Subgroups — #28377994

Order 13952.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295948	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295143	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	1	no	—
5	C₄×#11724786	—	2	1	no	—
6	#16812706	—	2	1	no	—
7	#16812635	C₄	1	1	yes	characteristic
8	#16812626	—	2	1	no	—
9	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	#135×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#11724786	—	2	2	no	—
12	#13182032	—	4	1	no	—
13	#13182013	—	2	1	no	—
14	#13182013	—	4	1	no	—
15	C₂×#11724784	—	4	1	no	—
16	C₂×#11724784	D₄	1	1	yes	characteristic
17	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
18	#41×C₁₀₉	—	2	1	no	—
19	Q₁₆×C₁₀₉	C₈	1	1	yes	characteristic, complemented
20	SD₁₆×C₁₀₉	—	2	1	no	—
21	D₈×C₁₀₉	C₈	1	1	yes	characteristic, complemented
22	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	Q₈×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	—	2	1	no	—
25	D₄×C₁₀₉	—	4	1	no	—
26	C₂×C₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	1	no	—
29	#11724786	—	2	4	no	—
30	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
32	#11724784	—	4	1	no	—
33	#11724784	D₈	1	1	yes	characteristic
34	#11724784	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₀₉	—	4	1	no	—
36	C₂²×C₁₀₉	#104	1	1	yes	characteristic
37	C₄×C₁₀₉	—	4	1	no	—
38	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	#102	1	1	yes	characteristic
40	Dic₁₀₉	—	2	2	no	—
41	C₂×C₁₀₉	—	2	1	no	—
42	C₂×C₁₀₉	—	8	1	no	—
43	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
44	#1335	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	109	1	no	—
47	#364	—	109	1	no	—
48	C₈²	—	109	1	no	—
49	#135	#89137	1	1	yes	characteristic
50	#131	—	218	1	no	—
51	M₃₂	—	218	1	no	—
52	#108	—	109	1	no	—
53	#104	—	218	1	no	—
54	C₄×C₈	—	109	1	no	—
55	C₄×C₈	—	218	1	no	—
56	#41	—	2	1	no	—
57	Q₁₆	#11724786	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#11724786	1	1	yes	characteristic, complemented
60	M₁₆	—	436	1	no	—
61	M₁₆	—	218	1	no	—
62	C₂×C₈	—	109	1	no	—
63	C₂×C₈	—	436	1	no	—
64	C₂×C₈	—	218	2	no	—
65	C₂×C₈	C₂×#89137	1	1	yes	characteristic
66	C₄²	—	109	1	no	—
67	C₁₆	—	436	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	218	1	no	—
73	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
74	C₈	—	109	1	no	—
75	C₈	—	109	1	no	—
76	C₈	—	436	1	no	—
77	C₈	—	218	4	no	—
78	C₈	C₂×#11724786	1	1	yes	characteristic
79	C₈	#13182038	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#16812763	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	218	2	no	—
84	C₄	C₂²⋊#11724786	1	1	yes	characteristic
85	C₄	#16812761	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#11724786	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812761
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting, prime radical	#135×C₁₀₉	C₄
socle	C₂×C₁₀₉	D₄⋊C₈
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	#135	#89137
hypocenter	C₁₀₉	#1335
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #20295948	C₂	semidirect
2. C₈×#11724786	C₂	semidirect
19. Q₁₆×C₁₀₉	C₈	semidirect
21. D₈×C₁₀₉	C₈	semidirect
45. C₁₀₉	#1335	semidirect
57. Q₁₆	#11724786	semidirect
59. D₈	#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20295948	2
C₈×#11724786	2
#20295143	2
#1335	109