Subgroups — #20261465

Order 6208.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15690809	C₂	1	4	yes	maximal
2	#15690808	C₂	1	4	yes	maximal
3	C₄×Dic₁₉₄	C₂	1	2	yes	maximal
4	#15690775	C₂	1	4	yes	maximal
5	C₄⋊Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₉₄	C₂²	1	4	yes	—
7	C₂×Dic₁₉₄	C₂²	1	2	yes	—
8	C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	#13095490	C₂²	1	8	yes	—
10	#13095490	C₂²	1	4	yes	—
11	C₄×Dic₉₇	C₂²	1	4	yes	—
12	C₄×Dic₉₇	C₂²	1	4	yes	—
13	C₂×Q₈×C₉₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
15	C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	Q₈×C₉₇	—	4	2	no	—
17	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₉₇	C₂³	1	4	yes	—
19	C₂×C₄×C₉₇	C₂³	1	2	yes	—
20	C₂×Dic₉₇	C₂³	1	4	yes	—
21	C₂×Dic₉₇	C₂³	1	4	yes	—
22	Dic₁₉₄	—	4	4	no	—
23	Dic₁₉₄	Q₈	1	8	yes	complemented
24	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
25	C₄×C₉₇	—	2	4	no	—
26	C₄×C₉₇	—	2	1	no	—
27	C₄×C₉₇	C₂×Q₈	1	4	yes	—
28	Dic₉₇	—	2	4	no	—
29	Dic₉₇	C₂×Q₈	1	8	yes	—
30	C₂×C₉₇	C₂²×Q₈	1	2	yes	—
31	C₂×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
32	C₉₇	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	97	1	no	maximal, Sylow
34	C₄⋊Q₈	—	97	4	no	—
35	C₄⋊Q₈	—	97	4	no	—
36	C₄⋊Q₈	D₉₇	1	1	yes	characteristic
37	C₄×Q₈	—	97	4	no	—
38	C₄×Q₈	—	97	2	no	—
39	C₂×Q₈	—	97	2	no	—
40	C₂×Q₈	—	97	4	no	—
41	C₂×Q₈	C₂×D₉₇	1	2	yes	—
42	C₄⋊C₄	—	97	8	no	—
43	C₄⋊C₄	—	97	4	no	—
44	C₄⋊C₄	—	97	2	no	—
45	C₄⋊C₄	C₂×D₉₇	1	4	yes	—
46	C₄²	—	97	4	no	—
47	C₄²	—	97	4	no	—
48	C₄²	C₂×D₉₇	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	Q₈	—	97	8	no	—
51	Q₈	—	388	4	no	—
52	C₂×C₄	—	97	4	no	—
53	C₂×C₄	—	97	4	no	—
54	C₂×C₄	C₂²×D₉₇	1	4	yes	—
55	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₉₇	1	2	yes	—
57	C₂²	C₂³×D₉₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	4	no	—
60	C₄	—	97	8	no	—
61	C₄	—	194	4	no	—
62	C₄	Q₈×D₉₇	1	4	yes	—
63	C₂	C₂×Q₈×D₉₇	1	2	yes	—
64	C₂	#15690855	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂⁴
Fitting	C₄⋊Q₈×C₉₇	C₂
hypercenter	C₄⋊Q₈	D₉₇
hypocenter	C₉₇	Q₈²
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
23. Dic₁₉₄	Q₈	semidirect
32. C₉₇	Q₈²	semidirect