₄	—	2	16	no	—
331	C₄×D₄	—	4	8	no	—
332	#117	—	4	4	no	—
333	C₂×C₄⋊C₄	—	2	8	no	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
339	C₂×C₂²⋊C₄	—	2	8	no	—
340	C₂×C₂²⋊C₄	—	2	8	no	—
341	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
352	C₂×C₄²	—	2	8	no	—
353	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
354	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
356	#95	C₂³×C₆₇	1	8	yes	—
357	#95	C₂³×C₆₇	1	8	yes	—
358	#95	C₂³×C₆₇	1	8	yes	—
359	#95	C₂³×C₆₇	1	4	yes	—
360	#95	C₂³×C₆₇	1	4	yes	—
361	C₂⁴	—	2	4	no	—
362	C₂⁴	—	2	8	no	—
363	C₂⁴	—	2	8	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	4	2	no	—
366	C₂⁴	—	4	4	no	—
367	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
368	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
369	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
370	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
371	C₂×D₄	—	2	64	no	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	8	no	—
376	C₂×D₄	—	2	16	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	#41×C₆₇	1	16	yes	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	16	no	—
390	C₂²×C₄	—	4	16	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	8	no	—
394	C₂²×C₄	—	4	4	no	—
395	C₂²×C₄	—	4	32	no	—
396	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
397	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
399	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	16	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	16	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	16	no	—
421	C₂²⋊C₄	—	4	2	no	—
422	C₂²⋊C₄	—	4	2	no	—
423	C₂²⋊C₄	—	4	8	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	8	no	—
426	C₄²	—	2	8	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	16	no	—
430	C₂³	—	2	16	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	8	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	8	no	—
437	C₂³	—	4	4	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	32	no	—
440	C₂³	—	4	8	no	—
441	C₂³	—	4	8	no	—
442	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
443	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
444	C₂³	C₂×#41×C₆₇	1	8	yes	—
445	C₂³	C₂×#41×C₆₇	1	8	yes	—
446	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
447	D₄	—	2	32	no	—
448	D₄	—	2	32	no	—
449	D₄	—	4	16	no	—
450	C₂×C₄	—	2	16	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	16	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	8	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	8	no	—
467	C₂×C₄	—	4	16	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
472	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
475	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	16	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	8	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	16	no	—
483	C₂²	—	4	16	no	—
484	C₂²	—	4	32	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	8	no	—
488	C₂²	—	4	4	no	—
489	C₂²	#525×C₆₇	1	8	yes	—
490	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
491	C₂²	C₂²×#41×C₆₇	1	4	yes	—
492	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
493	C₄	—	2	8	no	—
494	C₄	—	2	4	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	4	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	4	no	—
500	C₄	#534×C₆₇	1	4	yes	—
501	C₂	—	2	8	no	—
502	C₂	—	4	2	no	—
503	C₂	—	4	8	no	—
504	C₂	C₂×#525×C₆₇	1	2	yes	—
505	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
506	C₂	C₆₇×#3481	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₆₇×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×D₄×C₆₇	C₂	semidirect
2. C₆₇×#2600	C₂	semidirect
3. C₆₇×#2579	C₂	semidirect
4. C₆₇×#2534	C₂	semidirect
5. C₆₇×#2483	C₂	semidirect
6. C₆₇×#2405	C₂	semidirect
8. C₆₇×#2392	C₂	semidirect
9. C₄×#123×C₆₇	C₂	semidirect
10. C₄×#120×C₆₇	C₂	semidirect
11. C₆₇×#2280	C₂	semidirect
14. C₂⁴×C₄×C₆₇	C₂²	semidirect
20. C₂×C₄×D₄×C₆₇	C₂²	semidirect
21. C₂×C₄×D₄×C₆₇	C₂²	semidirect
22. C₂×C₄×D₄×C₆₇	C₂²	semidirect
23. C₂×#117×C₆₇	C₂²	semidirect
24. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
26. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
27. C₂²×C₄²×C₆₇	C₂²	semidirect
28. #398×C₆₇	C₂²	semidirect
29. #397×C₆₇	C₂²	semidirect
30. #396×C₆₇	C₂²	semidirect
31. #394×C₆₇	C₂²	semidirect
32. #394×C₆₇	C₂²	semidirect
34. #388×C₆₇	C₂²	semidirect
35. #387×C₆₇	C₂²	semidirect
38. #386×C₆₇	C₂²	semidirect
39. #385×C₆₇	C₂²	semidirect
40. #385×C₆₇	C₂²	semidirect
41. #385×C₆₇	C₂²	semidirect
42. #381×C₆₇	C₂²	semidirect
43. #381×C₆₇	C₂²	semidirect
44. #380×C₆₇	C₂²	semidirect
47. #379×C₆₇	C₂²	semidirect
48. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
61. C₂³×C₄×C₆₇	D₄	semidirect
247. #34956	C₆₇	direct (N × H)
254. C₂²×C₄×D₄	C₂×C₆₇	semidirect
255. #2600	C₂×C₆₇	semidirect
256. #2579	C₂×C₆₇	semidirect
257. #2534	C₂×C₆₇	semidirect
258. #2483	C₂×C₆₇	semidirect
259. #2405	C₂×C₆₇	semidirect
261. #2392	C₂×C₆₇	semidirect
262. C₄×#123	C₂×C₆₇	semidirect
263. C₄×#120	C₂×C₆₇	semidirect
264. #2280	C₂×C₆₇	semidirect
265. C₆₇	#34956	direct (N × H)
268. C₂⁴×C₄	C₂²×C₆₇	semidirect
274. C₂×C₄×D₄	C₂²×C₆₇	semidirect
275. C₂×C₄×D₄	C₂²×C₆₇	semidirect
276. C₂×C₄×D₄	C₂²×C₆₇	semidirect
277. C₂×#117	C₂²×C₆₇	semidirect
278. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
280. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
281. C₂²×C₄²	C₂²×C₆₇	semidirect
282. #398	C₂²×C₆₇	semidirect
283. #397	C₂²×C₆₇	semidirect
284. #396	C₂²×C₆₇	semidirect
285. #394	C₂²×C₆₇	semidirect
286. #394	C₂²×C₆₇	semidirect
288. #388	C₂²×C₆₇	semidirect
289. #387	C₂²×C₆₇	semidirect
292. #386	C₂²×C₆₇	semidirect
293. #385	C₂²×C₆₇	semidirect
294. #385	C₂²×C₆₇	semidirect
295. #385	C₂²×C₆₇	semidirect
296. #381	C₂²×C₆₇	semidirect
297. #381	C₂²×C₆₇	semidirect
298. #380	C₂²×C₆₇	semidirect
301. #379	C₂²×C₆₇	semidirect
302. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
303. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
304. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
315. 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
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2600	2
C₆₇×#2579	2
C₆₇×#2534	2
C₆₇×#2483	2
C₆₇×#2405	2
C₆₇×#2403	2
C₆₇×#2392	2
C₄×#123×C₆₇	2
C₄×#120×C₆₇	2
C₆₇×#2280	2
#34956	67

Subgroups — C67×#34956 #30714207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34956 #30714207