Subgroups — #28334429

Order 13696.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₈₅₆	C₂	1	1	yes	maximal, characteristic
2	#20292378	C₂	1	1	yes	maximal, characteristic, complemented
3	#346×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₅₆	C₄	1	1	yes	characteristic
5	C₄×D₄₂₈	—	2	1	no	—
6	#16812043	C₄	1	1	yes	characteristic
7	C₄⋊(C₁₀₇⋊C₈)	—	2	1	no	—
8	#16812031	—	2	1	no	—
9	C₂×C₁₆×C₁₀₇	—	2	1	no	—
10	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×D₄₂₈	—	2	1	no	—
12	C₂×C₄×D₁₀₇	—	4	1	no	—
13	D₈₅₆	—	2	1	no	—
14	D₈₅₆	C₈	1	2	yes	complemented
15	C₂×C₁₀₇⋊C₈	—	4	1	no	—
16	C₄⋊Dic₁₀₇	—	2	1	no	—
17	(C₂×D₁₀₇)⋊C₄	—	4	1	no	—
18	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
20	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₀₇	—	2	2	no	—
22	C₂²×D₁₀₇	—	4	1	no	—
23	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₇	SD₁₆	1	1	yes	characteristic
26	C₂×Dic₁₀₇	—	4	1	no	—
27	D₄₂₈	—	2	2	no	—
28	D₄₂₈	—	4	1	no	—
29	C₄×D₁₀₇	—	8	2	no	—
30	C₈×C₁₀₇	—	2	1	no	—
31	C₈×C₁₀₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₀₇	M₁₆	1	1	yes	characteristic
33	C₁₀₇⋊C₈	—	8	1	no	—
34	C₂²×C₁₀₇	#102	1	1	yes	characteristic
35	C₂×D₁₀₇	—	4	2	no	—
36	C₂×D₁₀₇	—	8	1	no	—
37	C₂×D₁₀₇	—	8	1	no	—
38	C₄×C₁₀₇	—	2	1	no	—
39	C₄×C₁₀₇	—	2	1	no	—
40	C₄×C₁₀₇	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₁₀₇	#104	1	1	yes	characteristic
42	Dic₁₀₇	—	8	1	no	—
43	C₂×C₁₀₇	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₁₀₇	#360	1	1	yes	characteristic
45	C₂×C₁₀₇	#361	1	1	yes	characteristic
46	D₁₀₇	—	8	2	no	—
47	#1333	—	107	1	no	maximal, Sylow
48	C₁₀₇	#1333	1	1	yes	characteristic, complemented, Sylow
49	C₄×D₈	—	107	1	no	—
50	#346	D₁₀₇	1	1	yes	characteristic, complemented
51	#334	—	107	1	no	—
52	C₂×D₈	—	107	1	no	—
53	C₄×D₄	—	214	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	#107	—	107	1	no	—
56	C₄⋊C₈	—	214	1	no	—
57	#102	—	214	1	no	—
58	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
59	C₂×D₄	—	214	1	no	—
60	C₂²×C₄	—	428	1	no	—
61	D₈	—	107	2	no	—
62	D₈	—	214	1	no	—
63	C₂×C₈	—	428	1	no	—
64	C₂×C₈	C₄×D₁₀₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
66	C₄⋊C₄	—	214	1	no	—
67	C₂²⋊C₄	—	428	1	no	—
68	C₄²	D₄₂₈	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₂³	—	428	1	no	—
71	D₄	—	428	1	no	—
72	D₄	—	214	2	no	—
73	C₂×C₄	—	856	2	no	—
74	C₂×C₄	—	428	1	no	—
75	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	D₈₅₆	1	1	yes	characteristic
77	C₂×C₄	#13133807	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	856	1	no	—
80	C₈	C₈×D₁₀₇	1	1	yes	characteristic
81	C₈	#13133801	1	1	yes	characteristic
82	C₂²	—	856	1	no	—
83	C₂²	—	856	1	no	—
84	C₂²	—	428	2	no	—
85	C₂²	#16812031	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	856	1	no	—
89	C₄	(C₂×D₁₀₇)⋊C₈	1	1	yes	characteristic
90	C₄	#16812037	1	1	yes	characteristic
91	C₂	—	856	2	no	—
92	C₂	#20292356	1	1	yes	characteristic
93	C₂	#20292438	1	1	yes	characteristic
94	C₂	#20292441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812031
commutator	C₈×C₁₀₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	#346×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#102
prime radical	C₂×D₈₅₆	C₄
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	#346	D₁₀₇
hypocenter	C₁₀₇	#1333
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #20292378	C₂	semidirect
3. #346×C₁₀₇	C₂	semidirect
14. D₈₅₆	C₈	semidirect
48. C₁₀₇	#1333	semidirect
50. #346	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₈₅₆	2
#20292378	2
#346×C₁₀₇	2
#1333	107