Subgroups — #17986394

Order 4544.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11765855	C₂	1	2	yes	maximal, complemented
2	C₂×#11765853	C₂	1	2	yes	maximal
3	#14556128	C₂	1	1	yes	maximal, characteristic, complemented
4	#14556039	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄₂	C₂²	1	2	yes	—
7	#11765855	—	2	4	no	—
8	#11765853	—	2	4	no	—
9	C₂×C₇₁⋊C₈	C₂²	1	2	yes	—
10	C₄⋊Dic₇₁	—	2	2	no	—
11	C₂×Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₇₁	—	2	2	no	—
14	C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂³×C₇₁	—	2	1	no	—
16	Q₈×C₇₁	—	2	2	no	—
17	D₄×C₇₁	—	2	2	no	—
18	C₂×C₄×C₇₁	—	2	1	no	—
19	C₂×C₄×C₇₁	D₄	1	2	yes	—
20	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₇₁	—	2	2	no	—
22	Dic₁₄₂	—	2	2	no	—
23	Dic₁₄₂	—	2	2	no	—
24	C₇₁⋊C₈	D₄	1	4	yes	—
25	C₂²×C₇₁	—	4	1	no	—
26	C₂²×C₇₁	—	4	2	no	—
27	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
28	C₄×C₇₁	—	2	2	no	—
29	C₄×C₇₁	—	4	1	no	—
30	C₄×C₇₁	C₂×D₄	1	2	yes	—
31	Dic₇₁	—	4	2	no	—
32	C₂×C₇₁	—	8	1	no	—
33	C₂×C₇₁	#127	1	1	yes	characteristic
34	C₂×C₇₁	#137	1	2	yes	—
35	C₇₁	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	71	1	no	maximal, Sylow
37	C₂×Q₁₆	—	71	2	no	—
38	C₂×SD₁₆	—	71	2	no	—
39	C₄⋊Q₈	—	71	1	no	—
40	#124	D₇₁	1	1	yes	characteristic
41	#97	—	71	1	no	—
42	C₂×Q₈	—	71	2	no	—
43	C₂×Q₈	C₂×D₇₁	1	1	yes	characteristic
44	C₂×D₄	C₂×D₇₁	1	1	yes	characteristic
45	Q₁₆	—	142	4	no	—
46	SD₁₆	—	142	4	no	—
47	C₂×C₈	—	71	2	no	—
48	C₄⋊C₄	—	142	2	no	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₄²	C₂×D₇₁	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	142	2	no	—
54	Q₈	—	142	2	no	—
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	142	2	no	—
58	C₂×C₄	C₂²⋊D₇₁	1	2	yes	—
59	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
60	C₈	—	71	4	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	284	2	no	—
67	C₄	D₄×D₇₁	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	#14556124	1	1	yes	characteristic
70	C₂	#14556158	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#124×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₄
prime radical	C₂×D₄×C₇₁	C₂²
hypercenter	#124	D₇₁
hypocenter	C₇₁	#497
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#11765855	C₂	semidirect
3. #14556128	C₂	semidirect
4. #14556039	C₂	semidirect
35. C₇₁	#497	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11765855	2
C₂×#11765853	2
#14556128	2
#14556039	2
#124×C₇₁	2
#497	71