Subgroups — #20268068

Order 6592.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄₁₂	C₂	1	1	yes	maximal, characteristic
2	#15692954	C₂	1	2	yes	maximal, complemented
3	C₄⋊(C₁₀₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692941	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₁₀₃	—	2	1	no	—
8	C₂×C₁₀₃⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₀₃	C₂²	1	1	yes	characteristic
10	(C₂×D₁₀₃)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₀₃	—	2	1	no	—
12	C₄⋊C₄×C₁₀₃	—	2	1	no	—
13	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
14	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₀₃	—	2	1	no	—
16	Q₈×C₁₀₃	—	4	2	no	—
17	C₂×C₄×C₁₀₃	—	2	2	no	—
18	C₂×C₄×C₁₀₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₀₃	—	2	1	no	—
22	D₄₁₂	—	2	1	no	—
23	D₄₁₂	Q₈	1	2	yes	complemented
24	C₄×D₁₀₃	—	4	2	no	—
25	C₁₀₃⋊C₈	—	2	2	no	—
26	C₂²×C₁₀₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₀₃	—	2	2	no	—
28	C₂×D₁₀₃	—	4	1	no	—
29	C₂×D₁₀₃	—	4	1	no	—
30	C₄×C₁₀₃	—	2	1	no	—
31	C₄×C₁₀₃	—	4	2	no	—
32	C₄×C₁₀₃	D₈	1	2	yes	—
33	C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
34	C₄×C₁₀₃	#41	1	1	yes	characteristic
35	Dic₁₀₃	—	4	1	no	—
36	C₂×C₁₀₃	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₁₀₃	C₂×D₈	1	1	yes	characteristic
38	C₂×C₁₀₃	#137	1	1	yes	characteristic
39	D₁₀₃	—	4	2	no	—
40	C₁₀₃	#474	1	1	yes	characteristic, complemented, Sylow
41	#474	—	103	1	no	maximal, Sylow
42	C₄⋊Q₈	D₁₀₃	1	1	yes	characteristic, complemented
43	C₄×D₄	—	103	1	no	—
44	#107	—	103	2	no	—
45	C₄⋊C₈	—	103	1	no	—
46	#102	—	103	2	no	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×D₄	—	103	1	no	—
49	C₂²×C₄	—	206	1	no	—
50	C₂×C₈	—	103	2	no	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	—	103	1	no	—
53	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
54	C₂²⋊C₄	—	206	1	no	—
55	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
56	C₂³	—	206	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	103	2	no	—
59	D₄	—	206	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	412	2	no	—
62	C₂×C₄	—	206	1	no	—
63	C₂×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₀₃	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
66	C₈	—	206	2	no	—
67	C₂²	—	412	1	no	—
68	C₂²	—	412	1	no	—
69	C₂²	—	206	2	no	—
70	C₂²	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	412	1	no	—
74	C₄	#13110090	1	2	yes	—
75	C₄	Q₈×D₁₀₃	1	1	yes	characteristic
76	C₄	#13110106	1	1	yes	characteristic
77	C₂	—	412	2	no	—
78	C₂	#15692997	1	1	yes	characteristic
79	C₂	C₂×#13110090	1	1	yes	characteristic
80	C₂	#15693052	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₃
Fitting	C₄⋊Q₈×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₀₃
hypocenter	C₁₀₃	#474
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #15692954	C₂	semidirect
3. C₄⋊(C₁₀₃⋊C₈)	C₂	semidirect
5. C₄⋊Q₈×C₁₀₃	C₂	semidirect
23. D₄₁₂	Q₈	semidirect
40. C₁₀₃	#474	semidirect
42. C₄⋊Q₈	D₁₀₃	semidirect