Subgroups — #31725473

Order 17152.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573614	C₂	1	2	yes	maximal
2	#22570748	C₂	1	8	yes	maximal, complemented
3	#22570740	C₂	1	8	yes	maximal, complemented
4	#22570733	C₂	1	4	yes	maximal, complemented
5	#22570723	C₂	1	4	yes	maximal, complemented
6	#22568799	C₂	1	2	yes	maximal
7	C₂×#16863434	C₂	1	2	yes	maximal
8	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16864027	—	2	4	no	—
10	#16864024	C₂²	1	16	yes	—
11	#16864022	—	2	4	no	—
12	#16863953	C₂²	1	8	yes	—
13	#16863949	—	2	8	no	—
14	C₂×#14524002	C₂²	1	4	yes	—
15	#16863805	C₂²	1	32	yes	complemented
16	#16863801	C₂²	1	8	yes	complemented
17	#16863798	C₂²	1	8	yes	complemented
18	#16863798	C₂²	1	8	yes	complemented
19	#16863794	C₂²	1	8	yes	complemented
20	#16863438	C₂²	1	16	yes	—
21	#16863434	C₂²	1	4	yes	—
22	#16863434	C₂²	1	4	yes	—
23	#16863434	C₂²	1	4	yes	—
24	#16863434	C₂²	1	4	yes	—
25	#16863434	C₂²	1	8	yes	—
26	C₂×#14523977	C₂²	1	4	yes	—
27	C₂×#14523975	C₂²	1	4	yes	—
28	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
31	#405×C₆₇	C₂²	1	4	yes	—
32	#405×C₆₇	C₂²	1	4	yes	—
33	C₂×#97×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
35	#14524043	C₂³	1	16	yes	—
36	#14524042	—	2	16	no	—
37	#14524035	C₂³	1	16	yes	—
38	#14524034	—	2	16	no	—
39	#14524022	—	2	16	no	—
40	#14524020	—	2	32	no	—
41	C₄×Dic₁₃₄	—	2	16	no	—
42	#14524002	—	2	4	no	—
43	#14524002	—	2	4	no	—
44	#14524002	C₂³	1	8	yes	—
45	#14524002	C₂³	1	8	yes	—
46	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
48	C₂×C₄×Dic₆₇	—	2	4	no	—
49	#14523977	C₂³	1	8	yes	—
50	#14523977	C₂³	1	16	yes	—
51	#14523977	C₂³	1	8	yes	—
52	#14523975	C₂³	1	8	yes	—
53	#14523975	C₂³	1	16	yes	—
54	#14523975	C₂³	1	8	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₆₇	Q₈	1	8	yes	complemented
57	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₆₇	Q₈	1	8	yes	complemented
59	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	#117×C₆₇	D₄	1	4	yes	—
65	#117×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	#97×C₆₇	C₂³	1	8	yes	—
70	C₄×C₈×C₆₇	C₂³	1	4	yes	—
71	C₄×C₈×C₆₇	C₂³	1	4	yes	—
72	C₂×Dic₁₃₄	—	2	16	no	—
73	C₂²×Dic₆₇	—	2	8	no	—
74	C₄⋊Dic₆₇	—	2	8	no	—
75	C₄⋊Dic₆₇	—	2	16	no	—
76	C₄⋊Dic₆₇	C₂⁴	1	16	yes	—
77	Dic₆₇⋊C₄	—	2	32	no	—
78	C₂²⋊Dic₆₇	—	2	8	no	—
79	C₂²⋊Dic₆₇	—	2	8	no	—
80	C₄×Dic₆₇	—	2	8	no	—
81	C₄×Dic₆₇	—	2	8	no	—
82	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
90	M₁₆×C₆₇	C₂×Q₈	1	16	yes	—
91	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
92	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
93	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
94	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
96	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
100	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
102	C₄²×C₆₇	C₂×D₄	1	4	yes	—
103	C₄²×C₆₇	C₂⁴	1	4	yes	—
104	C₂³×C₆₇	—	4	2	no	—
105	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
106	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
107	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	—	4	4	no	—
109	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
111	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
116	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₆₇	—	2	16	no	—
120	C₂×Dic₆₇	—	8	8	no	—
121	Dic₁₃₄	—	8	16	no	—
122	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
123	C₈×C₆₇	C₂²×Q₈	1	4	yes	—
124	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
125	C₈×C₆₇	C₂²×Q₈	1	8	yes	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	4	4	no	—
129	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
130	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
131	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₆₇	—	2	8	no	—
133	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
134	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
135	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
136	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
137	Dic₆₇	—	4	16	no	—
138	#60660	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	—	2	4	no	—
140	C₂×C₆₇	—	2	2	no	—
141	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₆₇	#3584	1	2	yes	—
143	#3458	—	67	2	no	—
144	#3164	—	67	8	no	—
145	#3163	—	67	8	no	—
146	#3162	—	67	4	no	—
147	#3160	—	67	4	no	—
148	#2911	—	67	2	no	—
149	C₂×#427	—	67	2	no	—
150	C₂×#405	D₆₇	1	1	yes	characteristic
151	C₆₇	#60660	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
154	#544	—	67	16	no	—
155	#544	—	134	4	no	—
156	#543	—	134	4	no	—
157	#533	—	67	8	no	—
158	#533	—	134	8	no	—
159	C₂×#117	—	67	4	no	—
160	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
161	#501	—	67	32	no	—
162	#500	—	67	8	no	—
163	#499	—	67	8	no	—
164	#499	—	67	8	no	—
165	#498	—	67	8	no	—
166	#428	—	67	16	no	—
167	#427	—	67	4	no	—
168	#427	—	67	8	no	—
169	#427	—	67	4	no	—
170	#427	—	67	4	no	—
171	#427	—	67	4	no	—
172	C₂×#107	—	67	4	no	—
173	C₂×#106	—	67	4	no	—
174	#405	C₂×D₆₇	1	4	yes	—
175	#405	C₂×D₆₇	1	4	yes	—
176	C₂×#97	C₂×D₆₇	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
178	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
179	C₂×M₁₆	Dic₁₃₄	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₆₇	1	4	yes	—
181	C₂²×C₈	Dic₁₃₄	1	8	yes	complemented
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
185	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
186	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
187	C₄⋊Q₈	—	67	16	no	—
188	C₄⋊Q₈	—	134	16	no	—
189	#125	—	67	16	no	—
190	#125	—	134	16	no	—
191	C₂²⋊Q₈	—	134	16	no	—
192	C₂²⋊Q₈	—	134	32	no	—
193	C₄×Q₈	—	134	16	no	—
194	#117	—	67	8	no	—
195	#117	—	67	8	no	—
196	#117	—	134	4	no	—
197	#117	—	134	4	no	—
198	#117	D₂₆₈	1	4	yes	—
199	#117	C₂²×D₆₇	1	4	yes	—
200	C₂×C₄⋊C₄	—	67	8	no	—
201	C₂×C₄⋊C₄	D₂₆₈	1	2	yes	—
202	C₂×C₂²⋊C₄	—	134	4	no	—
203	C₂×C₂²⋊C₄	D₂₆₈	1	2	yes	—
204	C₂×C₄²	—	134	4	no	—
205	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
206	#107	—	67	8	no	—
207	#107	—	67	8	no	—
208	#107	—	67	16	no	—
209	#106	—	67	8	no	—
210	#106	—	67	8	no	—
211	#106	—	67	16	no	—
212	#97	C₂²×D₆₇	1	8	yes	—
213	C₄×C₈	C₂²×D₆₇	1	4	yes	—
214	C₄×C₈	C₂²×D₆₇	1	4	yes	—
215	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
216	C₂×Q₈	—	134	16	no	—
217	C₂²×C₄	—	134	8	no	—
218	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
219	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
222	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
223	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
224	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
225	M₁₆	C₂×Dic₁₃₄	1	16	yes	—
226	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
227	C₂×C₈	C₂×Dic₁₃₄	1	4	yes	—
228	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
229	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
230	C₂×C₈	C₂×Dic₁₃₄	1	8	yes	—
231	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₆₇	1	4	yes	—
234	C₂×C₈	C₂³×D₆₇	1	2	yes	—
235	C₄⋊C₄	—	67	16	no	—
236	C₄⋊C₄	—	134	8	no	—
237	C₄⋊C₄	—	134	32	no	—
238	C₄⋊C₄	—	134	16	no	—
239	C₄⋊C₄	C₂×D₂₆₈	1	8	yes	—
240	C₂²⋊C₄	—	134	8	no	—
241	C₂²⋊C₄	—	134	8	no	—
242	C₂²⋊C₄	C₂×D₂₆₈	1	8	yes	—
243	C₄²	—	134	8	no	—
244	C₄²	—	134	8	no	—
245	C₄²	C₂×D₂₆₈	1	4	yes	—
246	C₄²	C₂³×D₆₇	1	4	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#14524043	1	4	yes	—
249	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
250	C₂³	C₂²×D₂₆₈	1	2	yes	—
251	Q₈	—	536	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	536	8	no	—
254	C₂×C₄	—	134	16	no	—
255	C₂×C₄	#14524043	1	4	yes	—
256	C₂×C₄	#14524043	1	8	yes	—
257	C₂×C₄	C₂²×D₂₆₈	1	8	yes	—
258	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
259	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
262	C₂×C₄	C₂²×Dic₁₃₄	1	2	yes	—
263	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
264	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
265	C₈	C₂²×Dic₁₃₄	1	2	yes	—
266	C₈	C₂²×Dic₁₃₄	1	2	yes	—
267	C₈	C₂²×Dic₁₃₄	1	4	yes	—
268	C₈	C₂²×Dic₁₃₄	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#14524043	1	2	yes	—
273	C₂²	C₂×#14524043	1	4	yes	—
274	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	268	16	no	—
277	C₄	C₂×#14524043	1	4	yes	—
278	C₄	C₂×#14524043	1	2	yes	—
279	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
280	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#14524043	1	1	yes	characteristic
284	C₂	#22574983	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#60660
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #22570748	C₂	semidirect
3. #22570740	C₂	semidirect
4. #22570733	C₂	semidirect
5. #22570723	C₂	semidirect
15. #16863805	C₂²	semidirect
16. #16863801	C₂²	semidirect
17. #16863798	C₂²	semidirect
18. #16863798	C₂²	semidirect
19. #16863794	C₂²	semidirect
56. C₂×M₁₆×C₆₇	Q₈	semidirect
58. C₂²×C₈×C₆₇	Q₈	semidirect
151. C₆₇	#60660	semidirect
179. C₂×M₁₆	Dic₁₃₄	semidirect
181. C₂²×C₈	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22573614	2
#22570748	2
#22570740	2
#22570733	2
#22570723	2
#22568799	2
C₂×#16863434	2
C₂×#405×C₆₇	2
#60660	67