Subgroups — #20261211

Order 6208.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₉₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×Dic₁₉₄	C₂	1	3	yes	maximal, complemented
4	#15690775	C₂	1	3	yes	maximal, complemented
5	#15690768	C₂	1	3	yes	maximal
6	#15690767	C₂	1	3	yes	maximal
7	C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₉₄	C₂²	1	3	yes	—
9	Q₈×D₉₇	C₄	1	8	yes	complemented
10	C₂×C₄×D₉₇	C₂²	1	3	yes	—
11	C₄⋊Dic₉₇	C₂²	1	3	yes	—
12	#13095490	C₂²	1	3	yes	—
13	#13095490	C₂²	1	3	yes	—
14	#13095488	C₂²	1	1	yes	characteristic
15	#13095488	C₂²	1	3	yes	—
16	C₄×Dic₉₇	C₂²	1	3	yes	—
17	C₄×Dic₉₇	C₂²	1	6	yes	—
18	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₉₇	C₂²	1	3	yes	—
20	C₄²×C₉₇	C₂²	1	3	yes	—
21	C₂²×D₉₇	C₂³	1	1	yes	characteristic
22	Q₈×C₉₇	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂³	1	3	yes	—
25	C₂×C₄×C₉₇	C₂³	1	3	yes	—
26	C₂×Dic₉₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₉₇	C₂³	1	3	yes	—
28	C₂×Dic₉₇	C₂³	1	3	yes	—
29	C₄×D₉₇	C₂×C₄	1	12	yes	—
30	Dic₁₉₄	C₂×C₄	1	12	yes	—
31	C₂²×C₉₇	C₂⁴	1	1	yes	characteristic
32	C₂×D₉₇	—	2	1	no	—
33	C₂×D₉₇	—	2	1	no	—
34	C₂×D₉₇	C₂²×C₄	1	2	yes	—
35	C₄×C₉₇	—	2	3	no	—
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	C₂²×C₄	1	6	yes	—
38	Dic₉₇	—	2	1	no	—
39	Dic₉₇	—	2	3	no	—
40	Dic₉₇	C₂²×C₄	1	6	yes	—
41	C₂×C₉₇	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
43	C₂×C₉₇	2₋¹⁺⁴	1	1	yes	characteristic
44	D₉₇	—	2	2	no	—
45	C₉₇	#519	1	1	yes	characteristic, complemented, Sylow
46	#519	—	97	1	no	maximal, Sylow
47	C₂²×Q₈	—	97	1	no	—
48	C₄×Q₈	—	97	1	no	—
49	C₄×Q₈	—	97	3	no	—
50	C₄×Q₈	—	97	3	no	—
51	C₄×Q₈	D₉₇	1	1	yes	characteristic, complemented
52	#117	—	97	3	no	—
53	#117	—	97	3	no	—
54	C₂×Q₈	—	97	8	no	—
55	C₂×Q₈	—	97	3	no	—
56	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
57	C₂²×C₄	—	97	3	no	—
58	C₄⋊C₄	—	97	3	no	—
59	C₄⋊C₄	—	97	3	no	—
60	C₄⋊C₄	—	97	3	no	—
61	C₄⋊C₄	C₂×D₉₇	1	3	yes	—
62	C₂²⋊C₄	—	97	3	no	—
63	C₂²⋊C₄	—	97	1	no	—
64	C₄²	—	97	6	no	—
65	C₄²	—	97	3	no	—
66	C₄²	C₂×D₉₇	1	3	yes	—
67	C₂³	—	97	1	no	—
68	Q₈	—	97	12	no	—
69	Q₈	C₄×D₉₇	1	4	yes	—
70	C₂×C₄	—	97	3	no	—
71	C₂×C₄	—	97	12	no	—
72	C₂×C₄	—	97	1	no	—
73	C₂×C₄	—	97	3	no	—
74	C₂×C₄	C₂²×D₉₇	1	3	yes	—
75	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₉₇	1	3	yes	—
77	C₂²	—	97	2	no	—
78	C₂²	—	194	1	no	—
79	C₂²	—	194	1	no	—
80	C₂²	C₂³×D₉₇	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	1	no	—
83	C₄	—	97	6	no	—
84	C₄	—	194	1	no	—
85	C₄	—	194	3	no	—
86	C₄	C₂×C₄×D₉₇	1	6	yes	—
87	C₂	—	194	2	no	—
88	C₂	C₂²×C₄×D₉₇	1	1	yes	characteristic
89	C₂	#15690857	1	1	yes	characteristic
90	C₂	#15690858	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₉₇
commutator	C₂×C₉₇	C₂³×C₄
Fitting	C₄×Q₈×C₉₇	C₂
socle	C₂²×C₉₇	C₂⁴
hypercenter	C₄×Q₈	D₉₇
hypocenter	C₉₇	#519
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₉₇	C₂	semidirect
3. C₄×Dic₁₉₄	C₂	semidirect
4. #15690775	C₂	semidirect
7. C₄×Q₈×C₉₇	C₂	semidirect
9. Q₈×D₉₇	C₄	semidirect
45. C₉₇	#519	semidirect
51. C₄×Q₈	D₉₇	semidirect