Subgroups — #27143766

Order 12416.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260917	C₂	1	3	yes	maximal
2	#20260534	C₂	1	3	yes	maximal, complemented
3	Q₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690858	C₂²	1	1	yes	characteristic
5	#15690841	—	2	3	no	—
6	#15690738	—	2	3	no	—
7	#15690729	C₂²	1	3	yes	—
8	C₄⋊Q₈×C₉₇	—	2	3	no	—
9	C₄⋊Q₈×C₉₇	C₂²	1	3	yes	—
10	C₄×Q₈×C₉₇	—	2	3	no	—
11	#13095512	—	4	1	no	—
12	#13095511	—	2	3	no	—
13	Q₈×D₉₇	—	4	1	no	—
14	#13095501	—	4	3	no	—
15	#13095498	—	4	3	no	—
16	#13095495	—	2	3	no	—
17	C₂×Q₈×C₉₇	—	2	1	no	—
18	C₂×Q₈×C₉₇	—	2	1	no	—
19	C₂×Q₈×C₉₇	D₄	1	3	yes	—
20	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₉₇	—	2	3	no	—
22	C₄⋊C₄×C₉₇	—	2	6	no	—
23	C₄²×C₉₇	—	2	3	no	—
24	C₄²×C₉₇	D₄	1	3	yes	—
25	Q₈×C₉₇	—	2	4	no	—
26	Q₈×C₉₇	—	4	1	no	—
27	Q₈×C₉₇	—	4	3	no	—
28	Q₈×C₉₇	—	8	1	no	—
29	C₂×C₄×C₉₇	—	2	3	no	—
30	C₂×C₄×C₉₇	—	2	3	no	—
31	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
32	C₂²⋊D₉₇	—	4	1	no	—
33	D₃₈₈	—	2	3	no	—
34	C₄×D₉₇	—	4	3	no	—
35	Dic₁₉₄	—	2	3	no	—
36	#11715895	—	4	3	no	—
37	C₂²×C₉₇	#120	1	1	yes	characteristic
38	C₂×D₉₇	—	4	1	no	—
39	C₄×C₉₇	—	2	6	no	—
40	C₄×C₉₇	—	2	3	no	—
41	C₄×C₉₇	—	4	3	no	—
42	Dic₉₇	—	4	1	no	—
43	C₂×C₉₇	—	2	1	no	—
44	C₂×C₉₇	B(4,𝔽₂)	1	1	yes	characteristic
45	D₉₇	—	8	1	no	—
46	#2205	—	97	1	no	maximal, Sylow
47	C₉₇	#2205	1	1	yes	characteristic, complemented, Sylow
48	Q₈²	D₉₇	1	1	yes	characteristic, complemented
49	#456	—	97	3	no	—
50	#356	—	97	3	no	—
51	2₋¹⁺⁴	—	97	1	no	—
52	#137	—	194	3	no	—
53	C₄⋊Q₈	—	2	3	no	—
54	C₄⋊Q₈	C₂×D₉₇	1	3	yes	—
55	C₄×Q₈	—	2	3	no	—
56	#104	—	194	3	no	—
57	#101	—	97	3	no	—
58	#41	—	388	1	no	—
59	#41	—	194	3	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	388	1	no	—
63	C₂×Q₈	C₂²⋊D₉₇	1	3	yes	—
64	C₂×Q₈	C₂²×D₉₇	1	1	yes	characteristic
65	Q₁₆	—	388	3	no	—
66	SD₁₆	—	388	3	no	—
67	M₁₆	—	194	3	no	—
68	C₄⋊C₄	—	2	3	no	—
69	C₄⋊C₄	—	2	6	no	—
70	C₄²	—	2	3	no	—
71	C₄²	C₂²⋊D₉₇	1	3	yes	—
72	Q₈	—	2	4	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	3	no	—
75	Q₈	—	8	1	no	—
76	Q₈	—	194	3	no	—
77	D₄	—	388	1	no	—
78	D₄	—	194	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	388	3	no	—
82	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
83	C₈	—	388	3	no	—
84	C₂²	—	388	1	no	—
85	C₂²	#15690779	1	1	yes	characteristic
86	C₄	—	2	6	no	—
87	C₄	—	2	3	no	—
88	C₄	—	4	3	no	—
89	C₄	—	388	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	776	1	no	—
92	C₂	#20260922	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260922
commutator, cosocle	C₂×Q₈×C₉₇	C₂³
Frattini	C₂×Q₈	C₂²×D₉₇
Fitting	Q₈²×C₉₇	C₂
socle	C₂×C₉₇	B(4,𝔽₂)
prime radical	#15690858	C₂²
hypercenter	Q₈²	D₉₇
hypocenter	C₉₇	#2205
LCS	G ▷ C₂×Q₈×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ Q₈²

Complements

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

N	H ≅ G/N	type
2. #20260534	C₂	semidirect
3. Q₈²×C₉₇	C₂	semidirect
47. C₉₇	#2205	semidirect
48. Q₈²	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260917	2
#20260534	2
Q₈²×C₉₇	2
#2205	97