Subgroups — #27139776

Order 12416.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261108	C₂	1	1	yes	maximal, characteristic
2	#20260556	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13095500	—	2	1	no	—
5	C₂×#13095500	—	2	1	no	—
6	#15690810	C₄	1	1	yes	characteristic
7	#15690807	C₄	1	1	yes	characteristic
8	#15690750	—	2	1	no	—
9	M₃₂×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	—	2	1	no	—
12	C₂×D₃₈₈	—	2	1	no	—
13	C₂×D₃₈₈	—	4	1	no	—
14	#13095500	—	4	1	no	—
15	#13095500	—	4	1	no	—
16	#13095500	—	4	1	no	—
17	#13095500	—	4	1	no	—
18	C₄⋊Dic₉₇	—	4	1	no	—
19	#13095486	—	4	1	no	—
20	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
22	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₉₇	—	4	1	no	—
24	C₂²×D₉₇	—	4	1	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₉₇	—	4	1	no	—
28	D₃₈₈	—	4	1	no	—
29	D₃₈₈	—	4	1	no	—
30	D₃₈₈	—	8	2	no	—
31	Dic₁₉₄	—	4	1	no	—
32	Dic₁₉₄	—	4	1	no	—
33	C₈×C₉₇	D₈	1	1	yes	characteristic
34	C₈×C₉₇	D₈	1	1	yes	characteristic
35	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
36	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
37	C₂²×C₉₇	#100	1	1	yes	characteristic
38	C₂×D₉₇	—	8	2	no	—
39	C₂×D₉₇	—	8	1	no	—
40	C₄×C₉₇	—	2	2	no	—
41	C₄×C₉₇	#102	1	1	yes	characteristic
42	C₄×C₉₇	#102	1	1	yes	characteristic
43	Dic₉₇	—	8	1	no	—
44	C₂×C₉₇	—	2	1	no	—
45	C₂×C₉₇	#331	1	1	yes	characteristic
46	D₉₇	—	16	1	no	—
47	#1359	—	97	1	no	maximal, Sylow
48	C₉₇	#1359	1	1	yes	characteristic, complemented, Sylow
49	#492	—	97	1	no	—
50	#364	—	97	1	no	—
51	#364	D₉₇	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	194	1	no	—
53	C₂×SD₁₆	—	194	1	no	—
54	C₄⋊Q₈	—	97	1	no	—
55	#127	—	97	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	194	1	no	—
58	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
59	C₂×Q₈	—	194	1	no	—
60	C₂×D₄	—	388	1	no	—
61	C₂×D₄	—	194	1	no	—
62	SD₁₆	—	388	1	no	—
63	SD₁₆	—	388	1	no	—
64	SD₁₆	—	388	1	no	—
65	SD₁₆	—	388	1	no	—
66	C₂×C₈	D₃₈₈	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
68	C₄⋊C₄	—	388	1	no	—
69	C₄²	C₄×D₉₇	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	388	1	no	—
72	C₂³	—	388	1	no	—
73	Q₈	—	388	1	no	—
74	Q₈	—	388	1	no	—
75	D₄	—	776	2	no	—
76	D₄	—	388	1	no	—
77	D₄	—	388	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	388	1	no	—
80	C₂×C₄	#13095488	1	1	yes	characteristic
81	C₈	#13095496	1	1	yes	characteristic
82	C₈	D₇₇₆	1	1	yes	characteristic
83	C₈	#13095498	1	1	yes	characteristic
84	C₈	#13095500	1	1	yes	characteristic
85	C₂²	—	776	2	no	—
86	C₂²	—	776	1	no	—
87	C₂²	#15690726	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	776	1	no	—
90	C₄	#15690730	1	1	yes	characteristic
91	C₄	#15690732	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	1552	1	no	—
94	C₂	#20260478	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260478
commutator	C₄²×C₉₇	C₂×C₄
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#364×C₉₇	C₂
socle	C₂×C₉₇	#331
prime radical	#15690807	C₄
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#364	D₉₇
hypocenter	C₉₇	#1359
LCS	G ▷ C₄²×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #20260556	C₂	semidirect
3. #364×C₉₇	C₂	semidirect
48. C₉₇	#1359	semidirect
51. #364	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261108	2
#20260556	2
#364×C₉₇	2
#1359	97