Subgroups — #20261346

Order 6208.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15690805	C₂	1	4	yes	maximal
2	#15690802	C₂	1	1	yes	maximal, characteristic, complemented
3	#15690800	C₂	1	4	yes	maximal, complemented
4	#15690786	C₂	1	2	yes	maximal
5	Q₈×Dic₉₇	C₂	1	2	yes	maximal, complemented
6	#15690767	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₉₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₄⋊Dic₉₇	C₂²	1	4	yes	—
11	#13095490	C₂²	1	2	yes	—
12	#13095490	C₂²	1	4	yes	—
13	#13095490	C₂²	1	4	yes	—
14	#13095488	C₂²	1	2	yes	—
15	#13095488	C₂²	1	4	yes	—
16	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
17	C₄×Dic₉₇	C₂²	1	4	yes	—
18	C₂×Q₈×C₉₇	C₂²	1	2	yes	—
19	C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
20	C₄²×C₉₇	C₂²	1	1	yes	characteristic
21	C₂²×D₉₇	C₂³	1	1	yes	characteristic
22	Q₈×C₉₇	—	2	4	no	—
23	C₂×C₄×C₉₇	C₂³	1	2	yes	—
24	C₂×C₄×C₉₇	C₂³	1	4	yes	—
25	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₉₇	C₂³	1	2	yes	—
27	C₂×Dic₉₇	C₂³	1	4	yes	—
28	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
29	C₄×D₉₇	—	2	2	no	—
30	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₉₇	—	2	1	no	—
32	C₂×D₉₇	—	2	2	no	—
33	C₄×C₉₇	—	2	4	no	—
34	C₄×C₉₇	—	2	2	no	—
35	C₄×C₉₇	#41	1	2	yes	—
36	Dic₉₇	—	2	1	no	—
37	Dic₉₇	—	2	2	no	—
38	Dic₉₇	—	2	4	no	—
39	C₂×C₉₇	C₂×#41	1	1	yes	characteristic
40	C₂×C₉₇	2₋¹⁺⁴	1	2	yes	—
41	D₉₇	—	4	1	no	—
42	C₉₇	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	97	1	no	maximal, Sylow
44	C₄⋊Q₈	D₉₇	1	1	yes	characteristic, complemented
45	#126	—	97	4	no	—
46	#125	—	97	4	no	—
47	#125	—	97	1	no	—
48	C₂²⋊Q₈	—	97	2	no	—
49	C₄×Q₈	—	97	2	no	—
50	#117	—	97	1	no	—
51	C₂×Q₈	C₂×D₉₇	1	2	yes	—
52	C₂²×C₄	—	97	1	no	—
53	C₄⋊C₄	—	97	2	no	—
54	C₄⋊C₄	—	97	4	no	—
55	C₄⋊C₄	—	97	4	no	—
56	C₄⋊C₄	—	97	2	no	—
57	C₄⋊C₄	—	97	4	no	—
58	C₄⋊C₄	C₂×D₉₇	1	4	yes	—
59	C₂²⋊C₄	—	97	2	no	—
60	C₂²⋊C₄	—	97	4	no	—
61	C₄²	—	97	1	no	—
62	C₄²	—	97	4	no	—
63	C₄²	C₂×D₉₇	1	1	yes	characteristic
64	C₂³	—	97	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	97	2	no	—
67	C₂×C₄	—	97	4	no	—
68	C₂×C₄	—	97	1	no	—
69	C₂×C₄	—	194	2	no	—
70	C₂×C₄	C₂²×D₉₇	1	4	yes	—
71	C₂×C₄	C₂²×D₉₇	1	2	yes	—
72	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
73	C₂²	—	194	1	no	—
74	C₂²	—	194	2	no	—
75	C₂²	C₂³×D₉₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	194	1	no	—
79	C₄	—	194	2	no	—
80	C₄	—	194	4	no	—
81	C₄	#13095510	1	2	yes	—
82	C₂	—	388	1	no	—
83	C₂	C₂×#13095510	1	1	yes	characteristic
84	C₂	#15690858	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₄⋊Q₈×C₉₇	C₂
hypercenter	C₄⋊Q₈	D₉₇
hypocenter	C₉₇	#541
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #15690802	C₂	semidirect
3. #15690800	C₂	semidirect
5. Q₈×Dic₉₇	C₂	semidirect
7. C₄⋊Q₈×C₉₇	C₂	semidirect
42. C₉₇	#541	semidirect
44. C₄⋊Q₈	D₉₇	semidirect