Subgroups — C₆₇×#33604 #30712855

Order 17152.

690 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3050	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2220	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2219	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2218	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₆₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#525×C₆₇	—	2	2	no	—
12	#508×C₆₇	—	2	4	no	—
13	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₆₇	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	C₂×M₃₂×C₆₇	—	2	4	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	#467×C₆₇	C₂²	1	4	yes	—
22	#466×C₆₇	C₂²	1	4	yes	—
23	#465×C₆₇	—	2	4	no	—
24	#427×C₆₇	C₂²	1	2	yes	—
25	#416×C₆₇	—	2	2	no	—
26	#367×C₆₇	C₂²	1	2	yes	complemented
27	#366×C₆₇	C₂²	1	2	yes	complemented
28	#358×C₆₇	C₂²	1	4	yes	complemented
29	#357×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#41×C₆₇	—	2	2	no	—
31	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
32	#135×C₆₇	—	2	4	no	—
33	#135×C₆₇	—	2	4	no	—
34	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₆₇	—	2	2	no	—
36	C₂×D₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#123×C₆₇	—	4	2	no	—
44	C₂²⋊Q₈×C₆₇	—	2	4	no	—
45	#121×C₆₇	—	2	4	no	—
46	#120×C₆₇	—	4	2	no	—
47	C₄×D₄×C₆₇	—	4	4	no	—
48	#117×C₆₇	—	2	2	no	—
49	Q₃₂×C₆₇	—	2	2	no	—
50	SD₃₂×C₆₇	—	2	4	no	—
51	D₁₆×C₆₇	—	2	2	no	—
52	M₃₂×C₆₇	—	2	4	no	—
53	M₃₂×C₆₇	—	2	4	no	—
54	C₂×C₁₆×C₆₇	—	2	2	no	—
55	C₂×C₁₆×C₆₇	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
57	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
59	#107×C₆₇	C₂³	1	4	yes	—
60	#106×C₆₇	—	2	2	no	—
61	#103×C₆₇	—	2	4	no	—
62	#102×C₆₇	—	2	4	no	—
63	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
64	#41×C₆₇	—	4	4	no	—
65	#41×C₆₇	—	4	4	no	—
66	C₂×Q₈×C₆₇	—	2	2	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	4	4	no	—
69	C₂×D₄×C₆₇	—	4	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	4	2	no	—
73	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	D₈	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₆₇	—	2	2	no	—
78	Q₁₆×C₆₇	—	2	2	no	—
79	SD₁₆×C₆₇	—	4	4	no	—
80	D₈×C₆₇	—	2	2	no	—
81	D₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	4	4	no	—
93	C₂²⋊C₄×C₆₇	—	4	2	no	—
94	C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₂²⋊C₄×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	—	4	2	no	—
97	C₁₆×C₆₇	—	2	2	no	—
98	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
99	C₁₆×C₆₇	C₂×D₄	1	2	yes	—
100	C₂³×C₆₇	—	2	2	no	—
101	C₂³×C₆₇	—	4	2	no	—
102	C₂³×C₆₇	—	4	2	no	—
103	C₂³×C₆₇	C₂×D₈	1	2	yes	—
104	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₆₇	—	4	2	no	—
106	Q₈×C₆₇	—	4	2	no	—
107	D₄×C₆₇	—	4	2	no	—
108	D₄×C₆₇	—	4	2	no	—
109	D₄×C₆₇	—	8	8	no	—
110	D₄×C₆₇	—	8	4	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	4	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	8	4	no	—
118	C₂×C₄×C₆₇	#121	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₆₇	#135	1	2	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₆₇	—	2	2	no	—
124	C₈×C₆₇	#121	1	2	yes	—
125	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	4	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	4	4	no	—
131	C₂²×C₆₇	—	8	4	no	—
132	C₂²×C₆₇	—	8	2	no	—
133	C₂²×C₆₇	#466	1	2	yes	—
134	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₆₇	—	2	2	no	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	—	8	2	no	—
138	C₄×C₆₇	#466	1	2	yes	—
139	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
140	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
141	#33604	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	—	2	2	no	—
143	C₂×C₆₇	—	4	2	no	—
144	C₂×C₆₇	—	16	2	no	—
145	C₂×C₆₇	C₂×#466	1	1	yes	characteristic
146	C₂×C₆₇	#3414	1	2	yes	—
147	C₂×#508	C₂×C₆₇	1	1	yes	characteristic, complemented
148	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
149	#3050	C₂×C₆₇	1	2	yes	complemented
150	#2220	C₂×C₆₇	1	4	yes	complemented
151	#2219	C₂×C₆₇	1	2	yes	complemented
152	#2218	C₂×C₆₇	1	2	yes	complemented
153	#2158	C₂×C₆₇	1	1	yes	characteristic, complemented
154	#2138	C₂×C₆₇	1	2	yes	complemented
155	C₆₇	#33604	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×#135	C₂²×C₆₇	1	2	yes	—
157	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
158	#525	—	2	2	no	—
159	#508	—	2	4	no	—
160	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic, complemented
161	C₂×SD₃₂	C₂²×C₆₇	1	2	yes	complemented
162	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
163	C₂×M₃₂	—	2	4	no	—
164	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
165	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
166	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
167	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
168	#467	C₂²×C₆₇	1	4	yes	—
169	#466	C₂²×C₆₇	1	4	yes	—
170	#465	—	2	4	no	—
171	#427	C₂²×C₆₇	1	2	yes	—
172	#416	—	2	2	no	—
173	#367	C₂²×C₆₇	1	2	yes	complemented
174	#366	C₂²×C₆₇	1	2	yes	complemented
175	#358	C₂²×C₆₇	1	4	yes	complemented
176	#357	C₂²×C₆₇	1	4	yes	complemented
177	C₂×#41	—	2	2	no	—
178	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
179	#135	—	2	4	no	—
180	#135	—	2	4	no	—
181	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×D₈	C₂³×C₆₇	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₆₇	1	2	yes	—
188	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
189	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
190	#123	—	4	2	no	—
191	C₂²⋊Q₈	—	2	4	no	—
192	#121	—	2	4	no	—
193	#120	—	4	2	no	—
194	C₄×D₄	—	4	4	no	—
195	#117	—	2	2	no	—
196	Q₃₂	—	2	2	no	—
197	SD₃₂	—	2	4	no	—
198	D₁₆	—	2	2	no	—
199	M₃₂	—	2	4	no	—
200	M₃₂	—	2	4	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	D₄×C₆₇	1	4	yes	complemented
203	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
204	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
205	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
206	#107	C₂³×C₆₇	1	4	yes	—
207	#106	—	2	2	no	—
208	#103	—	2	4	no	—
209	#102	—	2	4	no	—
210	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
211	#41	—	4	4	no	—
212	#41	—	4	4	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	4	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	D₈×C₆₇	1	2	yes	—
221	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
224	Q₁₆	—	2	2	no	—
225	Q₁₆	—	2	2	no	—
226	SD₁₆	—	4	4	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
236	C₂×C₈	#41×C₆₇	1	2	yes	—
237	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	4	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
246	C₁₆	C₂×D₄×C₆₇	1	2	yes	—
247	C₂³	—	2	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×D₈×C₆₇	1	2	yes	—
251	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	8	no	—
257	D₄	—	8	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	#121×C₆₇	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
267	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
268	C₂×C₄	#135×C₆₇	1	2	yes	—
269	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	#121×C₆₇	1	2	yes	—
272	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
273	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	#466×C₆₇	1	2	yes	—
281	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	#466×C₆₇	1	2	yes	—
286	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
287	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×#466×C₆₇	1	1	yes	characteristic
292	C₂	C₆₇×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₆₇	C₂	semidirect
2. C₂²×M₃₂×C₆₇	C₂	semidirect
3. C₆₇×#3050	C₂	semidirect
4. C₆₇×#2220	C₂	semidirect
5. C₆₇×#2219	C₂	semidirect
6. C₆₇×#2218	C₂	semidirect
7. C₆₇×#2158	C₂	semidirect
8. C₆₇×#2138	C₂	semidirect
13. C₂×Q₃₂×C₆₇	C₂²	semidirect
14. C₂×SD₃₂×C₆₇	C₂²	semidirect
15. C₂×D₁₆×C₆₇	C₂²	semidirect
26. #367×C₆₇	C₂²	semidirect
27. #366×C₆₇	C₂²	semidirect
28. #358×C₆₇	C₂²	semidirect
29. #357×C₆₇	C₂²	semidirect
55. C₂×C₁₆×C₆₇	D₄	semidirect
141. #33604	C₆₇	direct (N × H)
147. C₂×#508	C₂×C₆₇	semidirect
148. C₂²×M₃₂	C₂×C₆₇	semidirect
149. #3050	C₂×C₆₇	semidirect
150. #2220	C₂×C₆₇	semidirect
151. #2219	C₂×C₆₇	semidirect
152. #2218	C₂×C₆₇	semidirect
153. #2158	C₂×C₆₇	semidirect
154. #2138	C₂×C₆₇	semidirect
155. C₆₇	#33604	direct (N × H)
160. C₂×Q₃₂	C₂²×C₆₇	semidirect
161. C₂×SD₃₂	C₂²×C₆₇	semidirect
162. C₂×D₁₆	C₂²×C₆₇	semidirect
173. #367	C₂²×C₆₇	semidirect
174. #366	C₂²×C₆₇	semidirect
175. #358	C₂²×C₆₇	semidirect
176. #357	C₂²×C₆₇	semidirect
202. C₂×C₁₆	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₆₇×#3050	2
C₆₇×#2220	2
C₆₇×#2219	2
C₆₇×#2218	2
C₆₇×#2158	2
C₆₇×#2138	2
#33604	67

Subgroups — C67×#33604 #30712855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33604 #30712855