Subgroups — #32911311

Order 18176.

10894 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22603151	C₂	1	1	yes	maximal, characteristic, complemented
2	#22603120	C₂	1	15	yes	maximal, complemented
3	#22603119	C₂	1	15	yes	maximal, complemented
4	#22603111	C₂	1	15	yes	maximal, complemented
5	C₂×#17986802	C₂	1	15	yes	maximal, complemented
6	#22603052	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×H(7,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986836	—	2	6	no	—
9	#17986835	C₂²	1	15	yes	complemented
10	C₂×#14556174	C₂²	1	15	yes	complemented
11	H(5,𝔽₂)×D₇₁	—	2	10	no	—
12	#17986814	C₂²	1	20	yes	complemented
13	#17986810	C₂²	1	60	yes	complemented
14	#17986807	C₂²	1	60	yes	complemented
15	#17986802	C₂²	1	60	yes	complemented
16	#17986802	C₂²	1	120	yes	complemented
17	#17986797	C₂²	1	90	yes	complemented
18	C₂×#14556154	C₂²	1	45	yes	complemented
19	C₂×#14556153	C₂²	1	45	yes	complemented
20	C₂×#14556148	C₂²	1	45	yes	complemented
21	C₂²×#11765850	C₂²	1	15	yes	complemented
22	#17986766	C₂²	1	15	yes	complemented
23	C₂×#14556135	C₂²	1	15	yes	complemented
24	#585×C₇₁	—	2	6	no	—
25	#585×C₇₁	C₂²	1	1	yes	characteristic
26	#585×C₇₁	C₂²	1	15	yes	complemented
27	C₂×H(5,𝔽₂)×C₇₁	—	2	10	no	—
28	C₂×H(5,𝔽₂)×C₇₁	C₂²	1	15	yes	complemented
29	#14556176	—	2	15	no	—
30	#14556175	C₂³	1	20	yes	complemented
31	#14556174	—	2	60	no	—
32	#14556174	C₂³	1	60	yes	complemented
33	#14556173	—	2	45	no	—
34	C₂×#11765866	—	2	15	no	—
35	C₂×#11765865	C₂³	1	45	yes	complemented
36	#41×D₇₁	—	2	60	no	—
37	C₂²×D₂₈₄	C₂³	1	15	yes	complemented
38	C₂×D₄×D₇₁	—	2	45	no	—
39	#14556159	C₂³	1	60	yes	complemented
40	#14556158	C₂³	1	60	yes	complemented
41	#14556154	C₂³	1	180	yes	complemented
42	#14556153	C₂³	1	180	yes	complemented
43	#14556148	C₂³	1	60	yes	complemented
44	#14556148	C₂³	1	180	yes	complemented
45	C₂×#11765855	C₂³	1	15	yes	complemented
46	C₂×#11765853	C₂³	1	45	yes	complemented
47	C₂×#11765852	C₂³	1	45	yes	complemented
48	C₂×#11765850	C₂³	1	45	yes	complemented
49	C₂×#11765850	C₂³	1	90	yes	complemented
50	#14556135	C₂³	1	20	yes	complemented
51	#14556135	C₂³	1	60	yes	complemented
52	C₂×#11765849	C₂³	1	45	yes	complemented
53	C₂²×C₇₁⋊C₈	C₂³	1	15	yes	complemented
54	2₋¹⁺⁴×C₇₁	—	2	15	no	—
55	2₋¹⁺⁴×C₇₁	D₄	1	6	yes	—
56	2₋¹⁺⁴×C₇₁	C₂³	1	20	yes	complemented
57	H(5,𝔽₂)×C₇₁	—	2	60	no	—
58	H(5,𝔽₂)×C₇₁	—	2	45	no	—
59	H(5,𝔽₂)×C₇₁	D₄	1	10	yes	—
60	H(5,𝔽₂)×C₇₁	C₂³	1	60	yes	complemented
61	C₂×#41×C₇₁	—	2	60	no	—
62	C₂×#41×C₇₁	—	2	15	no	—
63	C₂×#41×C₇₁	C₂³	1	15	yes	—
64	C₂×#41×C₇₁	C₂³	1	45	yes	complemented
65	C₂²×D₄×C₇₁	—	2	45	no	—
66	C₂²×D₄×C₇₁	C₂³	1	15	yes	complemented
67	C₂³×D₇₁	—	2	15	no	—
68	#11765866	—	2	60	no	—
69	#11765865	—	2	90	no	—
70	#11765865	C₂⁴	1	60	yes	—
71	#11765864	—	2	60	no	—
72	C₂×Dic₁₄₂	C₂⁴	1	15	yes	—
73	Q₈×D₇₁	—	2	20	no	—
74	C₂×D₂₈₄	—	2	45	no	—
75	C₂×D₂₈₄	C₂⁴	1	45	yes	—
76	C₂×C₂²⋊D₇₁	—	2	45	no	—
77	D₄×D₇₁	—	2	180	no	—
78	C₂×C₄×D₇₁	—	2	45	no	—
79	#11765855	C₂⁴	1	6	yes	—
80	#11765855	C₂⁴	1	30	yes	—
81	#11765853	C₂⁴	1	60	yes	—
82	#11765852	C₂⁴	1	60	yes	—
83	#11765850	C₂⁴	1	10	yes	—
84	#11765850	C₂⁴	1	90	yes	—
85	#11765849	C₂⁴	1	60	yes	—
86	C₂×C₇₁⋊C₈	C₂⁴	1	15	yes	—
87	C₂×C₇₁⋊C₈	C₂⁴	1	45	yes	—
88	C₂⁴×C₇₁	—	2	15	no	—
89	#41×C₇₁	—	2	60	no	—
90	#41×C₇₁	—	2	60	no	—
91	#41×C₇₁	—	2	90	no	—
92	#41×C₇₁	C₂×D₄	1	60	yes	—
93	#41×C₇₁	C₂⁴	1	60	yes	—
94	#41×C₇₁	C₂⁴	1	20	yes	—
95	C₂×Q₈×C₇₁	—	2	20	no	—
96	C₂×Q₈×C₇₁	C₂×D₄	1	15	yes	—
97	C₂×Q₈×C₇₁	C₂⁴	1	15	yes	—
98	C₂×D₄×C₇₁	—	2	45	no	—
99	C₂×D₄×C₇₁	—	2	180	no	—
100	C₂×D₄×C₇₁	—	2	45	no	—
101	C₂×D₄×C₇₁	C₂×D₄	1	45	yes	—
102	C₂×D₄×C₇₁	C₂⁴	1	45	yes	—
103	C₂²×C₄×C₇₁	—	2	45	no	—
104	C₂²×C₄×C₇₁	C₂⁴	1	15	yes	—
105	C₂³×C₇₁	—	2	45	no	—
106	C₂³×C₇₁	—	16	15	no	—
107	C₂³×C₇₁	C₂²×D₄	1	15	yes	—
108	C₂²×D₇₁	—	2	45	no	—
109	C₂²×D₇₁	—	16	15	no	—
110	Q₈×C₇₁	—	2	15	no	—
111	Q₈×C₇₁	C₂²×D₄	1	20	yes	—
112	Q₈×C₇₁	C₂⁵	1	6	yes	—
113	D₄×C₇₁	—	2	60	no	—
114	D₄×C₇₁	—	2	45	no	—
115	D₄×C₇₁	C₂²×D₄	1	60	yes	—
116	D₄×C₇₁	C₂⁵	1	10	yes	—
117	C₂×C₄×C₇₁	—	2	15	no	—
118	C₂×C₄×C₇₁	—	2	60	no	—
119	C₂×C₄×C₇₁	C₂²×D₄	1	45	yes	—
120	C₂×C₄×C₇₁	C₂⁵	1	15	yes	—
121	C₂²⋊D₇₁	—	2	60	no	—
122	C₂×Dic₇₁	—	2	15	no	—
123	D₂₈₄	—	2	45	no	—
124	D₂₈₄	C₂⁵	1	10	yes	—
125	C₄×D₇₁	—	2	60	no	—
126	Dic₁₄₂	—	2	15	no	—
127	Dic₁₄₂	C₂⁵	1	6	yes	—
128	C₇₁⋊C₈	C₂⁵	1	1	yes	characteristic
129	C₇₁⋊C₈	C₂⁵	1	15	yes	—
130	C₂²×C₇₁	—	2	10	no	—
131	C₂²×C₇₁	—	4	15	no	—
132	C₂²×C₇₁	—	8	45	no	—
133	C₂²×C₇₁	C₂³×D₄	1	15	yes	—
134	C₂×D₇₁	—	2	10	no	—
135	C₂×D₇₁	—	8	45	no	—
136	C₄×C₇₁	—	2	6	no	—
137	C₄×C₇₁	C₂³×D₄	1	15	yes	—
138	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
139	Dic₇₁	—	2	6	no	—
140	#63091	—	71	1	no	maximal, Sylow
141	C₂×C₇₁	—	2	15	no	—
142	C₂×C₇₁	—	4	10	no	—
143	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
144	D₇₁	—	4	10	no	—
145	H(7,𝔽₂)	—	71	1	no	—
146	H(7,𝔽₂)	D₇₁	1	1	yes	characteristic, complemented
147	#3585	—	71	15	no	—
148	#3585	—	71	15	no	—
149	#3584	—	71	15	no	—
150	C₂×#576	—	71	15	no	—
151	#3573	—	71	1	no	—
152	C₇₁	#63091	1	1	yes	characteristic, complemented, Sylow
153	#585	—	2	6	no	—
154	#585	—	71	15	no	—
155	#585	—	142	6	no	—
156	#585	C₂×D₇₁	1	1	yes	characteristic
157	#585	C₂×D₇₁	1	15	yes	complemented
158	C₂×H(5,𝔽₂)	—	2	10	no	—
159	C₂×H(5,𝔽₂)	—	71	15	no	—
160	C₂×H(5,𝔽₂)	—	142	10	no	—
161	C₂×H(5,𝔽₂)	C₂×D₇₁	1	15	yes	complemented
162	#578	—	71	20	no	—
163	#577	—	71	60	no	—
164	#577	—	71	60	no	—
165	#576	—	71	60	no	—
166	#576	—	71	120	no	—
167	#575	—	71	90	no	—
168	C₂×AGL(1,ℤ/8)	—	71	45	no	—
169	C₂×AGL(1,ℤ/8)	—	71	45	no	—
170	C₂×#135	—	71	45	no	—
171	C₂²×D₈	—	71	15	no	—
172	#568	—	71	15	no	—
173	C₂×#131	—	71	15	no	—
174	2₋¹⁺⁴	—	2	15	no	—
175	2₋¹⁺⁴	—	71	20	no	—
176	2₋¹⁺⁴	—	142	15	no	—
177	2₋¹⁺⁴	C₂²⋊D₇₁	1	6	yes	—
178	2₋¹⁺⁴	C₂²×D₇₁	1	20	yes	complemented
179	H(5,𝔽₂)	—	2	60	no	—
180	H(5,𝔽₂)	—	2	45	no	—
181	H(5,𝔽₂)	—	71	60	no	—
182	H(5,𝔽₂)	—	142	60	no	—
183	H(5,𝔽₂)	—	142	45	no	—
184	H(5,𝔽₂)	C₂²⋊D₇₁	1	10	yes	—
185	H(5,𝔽₂)	C₂²×D₇₁	1	60	yes	complemented
186	C₂×#41	—	2	60	no	—
187	C₂×#41	—	2	15	no	—
188	C₂×#41	—	71	45	no	—
189	C₂×#41	—	142	60	no	—
190	C₂×#41	—	142	15	no	—
191	C₂×#41	C₂²×D₇₁	1	15	yes	—
192	C₂×#41	C₂²×D₇₁	1	45	yes	complemented
193	C₂²×D₄	—	2	45	no	—
194	C₂²×D₄	—	71	15	no	—
195	C₂²×D₄	—	142	45	no	—
196	C₂²×D₄	C₂²×D₇₁	1	15	yes	complemented
197	#137	—	71	60	no	—
198	#137	—	71	60	no	—
199	AGL(1,ℤ/8)	—	71	180	no	—
200	AGL(1,ℤ/8)	—	71	180	no	—
201	#135	—	71	60	no	—
202	#135	—	71	180	no	—
203	C₂×Q₁₆	—	71	15	no	—
204	C₂×SD₁₆	—	71	45	no	—
205	C₂×SD₁₆	—	71	45	no	—
206	C₂×D₈	—	71	45	no	—
207	C₂×D₈	—	71	90	no	—
208	#131	—	71	20	no	—
209	#131	—	71	60	no	—
210	C₂×M₁₆	—	71	45	no	—
211	C₂²×C₈	—	71	15	no	—
212	C₂⁴	—	2	15	no	—
213	C₂⁴	—	142	15	no	—
214	#41	—	2	60	no	—
215	#41	—	2	60	no	—
216	#41	—	2	90	no	—
217	#41	—	71	60	no	—
218	#41	—	142	60	no	—
219	#41	—	142	60	no	—
220	#41	—	142	90	no	—
221	#41	C₂×C₂²⋊D₇₁	1	60	yes	—
222	#41	C₂³×D₇₁	1	60	yes	—
223	#41	C₂³×D₇₁	1	20	yes	—
224	C₂×Q₈	—	2	20	no	—
225	C₂×Q₈	—	71	15	no	—
226	C₂×Q₈	—	142	20	no	—
227	C₂×Q₈	C₂×C₂²⋊D₇₁	1	15	yes	—
228	C₂×Q₈	C₂³×D₇₁	1	15	yes	—
229	C₂×D₄	—	2	45	no	—
230	C₂×D₄	—	2	180	no	—
231	C₂×D₄	—	2	45	no	—
232	C₂×D₄	—	71	45	no	—
233	C₂×D₄	—	142	45	no	—
234	C₂×D₄	—	142	180	no	—
235	C₂×D₄	—	142	45	no	—
236	C₂×D₄	C₂×C₂²⋊D₇₁	1	45	yes	—
237	C₂×D₄	C₂³×D₇₁	1	45	yes	—
238	C₂²×C₄	—	2	45	no	—
239	C₂²×C₄	—	142	45	no	—
240	C₂²×C₄	C₂³×D₇₁	1	15	yes	—
241	Q₁₆	—	71	6	no	—
242	Q₁₆	—	71	30	no	—
243	SD₁₆	—	71	60	no	—
244	SD₁₆	—	71	60	no	—
245	D₈	—	71	10	no	—
246	D₈	—	71	90	no	—
247	M₁₆	—	71	60	no	—
248	C₂×C₈	—	71	15	no	—
249	C₂×C₈	—	71	45	no	—
250	C₂³	—	2	45	no	—
251	C₂³	—	16	15	no	—
252	C₂³	—	1136	15	no	—
253	C₂³	—	142	45	no	—
254	C₂³	C₂²×C₂²⋊D₇₁	1	15	yes	—
255	Q₈	—	2	15	no	—
256	Q₈	—	71	6	no	—
257	Q₈	—	142	15	no	—
258	Q₈	C₂²×C₂²⋊D₇₁	1	20	yes	—
259	Q₈	C₂⁴×D₇₁	1	6	yes	—
260	D₄	—	2	60	no	—
261	D₄	—	2	45	no	—
262	D₄	—	71	10	no	—
263	D₄	—	142	60	no	—
264	D₄	—	142	45	no	—
265	D₄	C₂²×C₂²⋊D₇₁	1	60	yes	—
266	D₄	C₂⁴×D₇₁	1	10	yes	—
267	C₂×C₄	—	2	15	no	—
268	C₂×C₄	—	2	60	no	—
269	C₂×C₄	—	142	15	no	—
270	C₂×C₄	—	142	60	no	—
271	C₂×C₄	C₂²×C₂²⋊D₇₁	1	45	yes	—
272	C₂×C₄	C₂⁴×D₇₁	1	15	yes	—
273	C₈	—	71	1	no	—
274	C₈	—	71	15	no	—
275	C₂²	—	2	10	no	—
276	C₂²	—	4	15	no	—
277	C₂²	—	8	45	no	—
278	C₂²	—	568	45	no	—
279	C₂²	—	142	10	no	—
280	C₂²	C₂³×C₂²⋊D₇₁	1	15	yes	—
281	C₄	—	2	6	no	—
282	C₄	—	142	6	no	—
283	C₄	C₂³×C₂²⋊D₇₁	1	15	yes	—
284	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
285	C₂	—	2	15	no	—
286	C₂	—	4	10	no	—
287	C₂	—	284	10	no	—
288	C₂	C₂⁴×C₂²⋊D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂⁴×C₂²⋊D₇₁
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₇₁×H(7,𝔽₂)	C₂
socle	C₂×C₇₁	C₂⁴×D₄
hypercenter	H(7,𝔽₂)	D₇₁
hypocenter	C₇₁	#63091
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ H(7,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #22603151	C₂	semidirect
2. #22603120	C₂	semidirect
3. #22603119	C₂	semidirect
4. #22603111	C₂	semidirect
5. C₂×#17986802	C₂	semidirect
6. #22603052	C₂	semidirect
7. C₇₁×H(7,𝔽₂)	C₂	semidirect
9. #17986835	C₂²	semidirect
10. C₂×#14556174	C₂²	semidirect
12. #17986814	C₂²	semidirect
13. #17986810	C₂²	semidirect
14. #17986807	C₂²	semidirect
15. #17986802	C₂²	semidirect
16. #17986802	C₂²	semidirect
17. #17986797	C₂²	semidirect
18. C₂×#14556154	C₂²	semidirect
19. C₂×#14556153	C₂²	semidirect
20. C₂×#14556148	C₂²	semidirect
21. C₂²×#11765850	C₂²	semidirect
22. #17986766	C₂²	semidirect
23. C₂×#14556135	C₂²	semidirect
26. #585×C₇₁	C₂²	semidirect
28. C₂×H(5,𝔽₂)×C₇₁	C₂²	semidirect
30. #14556175	C₂³	semidirect
32. #14556174	C₂³	semidirect
35. C₂×#11765865	C₂³	semidirect
37. C₂²×D₂₈₄	C₂³	semidirect
39. #14556159	C₂³	semidirect
40. #14556158	C₂³	semidirect
41. #14556154	C₂³	semidirect
42. #14556153	C₂³	semidirect
43. #14556148	C₂³	semidirect
44. #14556148	C₂³	semidirect
45. C₂×#11765855	C₂³	semidirect
46. C₂×#11765853	C₂³	semidirect
47. C₂×#11765852	C₂³	semidirect
48. C₂×#11765850	C₂³	semidirect
49. C₂×#11765850	C₂³	semidirect
50. #14556135	C₂³	semidirect
51. #14556135	C₂³	semidirect
52. C₂×#11765849	C₂³	semidirect
53. C₂²×C₇₁⋊C₈	C₂³	semidirect
56. 2₋¹⁺⁴×C₇₁	C₂³	semidirect
60. H(5,𝔽₂)×C₇₁	C₂³	semidirect
64. C₂×#41×C₇₁	C₂³	semidirect
66. C₂²×D₄×C₇₁	C₂³	semidirect
146. H(7,𝔽₂)	D₇₁	semidirect
152. C₇₁	#63091	semidirect
157. #585	C₂×D₇₁	semidirect
161. C₂×H(5,𝔽₂)	C₂×D₇₁	semidirect
178. 2₋¹⁺⁴	C₂²×D₇₁	semidirect
185. H(5,𝔽₂)	C₂²×D₇₁	semidirect
192. C₂×#41	C₂²×D₇₁	semidirect
196. C₂²×D₄	C₂²×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22603151	2
#22603120	2
#22603119	2
#22603111	2
C₂×#17986802	2
#22603052	2
C₇₁×H(7,𝔽₂)	2
#63091	71