Subgroups — #20298592

Order 7232.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268647	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11744751	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813545	C₂	1	2	yes	maximal, complemented
4	#16813542	C₂	1	2	yes	maximal, complemented
5	#117×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268647	—	2	2	no	—
7	#14268647	C₄	1	4	yes	complemented
8	C₂×Dic₂₂₆	C₂²	1	1	yes	characteristic
9	C₂×D₄₅₂	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₁₃	—	2	1	no	—
11	C₂×C₄×D₁₁₃	—	2	1	no	—
12	C₂×#11744751	—	2	2	no	—
13	C₂×#11744751	C₂²	1	2	yes	—
14	C₂²×C₄×C₁₁₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₁₁₃	—	2	1	no	—
17	C₄²×C₁₁₃	—	2	1	no	—
18	C₂³×C₁₁₃	D₄	1	1	yes	characteristic
19	C₂²×D₁₁₃	—	2	1	no	—
20	C₂×C₄×C₁₁₃	—	2	2	no	—
21	C₂×C₄×C₁₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₁₁₃	D₄	1	2	yes	—
23	C₂×C₄×C₁₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	C₂²⋊D₁₁₃	—	2	4	no	—
26	C₂×Dic₁₁₃	—	2	1	no	—
27	D₄₅₂	—	2	1	no	—
28	D₄₅₂	C₂×C₄	1	2	yes	—
29	C₄×D₁₁₃	—	2	4	no	—
30	Dic₂₂₆	—	2	1	no	—
31	Dic₂₂₆	C₂×C₄	1	2	yes	—
32	#11744751	—	2	2	no	—
33	C₂²×C₁₁₃	—	2	2	no	—
34	C₂²×C₁₁₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
36	C₂×D₁₁₃	—	2	2	no	—
37	C₂×D₁₁₃	—	4	2	no	—
38	C₄×C₁₁₃	—	4	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	—	2	2	no	—
43	C₂×C₁₁₃	—	2	2	no	—
44	C₂×C₁₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₁₃	#135	1	2	yes	—
46	D₁₁₃	—	4	2	no	—
47	C₁₁₃	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	113	1	no	maximal, Sylow
49	C₂×#41	—	113	1	no	—
50	C₂²×C₈	—	113	1	no	—
51	#117	D₁₁₃	1	1	yes	characteristic, complemented
52	#103	—	113	2	no	—
53	#102	—	113	2	no	—
54	#41	—	113	4	no	—
55	#41	—	226	2	no	—
56	C₂×Q₈	—	113	1	no	—
57	C₂×D₄	—	113	1	no	—
58	C₂×D₄	—	226	1	no	—
59	C₂²×C₄	—	226	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	C₂×D₁₁₃	1	2	yes	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₂³	—	226	1	no	—
67	C₂³	C₂²⋊D₁₁₃	1	1	yes	characteristic
68	Q₈	—	113	2	no	—
69	Q₈	—	226	1	no	—
70	D₄	—	113	2	no	—
71	D₄	—	226	4	no	—
72	D₄	—	226	1	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	226	1	no	—
75	C₂×C₄	—	226	4	no	—
76	C₂×C₄	C₄×D₁₁₃	1	2	yes	—
77	C₂×C₄	D₄₅₂	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	—	452	2	no	—
83	C₂²	—	226	2	no	—
84	C₂²	#14268624	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₁₁₃	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	226	2	no	—
88	C₄	#14268624	1	2	yes	—
89	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
90	C₄	C₂×D₄₅₂	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	452	2	no	—
93	C₂	C₂×#14268624	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	#117×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#117	D₁₁₃
hypocenter	C₁₁₃	#416
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
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×#11744751	C₂	semidirect
3. #16813545	C₂	semidirect
4. #16813542	C₂	semidirect
5. #117×C₁₁₃	C₂	semidirect
7. #14268647	C₄	semidirect
47. C₁₁₃	#416	semidirect
51. #117	D₁₁₃	semidirect