Subgroups — #20292594

Order 6848.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133802	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₁₀₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×#41×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#13133802	—	2	2	no	—
6	C₂×C₁₀₇⋊C₈	—	2	2	no	—
7	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₀₇⋊C₈	C₂²	1	2	yes	—
9	#41×C₁₀₇	—	2	2	no	—
10	#41×C₁₀₇	—	2	2	no	—
11	C₂×Q₈×C₁₀₇	C₄	1	1	yes	characteristic
12	C₂×D₄×C₁₀₇	C₄	1	2	yes	—
13	C₂×D₄×C₁₀₇	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₇	C₂²	1	2	yes	—
15	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₂³×C₁₀₇	C₂×C₄	1	2	yes	—
17	C₂³×C₁₀₇	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₁₀₇	—	2	2	no	—
19	D₄×C₁₀₇	—	2	4	no	—
20	D₄×C₁₀₇	—	2	2	no	—
21	C₂×C₄×C₁₀₇	—	2	2	no	—
22	C₂×C₄×C₁₀₇	—	2	2	no	—
23	C₂×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₇	D₄	1	2	yes	—
26	C₂×C₄×C₁₀₇	D₄	1	2	yes	—
27	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	C₁₀₇⋊C₈	—	2	2	no	—
29	C₁₀₇⋊C₈	—	2	2	no	—
30	C₂²×C₁₀₇	—	2	2	no	—
31	C₂²×C₁₀₇	—	2	2	no	—
32	C₂²×C₁₀₇	—	2	4	no	—
33	C₂²×C₁₀₇	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₁₀₇	—	2	2	no	—
36	C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
37	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₁₀₇	—	2	2	no	—
40	C₂×C₁₀₇	—	4	2	no	—
41	C₂×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₁₀₇	#131	1	2	yes	—
43	C₁₀₇	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	107	1	no	maximal, Sylow
45	C₂×#41	D₁₀₇	1	1	yes	characteristic
46	C₂×M₁₆	—	107	1	no	—
47	C₂²×C₈	—	107	1	no	—
48	C₂²⋊C₈	—	107	4	no	—
49	#41	—	2	2	no	—
50	#41	—	2	2	no	—
51	C₂×Q₈	Dic₁₀₇	1	1	yes	characteristic
52	C₂×D₄	Dic₁₀₇	1	2	yes	—
53	C₂×D₄	Dic₁₀₇	1	1	yes	characteristic
54	C₂²×C₄	C₂×D₁₀₇	1	2	yes	—
55	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
56	M₁₆	—	214	2	no	—
57	C₂×C₈	—	107	2	no	—
58	C₂×C₈	—	107	2	no	—
59	C₂×C₈	—	214	2	no	—
60	C₂³	C₂×Dic₁₀₇	1	2	yes	—
61	C₂³	C₂×Dic₁₀₇	1	1	yes	characteristic
62	Q₈	—	2	2	no	—
63	D₄	—	2	4	no	—
64	D₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₂×Dic₁₀₇	1	2	yes	—
68	C₂×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₁₀₇	1	2	yes	—
70	C₂×C₄	C₂²⋊D₁₀₇	1	2	yes	—
71	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
72	C₈	—	214	2	no	—
73	C₈	—	214	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	C₂²⋊Dic₁₀₇	1	2	yes	—
78	C₂²	C₂²×Dic₁₀₇	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
81	C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	C₂	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
86	C₂	#16812116	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂²×C₄
Fitting	C₂×#41×C₁₀₇	C₂
hypercenter	C₂×#41	D₁₀₇
hypocenter	C₁₀₇	#408
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13133802	C₂	semidirect
2. C₂²×C₁₀₇⋊C₈	C₂	semidirect
3. C₂²⋊(C₁₀₇⋊C₈)	C₂	semidirect
43. C₁₀₇	#408	semidirect