Subgroups — #31917966

Order 18176.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985829	C₂	1	2	yes	maximal
2	#22586544	C₂	1	4	yes	maximal, complemented
3	C₂×M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	—	2	2	no	—
5	#17985829	—	2	2	no	—
6	#17985829	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	M₆₄×C₇₁	C₄	1	4	yes	—
9	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
10	C₂×#11765849	—	4	2	no	—
11	#14556064	—	2	4	no	—
12	#14556064	—	4	2	no	—
13	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
17	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
18	C₃₂×C₇₁	C₂×C₄	1	4	yes	—
19	#11765849	—	4	4	no	—
20	#11765849	—	8	2	no	—
21	C₂×C₇₁⋊C₈	—	8	2	no	—
22	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
23	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
26	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₇₁	Q₃₂	1	1	yes	characteristic
31	C₂×C₄×C₇₁	#107	1	2	yes	—
32	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
33	C₂×C₄×C₇₁	Q₃₂	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
35	C₈×C₇₁	#107	1	2	yes	—
36	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
38	C₇₁⋊C₈	—	8	4	no	—
39	C₂²×C₇₁	—	4	1	no	—
40	C₂²×C₇₁	#366	1	2	yes	—
41	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
42	C₄×C₇₁	#366	1	2	yes	—
43	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
44	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
45	#13636	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	—	2	2	no	—
47	C₂×C₇₁	—	2	1	no	—
48	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
49	C₂×M₆₄	D₇₁	1	1	yes	characteristic
50	C₂×#368	—	71	2	no	—
51	#1425	—	71	4	no	—
52	C₇₁	#13636	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
54	C₂×#108	—	142	2	no	—
55	M₆₄	Dic₇₁	1	4	yes	—
56	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
57	#368	—	71	4	no	—
58	#368	—	142	2	no	—
59	C₂×M₁₆	—	284	2	no	—
60	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
61	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
62	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
64	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
65	#108	—	284	2	no	—
66	#108	—	142	4	no	—
67	C₃₂	C₂×Dic₇₁	1	4	yes	—
68	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
69	M₁₆	—	284	4	no	—
70	M₁₆	—	568	2	no	—
71	C₂×C₈	—	568	2	no	—
72	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
73	C₂×C₈	D₅₆₈	1	1	yes	characteristic
74	C₂×C₈	Dic₂₈₄	1	2	yes	—
75	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
76	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
77	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
78	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
79	C₂³	Dic₅₆₈	1	1	yes	characteristic
80	C₂×C₄	#14556062	1	2	yes	—
81	C₂×C₄	D₁₁₃₆	1	2	yes	—
82	C₂×C₄	Dic₅₆₈	1	1	yes	characteristic
83	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
84	C₈	—	568	4	no	—
85	C₈	#14556062	1	2	yes	—
86	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
87	C₈	C₂×Dic₂₈₄	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	#17985825	1	2	yes	—
90	C₂²	C₂×Dic₅₆₈	1	1	yes	characteristic
91	C₄	#17985825	1	2	yes	—
92	C₄	C₂×#14556062	1	1	yes	characteristic
93	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₂×#17985825	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₁₃₆
commutator	C₁₆×C₇₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂×M₆₄×C₇₁	C₂
socle	C₂×C₇₁	C₂×#366
prime radical	C₂³×C₇₁	Q₃₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×M₆₄	D₇₁
hypocenter	C₇₁	#13636
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #22586544	C₂	semidirect
52. C₇₁	#13636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985829	2
#22586544	2
C₂×M₆₄×C₇₁	2
#13636	71