Subgroups — #28334627

Order 13696.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20293001	C₂	1	1	yes	maximal, characteristic
2	#20292426	C₂	1	1	yes	maximal, characteristic, complemented
3	#356×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13133805	—	2	1	no	—
5	C₂×#13133805	—	2	1	no	—
6	#16812106	C₄	1	1	yes	characteristic
7	#16812028	—	2	1	no	—
8	C₄×C₁₀₇⋊C₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
10	#101×C₁₀₇	—	2	1	no	—
11	C₂×D₄₂₈	—	2	1	no	—
12	C₂×D₄₂₈	—	4	1	no	—
13	#13133805	—	4	2	no	—
14	#13133805	—	4	2	no	—
15	#13133802	—	4	1	no	—
16	C₂×C₁₀₇⋊C₈	—	2	1	no	—
17	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₀₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₀₇	—	4	1	no	—
20	C₄⋊C₄×C₁₀₇	—	4	1	no	—
21	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₇	—	4	1	no	—
23	Q₈×C₁₀₇	—	4	1	no	—
24	Q₈×C₁₀₇	—	4	1	no	—
25	C₂×C₄×C₁₀₇	—	2	1	no	—
26	C₂×C₄×C₁₀₇	—	2	1	no	—
27	C₂×C₄×C₁₀₇	—	2	1	no	—
28	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
29	D₄₂₈	—	4	1	no	—
30	D₄₂₈	—	4	1	no	—
31	D₄₂₈	—	8	2	no	—
32	C₈×C₁₀₇	—	8	1	no	—
33	C₁₀₇⋊C₈	—	2	2	no	—
34	C₁₀₇⋊C₈	—	8	1	no	—
35	C₂²×C₁₀₇	#99	1	1	yes	characteristic
36	C₂×D₁₀₇	—	8	2	no	—
37	C₂×D₁₀₇	—	8	1	no	—
38	C₄×C₁₀₇	—	2	2	no	—
39	C₄×C₁₀₇	—	2	1	no	—
40	C₄×C₁₀₇	—	4	1	no	—
41	C₄×C₁₀₇	—	4	1	no	—
42	C₂×C₁₀₇	—	2	1	no	—
43	C₂×C₁₀₇	#353	1	1	yes	characteristic
44	D₁₀₇	—	16	1	no	—
45	#1410	—	107	1	no	maximal, Sylow
46	C₁₀₇	#1410	1	1	yes	characteristic, complemented, Sylow
47	#492	—	107	1	no	—
48	#356	—	107	1	no	—
49	#356	D₁₀₇	1	1	yes	characteristic, complemented
50	C₂×SD₁₆	—	214	1	no	—
51	C₂×SD₁₆	—	214	1	no	—
52	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
53	#127	—	107	1	no	—
54	#101	—	2	1	no	—
55	#101	—	214	1	no	—
56	C₄×C₈	—	107	1	no	—
57	C₂×Q₈	D₄₂₈	1	1	yes	characteristic
58	C₂×Q₈	C₂²⋊D₁₀₇	1	1	yes	characteristic
59	C₂×D₄	—	428	1	no	—
60	C₂×D₄	—	214	1	no	—
61	SD₁₆	—	428	2	no	—
62	SD₁₆	—	428	2	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	428	1	no	—
65	C₂×C₈	—	214	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄²	C₄×D₁₀₇	1	1	yes	characteristic
68	C₂³	—	428	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	856	2	no	—
72	D₄	—	428	1	no	—
73	D₄	—	428	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	856	1	no	—
80	C₈	—	214	2	no	—
81	C₂²	—	856	2	no	—
82	C₂²	—	856	1	no	—
83	C₂²	#16812023	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	1712	1	no	—
90	C₂	#20292419	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20292419
commutator	C₄²×C₁₀₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₀₇
Fitting	#356×C₁₀₇	C₂
socle	C₂×C₁₀₇	#353
prime radical	#16812106	C₄
cosocle	C₄⋊Q₈×C₁₀₇	C₂²
hypercenter	#356	D₁₀₇
hypocenter	C₁₀₇	#1410
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. #20292426	C₂	semidirect
3. #356×C₁₀₇	C₂	semidirect
46. C₁₀₇	#1410	semidirect
49. #356	D₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20293001	2
#20292426	2
#356×C₁₀₇	2
#1410	107