Subgroups — #27197430

Order 13184.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20268196	C₂	1	1	yes	maximal, characteristic
2	#20268193	C₂	1	1	yes	maximal, characteristic
3	#20267619	C₂	1	2	yes	maximal
4	#20267617	C₂	1	2	yes	maximal
5	#346×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	#15693021	—	2	1	no	—
7	#15693013	—	2	1	no	—
8	#15692955	—	2	1	no	—
9	#15692955	C₂²	1	1	yes	characteristic
10	#15692955	C₂²	1	2	yes	—
11	#15692955	C₂²	1	1	yes	characteristic
12	#15692953	—	2	1	no	—
13	C₂×C₁₆×C₁₀₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₆	—	4	1	no	—
16	C₄⋊Dic₁₀₃	—	2	1	no	—
17	C₄⋊Dic₁₀₃	—	2	1	no	—
18	C₄⋊Dic₁₀₃	—	2	2	no	—
19	C₄⋊Dic₁₀₃	—	4	1	no	—
20	Dic₁₀₃⋊C₄	—	4	1	no	—
21	Dic₁₀₃⋊C₄	—	4	1	no	—
22	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
24	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
25	C₁₆×C₁₀₃	Q₈	1	4	yes	complemented
26	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₁₀₃	—	4	1	no	—
30	C₂×Dic₁₀₃	—	4	1	no	—
31	C₂×Dic₁₀₃	—	4	2	no	—
32	Dic₂₀₆	—	8	2	no	—
33	C₈×C₁₀₃	Q₁₆	1	2	yes	—
34	C₈×C₁₀₃	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₀₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₀₃	—	2	1	no	—
37	C₄×C₁₀₃	—	2	1	no	—
38	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₄×C₁₀₃	C₂×Q₁₆	1	1	yes	characteristic
40	Dic₁₀₃	—	8	2	no	—
41	Dic₁₀₃	—	8	1	no	—
42	Dic₁₀₃	—	8	1	no	—
43	C₂×C₁₀₃	#500	1	1	yes	characteristic
44	C₂×C₁₀₃	#509	1	1	yes	characteristic
45	C₂×C₁₀₃	#510	1	1	yes	characteristic
46	#2255	—	103	1	no	maximal, Sylow
47	C₁₀₃	#2255	1	1	yes	characteristic, complemented, Sylow
48	#500	—	103	1	no	—
49	#499	—	103	1	no	—
50	#367	—	103	2	no	—
51	#366	—	103	2	no	—
52	#346	D₁₀₃	1	1	yes	characteristic
53	C₄⋊Q₈	—	206	1	no	—
54	#125	—	206	1	no	—
55	C₂×C₁₆	C₂×D₁₀₃	1	2	yes	—
56	#107	—	103	1	no	—
57	#107	—	103	2	no	—
58	#107	—	103	1	no	—
59	#107	—	206	1	no	—
60	#106	—	206	1	no	—
61	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
62	C₂×Q₈	—	412	1	no	—
63	C₂×C₈	D₄₁₂	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
65	C₄⋊C₄	—	412	1	no	—
66	C₄⋊C₄	—	412	1	no	—
67	C₄⋊C₄	—	412	1	no	—
68	C₄⋊C₄	—	206	1	no	—
69	C₄⋊C₄	—	206	1	no	—
70	C₄⋊C₄	—	206	2	no	—
71	C₄²	D₄₁₂	1	1	yes	characteristic
72	C₁₆	Dic₂₀₆	1	4	yes	complemented
73	Q₈	—	824	2	no	—
74	C₂×C₄	—	412	1	no	—
75	C₂×C₄	—	412	1	no	—
76	C₂×C₄	—	412	2	no	—
77	C₂×C₄	D₈₂₄	1	1	yes	characteristic
78	C₂×C₄	D₈₂₄	1	1	yes	characteristic
79	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
80	C₈	Dic₄₁₂	1	2	yes	—
81	C₈	C₂×Dic₂₀₆	1	2	yes	—
82	C₂²	C₂×D₈₂₄	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	824	2	no	—
86	C₄	—	824	1	no	—
87	C₄	—	824	1	no	—
88	C₄	#15693021	1	1	yes	characteristic
89	C₄	C₂×Dic₄₁₂	1	1	yes	characteristic
90	C₂	#20268196	1	1	yes	characteristic
91	C₂	#20268229	1	1	yes	characteristic
92	C₂	#20268234	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₂₄
commutator, cosocle	C₂×C₈×C₁₀₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₃
Fitting	#346×C₁₀₃	C₂
socle, prime radical	C₂²×C₁₀₃	C₂×D₈
hypercenter	#346	D₁₀₃
hypocenter	C₁₀₃	#2255
LCS	G ▷ C₂×C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
25. C₁₆×C₁₀₃	Q₈	semidirect
47. C₁₀₃	#2255	semidirect
72. C₁₆	Dic₂₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20268196	2
#20268193	2
#20267619	2
#20267617	2
#346×C₁₀₃	2
#2255	103