Subgroups — #28338718

Order 13696.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812052	C₂	1	4	yes	maximal, complemented
2	#20293004	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₀₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₈₅₆	—	2	1	no	—
6	C₂×D₈₅₆	C₂²	1	2	yes	—
7	#16812106	—	2	1	no	—
8	#16812052	—	2	8	no	—
9	C₂×C₁₀₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₀₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₀₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₇	—	2	1	no	—
13	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×D₄₂₈	—	2	2	no	—
15	C₂×D₄₂₈	—	4	2	no	—
16	D₈₅₆	—	2	4	no	—
17	D₈₅₆	—	4	2	no	—
18	C₁₀₇⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×Q₈×C₁₀₇	—	2	2	no	—
20	Q₁₆×C₁₀₇	—	2	4	no	—
21	Q₁₆×C₁₀₇	—	4	2	no	—
22	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₇	—	4	2	no	—
25	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
26	C₂²×D₁₀₇	—	4	2	no	—
27	Q₈×C₁₀₇	—	4	2	no	—
28	Q₈×C₁₀₇	—	4	2	no	—
29	C₂×C₄×C₁₀₇	—	4	2	no	—
30	C₂×C₄×C₁₀₇	D₈	1	2	yes	—
31	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	D₄₂₈	—	4	2	no	—
33	D₄₂₈	—	4	2	no	—
34	D₄₂₈	—	8	4	no	—
35	C₈×C₁₀₇	D₈	1	2	yes	—
36	C₈×C₁₀₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₀₇	C₂×D₈	1	1	yes	characteristic
38	C₂×D₁₀₇	—	8	2	no	—
39	C₂×D₁₀₇	—	8	4	no	—
40	C₄×C₁₀₇	—	8	2	no	—
41	C₄×C₁₀₇	SD₃₂	1	4	yes	complemented
42	C₄×C₁₀₇	#127	1	1	yes	characteristic
43	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₁₀₇	#493	1	1	yes	characteristic
45	C₂×C₁₀₇	C₂×SD₃₂	1	2	yes	—
46	D₁₀₇	—	16	2	no	—
47	#2248	—	107	1	no	maximal, Sylow
48	C₁₀₇	#2248	1	1	yes	characteristic, complemented, Sylow
49	C₂×SD₃₂	—	107	4	no	—
50	#494	D₁₀₇	1	1	yes	characteristic, complemented
51	#493	—	107	1	no	—
52	C₄×C₁₆	—	107	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₁₀₇	1	2	yes	—
55	C₂×D₈	—	107	2	no	—
56	C₂×D₈	—	214	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	214	1	no	—
59	SD₃₂	—	214	8	no	—
60	C₂×C₁₆	—	107	2	no	—
61	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	428	2	no	—
64	C₂×D₄	—	214	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	428	2	no	—
68	D₈	—	214	4	no	—
69	C₂×C₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
73	C₁₆	—	107	4	no	—
74	C₂³	—	428	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	856	4	no	—
78	D₄	—	428	2	no	—
79	D₄	—	428	2	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#13133803	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
83	C₈	#13133803	1	2	yes	—
84	C₈	D₄×D₁₀₇	1	2	yes	—
85	C₂²	—	856	2	no	—
86	C₂²	—	856	4	no	—
87	C₂²	C₂×#13133803	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#16812052	1	4	yes	complemented
90	C₄	#16812105	1	1	yes	characteristic
91	C₄	C₂×#13133803	1	1	yes	characteristic
92	C₂	—	1712	2	no	—
93	C₂	#20293003	1	1	yes	characteristic
94	C₂	C₂×#16812052	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133803
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	#494×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₈
prime radical	#20293004	C₂
hypercenter	#494	D₁₀₇
hypocenter	C₁₀₇	#2248
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#16812052	C₂	semidirect
3. C₄×C₁₀₇⋊C₁₆	C₂	semidirect
4. #494×C₁₀₇	C₂	semidirect
18. C₁₀₇⋊C₁₆	D₄	semidirect
41. C₄×C₁₀₇	SD₃₂	semidirect
48. C₁₀₇	#2248	semidirect
50. #494	D₁₀₇	semidirect
89. C₄	#16812052	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812052	2
#20293004	2
C₄×C₁₀₇⋊C₁₆	2
#494×C₁₀₇	2
#2248	107