Subgroups — #28338347

Order 13696.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20292744	C₂	1	2	yes	maximal, complemented
2	#20292457	C₂	1	1	yes	maximal, characteristic, complemented
3	#20292434	C₂	1	2	yes	maximal, complemented
4	#20292402	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	#16812116	—	2	2	no	—
7	C₂×C₁₀₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812045	C₂²	1	2	yes	—
9	#16812036	—	2	2	no	—
10	C₈×Dic₁₀₇	C₂²	1	1	yes	characteristic
11	#135×C₁₀₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₀₇	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₁₀₇	C₄	1	2	yes	complemented
14	D₁₆×C₁₀₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
16	#13133802	—	2	2	no	—
17	#13133802	—	4	2	no	—
18	C₂×C₁₀₇⋊C₈	—	4	2	no	—
19	C₂×C₁₀₇⋊C₈	D₄	1	1	yes	characteristic
20	C₄×Dic₁₀₇	D₄	1	1	yes	characteristic
21	C₁₀₇⋊C₁₆	—	2	1	no	—
22	#41×C₁₀₇	—	2	2	no	—
23	Q₁₆×C₁₀₇	C₂×C₄	1	2	yes	—
24	SD₁₆×C₁₀₇	—	2	2	no	—
25	D₈×C₁₀₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₀₇	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₁₀₇	—	2	2	no	—
30	D₄×C₁₀₇	—	2	2	no	—
31	D₄×C₁₀₇	—	4	2	no	—
32	C₂×C₄×C₁₀₇	—	4	2	no	—
33	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₀₇	—	2	1	no	—
35	C₈×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₁₀₇	#41	1	1	yes	characteristic
37	C₁₀₇⋊C₈	—	4	2	no	—
38	C₁₀₇⋊C₈	D₈	1	2	yes	—
39	C₂²×C₁₀₇	—	4	2	no	—
40	C₂²×C₁₀₇	#135	1	1	yes	characteristic
41	C₄×C₁₀₇	—	4	2	no	—
42	C₄×C₁₀₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₀₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₀₇	—	4	1	no	—
45	C₂×C₁₀₇	—	2	1	no	—
46	C₂×C₁₀₇	—	8	2	no	—
47	C₂×C₁₀₇	C₄×D₈	1	1	yes	characteristic
48	#2179	—	107	1	no	maximal, Sylow
49	C₁₀₇	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	D₁₀₇	1	1	yes	characteristic
51	#443	—	107	2	no	—
52	#368	—	107	1	no	—
53	#359	—	107	2	no	—
54	#346	—	107	1	no	—
55	#135	C₂×D₁₀₇	1	2	yes	—
56	#131	—	214	2	no	—
57	Q₃₂	Dic₁₀₇	1	1	yes	characteristic, complemented
58	SD₃₂	Dic₁₀₇	1	2	yes	complemented
59	D₁₆	Dic₁₀₇	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	107	1	no	—
61	C₂×C₁₆	C₂×D₁₀₇	1	1	yes	characteristic
62	#108	—	107	2	no	—
63	#104	—	214	2	no	—
64	C₄×C₈	—	107	1	no	—
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₁₀₇	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₁₀₇	1	2	yes	—
69	M₁₆	—	428	2	no	—
70	M₁₆	—	214	2	no	—
71	C₂×C₈	—	107	1	no	—
72	C₂×C₈	—	428	2	no	—
73	C₂×C₈	C₂²×D₁₀₇	1	1	yes	characteristic
74	C₄²	—	107	1	no	—
75	C₁₆	—	214	1	no	—
76	C₁₆	C₂×Dic₁₀₇	1	1	yes	characteristic
77	C₁₆	C₂×Dic₁₀₇	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	214	1	no	—
83	C₂×C₄	D₄×D₁₀₇	1	1	yes	characteristic
84	C₈	—	107	2	no	—
85	C₈	—	428	2	no	—
86	C₈	C₂²×Dic₁₀₇	1	1	yes	characteristic
87	C₈	#13133817	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#16812128	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	428	1	no	—
92	C₄	D₄×Dic₁₀₇	1	1	yes	characteristic
93	C₄	D₈×D₁₀₇	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₁₀₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₀₇
commutator	C₈×C₁₀₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₇
Fitting, prime radical	#508×C₁₀₇	C₂
socle	C₂×C₁₀₇	C₄×D₈
cosocle	C₂×C₈×C₁₀₇	C₂³
hypercenter	#508	D₁₀₇
hypocenter	C₁₀₇	#2179
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #20292744	C₂	semidirect
2. #20292457	C₂	semidirect
3. #20292434	C₂	semidirect
4. #20292402	C₂	semidirect
12. Q₃₂×C₁₀₇	C₄	semidirect
13. SD₃₂×C₁₀₇	C₄	semidirect
14. D₁₆×C₁₀₇	C₄	semidirect
49. C₁₀₇	#2179	semidirect
57. Q₃₂	Dic₁₀₇	semidirect
58. SD₃₂	Dic₁₀₇	semidirect
59. D₁₆	Dic₁₀₇	semidirect

Maximal subgroups

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

subgroup	index
#20292744	2
#20292457	2
#20292434	2
#20292402	2
#508×C₁₀₇	2
#2179	107