Subgroups — #28338717

Order 13696.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812054	C₂	1	4	yes	maximal, complemented
2	#20293007	C₂	1	1	yes	maximal, characteristic, complemented
3	#20293004	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₄₂₈	—	2	1	no	—
6	C₂×Dic₄₂₈	C₂²	1	2	yes	—
7	C₂×D₈₅₆	—	2	1	no	—
8	C₂×D₈₅₆	C₂²	1	2	yes	—
9	#16812109	—	2	1	no	—
10	#16812106	—	2	1	no	—
11	#16812054	—	2	8	no	—
12	C₂×C₁₆×C₁₀₇	C₂²	1	2	yes	—
13	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₄	—	2	2	no	—
15	C₂×D₄₂₈	—	2	2	no	—
16	C₂×D₄₂₈	—	4	2	no	—
17	Dic₄₂₈	—	2	4	no	—
18	Dic₄₂₈	—	4	2	no	—
19	D₈₅₆	—	2	4	no	—
20	D₈₅₆	—	4	2	no	—
21	C₄⋊Dic₁₀₇	—	4	2	no	—
22	C₂×C₈×C₁₀₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
24	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
25	C₁₆×C₁₀₇	D₄	1	4	yes	complemented
26	C₂²×D₁₀₇	—	4	2	no	—
27	C₂×C₄×C₁₀₇	D₈	1	2	yes	—
28	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₇	—	4	2	no	—
30	D₄₂₈	—	4	2	no	—
31	D₄₂₈	—	4	2	no	—
32	D₄₂₈	—	8	4	no	—
33	Dic₂₁₄	—	4	2	no	—
34	Dic₂₁₄	—	4	2	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	SD₃₂	1	4	yes	complemented
41	C₄×C₁₀₇	#127	1	1	yes	characteristic
42	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
43	Dic₁₀₇	—	8	2	no	—
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	—	107	1	no	—
51	#493	—	107	1	no	—
52	C₄×C₁₆	D₁₀₇	1	1	yes	characteristic, complemented
53	C₂×Q₁₆	—	107	2	no	—
54	C₂×Q₁₆	—	214	1	no	—
55	C₂×D₈	—	107	2	no	—
56	C₂×D₈	—	214	1	no	—
57	C₄⋊Q₈	—	214	1	no	—
58	#127	—	214	1	no	—
59	SD₃₂	—	214	8	no	—
60	C₂×C₁₆	C₂×D₁₀₇	1	2	yes	—
61	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
62	C₂×Q₈	—	214	2	no	—
63	C₂×D₄	—	428	2	no	—
64	C₂×D₄	—	214	2	no	—
65	Q₁₆	—	428	2	no	—
66	Q₁₆	—	214	4	no	—
67	D₈	—	428	2	no	—
68	D₈	—	214	4	no	—
69	C₂×C₈	D₄₂₈	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
71	C₄⋊C₄	—	428	2	no	—
72	C₄²	D₄₂₈	1	1	yes	characteristic
73	C₁₆	D₄₂₈	1	4	yes	complemented
74	C₂³	—	428	2	no	—
75	Q₈	—	428	2	no	—
76	Q₈	—	428	2	no	—
77	D₄	—	856	4	no	—
78	D₄	—	428	2	no	—
79	D₄	—	428	2	no	—
80	C₂×C₄	—	428	2	no	—
81	C₂×C₄	D₈₅₆	1	2	yes	—
82	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
83	C₈	D₈₅₆	1	2	yes	—
84	C₈	C₂×D₄₂₈	1	2	yes	—
85	C₂²	—	856	2	no	—
86	C₂²	—	856	4	no	—
87	C₂²	C₂×D₈₅₆	1	1	yes	characteristic
88	C₄	—	856	2	no	—
89	C₄	#16812054	1	4	yes	complemented
90	C₄	#16812106	1	1	yes	characteristic
91	C₄	C₂×D₈₅₆	1	1	yes	characteristic
92	C₂	—	1712	2	no	—
93	C₂	#20293004	1	1	yes	characteristic
94	C₂	C₂×#16812054	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₅₆
commutator, cosocle	C₂×C₈×C₁₀₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting	C₄×C₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₈
prime radical	#20293004	C₂
hypercenter	C₄×C₁₆	D₁₀₇
hypocenter	C₁₀₇	#2248
LCS	G ▷ C₂×C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16812054	C₂	semidirect
2. #20293007	C₂	semidirect
4. C₄×C₁₆×C₁₀₇	C₂	semidirect
25. C₁₆×C₁₀₇	D₄	semidirect
40. C₄×C₁₀₇	SD₃₂	semidirect
48. C₁₀₇	#2248	semidirect
52. C₄×C₁₆	D₁₀₇	semidirect
73. C₁₆	D₄₂₈	semidirect
89. C₄	#16812054	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812054	2
#20293007	2
#20293004	2
C₄×C₁₆×C₁₀₇	2
#2248	107