Subgroups — #27143253

Order 12416.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260784	C₂	1	3	yes	maximal, complemented
2	#20260698	C₂	1	3	yes	maximal
3	#338×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
5	C₂×#13095495	C₂²	1	3	yes	—
6	#15690746	—	2	6	no	—
7	#15690722	—	2	6	no	—
8	C₂×M₁₆×C₉₇	C₂²	1	3	yes	—
9	C₂³×D₉₇	Q₈	1	1	yes	characteristic
10	C₂²×Dic₉₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×D₉₇	—	2	3	no	—
12	C₂×C₄×D₉₇	—	2	3	no	—
13	#13095495	—	2	6	no	—
14	C₂×#11715895	—	2	3	no	—
15	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₉₇	—	2	6	no	—
17	C₂×C₈×C₉₇	D₄	1	6	yes	—
18	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
19	C₂²×D₉₇	—	2	3	no	—
20	C₂²×D₉₇	—	4	2	no	—
21	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
22	C₂×C₄×C₉₇	#41	1	3	yes	—
23	C₂×Dic₉₇	—	2	3	no	—
24	C₄×D₉₇	—	4	3	no	—
25	C₄×D₉₇	—	4	1	no	—
26	C₈×C₉₇	—	2	6	no	—
27	#11715895	—	4	3	no	—
28	C₂²×C₉₇	—	4	1	no	—
29	C₂²×C₉₇	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₉₇	—	2	6	no	—
31	C₂×D₉₇	—	4	3	no	—
32	C₂×D₉₇	—	4	1	no	—
33	C₄×C₉₇	#123	1	3	yes	—
34	C₄×C₉₇	#127	1	1	yes	characteristic
35	Dic₉₇	—	4	1	no	—
36	C₂×C₉₇	—	2	3	no	—
37	C₂×C₉₇	#399	1	1	yes	characteristic
38	D₉₇	—	4	2	no	—
39	#2091	—	97	1	no	maximal, Sylow
40	C₉₇	#2091	1	1	yes	characteristic, complemented, Sylow
41	#430	—	97	3	no	—
42	#407	—	97	3	no	—
43	#338	D₉₇	1	1	yes	characteristic, complemented
44	C₂³×C₄	—	97	1	no	—
45	C₂×M₁₆	—	97	3	no	—
46	C₂×M₁₆	C₂×D₉₇	1	3	yes	—
47	#108	—	194	6	no	—
48	C₂²⋊C₈	—	194	6	no	—
49	C₂⁴	—	97	1	no	—
50	C₂²×C₄	—	97	1	no	—
51	C₂²×C₄	—	194	3	no	—
52	C₂²×C₄	—	194	3	no	—
53	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	194	6	no	—
56	C₂×C₈	—	194	3	no	—
57	C₂×C₈	D₃₈₈	1	6	yes	—
58	C₂³	—	388	2	no	—
59	C₂³	—	194	3	no	—
60	C₂³	Q₈×D₉₇	1	1	yes	characteristic
61	C₂×C₄	—	388	3	no	—
62	C₂×C₄	—	388	1	no	—
63	C₂×C₄	—	194	3	no	—
64	C₂×C₄	C₂×D₃₈₈	1	3	yes	—
65	C₂×C₄	#13095510	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	388	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	388	3	no	—
70	C₂²	—	388	1	no	—
71	C₂²	—	194	6	no	—
72	C₂²	#15690790	1	3	yes	—
73	C₄	—	388	1	no	—
74	C₄	#15690794	1	3	yes	—
75	C₄	#15690807	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	388	2	no	—
78	C₂	#20260672	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690807
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#338×C₉₇	C₂
socle	C₂×C₉₇	#399
prime radical	C₂³×D₉₇	Q₈
hypercenter	#338	D₉₇
hypocenter	C₉₇	#2091
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #20260784	C₂	semidirect
3. #338×C₉₇	C₂	semidirect
40. C₉₇	#2091	semidirect
43. #338	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260784	2
#20260698	2
#338×C₉₇	2
#2091	97