Subgroups — #27141025

Order 12416.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261125	C₂	1	3	yes	maximal, complemented
2	#20260475	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095498	—	2	3	no	—
5	C₂×#13095496	—	2	3	no	—
6	#15690807	C₂²	1	1	yes	characteristic, complemented
7	#15690721	C₂²	1	3	yes	complemented
8	#124×C₉₇	—	2	3	no	—
9	#124×C₉₇	C₂²	1	3	yes	—
10	#120×C₉₇	—	2	3	no	—
11	C₂×D₃₈₈	—	2	3	no	—
12	#13095498	—	4	6	no	—
13	#13095496	—	4	6	no	—
14	C₂×#11715895	—	2	3	no	—
15	C₂⁴×C₉₇	—	2	1	no	—
16	C₂×Q₈×C₉₇	D₄	1	3	yes	—
17	C₂×D₄×C₉₇	—	2	3	no	—
18	C₂×D₄×C₉₇	D₄	1	3	yes	—
19	C₂²⋊C₄×C₉₇	—	2	3	no	—
20	C₂²⋊C₄×C₉₇	—	2	6	no	—
21	C₄²×C₉₇	—	2	1	no	—
22	C₄²×C₉₇	C₂³	1	1	yes	characteristic
23	C₂³×C₉₇	—	2	3	no	—
24	C₂³×C₉₇	—	8	3	no	—
25	C₂²×D₉₇	—	4	1	no	—
26	Q₈×C₉₇	—	4	3	no	—
27	D₄×C₉₇	—	4	3	no	—
28	D₄×C₉₇	—	8	3	no	—
29	C₂×C₄×C₉₇	—	2	3	no	—
30	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
31	D₃₈₈	—	4	3	no	—
32	D₃₈₈	—	4	3	no	—
33	#11715895	—	2	6	no	—
34	C₂²×C₉₇	—	4	3	no	—
35	C₂²×C₉₇	—	4	6	no	—
36	C₂²×C₉₇	—	8	4	no	—
37	C₂²×C₉₇	#120	1	1	yes	characteristic
38	C₂×D₉₇	—	8	3	no	—
39	C₄×C₉₇	—	2	3	no	—
40	C₄×C₉₇	—	4	3	no	—
41	C₂×C₉₇	—	8	3	no	—
42	C₂×C₉₇	#453	1	3	yes	—
43	D₉₇	—	16	1	no	—
44	#1655	—	97	1	no	maximal, Sylow
45	C₉₇	#1655	1	1	yes	characteristic, complemented, Sylow
46	H(3,𝔽₄)	D₉₇	1	1	yes	characteristic, complemented
47	#496	—	97	3	no	—
48	#329	—	97	3	no	—
49	C₂×SD₁₆	—	194	3	no	—
50	C₂×D₈	—	194	3	no	—
51	#127	—	97	1	no	—
52	#124	—	2	3	no	—
53	#124	C₂×D₉₇	1	3	yes	—
54	#120	—	2	3	no	—
55	#97	—	97	3	no	—
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	C₂²⋊D₉₇	1	3	yes	—
58	C₂×D₄	—	2	3	no	—
59	C₂×D₄	—	194	3	no	—
60	C₂×D₄	C₂²⋊D₉₇	1	3	yes	—
61	SD₁₆	—	388	6	no	—
62	D₈	—	388	6	no	—
63	C₂×C₈	—	194	3	no	—
64	C₂²⋊C₄	—	2	3	no	—
65	C₂²⋊C₄	—	2	6	no	—
66	C₄²	—	2	1	no	—
67	C₄²	C₂²×D₉₇	1	1	yes	characteristic
68	C₂³	—	2	3	no	—
69	C₂³	—	8	3	no	—
70	C₂³	—	388	1	no	—
71	Q₈	—	4	3	no	—
72	D₄	—	4	3	no	—
73	D₄	—	8	3	no	—
74	D₄	—	388	3	no	—
75	D₄	—	388	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
78	C₈	—	194	6	no	—
79	C₂²	—	4	3	no	—
80	C₂²	—	4	6	no	—
81	C₂²	—	8	4	no	—
82	C₂²	—	776	3	no	—
83	C₂²	#15690779	1	1	yes	characteristic
84	C₄	—	2	3	no	—
85	C₄	—	4	3	no	—
86	C₂	—	8	3	no	—
87	C₂	—	1552	1	no	—
88	C₂	#20260903	1	3	yes	—

	subgroup	G/H
center	C₂²	#15690779
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	H(3,𝔽₄)×C₉₇	C₂
socle	C₂²×C₉₇	#120
hypercenter	H(3,𝔽₄)	D₉₇
hypocenter	C₉₇	#1655
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #20261125	C₂	semidirect
2. #20260475	C₂	semidirect
3. H(3,𝔽₄)×C₉₇	C₂	semidirect
6. #15690807	C₂²	semidirect
7. #15690721	C₂²	semidirect
45. C₉₇	#1655	semidirect
46. H(3,𝔽₄)	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261125	2
#20260475	2
H(3,𝔽₄)×C₉₇	2
#1655	97