Subgroups — C₆₇×#17895 #30697146

Order 17152.

710 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2975	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1594	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1589	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1478	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
10	C₄×D₈×C₆₇	C₄	1	2	yes	—
11	C₄×D₈×C₆₇	C₄	1	2	yes	—
12	C₄×D₈×C₆₇	C₂²	1	2	yes	—
13	C₄×D₈×C₆₇	C₂²	1	2	yes	—
14	#436×C₆₇	—	2	4	no	—
15	#436×C₆₇	C₂²	1	8	yes	complemented
16	#431×C₆₇	C₂²	1	1	yes	characteristic
17	#431×C₆₇	C₂²	1	2	yes	—
18	#431×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#102×C₆₇	C₄	1	2	yes	—
21	#407×C₆₇	—	2	4	no	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#335×C₆₇	C₂²	1	2	yes	complemented
24	#334×C₆₇	C₂²	1	2	yes	complemented
25	D₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
26	#322×C₆₇	C₂²	1	2	yes	complemented
27	C₈²×C₆₇	C₂²	1	2	yes	complemented
28	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	—	2	2	no	—
30	C₂×D₈×C₆₇	—	2	4	no	—
31	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₆₇	—	2	8	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	—	2	2	no	—
38	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₄×D₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	C₂×C₄	1	2	yes	—
44	#107×C₆₇	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₆₇	—	2	1	no	—
46	C₄⋊C₈×C₆₇	D₄	1	2	yes	—
47	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
48	#102×C₆₇	C₂×C₄	1	4	yes	—
49	#102×C₆₇	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₆₇	—	2	2	no	—
51	C₂²⋊C₈×C₆₇	—	2	1	no	—
52	C₂²⋊C₈×C₆₇	—	2	8	no	—
53	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
54	C₄×C₈×C₆₇	—	2	2	no	—
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂³	1	2	yes	—
58	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂⁴×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	—	2	4	no	—
65	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	4	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	8	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
73	D₈×C₆₇	—	2	4	no	—
74	D₈×C₆₇	—	2	4	no	—
75	M₁₆×C₆₇	—	8	8	no	—
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	4	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
80	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₆₇	—	2	4	no	—
88	C₂²⋊C₄×C₆₇	—	2	4	no	—
89	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₆₇	#41	1	1	yes	characteristic
92	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	—	4	4	no	—
95	C₂³×C₆₇	—	8	2	no	—
96	C₂³×C₆₇	—	8	2	no	—
97	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
98	D₄×C₆₇	—	2	4	no	—
99	D₄×C₆₇	—	2	4	no	—
100	D₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	4	1	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	—	8	4	no	—
106	C₂×C₄×C₆₇	—	8	2	no	—
107	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₆₇	#131	1	2	yes	—
112	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₆₇	—	2	1	no	—
116	C₈×C₆₇	—	2	1	no	—
117	C₈×C₆₇	—	2	4	no	—
118	C₈×C₆₇	—	4	2	no	—
119	C₈×C₆₇	—	4	4	no	—
120	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
121	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
122	C₂²×C₆₇	—	2	1	no	—
123	C₂²×C₆₇	—	2	1	no	—
124	C₂²×C₆₇	—	4	4	no	—
125	C₂²×C₆₇	—	4	4	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	—	8	8	no	—
128	C₂²×C₆₇	#436	1	2	yes	—
129	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₆₇	—	2	1	no	—
131	C₄×C₆₇	—	2	1	no	—
132	C₄×C₆₇	—	2	1	no	—
133	C₄×C₆₇	—	4	4	no	—
134	C₄×C₆₇	#436	1	2	yes	—
135	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
136	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₆₇	#576	1	2	yes	—
138	#17895	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₆₇	—	2	2	no	—
140	C₂×C₆₇	—	8	4	no	—
141	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
142	C₂×C₆₇	#2948	1	1	yes	characteristic
143	C₂×C₆₇	#2957	1	1	yes	characteristic
144	#2975	C₂×C₆₇	1	2	yes	complemented
145	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
146	#1594	C₂×C₆₇	1	4	yes	complemented
147	#1589	C₂×C₆₇	1	4	yes	complemented
148	#1564	C₂×C₆₇	1	1	yes	characteristic, complemented
149	#1478	C₂×C₆₇	1	2	yes	complemented
150	#1450	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₆₇	#17895	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
153	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
154	C₄×D₈	C₄×C₆₇	1	2	yes	—
155	C₄×D₈	C₄×C₆₇	1	2	yes	—
156	C₄×D₈	C₂²×C₆₇	1	2	yes	—
157	C₄×D₈	C₂²×C₆₇	1	2	yes	—
158	#436	—	2	4	no	—
159	#436	C₂²×C₆₇	1	8	yes	complemented
160	#431	C₂²×C₆₇	1	2	yes	—
161	#431	C₂²×C₆₇	1	1	yes	characteristic
162	#431	C₂²×C₆₇	1	1	yes	characteristic
163	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
164	C₂×#102	C₄×C₆₇	1	2	yes	—
165	#407	—	2	4	no	—
166	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
167	#335	C₂²×C₆₇	1	2	yes	complemented
168	#334	C₂²×C₆₇	1	2	yes	complemented
169	D₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
170	#322	C₂²×C₆₇	1	2	yes	complemented
171	C₈²	C₂²×C₆₇	1	2	yes	complemented
172	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
173	C₂³×C₄	—	2	2	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
177	C₂×M₁₆	—	2	8	no	—
178	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
179	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
183	C₄×D₄	C₂³×C₆₇	1	4	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
187	#107	C₂×C₄×C₆₇	1	2	yes	—
188	#107	C₂×C₄×C₆₇	1	2	yes	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	D₄×C₆₇	1	2	yes	—
191	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
192	#102	C₂×C₄×C₆₇	1	4	yes	—
193	#102	C₂×C₄×C₆₇	1	4	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	1	no	—
196	C₂²⋊C₈	—	2	8	no	—
197	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
200	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₆₇	1	2	yes	—
202	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
204	C₄×C₈	C₂³×C₆₇	1	2	yes	—
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
217	D₈	—	2	4	no	—
218	D₈	—	2	4	no	—
219	M₁₆	—	8	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	M₁₆×C₆₇	1	4	yes	complemented
224	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₄²	#41×C₆₇	1	1	yes	characteristic
236	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
242	D₄	—	2	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
255	C₂×C₄	#131×C₆₇	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	4	no	—
264	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
265	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	#436×C₆₇	1	2	yes	—
273	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	4	no	—
278	C₄	#436×C₆₇	1	2	yes	—
279	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
280	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
281	C₄	#576×C₆₇	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
285	C₂	C₆₇×#2948	1	1	yes	characteristic
286	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ #431×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2975	C₂	semidirect
3. C₆₇×#1594	C₂	semidirect
4. C₆₇×#1589	C₂	semidirect
5. C₆₇×#1564	C₂	semidirect
6. C₆₇×#1478	C₂	semidirect
7. C₆₇×#1450	C₂	semidirect
15. #436×C₆₇	C₂²	semidirect
23. #335×C₆₇	C₂²	semidirect
24. #334×C₆₇	C₂²	semidirect
25. D₄⋊C₈×C₆₇	C₂²	semidirect
26. #322×C₆₇	C₂²	semidirect
27. C₈²×C₆₇	C₂²	semidirect
79. C₂×C₈×C₆₇	M₁₆	semidirect
138. #17895	C₆₇	direct (N × H)
144. #2975	C₂×C₆₇	semidirect
146. #1594	C₂×C₆₇	semidirect
147. #1589	C₂×C₆₇	semidirect
148. #1564	C₂×C₆₇	semidirect
149. #1478	C₂×C₆₇	semidirect
150. #1450	C₂×C₆₇	semidirect
151. C₆₇	#17895	direct (N × H)
159. #436	C₂²×C₆₇	semidirect
167. #335	C₂²×C₆₇	semidirect
168. #334	C₂²×C₆₇	semidirect
169. D₄⋊C₈	C₂²×C₆₇	semidirect
170. #322	C₂²×C₆₇	semidirect
171. C₈²	C₂²×C₆₇	semidirect
223. C₂×C₈	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2975	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#1594	2
C₆₇×#1589	2
C₆₇×#1564	2
C₆₇×#1478	2
C₆₇×#1450	2
#17895	67

Subgroups — C67×#17895 #30697146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17895 #30697146