Subgroups — C₆₇×#22654 #30701905

Order 17152.

1010 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2352	C₂	1	2	yes	maximal
5	C₆₇×#1949	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1854	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1852	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₈×D₄×C₆₇	C₄	1	8	yes	complemented
18	#427×C₆₇	—	2	4	no	—
19	C₂×#107×C₆₇	C₂²	1	2	yes	—
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×#106×C₆₇	—	2	4	no	—
22	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	#388×C₆₇	—	2	2	no	—
32	#387×C₆₇	—	2	2	no	—
33	#386×C₆₇	—	2	2	no	—
34	#382×C₆₇	C₂²	1	2	yes	—
35	#380×C₆₇	C₂²	1	2	yes	—
36	#380×C₆₇	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	—	2	2	no	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	—	2	2	no	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₆₇	Q₈	1	4	yes	—
58	#117×C₆₇	—	2	4	no	—
59	#117×C₆₇	—	4	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	—	2	4	no	—
77	#107×C₆₇	—	4	2	no	—
78	#106×C₆₇	—	2	2	no	—
79	#106×C₆₇	—	2	2	no	—
80	#106×C₆₇	—	2	2	no	—
81	#106×C₆₇	—	2	4	no	—
82	#106×C₆₇	D₄	1	4	yes	complemented
83	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
84	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
85	C₂²⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
86	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
87	#95×C₆₇	—	2	2	no	—
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂×D₄×C₆₇	—	2	4	no	—
95	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	4	4	no	—
109	C₂²×C₄×C₆₇	—	4	4	no	—
110	C₂²×C₄×C₆₇	—	8	2	no	—
111	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₆₇	—	2	1	no	—
121	C₂×C₈×C₆₇	—	2	1	no	—
122	C₂×C₈×C₆₇	—	2	1	no	—
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	8	no	—
127	C₂×C₈×C₆₇	—	2	8	no	—
128	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₆₇	#41	1	2	yes	—
134	C₄⋊C₄×C₆₇	—	2	2	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	2	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	4	4	no	—
142	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
143	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₂²⋊C₄×C₆₇	—	4	4	no	—
146	C₂²⋊C₄×C₆₇	—	8	2	no	—
147	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
148	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
149	C₄²×C₆₇	—	4	4	no	—
150	C₄²×C₆₇	—	8	2	no	—
151	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
152	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	1	no	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
161	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
162	C₂³×C₆₇	#135	1	4	yes	—
163	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	D₄×C₆₇	—	2	8	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	2	1	no	—
167	C₂×C₄×C₆₇	—	2	1	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	8	8	no	—
192	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
202	C₈×C₆₇	—	2	2	no	—
203	C₈×C₆₇	—	2	2	no	—
204	C₈×C₆₇	—	2	2	no	—
205	C₈×C₆₇	C₄×D₄	1	4	yes	—
206	C₂²×C₆₇	—	2	4	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	2	2	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	#427	1	4	yes	complemented
216	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
217	C₂²×C₆₇	D₄²	1	1	yes	characteristic
218	C₂²×C₆₇	#547	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#135	1	2	yes	—
220	C₂²×C₆₇	#577	1	2	yes	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	2	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₄×C₆₇	—	8	2	no	—
227	C₄×C₆₇	—	8	2	no	—
228	C₄×C₆₇	—	8	2	no	—
229	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₄×C₆₇	#520	1	1	yes	characteristic
231	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
232	C₄×C₆₇	#554	1	1	yes	characteristic
233	#22654	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₆₇	—	2	4	no	—
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
237	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
238	C₂×C₆₇	#2912	1	1	yes	characteristic
239	C₂×C₆₇	#3282	1	2	yes	—
240	C₂×C₆₇	#3337	1	2	yes	—
241	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
242	C₂×#427	C₂×C₆₇	1	1	yes	characteristic, complemented
243	C₂²×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#2352	C₂×C₆₇	1	2	yes	—
245	#1949	C₂×C₆₇	1	2	yes	complemented
246	#1854	C₂×C₆₇	1	2	yes	complemented
247	#1852	C₂×C₆₇	1	2	yes	complemented
248	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1807	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₄×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₆₇	#22654	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
256	C₂×#117	C₂²×C₆₇	1	2	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
258	C₈×D₄	C₄×C₆₇	1	8	yes	complemented
259	#427	—	2	4	no	—
260	C₂×#107	C₂²×C₆₇	1	2	yes	—
261	C₂×#107	C₂²×C₆₇	1	2	yes	—
262	C₂×#106	—	2	4	no	—
263	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
266	C₂×#106	C₂²×C₆₇	1	2	yes	—
267	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
272	#388	—	2	2	no	—
273	#387	—	2	2	no	—
274	#386	—	2	2	no	—
275	#382	C₂²×C₆₇	1	2	yes	—
276	#380	C₂²×C₆₇	1	2	yes	—
277	#380	C₂²×C₆₇	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
279	C₄×C₂²⋊C₄	—	2	2	no	—
280	#340	C₂²×C₆₇	1	2	yes	—
281	#340	C₂²×C₆₇	1	2	yes	—
282	#340	C₂²×C₆₇	1	2	yes	—
283	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	2	no	—
291	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
292	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
293	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
298	C₄×D₄	Q₈×C₆₇	1	4	yes	—
299	#117	—	2	4	no	—
300	#117	—	4	2	no	—
301	C₂×C₄⋊C₄	—	2	4	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	4	2	no	—
304	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	—	2	2	no	—
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
317	#107	—	2	4	no	—
318	#107	—	4	2	no	—
319	#106	—	2	2	no	—
320	#106	—	2	2	no	—
321	#106	—	2	2	no	—
322	#106	—	2	4	no	—
323	#106	D₄×C₆₇	1	4	yes	complemented
324	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
325	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	—
326	C₂²⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
327	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
328	#95	—	2	2	no	—
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	#95	—	2	2	no	—
333	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
334	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	4	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	8	2	no	—
352	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	8	no	—
368	C₂×C₈	—	2	8	no	—
369	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
372	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
373	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
374	C₂×C₈	#41×C₆₇	1	2	yes	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	4	4	no	—
383	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
384	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
385	C₂²⋊C₄	—	4	4	no	—
386	C₂²⋊C₄	—	4	4	no	—
387	C₂²⋊C₄	—	8	2	no	—
388	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
389	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
390	C₄²	—	4	4	no	—
391	C₄²	—	8	2	no	—
392	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
393	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
402	C₂³	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
403	C₂³	#135×C₆₇	1	4	yes	—
404	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
405	D₄	—	2	8	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	8	8	no	—
433	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
434	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
435	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
436	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
438	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
439	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
441	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
442	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
443	C₈	—	2	2	no	—
444	C₈	—	2	2	no	—
445	C₈	—	2	2	no	—
446	C₈	C₄×D₄×C₆₇	1	4	yes	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	#427×C₆₇	1	4	yes	complemented
457	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
458	C₂²	D₄²×C₆₇	1	1	yes	characteristic
459	C₂²	#547×C₆₇	1	1	yes	characteristic
460	C₂²	C₂×#135×C₆₇	1	2	yes	—
461	C₂²	#577×C₆₇	1	2	yes	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	4	no	—
467	C₄	—	8	2	no	—
468	C₄	—	8	2	no	—
469	C₄	—	8	2	no	—
470	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
471	C₄	#520×C₆₇	1	1	yes	characteristic
472	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
473	C₄	#554×C₆₇	1	1	yes	characteristic
474	C₂	—	2	4	no	—
475	C₂	—	4	2	no	—
476	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
477	C₂	C₂×#427×C₆₇	1	1	yes	characteristic
478	C₂	C₆₇×#2912	1	1	yes	characteristic
479	C₂	C₆₇×#3282	1	2	yes	—
480	C₂	C₆₇×#3337	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₆₇	C₂	semidirect
3. C₂²×#106×C₆₇	C₂	semidirect
5. C₆₇×#1949	C₂	semidirect
6. C₆₇×#1854	C₂	semidirect
7. C₆₇×#1852	C₂	semidirect
8. C₆₇×#1846	C₂	semidirect
9. C₆₇×#1807	C₂	semidirect
10. C₆₇×#1807	C₂	semidirect
11. C₄×#106×C₆₇	C₂	semidirect
17. C₈×D₄×C₆₇	C₄	semidirect
51. C₂²×C₈×C₆₇	C₂×C₄	semidirect
82. #106×C₆₇	D₄	semidirect
83. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
85. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
86. C₄×C₈×C₆₇	C₂×C₄	semidirect
215. C₂²×C₆₇	#427	semidirect
233. #22654	C₆₇	direct (N × H)
242. C₂×#427	C₂×C₆₇	semidirect
243. C₂²×#106	C₂×C₆₇	semidirect
245. #1949	C₂×C₆₇	semidirect
246. #1854	C₂×C₆₇	semidirect
247. #1852	C₂×C₆₇	semidirect
248. #1846	C₂×C₆₇	semidirect
249. #1807	C₂×C₆₇	semidirect
250. #1807	C₂×C₆₇	semidirect
251. C₄×#106	C₂×C₆₇	semidirect
252. C₆₇	#22654	direct (N × H)
258. C₈×D₄	C₄×C₆₇	semidirect
292. C₂²×C₈	C₂×C₄×C₆₇	semidirect
323. #106	D₄×C₆₇	semidirect
324. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
326. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
327. C₄×C₈	C₂×C₄×C₆₇	semidirect
456. C₂²	#427×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₆₇	2
C₂×#427×C₆₇	2
C₂²×#106×C₆₇	2
C₆₇×#2352	2
C₆₇×#1949	2
C₆₇×#1854	2
C₆₇×#1852	2
C₆₇×#1846	2
C₆₇×#1807	2
C₄×#106×C₆₇	2
#22654	67

Subgroups — C67×#22654 #30701905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22654 #30701905