Subgroups — #27140033

Order 12416.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260788	C₂	1	2	yes	maximal
2	#20260457	C₂	1	2	yes	maximal, complemented
3	#20260456	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11715895	C₄	1	4	yes	—
7	C₂²⋊#11715895	C₄	1	4	yes	—
8	C₄×#11715895	C₂²	1	2	yes	—
9	C₄×#11715895	C₂²	1	2	yes	—
10	C₂×M₁₆×C₉₇	C₄	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	#97×C₉₇	C₄	1	2	yes	—
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×#11715895	C₂×C₄	1	4	yes	—
16	C₂×#11715895	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	M₁₆×C₉₇	—	2	4	no	—
20	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
22	C₄²×C₉₇	C₂×C₄	1	2	yes	—
23	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉₇	C₂³	1	1	yes	characteristic
25	C₂³×C₉₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₉₇	—	2	2	no	—
27	C₂×C₄×C₉₇	—	2	4	no	—
28	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₄²	1	2	yes	—
30	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
31	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₉₇	—	2	2	no	—
33	C₈×C₉₇	M₁₆	1	4	yes	—
34	#11715895	—	2	4	no	—
35	#11715895	—	2	4	no	—
36	C₂²×C₉₇	—	2	1	no	—
37	C₂²×C₉₇	—	2	2	no	—
38	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	—	2	2	no	—
41	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
42	C₄×C₉₇	#131	1	4	yes	—
43	C₂×C₉₇	—	4	1	no	—
44	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
45	C₂×C₉₇	#405	1	2	yes	—
46	#1453	—	97	1	no	maximal, Sylow
47	C₉₇	#1453	1	1	yes	characteristic, complemented, Sylow
48	#431	—	97	2	no	—
49	C₄×M₁₆	D₉₇	1	1	yes	characteristic
50	#322	—	97	1	no	—
51	#322	—	97	2	no	—
52	C₈²	—	97	1	no	—
53	C₂×M₁₆	Dic₉₇	1	2	yes	—
54	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
55	C₄⋊C₈	—	97	4	no	—
56	C₂²⋊C₈	—	97	4	no	—
57	#97	Dic₉₇	1	2	yes	—
58	C₄×C₈	—	97	2	no	—
59	C₄×C₈	—	97	2	no	—
60	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
62	C₂²×C₄	C₄×D₉₇	1	2	yes	—
63	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	97	4	no	—
66	C₂×C₈	—	97	4	no	—
67	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
68	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
69	C₄²	C₄×D₉₇	1	2	yes	—
70	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
71	C₄²	C₂²×D₉₇	1	1	yes	characteristic
72	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
76	C₂×C₄	C₄×Dic₉₇	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
78	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	194	4	no	—
81	C₈	—	194	4	no	—
82	C₈	#13095495	1	4	yes	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄×Dic₉₇	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	2	no	—
88	C₄	C₂×#13095495	1	2	yes	—
89	C₄	#15690818	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₄×#13095495	1	1	yes	characteristic
92	C₂	#20260693	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₄×M₁₆×C₉₇	C₂
prime radical	C₂³×C₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	#1453
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20260457	C₂	semidirect
3. #20260456	C₂	semidirect
4. C₈×#11715895	C₂	semidirect
47. C₉₇	#1453	semidirect

Maximal subgroups

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

subgroup	index
#20260788	2
#20260457	2
#20260456	2
C₈×#11715895	2
C₄×M₁₆×C₉₇	2
#1453	97