Subgroups — #31918447

Order 18176.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₇₂	C₂	1	4	yes	maximal, complemented
2	#22590590	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₃₆	—	2	2	no	—
5	C₂×D₁₁₃₆	C₂²	1	4	yes	—
6	#17986376	—	2	2	no	—
7	D₂₂₇₂	—	2	8	no	—
8	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×D₅₆₈	—	2	4	no	—
11	C₂×D₅₆₈	—	4	2	no	—
12	#14556125	—	4	2	no	—
13	D₁₁₃₆	—	2	8	no	—
14	D₁₁₃₆	—	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	complemented
19	C₂×D₂₈₄	—	4	4	no	—
20	C₂×D₂₈₄	—	8	4	no	—
21	D₅₆₈	—	4	4	no	—
22	D₅₆₈	—	4	4	no	—
23	D₅₆₈	—	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	C₂²×D₇₁	—	8	4	no	—
30	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
31	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
32	D₂₈₄	—	8	4	no	—
33	D₂₈₄	—	8	4	no	—
34	D₂₈₄	—	16	8	no	—
35	C₈×C₇₁	D₁₆	1	2	yes	—
36	C₈×C₇₁	#127	1	1	yes	characteristic
37	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
38	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
39	C₂×D₇₁	—	16	4	no	—
40	C₂×D₇₁	—	16	8	no	—
41	C₄×C₇₁	D₃₂	1	4	yes	complemented
42	C₄×C₇₁	#493	1	1	yes	characteristic
43	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
44	#13725	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	#2246	1	1	yes	characteristic
46	C₂×C₇₁	C₂×D₃₂	1	2	yes	—
47	D₇₁	—	32	4	no	—
48	C₂×D₃₂	—	71	4	no	—
49	#2246	—	71	2	no	—
50	C₄×C₃₂	D₇₁	1	1	yes	characteristic, complemented
51	C₇₁	#13725	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₁₆	—	71	4	no	—
53	C₂×D₁₆	—	142	2	no	—
54	#493	—	142	2	no	—
55	D₃₂	—	142	8	no	—
56	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
57	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
58	C₂×D₈	—	284	2	no	—
59	C₂×D₈	—	142	4	no	—
60	#127	—	284	2	no	—
61	D₁₆	—	284	4	no	—
62	D₁₆	—	142	8	no	—
63	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
65	C₄×C₈	D₂₈₄	1	1	yes	characteristic
66	C₃₂	D₂₈₄	1	4	yes	complemented
67	C₂×D₄	—	284	4	no	—
68	C₂×D₄	—	568	4	no	—
69	D₈	—	284	4	no	—
70	D₈	—	284	4	no	—
71	D₈	—	568	4	no	—
72	C₂×C₈	D₅₆₈	1	1	yes	characteristic
73	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
74	C₄²	D₅₆₈	1	1	yes	characteristic
75	C₁₆	D₅₆₈	1	2	yes	—
76	C₁₆	C₂×D₂₈₄	1	2	yes	—
77	C₂³	—	568	4	no	—
78	D₄	—	568	4	no	—
79	D₄	—	568	4	no	—
80	D₄	—	1136	8	no	—
81	C₂×C₄	D₁₁₃₆	1	2	yes	—
82	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
83	C₈	D₁₁₃₆	1	2	yes	—
84	C₈	#14556125	1	1	yes	characteristic
85	C₈	C₂×D₅₆₈	1	1	yes	characteristic
86	C₂²	—	1136	4	no	—
87	C₂²	—	1136	8	no	—
88	C₂²	C₂×D₁₁₃₆	1	1	yes	characteristic
89	C₄	D₂₂₇₂	1	4	yes	complemented
90	C₄	#17986376	1	1	yes	characteristic
91	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
92	C₂	—	2272	4	no	—
93	C₂	#22590590	1	1	yes	characteristic
94	C₂	C₂×D₂₂₇₂	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	C₂²×C₇₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#13725
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
1. C₂×D₂₂₇₂	C₂	semidirect
2. #22590590	C₂	semidirect
3. C₄×C₃₂×C₇₁	C₂	semidirect
18. C₃₂×C₇₁	D₄	semidirect
41. C₄×C₇₁	D₃₂	semidirect
50. C₄×C₃₂	D₇₁	semidirect
51. C₇₁	#13725	semidirect
66. C₃₂	D₂₈₄	semidirect
89. C₄	D₂₂₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₂₇₂	2
#22590590	2
C₄×C₃₂×C₇₁	2
#13725	71