Subgroups — C₆₇×#20645 #30699896

Order 17152.

1514 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#416×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1804	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1803	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1802	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1735	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	C₄×#41×C₆₇	—	2	4	no	—
13	C₄×#41×C₆₇	C₄	1	8	yes	complemented
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	—	2	1	no	—
21	C₂×#117×C₆₇	C₂²	1	2	yes	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	—	2	1	no	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
28	#416×C₆₇	—	2	8	no	—
29	C₂×#103×C₆₇	C₂²	1	4	yes	—
30	C₂×#102×C₆₇	C₂²	1	4	yes	—
31	#384×C₆₇	C₂²	1	2	yes	—
32	#382×C₆₇	C₂²	1	2	yes	—
33	#380×C₆₇	—	2	1	no	—
34	#380×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
37	#340×C₆₇	C₂²	1	4	yes	—
38	#340×C₆₇	C₂²	1	4	yes	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	2	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×#41×C₆₇	D₄	1	4	yes	—
44	C₂×#41×C₆₇	Q₈	1	4	yes	—
45	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	—	2	1	no	—
47	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	—	2	1	no	—
49	C₂³×C₄×C₆₇	—	2	1	no	—
50	C₂³×C₄×C₆₇	—	2	1	no	—
51	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	—	2	4	no	—
55	C₂²×C₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	—	2	4	no	—
57	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
61	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
62	C₄×D₄×C₆₇	—	2	8	no	—
63	C₄×D₄×C₆₇	—	2	8	no	—
64	C₄×D₄×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
68	#117×C₆₇	—	2	2	no	—
69	#117×C₆₇	—	2	8	no	—
70	#117×C₆₇	—	2	4	no	—
71	#117×C₆₇	—	2	4	no	—
72	#117×C₆₇	C₂×C₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	—	2	8	no	—
88	C₂×C₄²×C₆₇	—	2	2	no	—
89	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
90	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	D₄	1	4	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₄⋊C₈×C₆₇	—	2	4	no	—
96	C₄⋊C₈×C₆₇	—	2	4	no	—
97	#103×C₆₇	—	2	8	no	—
98	#102×C₆₇	—	2	8	no	—
99	#95×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
103	#41×C₆₇	—	2	8	no	—
104	#41×C₆₇	—	2	8	no	—
105	#41×C₆₇	—	2	8	no	—
106	#41×C₆₇	C₄⋊C₄	1	16	yes	complemented
107	C₂×Q₈×C₆₇	—	2	1	no	—
108	C₂×Q₈×C₆₇	—	2	1	no	—
109	C₂×Q₈×C₆₇	—	2	1	no	—
110	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
111	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
112	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
113	C₂×D₄×C₆₇	—	2	4	no	—
114	C₂×D₄×C₆₇	—	2	4	no	—
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	1	no	—
117	C₂×D₄×C₆₇	—	2	1	no	—
118	C₂×D₄×C₆₇	—	2	1	no	—
119	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
120	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	4	2	no	—
133	C₂²×C₄×C₆₇	—	4	4	no	—
134	C₂²×C₄×C₆₇	—	4	4	no	—
135	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
146	C₂²×C₄×C₆₇	#41	1	2	yes	—
147	C₂²×C₄×C₆₇	#41	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂×C₈×C₆₇	—	2	4	no	—
150	C₂×C₈×C₆₇	—	2	4	no	—
151	C₂×C₈×C₆₇	—	2	4	no	—
152	C₂×C₈×C₆₇	—	4	8	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	1	no	—
156	C₄⋊C₄×C₆₇	—	2	1	no	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	4	4	no	—
160	C₄⋊C₄×C₆₇	—	4	4	no	—
161	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
162	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
163	C₂²⋊C₄×C₆₇	—	2	4	no	—
164	C₂²⋊C₄×C₆₇	—	2	4	no	—
165	C₂²⋊C₄×C₆₇	—	2	4	no	—
166	C₂²⋊C₄×C₆₇	—	4	4	no	—
167	C₄²×C₆₇	—	2	4	no	—
168	C₄²×C₆₇	—	2	4	no	—
169	C₄²×C₆₇	—	2	4	no	—
170	C₄²×C₆₇	—	4	4	no	—
171	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
172	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
173	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
174	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
175	C₄²×C₆₇	C₂×D₄	1	2	yes	—
176	C₄²×C₆₇	C₂×D₄	1	2	yes	—
177	C₂³×C₆₇	—	2	2	no	—
178	C₂³×C₆₇	—	2	2	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	—	4	2	no	—
184	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
185	C₂³×C₆₇	#121	1	2	yes	—
186	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
187	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₆₇	—	2	2	no	—
189	Q₈×C₆₇	—	2	2	no	—
190	Q₈×C₆₇	—	2	2	no	—
191	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
192	D₄×C₆₇	—	2	16	no	—
193	D₄×C₆₇	—	2	2	no	—
194	D₄×C₆₇	—	2	2	no	—
195	D₄×C₆₇	—	2	2	no	—
196	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	2	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	16	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	4	2	no	—
211	C₂×C₄×C₆₇	—	4	8	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	4	4	no	—
215	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
217	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
219	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
221	C₂×C₄×C₆₇	#117	1	4	yes	—
222	C₂×C₄×C₆₇	#121	1	4	yes	—
223	C₂×C₄×C₆₇	#121	1	2	yes	—
224	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
225	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
226	C₂×C₄×C₆₇	#135	1	4	yes	—
227	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
228	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
233	C₈×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	2	2	no	—
236	C₂²×C₆₇	—	2	2	no	—
237	C₂²×C₆₇	—	2	4	no	—
238	C₂²×C₆₇	—	2	4	no	—
239	C₂²×C₆₇	—	4	4	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	4	no	—
243	C₂²×C₆₇	#380	1	4	yes	—
244	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
246	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂²×C₆₇	C₂×#135	1	2	yes	—
248	C₂²×C₆₇	#575	1	2	yes	—
249	C₄×C₆₇	—	2	2	no	—
250	C₄×C₆₇	—	2	2	no	—
251	C₄×C₆₇	—	2	2	no	—
252	C₄×C₆₇	—	2	4	no	—
253	C₄×C₆₇	—	4	2	no	—
254	C₄×C₆₇	—	4	2	no	—
255	C₄×C₆₇	—	8	4	no	—
256	C₄×C₆₇	#380	1	4	yes	—
257	C₄×C₆₇	#416	1	4	yes	complemented
258	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
259	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
260	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
261	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
262	#20645	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
263	C₂×C₆₇	—	2	4	no	—
264	C₂×C₆₇	—	4	4	no	—
265	C₂×C₆₇	C₂×#380	1	1	yes	characteristic
266	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
267	C₂×C₆₇	#2896	1	1	yes	characteristic
268	C₂×C₆₇	#3035	1	2	yes	—
269	C₂×C₆₇	#3076	1	2	yes	—
270	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
271	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
272	C₂×#416	C₂×C₆₇	1	2	yes	—
273	#2285	C₂×C₆₇	1	1	yes	characteristic, complemented
274	#1805	C₂×C₆₇	1	2	yes	complemented
275	#1804	C₂×C₆₇	1	2	yes	complemented
276	#1803	C₂×C₆₇	1	2	yes	complemented
277	#1802	C₂×C₆₇	1	2	yes	complemented
278	#1735	C₂×C₆₇	1	2	yes	complemented
279	C₆₇	#20645	1	1	yes	characteristic, direct product factor, Sylow
280	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
281	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
282	C₄×#41	—	2	4	no	—
283	C₄×#41	C₄×C₆₇	1	8	yes	complemented
284	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
285	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₄×D₄	—	2	1	no	—
287	C₂×C₄×D₄	—	2	1	no	—
288	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
289	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#117	—	2	1	no	—
291	C₂×#117	C₂²×C₆₇	1	2	yes	—
292	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
293	C₂²×C₄²	—	2	1	no	—
294	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
295	C₂×C₄⋊C₈	—	2	4	no	—
296	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
297	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
298	#416	—	2	8	no	—
299	C₂×#103	C₂²×C₆₇	1	4	yes	—
300	C₂×#102	C₂²×C₆₇	1	4	yes	—
301	#384	C₂²×C₆₇	1	2	yes	—
302	#382	C₂²×C₆₇	1	2	yes	—
303	#380	—	2	1	no	—
304	#380	—	2	1	no	—
305	C₄×C₄⋊C₄	—	2	1	no	—
306	C₄×C₄⋊C₄	—	2	1	no	—
307	#340	C₂²×C₆₇	1	4	yes	—
308	#340	C₂²×C₆₇	1	4	yes	—
309	C₂×#41	—	2	2	no	—
310	C₂×#41	—	2	2	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
313	C₂×#41	D₄×C₆₇	1	4	yes	—
314	C₂×#41	Q₈×C₆₇	1	4	yes	—
315	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
318	C₂³×C₄	—	2	1	no	—
319	C₂³×C₄	—	2	1	no	—
320	C₂³×C₄	—	2	1	no	—
321	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
322	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	—	2	4	no	—
326	C₂²×C₈	—	2	4	no	—
327	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
331	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
332	C₄×D₄	—	2	8	no	—
333	C₄×D₄	—	2	8	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
337	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
338	#117	—	2	2	no	—
339	#117	—	2	8	no	—
340	#117	—	2	4	no	—
341	#117	—	2	4	no	—
342	#117	C₂×C₄×C₆₇	1	4	yes	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	2	no	—
347	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
348	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₂²⋊C₄	—	2	1	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	—	2	1	no	—
356	C₂×C₄²	—	2	1	no	—
357	C₂×C₄²	—	2	8	no	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
360	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
361	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
362	C₂×C₄²	D₄×C₆₇	1	4	yes	—
363	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
365	C₄⋊C₈	—	2	4	no	—
366	C₄⋊C₈	—	2	4	no	—
367	#103	—	2	8	no	—
368	#102	—	2	8	no	—
369	#95	—	2	2	no	—
370	#95	—	2	2	no	—
371	C₂⁴	—	2	1	no	—
372	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
373	#41	—	2	8	no	—
374	#41	—	2	8	no	—
375	#41	—	2	8	no	—
376	#41	C₄⋊C₄×C₆₇	1	16	yes	complemented
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
381	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
382	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
390	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
391	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
406	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
415	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
416	C₂²×C₄	#41×C₆₇	1	2	yes	—
417	C₂²×C₄	#41×C₆₇	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	—	4	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	4	4	no	—
430	C₄⋊C₄	—	4	4	no	—
431	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
432	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₄²	—	2	4	no	—
438	C₄²	—	2	4	no	—
439	C₄²	—	2	4	no	—
440	C₄²	—	4	4	no	—
441	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
442	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
443	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
444	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
445	C₄²	C₂×D₄×C₆₇	1	2	yes	—
446	C₄²	C₂×D₄×C₆₇	1	2	yes	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
455	C₂³	#121×C₆₇	1	2	yes	—
456	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
457	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	—	2	2	no	—
461	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
462	D₄	—	2	16	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	2	no	—
466	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	16	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
486	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
487	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
488	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
489	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
490	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
491	C₂×C₄	#117×C₆₇	1	4	yes	—
492	C₂×C₄	#121×C₆₇	1	4	yes	—
493	C₂×C₄	#121×C₆₇	1	2	yes	—
494	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
495	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
496	C₂×C₄	#135×C₆₇	1	4	yes	—
497	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
500	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
501	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
502	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
503	C₈	—	4	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	4	4	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	#380×C₆₇	1	4	yes	—
514	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
515	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
516	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
517	C₂²	C₂×#135×C₆₇	1	2	yes	—
518	C₂²	#575×C₆₇	1	2	yes	—
519	C₄	—	2	4	no	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	8	4	no	—
526	C₄	#380×C₆₇	1	4	yes	—
527	C₄	#416×C₆₇	1	4	yes	complemented
528	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
529	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
530	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
531	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
532	C₂	—	2	4	no	—
533	C₂	—	4	4	no	—
534	C₂	C₂×#380×C₆₇	1	1	yes	characteristic
535	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
536	C₂	C₆₇×#2896	1	1	yes	characteristic
537	C₂	C₆₇×#3035	1	2	yes	—
538	C₂	C₆₇×#3076	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
4. C₆₇×#2285	C₂	semidirect
5. C₆₇×#1805	C₂	semidirect
6. C₆₇×#1804	C₂	semidirect
7. C₆₇×#1803	C₂	semidirect
8. C₆₇×#1802	C₂	semidirect
9. C₆₇×#1735	C₂	semidirect
13. C₄×#41×C₆₇	C₄	semidirect
106. #41×C₆₇	C₄⋊C₄	semidirect
257. C₄×C₆₇	#416	semidirect
262. #20645	C₆₇	direct (N × H)
271. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
273. #2285	C₂×C₆₇	semidirect
274. #1805	C₂×C₆₇	semidirect
275. #1804	C₂×C₆₇	semidirect
276. #1803	C₂×C₆₇	semidirect
277. #1802	C₂×C₆₇	semidirect
278. #1735	C₂×C₆₇	semidirect
279. C₆₇	#20645	direct (N × H)
283. C₄×#41	C₄×C₆₇	semidirect
376. #41	C₄⋊C₄×C₆₇	semidirect
527. C₄	#416×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#2285	2
C₆₇×#1805	2
C₆₇×#1804	2
C₆₇×#1803	2
C₆₇×#1802	2
C₆₇×#1735	2
#20645	67

Subgroups — C67×#20645 #30699896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20645 #30699896