Subgroups — #32904075

Order 18176.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴×Dic₇₁	C₂	1	32	yes	maximal, complemented
2	#22601543	C₂	1	20	yes	maximal, complemented
3	C₂×#17986477	C₂	1	10	yes	maximal, complemented
4	C₂²×2₋¹⁺⁴×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986482	C₂²	1	160	yes	complemented
6	#17986477	C₂²	1	40	yes	—
7	#17986477	C₂²	1	80	yes	complemented
8	#17986474	C₂²	1	40	yes	—
9	#41×Dic₇₁	C₂²	1	160	yes	complemented
10	C₂×Q₈×Dic₇₁	C₂²	1	80	yes	complemented
11	C₂×#14556087	C₂²	1	60	yes	—
12	C₂×2₋¹⁺⁴×C₇₁	C₄	1	32	yes	complemented
13	C₂×2₋¹⁺⁴×C₇₁	C₂²	1	16	yes	—
14	C₂²×#41×C₇₁	C₂²	1	10	yes	—
15	C₂³×Q₈×C₇₁	C₂²	1	5	yes	—
16	Q₈×Dic₇₁	C₂³	1	320	yes	—
17	D₄×Dic₇₁	C₂³	1	320	yes	—
18	#14556087	C₂³	1	80	yes	—
19	#14556087	C₂³	1	240	yes	—
20	C₂×C₄⋊Dic₇₁	C₂³	1	120	yes	—
21	C₂×C₂²⋊Dic₇₁	C₂³	1	40	yes	—
22	C₂×C₄×Dic₇₁	C₂³	1	120	yes	—
23	2₋¹⁺⁴×C₇₁	C₂×C₄	1	256	yes	complemented
24	C₂×#41×C₇₁	C₂×C₄	1	160	yes	complemented
25	C₂×#41×C₇₁	C₂³	1	80	yes	—
26	C₂²×Q₈×C₇₁	C₂×C₄	1	80	yes	complemented
27	C₂²×Q₈×C₇₁	C₂³	1	40	yes	—
28	C₂²×Q₈×C₇₁	C₂³	1	10	yes	—
29	C₂²×D₄×C₇₁	C₂³	1	10	yes	—
30	C₂³×C₄×C₇₁	C₂³	1	15	yes	—
31	C₂²×Dic₇₁	C₂⁴	1	80	yes	—
32	C₄⋊Dic₇₁	C₂⁴	1	160	yes	—
33	C₂²⋊Dic₇₁	C₂⁴	1	16	yes	—
34	C₂²⋊Dic₇₁	C₂⁴	1	80	yes	—
35	C₄×Dic₇₁	C₂⁴	1	160	yes	—
36	C₂⁴×C₇₁	C₂⁴	1	5	yes	—
37	#41×C₇₁	C₂²×C₄	1	640	yes	—
38	C₂×Q₈×C₇₁	C₂²×C₄	1	80	yes	—
39	C₂×Q₈×C₇₁	C₂²×C₄	1	320	yes	—
40	C₂×Q₈×C₇₁	C₂⁴	1	40	yes	—
41	C₂×D₄×C₇₁	C₂²×C₄	1	80	yes	—
42	C₂×D₄×C₇₁	C₂⁴	1	40	yes	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	120	yes	—
44	C₂²×C₄×C₇₁	C₂⁴	1	10	yes	—
45	C₂²×C₄×C₇₁	C₂⁴	1	60	yes	—
46	C₂³×C₇₁	—	4	10	no	—
47	C₂³×C₇₁	C₂³×C₄	1	20	yes	—
48	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
49	C₂³×C₇₁	C₂⁵	1	10	yes	—
50	Q₈×C₇₁	C₂³×C₄	1	160	yes	—
51	D₄×C₇₁	C₂³×C₄	1	160	yes	—
52	C₂×C₄×C₇₁	C₂³×C₄	1	40	yes	—
53	C₂×C₄×C₇₁	C₂³×C₄	1	240	yes	—
54	C₂×C₄×C₇₁	C₂⁵	1	20	yes	—
55	C₂×Dic₇₁	—	4	80	no	—
56	C₂×Dic₇₁	C₂⁵	1	32	yes	—
57	C₂²×C₇₁	—	2	20	no	—
58	C₂²×C₇₁	—	2	2	no	—
59	C₂²×C₇₁	—	4	20	no	—
60	C₂²×C₇₁	C₂⁴×C₄	1	2	yes	—
61	C₂²×C₇₁	C₂⁴×C₄	1	20	yes	—
62	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
63	C₄×C₇₁	C₂⁴×C₄	1	40	yes	—
64	Dic₇₁	—	2	32	no	—
65	#62653	—	71	1	no	maximal, Sylow
66	C₂×C₇₁	—	2	2	no	—
67	C₂×C₇₁	—	2	20	no	—
68	C₂×C₇₁	C₂⁵×C₄	1	1	yes	characteristic
69	C₂×C₇₁	2₋¹⁺⁶	1	2	yes	—
70	C₂²×2₋¹⁺⁴	D₇₁	1	1	yes	characteristic
71	C₄×2₋¹⁺⁴	—	71	32	no	—
72	#3428	—	71	20	no	—
73	C₂×#519	—	71	10	no	—
74	C₇₁	#62653	1	1	yes	characteristic, complemented, Sylow
75	C₂×2₋¹⁺⁴	Dic₇₁	1	32	yes	complemented
76	C₂×2₋¹⁺⁴	C₂×D₇₁	1	16	yes	—
77	C₂²×#41	C₂×D₇₁	1	10	yes	—
78	C₂³×Q₈	C₂×D₇₁	1	5	yes	—
79	#520	—	71	160	no	—
80	#519	—	71	40	no	—
81	#519	—	71	80	no	—
82	#518	—	71	40	no	—
83	C₄×#41	—	71	160	no	—
84	C₂×C₄×Q₈	—	71	80	no	—
85	C₂×#117	—	71	60	no	—
86	2₋¹⁺⁴	C₂×Dic₇₁	1	256	yes	complemented
87	C₂×#41	C₂×Dic₇₁	1	160	yes	complemented
88	C₂×#41	C₂²×D₇₁	1	80	yes	—
89	C₂²×Q₈	C₂×Dic₇₁	1	80	yes	complemented
90	C₂²×Q₈	C₂²×D₇₁	1	40	yes	—
91	C₂²×Q₈	C₂²×D₇₁	1	10	yes	—
92	C₂²×D₄	C₂²×D₇₁	1	10	yes	—
93	C₂³×C₄	C₂²×D₇₁	1	15	yes	—
94	C₄×Q₈	—	71	320	no	—
95	C₄×D₄	—	71	320	no	—
96	#117	—	71	80	no	—
97	#117	—	71	240	no	—
98	C₂×C₄⋊C₄	—	71	120	no	—
99	C₂×C₂²⋊C₄	—	71	40	no	—
100	C₂×C₄²	—	71	120	no	—
101	C₂⁴	C₂³×D₇₁	1	5	yes	—
102	#41	C₂²×Dic₇₁	1	640	yes	—
103	C₂×Q₈	C₂²×Dic₇₁	1	80	yes	—
104	C₂×Q₈	C₂²×Dic₇₁	1	320	yes	—
105	C₂×Q₈	C₂³×D₇₁	1	40	yes	—
106	C₂×D₄	C₂²×Dic₇₁	1	80	yes	—
107	C₂×D₄	C₂³×D₇₁	1	40	yes	—
108	C₂²×C₄	—	71	80	no	—
109	C₂²×C₄	C₂²×Dic₇₁	1	120	yes	—
110	C₂²×C₄	C₂³×D₇₁	1	60	yes	—
111	C₂²×C₄	C₂³×D₇₁	1	10	yes	—
112	C₄⋊C₄	—	71	160	no	—
113	C₂²⋊C₄	—	71	16	no	—
114	C₂²⋊C₄	—	71	80	no	—
115	C₄²	—	71	160	no	—
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×Dic₇₁	1	20	yes	—
118	C₂³	C₂⁴×D₇₁	1	1	yes	characteristic
119	C₂³	C₂⁴×D₇₁	1	10	yes	—
120	Q₈	C₂³×Dic₇₁	1	160	yes	—
121	D₄	C₂³×Dic₇₁	1	160	yes	—
122	C₂×C₄	—	284	80	no	—
123	C₂×C₄	—	71	32	no	—
124	C₂×C₄	C₂³×Dic₇₁	1	40	yes	—
125	C₂×C₄	C₂³×Dic₇₁	1	240	yes	—
126	C₂×C₄	C₂⁴×D₇₁	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×Dic₇₁	1	2	yes	—
131	C₂²	C₂⁴×Dic₇₁	1	20	yes	—
132	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
133	C₄	—	142	32	no	—
134	C₄	C₂⁴×Dic₇₁	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×Dic₇₁	1	1	yes	characteristic
138	C₂	#22603152	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₁
commutator	C₂×C₇₁	C₂⁵×C₄
Fitting, prime radical	C₂²×2₋¹⁺⁴×C₇₁	C₂
socle, cosocle	C₂²×C₇₁	C₂⁶
hypercenter	C₂²×2₋¹⁺⁴	D₇₁
hypocenter	C₇₁	#62653
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. 2₋¹⁺⁴×Dic₇₁	C₂	semidirect
2. #22601543	C₂	semidirect
3. C₂×#17986477	C₂	semidirect
5. #17986482	C₂²	semidirect
7. #17986477	C₂²	semidirect
9. #41×Dic₇₁	C₂²	semidirect
10. C₂×Q₈×Dic₇₁	C₂²	semidirect
12. C₂×2₋¹⁺⁴×C₇₁	C₄	semidirect
23. 2₋¹⁺⁴×C₇₁	C₂×C₄	semidirect
24. C₂×#41×C₇₁	C₂×C₄	semidirect
26. C₂²×Q₈×C₇₁	C₂×C₄	semidirect
74. C₇₁	#62653	semidirect
75. C₂×2₋¹⁺⁴	Dic₇₁	semidirect
86. 2₋¹⁺⁴	C₂×Dic₇₁	semidirect
87. C₂×#41	C₂×Dic₇₁	semidirect
89. C₂²×Q₈	C₂×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
2₋¹⁺⁴×Dic₇₁	2
#22601543	2
C₂×#17986477	2
C₂²×2₋¹⁺⁴×C₇₁	2
#62653	71