Subgroups — #28334501

Order 13696.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293000	C₂	1	1	yes	maximal, characteristic
2	#20292448	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133807	—	2	1	no	—
5	C₂×#13133807	—	2	1	no	—
6	#16812109	C₄	1	1	yes	characteristic
7	#16812106	C₄	1	1	yes	characteristic
8	#16812049	—	2	1	no	—
9	M₃₂×C₁₀₇	—	2	1	no	—
10	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₄	—	2	1	no	—
12	C₂×D₄₂₈	—	2	1	no	—
13	C₂×D₄₂₈	—	4	1	no	—
14	#13133807	—	4	1	no	—
15	#13133807	—	4	1	no	—
16	#13133807	—	4	1	no	—
17	#13133807	—	4	1	no	—
18	C₄⋊Dic₁₀₇	—	4	1	no	—
19	C₁₀₇⋊C₁₆	—	4	1	no	—
20	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
22	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₀₇	—	4	1	no	—
24	C₂²×D₁₀₇	—	4	1	no	—
25	C₂×C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₁₀₇	—	4	1	no	—
28	D₄₂₈	—	4	1	no	—
29	D₄₂₈	—	4	1	no	—
30	D₄₂₈	—	8	2	no	—
31	Dic₂₁₄	—	4	1	no	—
32	Dic₂₁₄	—	4	1	no	—
33	C₈×C₁₀₇	D₈	1	1	yes	characteristic
34	C₈×C₁₀₇	D₈	1	1	yes	characteristic
35	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
36	C₈×C₁₀₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₁₀₇	#100	1	1	yes	characteristic
38	C₂×D₁₀₇	—	8	2	no	—
39	C₂×D₁₀₇	—	8	1	no	—
40	C₄×C₁₀₇	—	2	2	no	—
41	C₄×C₁₀₇	#102	1	1	yes	characteristic
42	C₄×C₁₀₇	#102	1	1	yes	characteristic
43	Dic₁₀₇	—	8	1	no	—
44	C₂×C₁₀₇	—	2	1	no	—
45	C₂×C₁₀₇	#331	1	1	yes	characteristic
46	D₁₀₇	—	16	1	no	—
47	#1359	—	107	1	no	maximal, Sylow
48	C₁₀₇	#1359	1	1	yes	characteristic, complemented, Sylow
49	#492	—	107	1	no	—
50	#364	—	107	1	no	—
51	#364	D₁₀₇	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	214	1	no	—
53	C₂×SD₁₆	—	214	1	no	—
54	C₄⋊Q₈	—	107	1	no	—
55	#127	—	107	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	214	1	no	—
58	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
59	C₂×Q₈	—	214	1	no	—
60	C₂×D₄	—	428	1	no	—
61	C₂×D₄	—	214	1	no	—
62	SD₁₆	—	428	1	no	—
63	SD₁₆	—	428	1	no	—
64	SD₁₆	—	428	1	no	—
65	SD₁₆	—	428	1	no	—
66	C₂×C₈	D₄₂₈	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
68	C₄⋊C₄	—	428	1	no	—
69	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	428	1	no	—
72	C₂³	—	428	1	no	—
73	Q₈	—	428	1	no	—
74	Q₈	—	428	1	no	—
75	D₄	—	856	2	no	—
76	D₄	—	428	1	no	—
77	D₄	—	428	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	428	1	no	—
80	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
81	C₈	#13133803	1	1	yes	characteristic
82	C₈	D₈₅₆	1	1	yes	characteristic
83	C₈	#13133805	1	1	yes	characteristic
84	C₈	#13133807	1	1	yes	characteristic
85	C₂²	—	856	2	no	—
86	C₂²	—	856	1	no	—
87	C₂²	#16812025	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	856	1	no	—
90	C₄	#16812029	1	1	yes	characteristic
91	C₄	#16812031	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	1712	1	no	—
94	C₂	#20292370	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292370
commutator	C₄²×C₁₀₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₇
Fitting	#364×C₁₀₇	C₂
socle	C₂×C₁₀₇	#331
prime radical	#16812106	C₄
cosocle	C₄×C₈×C₁₀₇	C₂²
hypercenter	#364	D₁₀₇
hypocenter	C₁₀₇	#1359
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #20292448	C₂	semidirect
3. #364×C₁₀₇	C₂	semidirect
48. C₁₀₇	#1359	semidirect
51. #364	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293000	2
#20292448	2
#364×C₁₀₇	2
#1359	107