Subgroups — #27139840

Order 12416.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690746	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095495	—	2	2	no	—
4	#15690746	—	2	2	no	—
5	#15690746	C₄	1	4	yes	—
6	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₉₇	—	2	2	no	—
8	C₂×C₁₆×C₉₇	C₄	1	2	yes	—
9	#13095495	—	2	4	no	—
10	#13095495	—	4	2	no	—
11	C₂×#11715895	—	4	2	no	—
12	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₉₇	D₄	1	2	yes	—
16	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₉₇	—	2	2	no	—
18	C₂³×C₉₇	Q₁₆	1	1	yes	characteristic
19	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
22	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
23	C₈×C₉₇	C₄²	1	1	yes	characteristic
24	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
26	#11715895	—	4	4	no	—
27	C₂²×C₉₇	—	4	1	no	—
28	C₂²×C₉₇	#103	1	2	yes	—
29	C₂²×C₉₇	#107	1	1	yes	characteristic
30	C₄×C₉₇	#95	1	1	yes	characteristic
31	C₄×C₉₇	#102	1	2	yes	—
32	C₄×C₉₇	#106	1	1	yes	characteristic
33	C₂×C₉₇	—	2	2	no	—
34	C₂×C₉₇	—	2	1	no	—
35	C₂×C₉₇	#340	1	1	yes	characteristic
36	#1385	—	97	1	no	maximal, Sylow
37	C₉₇	#1385	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₃₂	D₉₇	1	1	yes	characteristic
39	C₂×#108	—	97	2	no	—
40	C₂×M₁₆	—	194	2	no	—
41	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₉₇	1	2	yes	—
44	#108	—	97	4	no	—
45	#108	—	194	2	no	—
46	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
47	M₁₆	—	388	2	no	—
48	M₁₆	—	194	4	no	—
49	C₂×C₈	—	388	2	no	—
50	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
51	C₂×C₈	C₄×D₉₇	1	2	yes	—
52	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₃₈₈	1	1	yes	characteristic
56	C₂×C₄	#13095488	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
58	C₂×C₄	D₇₇₆	1	1	yes	characteristic
59	C₂×C₄	#13095500	1	2	yes	—
60	C₈	—	388	4	no	—
61	C₈	C₄×Dic₉₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
63	C₈	#13095490	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#15690735	1	2	yes	—
66	C₂²	#15690744	1	1	yes	characteristic
67	C₄	#15690717	1	1	yes	characteristic
68	C₄	#15690732	1	2	yes	—
69	C₄	#15690742	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#20260498	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690742
commutator	C₈×C₉₇	C₄²
Frattini	C₂²×C₈	C₂×D₉₇
Fitting	C₂×M₃₂×C₉₇	C₂
socle	C₂×C₉₇	#340
prime radical	C₂³×C₉₇	Q₁₆
cosocle	C₂²×C₈×C₉₇	C₂²
hypercenter	C₂×M₃₂	D₉₇
hypocenter	C₉₇	#1385
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
37. C₉₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690746	2
C₂×M₃₂×C₉₇	2
#1385	97