Subgroups — C₆₇×#34925 #30714176

Order 17152.

3436 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×#532×C₆₇	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2555	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2551	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2541	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2537	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2491	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2428	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2388	C₂	1	4	yes	maximal, complemented
9	C₄×#127×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
11	C₄²×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×#41×C₆₇	C₂²	1	4	yes	—
13	#532×C₆₇	—	2	32	no	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₇	C₂²	1	4	yes	—
17	C₂×#124×C₆₇	C₂²	1	8	yes	—
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×#121×C₆₇	C₂²	1	4	yes	—
20	C₂×#121×C₆₇	C₂²	1	4	yes	—
21	C₂×#121×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₇	C₂²	1	4	yes	—
29	#395×C₆₇	C₂²	1	4	yes	—
30	#394×C₆₇	C₂²	1	4	yes	—
31	#394×C₆₇	C₂²	1	8	yes	—
32	#393×C₆₇	C₂²	1	4	yes	—
33	#392×C₆₇	C₂²	1	8	yes	—
34	#392×C₆₇	C₂²	1	4	yes	—
35	#390×C₆₇	C₂²	1	4	yes	—
36	#390×C₆₇	C₂²	1	2	yes	—
37	#390×C₆₇	C₂²	1	4	yes	—
38	#389×C₆₇	C₂²	1	4	yes	—
39	#388×C₆₇	C₂²	1	8	yes	—
40	#386×C₆₇	C₂²	1	8	yes	—
41	#384×C₆₇	C₂²	1	1	yes	characteristic
42	#380×C₆₇	C₂²	1	4	yes	—
43	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
45	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
46	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
47	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
48	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
50	C₄³×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×#41×C₆₇	—	2	32	no	—
52	C₂×#41×C₆₇	—	4	8	no	—
53	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
55	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₄⋊Q₈×C₆₇	—	2	16	no	—
65	#127×C₆₇	—	2	16	no	—
66	#127×C₆₇	—	2	4	no	—
67	#124×C₆₇	—	2	32	no	—
68	#121×C₆₇	—	2	16	no	—
69	#121×C₆₇	—	2	16	no	—
70	#121×C₆₇	—	2	16	no	—
71	#121×C₆₇	—	2	32	no	—
72	C₄×D₄×C₆₇	—	2	32	no	—
73	C₄×D₄×C₆₇	—	2	8	no	—
74	C₄×D₄×C₆₇	—	2	16	no	—
75	C₄×D₄×C₆₇	—	2	8	no	—
76	C₄×D₄×C₆₇	—	4	4	no	—
77	C₄×D₄×C₆₇	D₄	1	16	yes	complemented
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
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	8	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₄²×C₆₇	—	2	16	no	—
93	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
95	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
101	#95×C₆₇	C₂³	1	4	yes	—
102	#95×C₆₇	C₂³	1	4	yes	—
103	#95×C₆₇	C₂³	1	4	yes	—
104	#95×C₆₇	C₂³	1	4	yes	—
105	#95×C₆₇	C₂³	1	4	yes	—
106	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
107	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
108	#41×C₆₇	—	4	64	no	—
109	C₂×Q₈×C₆₇	—	2	16	no	—
110	C₂×Q₈×C₆₇	—	4	4	no	—
111	C₂×D₄×C₆₇	—	2	16	no	—
112	C₂×D₄×C₆₇	—	2	16	no	—
113	C₂×D₄×C₆₇	—	2	16	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	2	16	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	2	16	no	—
119	C₂×D₄×C₆₇	—	2	8	no	—
120	C₂×D₄×C₆₇	—	2	4	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	4	2	no	—
123	C₂×D₄×C₆₇	—	4	2	no	—
124	C₂×D₄×C₆₇	—	4	8	no	—
125	C₂×D₄×C₆₇	—	4	8	no	—
126	C₂×D₄×C₆₇	—	4	4	no	—
127	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
128	C₂²×C₄×C₆₇	—	2	8	no	—
129	C₂²×C₄×C₆₇	—	2	8	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	8	no	—
132	C₂²×C₄×C₆₇	—	2	16	no	—
133	C₂²×C₄×C₆₇	—	2	8	no	—
134	C₂²×C₄×C₆₇	—	2	8	no	—
135	C₂²×C₄×C₆₇	—	2	8	no	—
136	C₂²×C₄×C₆₇	—	2	8	no	—
137	C₂²×C₄×C₆₇	—	4	4	no	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₄⋊C₄×C₆₇	—	2	16	no	—
149	C₄⋊C₄×C₆₇	—	2	8	no	—
150	C₄⋊C₄×C₆₇	—	2	4	no	—
151	C₄⋊C₄×C₆₇	—	4	2	no	—
152	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
153	C₄⋊C₄×C₆₇	#41	1	8	yes	complemented
154	C₂²⋊C₄×C₆₇	—	2	16	no	—
155	C₂²⋊C₄×C₆₇	—	2	16	no	—
156	C₂²⋊C₄×C₆₇	—	2	16	no	—
157	C₂²⋊C₄×C₆₇	—	2	8	no	—
158	C₂²⋊C₄×C₆₇	—	2	8	no	—
159	C₂²⋊C₄×C₆₇	—	2	16	no	—
160	C₂²⋊C₄×C₆₇	—	4	4	no	—
161	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
162	C₄²×C₆₇	—	2	8	no	—
163	C₄²×C₆₇	—	2	8	no	—
164	C₄²×C₆₇	—	2	8	no	—
165	C₄²×C₆₇	—	2	4	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	4	1	no	—
168	C₄²×C₆₇	C₂×D₄	1	8	yes	—
169	C₄²×C₆₇	#41	1	4	yes	complemented
170	C₂³×C₆₇	—	2	4	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	8	no	—
177	C₂³×C₆₇	—	2	8	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	4	no	—
181	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
182	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
183	Q₈×C₆₇	—	4	16	no	—
184	D₄×C₆₇	—	2	16	no	—
185	D₄×C₆₇	—	4	32	no	—
186	D₄×C₆₇	—	4	32	no	—
187	D₄×C₆₇	—	4	8	no	—
188	D₄×C₆₇	—	4	16	no	—
189	D₄×C₆₇	—	4	16	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	4	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	32	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	4	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	4	16	no	—
207	C₂×C₄×C₆₇	—	4	16	no	—
208	C₂×C₄×C₆₇	—	4	16	no	—
209	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
214	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
215	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
217	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
218	C₂²×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	2	8	no	—
220	C₂²×C₆₇	—	2	8	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	8	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	—	4	4	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	4	4	no	—
230	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
231	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
232	C₂²×C₆₇	C₂²×#41	1	2	yes	—
233	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂²×C₆₇	#585	1	1	yes	characteristic
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	2	8	no	—
237	C₄×C₆₇	—	2	4	no	—
238	C₄×C₆₇	—	2	8	no	—
239	C₄×C₆₇	—	4	4	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	1	no	—
242	C₄×C₆₇	#532	1	8	yes	complemented
243	C₄×C₆₇	D₄²	1	4	yes	—
244	#34925	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂×C₆₇	—	4	8	no	—
246	C₂×C₆₇	—	8	4	no	—
247	C₂×C₆₇	C₂×#532	1	2	yes	—
248	C₂×C₆₇	#3460	1	1	yes	characteristic
249	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
250	C₂×C₆₇	D₄×#41	1	2	yes	—
251	C₂×C₆₇	#3483	1	1	yes	characteristic
252	C₂×#532	C₂×C₆₇	1	4	yes	complemented
253	#2555	C₂×C₆₇	1	4	yes	complemented
254	#2551	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#2541	C₂×C₆₇	1	2	yes	complemented
256	#2537	C₂×C₆₇	1	1	yes	characteristic, complemented
257	#2491	C₂×C₆₇	1	4	yes	complemented
258	#2428	C₂×C₆₇	1	4	yes	complemented
259	#2388	C₂×C₆₇	1	4	yes	complemented
260	C₄×#127	C₂×C₆₇	1	1	yes	characteristic, complemented
261	C₄×#121	C₂×C₆₇	1	4	yes	complemented
262	C₄²×D₄	C₂×C₆₇	1	2	yes	complemented
263	C₆₇	#34925	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²×#41	C₂²×C₆₇	1	4	yes	—
265	#532	—	2	32	no	—
266	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
267	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×#127	C₂²×C₆₇	1	4	yes	—
269	C₂×#124	C₂²×C₆₇	1	8	yes	—
270	C₂×#121	C₂²×C₆₇	1	4	yes	—
271	C₂×#121	C₂²×C₆₇	1	8	yes	—
272	C₂×#121	C₂²×C₆₇	1	4	yes	—
273	C₂×#121	C₂²×C₆₇	1	4	yes	—
274	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
276	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
277	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
278	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
279	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
280	C₂²×C₄²	C₂²×C₆₇	1	4	yes	—
281	#395	C₂²×C₆₇	1	4	yes	—
282	#394	C₂²×C₆₇	1	8	yes	—
283	#394	C₂²×C₆₇	1	4	yes	—
284	#393	C₂²×C₆₇	1	4	yes	—
285	#392	C₂²×C₆₇	1	4	yes	—
286	#392	C₂²×C₆₇	1	8	yes	—
287	#390	C₂²×C₆₇	1	4	yes	—
288	#390	C₂²×C₆₇	1	2	yes	—
289	#390	C₂²×C₆₇	1	4	yes	—
290	#389	C₂²×C₆₇	1	4	yes	—
291	#388	C₂²×C₆₇	1	8	yes	—
292	#386	C₂²×C₆₇	1	8	yes	—
293	#384	C₂²×C₆₇	1	1	yes	characteristic
294	#380	C₂²×C₆₇	1	4	yes	—
295	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
297	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
298	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
299	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
300	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
301	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
302	C₄³	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#41	—	2	32	no	—
304	C₂×#41	—	4	8	no	—
305	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
306	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
307	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
308	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
309	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
310	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
311	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
312	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
313	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
314	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
315	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
316	C₄⋊Q₈	—	2	16	no	—
317	#127	—	2	16	no	—
318	#127	—	2	4	no	—
319	#124	—	2	32	no	—
320	#121	—	2	16	no	—
321	#121	—	2	16	no	—
322	#121	—	2	16	no	—
323	#121	—	2	32	no	—
324	C₄×D₄	—	2	32	no	—
325	C₄×D₄	—	2	8	no	—
326	C₄×D₄	—	2	16	no	—
327	C₄×D₄	—	2	8	no	—
328	C₄×D₄	—	4	4	no	—
329	C₄×D₄	D₄×C₆₇	1	16	yes	complemented
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
343	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₄²	—	2	16	no	—
345	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
347	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
348	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
349	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
350	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
352	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
353	#95	C₂³×C₆₇	1	4	yes	—
354	#95	C₂³×C₆₇	1	4	yes	—
355	#95	C₂³×C₆₇	1	4	yes	—
356	#95	C₂³×C₆₇	1	4	yes	—
357	#95	C₂³×C₆₇	1	4	yes	—
358	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
359	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
360	#41	—	4	64	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×D₄	—	2	16	no	—
364	C₂×D₄	—	2	16	no	—
365	C₂×D₄	—	2	16	no	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	16	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	16	no	—
371	C₂×D₄	—	2	8	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	4	2	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	—	4	4	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	8	no	—
379	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	16	no	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
400	C₄⋊C₄	—	2	16	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
405	C₄⋊C₄	#41×C₆₇	1	8	yes	complemented
406	C₂²⋊C₄	—	2	16	no	—
407	C₂²⋊C₄	—	2	16	no	—
408	C₂²⋊C₄	—	2	16	no	—
409	C₂²⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	16	no	—
412	C₂²⋊C₄	—	4	4	no	—
413	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
414	C₄²	—	2	8	no	—
415	C₄²	—	2	8	no	—
416	C₄²	—	2	8	no	—
417	C₄²	—	2	4	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	4	1	no	—
420	C₄²	C₂×D₄×C₆₇	1	8	yes	—
421	C₄²	#41×C₆₇	1	4	yes	complemented
422	C₂³	—	2	4	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	4	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	2	4	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	2	8	no	—
429	C₂³	—	2	8	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	4	no	—
433	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
434	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
435	Q₈	—	4	16	no	—
436	D₄	—	2	16	no	—
437	D₄	—	4	32	no	—
438	D₄	—	4	32	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	16	no	—
441	D₄	—	4	16	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	32	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	—	4	16	no	—
460	C₂×C₄	—	4	16	no	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
462	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
464	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
465	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
466	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
467	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
468	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
469	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
470	C₂²	—	2	8	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	8	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	8	no	—
481	C₂²	—	4	4	no	—
482	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
483	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
484	C₂²	C₂²×#41×C₆₇	1	2	yes	—
485	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
486	C₂²	#585×C₆₇	1	1	yes	characteristic
487	C₄	—	2	2	no	—
488	C₄	—	2	8	no	—
489	C₄	—	2	4	no	—
490	C₄	—	2	8	no	—
491	C₄	—	4	4	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	1	no	—
494	C₄	#532×C₆₇	1	8	yes	complemented
495	C₄	D₄²×C₆₇	1	4	yes	—
496	C₂	—	4	8	no	—
497	C₂	—	8	4	no	—
498	C₂	C₂×#532×C₆₇	1	2	yes	—
499	C₂	C₆₇×#3460	1	1	yes	characteristic
500	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
501	C₂	D₄×#41×C₆₇	1	2	yes	—
502	C₂	C₆₇×#3483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
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. C₂×#532×C₆₇	C₂	semidirect
2. C₆₇×#2555	C₂	semidirect
3. C₆₇×#2551	C₂	semidirect
4. C₆₇×#2541	C₂	semidirect
5. C₆₇×#2537	C₂	semidirect
6. C₆₇×#2491	C₂	semidirect
7. C₆₇×#2428	C₂	semidirect
8. C₆₇×#2388	C₂	semidirect
9. C₄×#127×C₆₇	C₂	semidirect
10. C₄×#121×C₆₇	C₂	semidirect
11. C₄²×D₄×C₆₇	C₂	semidirect
77. C₄×D₄×C₆₇	D₄	semidirect
153. C₄⋊C₄×C₆₇	#41	semidirect
169. C₄²×C₆₇	#41	semidirect
242. C₄×C₆₇	#532	semidirect
244. #34925	C₆₇	direct (N × H)
252. C₂×#532	C₂×C₆₇	semidirect
253. #2555	C₂×C₆₇	semidirect
254. #2551	C₂×C₆₇	semidirect
255. #2541	C₂×C₆₇	semidirect
256. #2537	C₂×C₆₇	semidirect
257. #2491	C₂×C₆₇	semidirect
258. #2428	C₂×C₆₇	semidirect
259. #2388	C₂×C₆₇	semidirect
260. C₄×#127	C₂×C₆₇	semidirect
261. C₄×#121	C₂×C₆₇	semidirect
262. C₄²×D₄	C₂×C₆₇	semidirect
263. C₆₇	#34925	direct (N × H)
329. C₄×D₄	D₄×C₆₇	semidirect
405. C₄⋊C₄	#41×C₆₇	semidirect
421. C₄²	#41×C₆₇	semidirect
494. C₄	#532×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₆₇	2
C₆₇×#2555	2
C₆₇×#2551	2
C₆₇×#2541	2
C₆₇×#2537	2
C₆₇×#2491	2
C₆₇×#2428	2
C₆₇×#2388	2
C₄×#127×C₆₇	2
C₄×#121×C₆₇	2
C₄²×D₄×C₆₇	2
#34925	67

Subgroups — C67×#34925 #30714176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34925 #30714176