Subgroups — #27139902

Order 12416.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261109	C₂	1	1	yes	maximal, characteristic
2	#20260534	C₂	1	1	yes	maximal, characteristic, complemented
3	#356×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095498	—	2	1	no	—
5	C₂×#13095498	—	2	1	no	—
6	#15690807	C₄	1	1	yes	characteristic
7	#15690729	—	2	1	no	—
8	C₄×#11715895	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
10	#101×C₉₇	—	2	1	no	—
11	C₂×D₃₈₈	—	2	1	no	—
12	C₂×D₃₈₈	—	4	1	no	—
13	#13095498	—	4	2	no	—
14	#13095498	—	4	2	no	—
15	#13095495	—	4	1	no	—
16	C₂×#11715895	—	2	1	no	—
17	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
19	M₁₆×C₉₇	—	4	1	no	—
20	C₄⋊C₄×C₉₇	—	4	1	no	—
21	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₉₇	—	4	1	no	—
23	Q₈×C₉₇	—	4	1	no	—
24	Q₈×C₉₇	—	4	1	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	—	2	1	no	—
28	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	D₃₈₈	—	4	1	no	—
30	D₃₈₈	—	4	1	no	—
31	D₃₈₈	—	8	2	no	—
32	C₈×C₉₇	—	8	1	no	—
33	#11715895	—	2	2	no	—
34	#11715895	—	8	1	no	—
35	C₂²×C₉₇	#99	1	1	yes	characteristic
36	C₂×D₉₇	—	8	2	no	—
37	C₂×D₉₇	—	8	1	no	—
38	C₄×C₉₇	—	2	2	no	—
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	—	4	1	no	—
41	C₄×C₉₇	—	4	1	no	—
42	C₂×C₉₇	—	2	1	no	—
43	C₂×C₉₇	#353	1	1	yes	characteristic
44	D₉₇	—	16	1	no	—
45	#1410	—	97	1	no	maximal, Sylow
46	C₉₇	#1410	1	1	yes	characteristic, complemented, Sylow
47	#492	—	97	1	no	—
48	#356	—	97	1	no	—
49	#356	D₉₇	1	1	yes	characteristic, complemented
50	C₂×SD₁₆	—	194	1	no	—
51	C₂×SD₁₆	—	194	1	no	—
52	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
53	#127	—	97	1	no	—
54	#101	—	2	1	no	—
55	#101	—	194	1	no	—
56	C₄×C₈	—	97	1	no	—
57	C₂×Q₈	D₃₈₈	1	1	yes	characteristic
58	C₂×Q₈	C₂²⋊D₉₇	1	1	yes	characteristic
59	C₂×D₄	—	388	1	no	—
60	C₂×D₄	—	194	1	no	—
61	SD₁₆	—	388	2	no	—
62	SD₁₆	—	388	2	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	388	1	no	—
65	C₂×C₈	—	194	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄²	C₄×D₉₇	1	1	yes	characteristic
68	C₂³	—	388	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	776	2	no	—
72	D₄	—	388	1	no	—
73	D₄	—	388	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	#13095488	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	776	1	no	—
80	C₈	—	194	2	no	—
81	C₂²	—	776	2	no	—
82	C₂²	—	776	1	no	—
83	C₂²	#15690724	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	1552	1	no	—
90	C₂	#20260527	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260527
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₉₇
Fitting	#356×C₉₇	C₂
socle	C₂×C₉₇	#353
prime radical	#15690807	C₄
cosocle	C₄⋊Q₈×C₉₇	C₂²
hypercenter	#356	D₉₇
hypocenter	C₉₇	#1410
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. #20260534	C₂	semidirect
3. #356×C₉₇	C₂	semidirect
46. C₉₇	#1410	semidirect
49. #356	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261109	2
#20260534	2
#356×C₉₇	2
#1410	97