Subgroups — C₆₇×#35012 #30714263

Order 17152.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2851	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2848	C₂	1	2	yes	maximal
3	C₆₇×#2659	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2659	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2626	C₂	1	2	yes	maximal
6	C₆₇×#2574	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2556	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2522	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2465	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2329	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2312	C₂	1	2	yes	maximal
12	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#399×C₆₇	C₂²	1	4	yes	—
26	#398×C₆₇	C₂²	1	4	yes	complemented
27	#398×C₆₇	C₂²	1	4	yes	complemented
28	#396×C₆₇	C₂²	1	4	yes	—
29	#396×C₆₇	C₂²	1	4	yes	—
30	#395×C₆₇	C₂²	1	4	yes	complemented
31	#395×C₆₇	C₂²	1	4	yes	complemented
32	#393×C₆₇	C₂²	1	4	yes	—
33	#391×C₆₇	C₂²	1	4	yes	complemented
34	#391×C₆₇	C₂²	1	2	yes	complemented
35	#389×C₆₇	C₂²	1	2	yes	complemented
36	#389×C₆₇	C₂²	1	4	yes	complemented
37	#389×C₆₇	C₂²	1	2	yes	complemented
38	#389×C₆₇	C₂²	1	2	yes	complemented
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	complemented
44	#387×C₆₇	C₂²	1	4	yes	complemented
45	#387×C₆₇	C₂²	1	4	yes	complemented
46	#387×C₆₇	C₂²	1	4	yes	complemented
47	#387×C₆₇	C₂²	1	4	yes	complemented
48	#385×C₆₇	C₂²	1	4	yes	—
49	#385×C₆₇	C₂²	1	4	yes	—
50	#385×C₆₇	C₂²	1	4	yes	—
51	#385×C₆₇	C₂²	1	4	yes	—
52	#382×C₆₇	C₂²	1	2	yes	complemented
53	#380×C₆₇	C₂²	1	2	yes	—
54	#380×C₆₇	C₂²	1	2	yes	—
55	#380×C₆₇	C₂²	1	2	yes	—
56	#380×C₆₇	C₂²	1	2	yes	—
57	#380×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	2	yes	—
59	#379×C₆₇	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
65	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
66	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
67	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	—	2	4	no	—
71	C₂³×C₄×C₆₇	—	4	2	no	—
72	C₂³×C₄×C₆₇	—	4	2	no	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	#125×C₆₇	—	4	4	no	—
82	C₂²⋊Q₈×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	8	no	—
84	C₂²⋊Q₈×C₆₇	—	4	4	no	—
85	C₂²⋊Q₈×C₆₇	—	4	4	no	—
86	C₄×Q₈×C₆₇	—	4	4	no	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
104	C₂×C₂²⋊C₄×C₆₇	Q₈	1	8	yes	complemented
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	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄²×C₆₇	—	4	2	no	—
114	C₂×C₄²×C₆₇	—	4	2	no	—
115	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
120	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
121	#95×C₆₇	C₂³	1	4	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	4	yes	—
124	#95×C₆₇	C₂³	1	4	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	4	yes	—
130	#95×C₆₇	C₂³	1	4	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	4	yes	—
133	C₂⁴×C₆₇	—	2	2	no	—
134	C₂⁴×C₆₇	—	2	2	no	—
135	C₂⁴×C₆₇	—	4	1	no	—
136	C₂⁴×C₆₇	—	4	1	no	—
137	C₂⁴×C₆₇	—	4	1	no	—
138	C₂⁴×C₆₇	—	4	1	no	—
139	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
140	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
141	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂×Q₈×C₆₇	—	2	4	no	—
144	C₂×Q₈×C₆₇	—	4	2	no	—
145	C₂×Q₈×C₆₇	—	4	2	no	—
146	C₂×Q₈×C₆₇	—	4	2	no	—
147	C₂×Q₈×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	4	1	no	—
157	C₂²×C₄×C₆₇	—	4	1	no	—
158	C₂²×C₄×C₆₇	—	4	1	no	—
159	C₂²×C₄×C₆₇	—	4	1	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	—	4	2	no	—
162	C₂²×C₄×C₆₇	—	4	1	no	—
163	C₂²×C₄×C₆₇	—	4	1	no	—
164	C₂²×C₄×C₆₇	—	4	1	no	—
165	C₂²×C₄×C₆₇	—	4	1	no	—
166	C₂²×C₄×C₆₇	—	4	1	no	—
167	C₂²×C₄×C₆₇	—	4	1	no	—
168	C₂²×C₄×C₆₇	—	4	16	no	—
169	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
170	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
171	C₂²×C₄×C₆₇	#41	1	4	yes	—
172	C₂²×C₄×C₆₇	#41	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	4	no	—
186	C₄⋊C₄×C₆₇	—	2	8	no	—
187	C₄⋊C₄×C₆₇	—	2	8	no	—
188	C₄⋊C₄×C₆₇	—	2	8	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	4	2	no	—
191	C₄⋊C₄×C₆₇	—	4	2	no	—
192	C₄⋊C₄×C₆₇	—	4	2	no	—
193	C₄⋊C₄×C₆₇	—	4	4	no	—
194	C₄⋊C₄×C₆₇	—	4	2	no	—
195	C₄⋊C₄×C₆₇	—	4	2	no	—
196	C₄⋊C₄×C₆₇	—	4	2	no	—
197	C₄⋊C₄×C₆₇	—	4	2	no	—
198	C₄⋊C₄×C₆₇	—	4	2	no	—
199	C₄⋊C₄×C₆₇	—	4	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	8	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	—	4	4	no	—
204	C₂²⋊C₄×C₆₇	—	4	4	no	—
205	C₂²⋊C₄×C₆₇	—	4	4	no	—
206	C₂²⋊C₄×C₆₇	—	4	4	no	—
207	C₂²⋊C₄×C₆₇	C₂×Q₈	1	16	yes	—
208	C₄²×C₆₇	—	2	4	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	—	4	2	no	—
211	C₄²×C₆₇	—	4	2	no	—
212	C₄²×C₆₇	—	4	4	no	—
213	C₄²×C₆₇	—	4	4	no	—
214	C₂³×C₆₇	—	2	1	no	—
215	C₂³×C₆₇	—	2	1	no	—
216	C₂³×C₆₇	—	2	1	no	—
217	C₂³×C₆₇	—	2	1	no	—
218	C₂³×C₆₇	—	2	8	no	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	2	2	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	4	4	no	—
226	C₂³×C₆₇	—	4	4	no	—
227	C₂³×C₆₇	—	4	4	no	—
228	C₂³×C₆₇	—	4	4	no	—
229	C₂³×C₆₇	—	4	4	no	—
230	C₂³×C₆₇	—	4	4	no	—
231	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
232	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
233	C₂³×C₆₇	C₂×#41	1	2	yes	—
234	C₂³×C₆₇	C₂×#41	1	2	yes	—
235	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
236	Q₈×C₆₇	—	4	8	no	—
237	Q₈×C₆₇	—	8	4	no	—
238	C₂×C₄×C₆₇	—	2	8	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	4	4	no	—
267	C₂×C₄×C₆₇	—	4	4	no	—
268	C₂×C₄×C₆₇	—	4	4	no	—
269	C₂×C₄×C₆₇	—	4	4	no	—
270	C₂×C₄×C₆₇	—	4	8	no	—
271	C₂×C₄×C₆₇	—	4	8	no	—
272	C₂×C₄×C₆₇	—	4	4	no	—
273	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
274	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
275	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
276	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
277	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
278	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
279	C₂²×C₆₇	—	2	2	no	—
280	C₂²×C₆₇	—	2	2	no	—
281	C₂²×C₆₇	—	2	2	no	—
282	C₂²×C₆₇	—	2	2	no	—
283	C₂²×C₆₇	—	2	2	no	—
284	C₂²×C₆₇	—	2	2	no	—
285	C₂²×C₆₇	—	2	4	no	—
286	C₂²×C₆₇	—	2	4	no	—
287	C₂²×C₆₇	—	2	4	no	—
288	C₂²×C₆₇	—	4	16	no	—
289	C₂²×C₆₇	—	4	2	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	—	4	2	no	—
292	C₂²×C₆₇	—	4	2	no	—
293	C₂²×C₆₇	#533	1	4	yes	—
294	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
295	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
296	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
297	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
299	C₂²×C₆₇	#585	1	1	yes	characteristic
300	C₂²×C₆₇	#585	1	1	yes	characteristic
301	C₄×C₆₇	—	2	4	no	—
302	C₄×C₆₇	—	2	2	no	—
303	C₄×C₆₇	—	2	2	no	—
304	C₄×C₆₇	—	2	4	no	—
305	C₄×C₆₇	—	2	4	no	—
306	C₄×C₆₇	—	2	4	no	—
307	C₄×C₆₇	—	4	4	no	—
308	C₄×C₆₇	—	4	4	no	—
309	C₄×C₆₇	—	4	2	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	—	4	4	no	—
312	#35012	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
313	C₂×C₆₇	—	2	4	no	—
314	C₂×C₆₇	—	4	4	no	—
315	C₂×C₆₇	C₂×#533	1	1	yes	characteristic
316	C₂×C₆₇	#3459	1	1	yes	characteristic
317	C₂×C₆₇	#3460	1	1	yes	characteristic
318	C₂×C₆₇	#3490	1	1	yes	characteristic
319	C₂×C₆₇	#3493	1	1	yes	characteristic
320	C₂×C₆₇	#3501	1	1	yes	characteristic
321	C₂×C₆₇	#3501	1	1	yes	characteristic
322	#2851	C₂×C₆₇	1	2	yes	complemented
323	#2848	C₂×C₆₇	1	2	yes	—
324	#2659	C₂×C₆₇	1	1	yes	characteristic, complemented
325	#2659	C₂×C₆₇	1	1	yes	characteristic, complemented
326	#2626	C₂×C₆₇	1	2	yes	—
327	#2574	C₂×C₆₇	1	4	yes	complemented
328	#2556	C₂×C₆₇	1	4	yes	complemented
329	#2522	C₂×C₆₇	1	4	yes	complemented
330	#2465	C₂×C₆₇	1	4	yes	complemented
331	#2329	C₂×C₆₇	1	2	yes	complemented
332	#2312	C₂×C₆₇	1	2	yes	—
333	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
334	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
335	C₆₇	#35012	1	1	yes	characteristic, direct product factor, Sylow
336	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
338	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
339	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
340	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
341	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
342	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
343	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
345	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
346	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
347	#399	C₂²×C₆₇	1	4	yes	—
348	#398	C₂²×C₆₇	1	4	yes	complemented
349	#398	C₂²×C₆₇	1	4	yes	complemented
350	#396	C₂²×C₆₇	1	4	yes	—
351	#396	C₂²×C₆₇	1	4	yes	—
352	#395	C₂²×C₆₇	1	4	yes	complemented
353	#395	C₂²×C₆₇	1	4	yes	complemented
354	#393	C₂²×C₆₇	1	4	yes	—
355	#391	C₂²×C₆₇	1	2	yes	complemented
356	#391	C₂²×C₆₇	1	4	yes	complemented
357	#389	C₂²×C₆₇	1	4	yes	complemented
358	#389	C₂²×C₆₇	1	2	yes	complemented
359	#389	C₂²×C₆₇	1	2	yes	complemented
360	#389	C₂²×C₆₇	1	2	yes	complemented
361	#388	C₂²×C₆₇	1	4	yes	—
362	#388	C₂²×C₆₇	1	4	yes	—
363	#388	C₂²×C₆₇	1	4	yes	—
364	#388	C₂²×C₆₇	1	4	yes	—
365	#387	C₂²×C₆₇	1	4	yes	complemented
366	#387	C₂²×C₆₇	1	4	yes	complemented
367	#387	C₂²×C₆₇	1	4	yes	complemented
368	#387	C₂²×C₆₇	1	4	yes	complemented
369	#387	C₂²×C₆₇	1	4	yes	complemented
370	#385	C₂²×C₆₇	1	4	yes	—
371	#385	C₂²×C₆₇	1	4	yes	—
372	#385	C₂²×C₆₇	1	4	yes	—
373	#385	C₂²×C₆₇	1	4	yes	—
374	#382	C₂²×C₆₇	1	2	yes	complemented
375	#380	C₂²×C₆₇	1	2	yes	—
376	#380	C₂²×C₆₇	1	2	yes	—
377	#380	C₂²×C₆₇	1	2	yes	—
378	#380	C₂²×C₆₇	1	2	yes	—
379	#380	C₂²×C₆₇	1	2	yes	—
380	#380	C₂²×C₆₇	1	2	yes	—
381	#379	C₂²×C₆₇	1	4	yes	—
382	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
384	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
387	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
388	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
389	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
390	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
391	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
392	C₂³×C₄	—	2	4	no	—
393	C₂³×C₄	—	4	2	no	—
394	C₂³×C₄	—	4	2	no	—
395	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
397	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
398	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
401	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
403	#125	—	4	4	no	—
404	C₂²⋊Q₈	—	2	8	no	—
405	C₂²⋊Q₈	—	2	8	no	—
406	C₂²⋊Q₈	—	4	4	no	—
407	C₂²⋊Q₈	—	4	4	no	—
408	C₄×Q₈	—	4	4	no	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₂²⋊C₄	—	2	4	no	—
426	C₂×C₂²⋊C₄	Q₈×C₆₇	1	8	yes	complemented
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
435	C₂×C₄²	—	4	2	no	—
436	C₂×C₄²	—	4	2	no	—
437	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
440	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
443	#95	C₂³×C₆₇	1	4	yes	—
444	#95	C₂³×C₆₇	1	4	yes	—
445	#95	C₂³×C₆₇	1	4	yes	—
446	#95	C₂³×C₆₇	1	4	yes	—
447	#95	C₂³×C₆₇	1	4	yes	—
448	#95	C₂³×C₆₇	1	4	yes	—
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	4	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	#95	C₂³×C₆₇	1	4	yes	—
453	#95	C₂³×C₆₇	1	4	yes	—
454	#95	C₂³×C₆₇	1	4	yes	—
455	C₂⁴	—	2	2	no	—
456	C₂⁴	—	2	2	no	—
457	C₂⁴	—	4	1	no	—
458	C₂⁴	—	4	1	no	—
459	C₂⁴	—	4	1	no	—
460	C₂⁴	—	4	1	no	—
461	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
462	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
463	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	—	4	2	no	—
467	C₂×Q₈	—	4	2	no	—
468	C₂×Q₈	—	4	2	no	—
469	C₂×Q₈	—	4	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	4	1	no	—
479	C₂²×C₄	—	4	1	no	—
480	C₂²×C₄	—	4	1	no	—
481	C₂²×C₄	—	4	1	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	1	no	—
485	C₂²×C₄	—	4	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	1	no	—
489	C₂²×C₄	—	4	1	no	—
490	C₂²×C₄	—	4	16	no	—
491	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
492	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
493	C₂²×C₄	#41×C₆₇	1	4	yes	—
494	C₂²×C₄	#41×C₆₇	1	4	yes	—
495	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
496	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
500	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
502	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	8	no	—
511	C₄⋊C₄	—	2	8	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	—	4	4	no	—
518	C₄⋊C₄	—	4	2	no	—
519	C₄⋊C₄	—	4	2	no	—
520	C₄⋊C₄	—	4	2	no	—
521	C₄⋊C₄	—	4	4	no	—
522	C₂²⋊C₄	—	2	8	no	—
523	C₂²⋊C₄	—	2	8	no	—
524	C₂²⋊C₄	—	2	8	no	—
525	C₂²⋊C₄	—	4	4	no	—
526	C₂²⋊C₄	—	4	4	no	—
527	C₂²⋊C₄	—	4	4	no	—
528	C₂²⋊C₄	—	4	4	no	—
529	C₂²⋊C₄	C₂×Q₈×C₆₇	1	16	yes	—
530	C₄²	—	2	4	no	—
531	C₄²	—	2	4	no	—
532	C₄²	—	4	2	no	—
533	C₄²	—	4	2	no	—
534	C₄²	—	4	4	no	—
535	C₄²	—	4	4	no	—
536	C₂³	—	2	1	no	—
537	C₂³	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	—	2	8	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	4	4	no	—
549	C₂³	—	4	4	no	—
550	C₂³	—	4	4	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	4	4	no	—
553	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
554	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
555	C₂³	C₂×#41×C₆₇	1	2	yes	—
556	C₂³	C₂×#41×C₆₇	1	2	yes	—
557	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
558	Q₈	—	4	8	no	—
559	Q₈	—	8	4	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	4	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	4	8	no	—
593	C₂×C₄	—	4	8	no	—
594	C₂×C₄	—	4	4	no	—
595	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
596	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
597	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
598	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
599	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
600	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	2	no	—
607	C₂²	—	2	4	no	—
608	C₂²	—	2	4	no	—
609	C₂²	—	2	4	no	—
610	C₂²	—	4	16	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	2	no	—
615	C₂²	#533×C₆₇	1	4	yes	—
616	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
617	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
618	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
619	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
620	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
621	C₂²	#585×C₆₇	1	1	yes	characteristic
622	C₂²	#585×C₆₇	1	1	yes	characteristic
623	C₄	—	2	4	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	4	no	—
627	C₄	—	2	4	no	—
628	C₄	—	2	4	no	—
629	C₄	—	4	4	no	—
630	C₄	—	4	2	no	—
631	C₄	—	4	4	no	—
632	C₄	—	4	4	no	—
633	C₄	—	4	4	no	—
634	C₂	—	2	4	no	—
635	C₂	—	4	4	no	—
636	C₂	C₂×#533×C₆₇	1	1	yes	characteristic
637	C₂	C₆₇×#3459	1	1	yes	characteristic
638	C₂	C₆₇×#3460	1	1	yes	characteristic
639	C₂	C₆₇×#3490	1	1	yes	characteristic
640	C₂	C₆₇×#3493	1	1	yes	characteristic
641	C₂	C₆₇×#3501	1	1	yes	characteristic
642	C₂	C₆₇×#3501	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2851	C₂	semidirect
3. C₆₇×#2659	C₂	semidirect
4. C₆₇×#2659	C₂	semidirect
6. C₆₇×#2574	C₂	semidirect
7. C₆₇×#2556	C₂	semidirect
8. C₆₇×#2522	C₂	semidirect
9. C₆₇×#2465	C₂	semidirect
10. C₆₇×#2329	C₂	semidirect
12. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
15. C₂×#125×C₆₇	C₂²	semidirect
20. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
26. #398×C₆₇	C₂²	semidirect
27. #398×C₆₇	C₂²	semidirect
30. #395×C₆₇	C₂²	semidirect
31. #395×C₆₇	C₂²	semidirect
33. #391×C₆₇	C₂²	semidirect
34. #391×C₆₇	C₂²	semidirect
35. #389×C₆₇	C₂²	semidirect
36. #389×C₆₇	C₂²	semidirect
37. #389×C₆₇	C₂²	semidirect
38. #389×C₆₇	C₂²	semidirect
43. #387×C₆₇	C₂²	semidirect
44. #387×C₆₇	C₂²	semidirect
45. #387×C₆₇	C₂²	semidirect
46. #387×C₆₇	C₂²	semidirect
47. #387×C₆₇	C₂²	semidirect
52. #382×C₆₇	C₂²	semidirect
60. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
61. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
104. C₂×C₂²⋊C₄×C₆₇	Q₈	semidirect
312. #35012	C₆₇	direct (N × H)
322. #2851	C₂×C₆₇	semidirect
324. #2659	C₂×C₆₇	semidirect
325. #2659	C₂×C₆₇	semidirect
327. #2574	C₂×C₆₇	semidirect
328. #2556	C₂×C₆₇	semidirect
329. #2522	C₂×C₆₇	semidirect
330. #2465	C₂×C₆₇	semidirect
331. #2329	C₂×C₆₇	semidirect
333. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
335. C₆₇	#35012	direct (N × H)
337. C₂×#125	C₂²×C₆₇	semidirect
342. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
348. #398	C₂²×C₆₇	semidirect
349. #398	C₂²×C₆₇	semidirect
352. #395	C₂²×C₆₇	semidirect
353. #395	C₂²×C₆₇	semidirect
355. #391	C₂²×C₆₇	semidirect
356. #391	C₂²×C₆₇	semidirect
357. #389	C₂²×C₆₇	semidirect
358. #389	C₂²×C₆₇	semidirect
359. #389	C₂²×C₆₇	semidirect
360. #389	C₂²×C₆₇	semidirect
365. #387	C₂²×C₆₇	semidirect
366. #387	C₂²×C₆₇	semidirect
367. #387	C₂²×C₆₇	semidirect
368. #387	C₂²×C₆₇	semidirect
369. #387	C₂²×C₆₇	semidirect
374. #382	C₂²×C₆₇	semidirect
382. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
383. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
426. C₂×C₂²⋊C₄	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2851	2
C₆₇×#2848	2
C₆₇×#2659	2
C₆₇×#2626	2
C₆₇×#2574	2
C₆₇×#2556	2
C₆₇×#2522	2
C₆₇×#2465	2
C₆₇×#2329	2
C₆₇×#2312	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
#35012	67

Subgroups — C67×#35012 #30714263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35012 #30714263