Subgroups — #31918450

Order 18176.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₃₆	C₂	1	4	yes	maximal
2	#22590593	C₂	1	2	yes	maximal
3	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₆₈	—	2	2	no	—
5	C₂×Dic₅₆₈	C₂²	1	4	yes	—
6	#17986379	—	2	2	no	—
7	Dic₁₁₃₆	—	2	8	no	—
8	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₈₄	—	2	4	no	—
11	C₂×Dic₂₈₄	—	4	2	no	—
12	#14556128	—	4	2	no	—
13	Dic₅₆₈	—	2	8	no	—
14	Dic₅₆₈	—	4	4	no	—
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	4	yes	—
19	C₂×Dic₁₄₂	—	4	4	no	—
20	Dic₂₈₄	—	4	4	no	—
21	Dic₂₈₄	—	4	4	no	—
22	Dic₂₈₄	—	8	4	no	—
23	C₄⋊Dic₇₁	—	8	4	no	—
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₇₁	D₈	1	1	yes	characteristic
27	C₁₆×C₇₁	D₈	1	2	yes	—
28	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
30	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
31	C₂×Dic₇₁	—	8	4	no	—
32	Dic₁₄₂	—	8	4	no	—
33	Dic₁₄₂	—	8	4	no	—
34	C₈×C₇₁	D₁₆	1	2	yes	—
35	C₈×C₇₁	#127	1	1	yes	characteristic
36	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
37	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
38	C₄×C₇₁	Q₆₄	1	4	yes	complemented
39	C₄×C₇₁	#493	1	1	yes	characteristic
40	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
41	Dic₇₁	—	16	4	no	—
42	#13726	—	71	1	no	maximal, Sylow
43	C₂×C₇₁	#2246	1	1	yes	characteristic
44	C₂×C₇₁	C₂×Q₆₄	1	2	yes	—
45	C₂×Q₆₄	—	71	4	no	—
46	#2247	—	71	2	no	—
47	C₄×C₃₂	D₇₁	1	1	yes	characteristic
48	C₇₁	#13726	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₃₂	—	71	4	no	—
50	C₂×Q₃₂	—	142	2	no	—
51	#494	—	142	2	no	—
52	Q₆₄	—	142	8	no	—
53	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
54	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
55	C₂×Q₁₆	—	284	2	no	—
56	C₂×Q₁₆	—	142	4	no	—
57	C₄⋊Q₈	—	284	2	no	—
58	Q₃₂	—	284	4	no	—
59	Q₃₂	—	142	8	no	—
60	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
62	C₄×C₈	D₂₈₄	1	1	yes	characteristic
63	C₃₂	D₂₈₄	1	4	yes	—
64	C₂×Q₈	—	284	4	no	—
65	Q₁₆	—	284	4	no	—
66	Q₁₆	—	284	4	no	—
67	Q₁₆	—	568	4	no	—
68	C₂×C₈	D₅₆₈	1	1	yes	characteristic
69	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
70	C₄⋊C₄	—	568	4	no	—
71	C₄²	D₅₆₈	1	1	yes	characteristic
72	C₁₆	D₅₆₈	1	2	yes	—
73	C₁₆	C₂×D₂₈₄	1	2	yes	—
74	Q₈	—	568	4	no	—
75	Q₈	—	568	4	no	—
76	C₂×C₄	—	568	4	no	—
77	C₂×C₄	D₁₁₃₆	1	2	yes	—
78	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
79	C₈	D₁₁₃₆	1	2	yes	—
80	C₈	#14556125	1	1	yes	characteristic
81	C₈	C₂×D₅₆₈	1	1	yes	characteristic
82	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
83	C₄	—	1136	4	no	—
84	C₄	Dic₁₁₃₆	1	4	yes	complemented
85	C₄	#17986376	1	1	yes	characteristic
86	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
87	C₂	#22590590	1	1	yes	characteristic
88	C₂	C₂×Dic₁₁₃₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₁₃₆
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₄×C₃₂×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#13726
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
38. C₄×C₇₁	Q₆₄	semidirect
48. C₇₁	#13726	semidirect
84. C₄	Dic₁₁₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₃₆	2
#22590593	2
C₄×C₃₂×C₇₁	2
#13726	71