Subgroups — #20298594

Order 7232.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813581	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813546	C₂	1	2	yes	maximal, complemented
4	#16813543	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₁₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11744751	—	2	2	no	—
7	C₂×#11744751	C₂²	1	2	yes	—
8	C₄⋊Dic₁₁₃	C₂²	1	2	yes	—
9	C₂²⋊Dic₁₁₃	—	2	1	no	—
10	C₄×Dic₁₁₃	—	2	1	no	—
11	#41×C₁₁₃	—	2	2	no	—
12	#41×C₁₁₃	C₄	1	4	yes	complemented
13	C₂×Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₁₃	—	2	1	no	—
15	C₂×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₁₃	—	2	1	no	—
17	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
18	C₂³×C₁₁₃	—	2	1	no	—
19	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
20	Q₈×C₁₁₃	—	2	1	no	—
21	Q₈×C₁₁₃	C₂×C₄	1	2	yes	—
22	D₄×C₁₁₃	—	2	4	no	—
23	D₄×C₁₁₃	—	2	1	no	—
24	D₄×C₁₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₁₃	—	2	1	no	—
26	C₂×C₄×C₁₁₃	—	2	4	no	—
27	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
29	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₁₃	—	2	2	no	—
32	#11744751	—	2	2	no	—
33	C₂²×C₁₁₃	—	2	2	no	—
34	C₂²×C₁₁₃	—	2	2	no	—
35	C₂²×C₁₁₃	—	4	2	no	—
36	C₂²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₁₃	—	2	2	no	—
39	C₄×C₁₁₃	C₂²⋊C₄	1	2	yes	—
40	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
42	Dic₁₁₃	—	4	2	no	—
43	C₂×C₁₁₃	—	2	2	no	—
44	C₂×C₁₁₃	—	4	2	no	—
45	C₂×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₁₁₃	#135	1	2	yes	—
47	C₁₁₃	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	113	1	no	maximal, Sylow
49	C₂×#41	D₁₁₃	1	1	yes	characteristic
50	C₂²×C₈	—	113	1	no	—
51	#117	—	113	1	no	—
52	#103	—	113	2	no	—
53	#102	—	113	2	no	—
54	#41	—	2	2	no	—
55	#41	Dic₁₁₃	1	4	yes	complemented
56	C₂×Q₈	C₂×D₁₁₃	1	1	yes	characteristic
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂×D₁₁₃	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂×D₁₁₃	1	1	yes	characteristic
61	C₂×C₈	—	113	2	no	—
62	C₂×C₈	—	226	2	no	—
63	C₄⋊C₄	—	113	2	no	—
64	C₂²⋊C₄	—	226	1	no	—
65	C₄²	—	226	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	C₂²⋊D₁₁₃	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	C₂×Dic₁₁₃	1	2	yes	—
70	D₄	—	2	4	no	—
71	D₄	—	2	1	no	—
72	D₄	C₂×Dic₁₁₃	1	2	yes	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	226	2	no	—
76	C₂×C₄	C₂×Dic₁₁₃	1	2	yes	—
77	C₂×C₄	C₂²⋊D₁₁₃	1	2	yes	—
78	C₂×C₄	C₂²⋊D₁₁₃	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
80	C₈	—	226	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊Dic₁₁₃	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	452	2	no	—
88	C₄	C₂²⋊Dic₁₁₃	1	2	yes	—
89	C₄	C₂²×Dic₁₁₃	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	C₂×C₂²⋊Dic₁₁₃	1	1	yes	characteristic
94	C₂	#16813642	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	C₂×#41×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	C₂×#41	D₁₁₃
hypocenter	C₁₁₃	#416
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#11744751	C₂	semidirect
2. #16813581	C₂	semidirect
3. #16813546	C₂	semidirect
4. #16813543	C₂	semidirect
12. #41×C₁₁₃	C₄	semidirect
47. C₁₁₃	#416	semidirect
55. #41	Dic₁₁₃	semidirect