Subgroups — #17985993

Order 4544.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11765865	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#14556051	C₂	1	2	yes	maximal, complemented
4	#14556048	C₂	1	2	yes	maximal, complemented
5	#117×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#11765865	—	2	2	no	—
7	#11765865	C₄	1	4	yes	complemented
8	C₂×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	C₂×D₂₈₄	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₇₁	—	2	1	no	—
11	C₂×C₄×D₇₁	—	2	1	no	—
12	C₂×C₇₁⋊C₈	—	2	2	no	—
13	C₂×C₇₁⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₄×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₇₁	—	2	1	no	—
17	C₄²×C₇₁	—	2	1	no	—
18	C₂³×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×D₇₁	—	2	1	no	—
20	C₂×C₄×C₇₁	—	2	2	no	—
21	C₂×C₄×C₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₇₁	D₄	1	2	yes	—
23	C₂×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
25	C₂²⋊D₇₁	—	2	4	no	—
26	C₂×Dic₇₁	—	2	1	no	—
27	D₂₈₄	—	2	1	no	—
28	D₂₈₄	C₂×C₄	1	2	yes	—
29	C₄×D₇₁	—	2	4	no	—
30	Dic₁₄₂	—	2	1	no	—
31	Dic₁₄₂	C₂×C₄	1	2	yes	—
32	C₇₁⋊C₈	—	2	2	no	—
33	C₂²×C₇₁	—	2	2	no	—
34	C₂²×C₇₁	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×D₇₁	—	2	2	no	—
37	C₂×D₇₁	—	4	2	no	—
38	C₄×C₇₁	—	4	2	no	—
39	C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
40	C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	Dic₇₁	—	2	2	no	—
43	C₂×C₇₁	—	2	2	no	—
44	C₂×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₇₁	#135	1	2	yes	—
46	D₇₁	—	4	2	no	—
47	C₇₁	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	71	1	no	maximal, Sylow
49	C₂×#41	—	71	1	no	—
50	C₂²×C₈	—	71	1	no	—
51	#117	D₇₁	1	1	yes	characteristic, complemented
52	#103	—	71	2	no	—
53	#102	—	71	2	no	—
54	#41	—	71	4	no	—
55	#41	—	142	2	no	—
56	C₂×Q₈	—	71	1	no	—
57	C₂×D₄	—	71	1	no	—
58	C₂×D₄	—	142	1	no	—
59	C₂²×C₄	—	142	1	no	—
60	C₂²×C₄	C₂×D₇₁	1	1	yes	characteristic
61	C₂×C₈	—	71	2	no	—
62	C₂×C₈	—	142	2	no	—
63	C₄⋊C₄	C₂×D₇₁	1	2	yes	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₂³	—	142	1	no	—
67	C₂³	C₂²⋊D₇₁	1	1	yes	characteristic
68	Q₈	—	71	2	no	—
69	Q₈	—	142	1	no	—
70	D₄	—	71	2	no	—
71	D₄	—	142	4	no	—
72	D₄	—	142	1	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	142	1	no	—
75	C₂×C₄	—	142	4	no	—
76	C₂×C₄	C₄×D₇₁	1	2	yes	—
77	C₂×C₄	D₂₈₄	1	2	yes	—
78	C₂×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
80	C₈	—	142	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	284	2	no	—
83	C₂²	—	142	2	no	—
84	C₂²	(C₂×D₇₁)⋊C₄	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	142	2	no	—
88	C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
89	C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
90	C₄	C₂×D₂₈₄	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	284	2	no	—
93	C₂	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
94	C₂	#14556148	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#117×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₄
prime radical	C₂×#11765865	C₂
cosocle	C₂×C₄×C₇₁	C₂³
hypercenter	#117	D₇₁
hypocenter	C₇₁	#416
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇₁⋊C₈	C₂	semidirect
3. #14556051	C₂	semidirect
4. #14556048	C₂	semidirect
5. #117×C₇₁	C₂	semidirect
7. #11765865	C₄	semidirect
47. C₇₁	#416	semidirect
51. #117	D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11765865	2
C₂²×C₇₁⋊C₈	2
#14556051	2
#14556048	2
#117×C₇₁	2
#416	71