Subgroups — #31918507

Order 18176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590650	C₂	1	2	yes	maximal, complemented
2	#22586554	C₂	1	2	yes	maximal, complemented
3	D₄₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₄×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986432	—	2	2	no	—
7	#17985836	—	2	2	no	—
8	D₂₂₇₂	C₂²	1	2	yes	—
9	C₃₂×D₇₁	C₂²	1	1	yes	characteristic
10	C₇₁⋊C₆₄	C₂²	1	1	yes	characteristic
11	Q₆₄×C₇₁	C₂²	1	2	yes	—
12	C₆₄×C₇₁	C₂²	1	1	yes	characteristic
13	#14556151	—	4	2	no	—
14	#14556071	—	4	2	no	—
15	D₁₁₃₆	—	2	2	no	—
16	C₁₆×D₇₁	D₄	1	1	yes	characteristic
17	C₇₁⋊C₃₂	D₄	1	1	yes	characteristic
18	Q₃₂×C₇₁	—	2	2	no	—
19	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
20	#11765866	—	8	2	no	—
21	#11765852	—	8	2	no	—
22	D₅₆₈	—	4	2	no	—
23	C₈×D₇₁	D₈	1	1	yes	characteristic
24	C₇₁⋊C₁₆	D₈	1	1	yes	characteristic
25	Q₁₆×C₇₁	—	4	2	no	—
26	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
27	Q₈×C₇₁	—	8	2	no	—
28	D₂₈₄	—	8	2	no	—
29	D₂₈₄	—	16	2	no	—
30	C₄×D₇₁	—	16	2	no	—
31	C₄×D₇₁	D₁₆	1	1	yes	characteristic
32	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
33	C₇₁⋊C₈	D₁₆	1	1	yes	characteristic
34	C₂×D₇₁	—	16	2	no	—
35	C₂×D₇₁	D₃₂	1	1	yes	characteristic
36	C₄×C₇₁	—	16	2	no	—
37	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
38	Dic₇₁	D₃₂	1	1	yes	characteristic
39	#13741	—	71	1	no	maximal, Sylow
40	C₂×C₇₁	C₂×D₃₂	1	1	yes	characteristic
41	D₇₁	—	2	1	no	—
42	D₇₁	—	32	2	no	—
43	#2262	—	71	2	no	—
44	Q₁₂₈	D₇₁	1	1	yes	characteristic, complemented
45	SD₁₂₈	—	71	2	no	—
46	D₆₄	—	71	1	no	—
47	C₂×C₆₄	—	71	1	no	—
48	C₇₁	#13741	1	1	yes	characteristic, complemented, Sylow
49	#508	—	142	2	no	—
50	Q₆₄	C₂×D₇₁	1	2	yes	—
51	SD₆₄	—	142	2	no	—
52	D₃₂	—	71	2	no	—
53	C₂×C₃₂	—	71	1	no	—
54	C₆₄	—	71	1	no	—
55	C₆₄	C₂×D₇₁	1	1	yes	characteristic
56	#135	—	284	2	no	—
57	Q₃₂	—	2	2	no	—
58	SD₃₂	—	284	2	no	—
59	D₁₆	—	142	2	no	—
60	C₂×C₁₆	—	71	1	no	—
61	C₃₂	—	71	1	no	—
62	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
63	#41	—	568	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	568	2	no	—
66	D₈	—	284	2	no	—
67	C₂×C₈	—	71	1	no	—
68	C₁₆	—	71	1	no	—
69	C₁₆	D₄×D₇₁	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	568	2	no	—
72	D₄	—	1136	2	no	—
73	C₂×C₄	—	71	1	no	—
74	C₂×C₄	—	1136	2	no	—
75	C₈	—	71	1	no	—
76	C₈	D₈×D₇₁	1	1	yes	characteristic
77	C₂²	—	71	1	no	—
78	C₂²	—	1136	2	no	—
79	C₄	—	16	2	no	—
80	C₄	—	71	1	no	—
81	C₄	D₁₆×D₇₁	1	1	yes	characteristic
82	C₂	—	142	1	no	—
83	C₂	—	2272	2	no	—
84	C₂	D₃₂×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₇₁
commutator, cosocle	C₃₂×C₇₁	C₂³
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₇₁×Q₁₂₈	C₂
socle	C₂×C₇₁	C₂×D₃₂
hypercenter	Q₁₂₈	D₇₁
hypocenter	C₇₁	#13741
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

Complements

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

N	H ≅ G/N	type
1. #22590650	C₂	semidirect
2. #22586554	C₂	semidirect
3. D₄₅₄₄	C₂	semidirect
4. C₆₄×D₇₁	C₂	semidirect
5. C₇₁×Q₁₂₈	C₂	semidirect
44. Q₁₂₈	D₇₁	semidirect
48. C₇₁	#13741	semidirect

Maximal subgroups

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

subgroup	index
#22590650	2
#22586554	2
D₄₅₄₄	2
C₆₄×D₇₁	2
C₇₁×Q₁₂₈	2
#13741	71