Subgroups — #27150347

Order 12416.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261502	C₂	1	2	yes	maximal
2	#20260784	C₂	1	6	yes	maximal, complemented
3	C₂×#15690746	C₂	1	6	yes	maximal, complemented
4	C₂×#131×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690817	—	2	8	no	—
6	C₂×#13095495	C₂²	1	12	yes	—
7	#15690746	C₂²	1	4	yes	—
8	#15690746	C₂²	1	12	yes	—
9	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
10	#131×C₉₇	C₄	1	8	yes	—
11	C₂×M₁₆×C₉₇	C₂²	1	3	yes	—
12	C₂²×C₈×C₉₇	C₂²	1	3	yes	—
13	#13095495	—	2	12	no	—
14	#13095495	C₂³	1	8	yes	—
15	C₂×#11715895	—	2	12	no	—
16	#41×C₉₇	C₂×C₄	1	4	yes	—
17	#41×C₉₇	D₄	1	4	yes	—
18	C₂×Q₈×C₉₇	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₉₇	Q₈	1	3	yes	—
20	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
21	M₁₆×C₉₇	C₂×C₄	1	12	yes	—
22	C₂×C₈×C₉₇	C₂×C₄	1	12	yes	—
23	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₉₇	C₂³	1	3	yes	—
25	C₂³×C₉₇	C₂×Q₈	1	3	yes	—
26	Q₈×C₉₇	C₄⋊C₄	1	4	yes	—
27	D₄×C₉₇	C₄⋊C₄	1	12	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₉₇	C₂×D₄	1	6	yes	—
30	C₂×C₄×C₉₇	C₂×Q₈	1	3	yes	—
31	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
32	C₈×C₉₇	C₂²×C₄	1	2	yes	—
33	C₈×C₉₇	C₂²×C₄	1	6	yes	—
34	#11715895	—	2	8	no	—
35	C₂²×C₉₇	—	4	3	no	—
36	C₂²×C₉₇	C₂×C₄⋊C₄	1	6	yes	—
37	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
38	C₄×C₉₇	C₂×C₄⋊C₄	1	6	yes	—
39	C₄×C₉₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	—	2	6	no	—
43	C₂×C₉₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
44	#2916	—	97	1	no	maximal, Sylow
45	C₉₇	#2916	1	1	yes	characteristic, complemented, Sylow
46	#568	—	97	2	no	—
47	C₂×#131	D₉₇	1	1	yes	characteristic
48	#430	—	97	6	no	—
49	C₂×#108	—	97	6	no	—
50	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
51	#131	—	194	8	no	—
52	#131	Dic₉₇	1	8	yes	—
53	C₂×M₁₆	—	97	12	no	—
54	C₂×M₁₆	C₂×D₉₇	1	3	yes	—
55	C₂²×C₈	C₂×D₉₇	1	3	yes	—
56	#108	—	97	4	no	—
57	#108	—	97	12	no	—
58	#41	D₃₈₈	1	4	yes	—
59	#41	C₂×Dic₉₇	1	4	yes	—
60	C₂×Q₈	Dic₁₉₄	1	1	yes	characteristic
61	C₂×D₄	Dic₁₉₄	1	3	yes	—
62	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
63	M₁₆	—	97	8	no	—
64	M₁₆	—	194	12	no	—
65	M₁₆	C₂×Dic₉₇	1	12	yes	—
66	C₂×C₈	—	194	12	no	—
67	C₂×C₈	C₂×Dic₉₇	1	12	yes	—
68	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₉₇	1	3	yes	—
70	C₂³	C₂×Dic₁₉₄	1	3	yes	—
71	Q₈	C₄⋊Dic₉₇	1	4	yes	—
72	D₄	C₄⋊Dic₉₇	1	12	yes	—
73	C₂×C₄	C₂²×Dic₉₇	1	6	yes	—
74	C₂×C₄	C₂×D₃₈₈	1	6	yes	—
75	C₂×C₄	C₂×Dic₁₉₄	1	3	yes	—
76	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
77	C₈	—	194	8	no	—
78	C₈	C₂²×Dic₉₇	1	2	yes	—
79	C₈	C₂²×Dic₉₇	1	6	yes	—
80	C₂²	—	4	3	no	—
81	C₂²	C₂×C₄⋊Dic₉₇	1	6	yes	—
82	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
83	C₄	C₂×C₄⋊Dic₉₇	1	6	yes	—
84	C₄	C₂³×Dic₉₇	1	1	yes	characteristic
85	C₄	C₂²×D₃₈₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	6	no	—
88	C₂	C₂²×C₄⋊Dic₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₈₈
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#131×C₉₇	C₂
socle	C₂×C₉₇	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₉₇	C₂²
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×#131	D₉₇
hypocenter	C₉₇	#2916
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #20260784	C₂	semidirect
3. C₂×#15690746	C₂	semidirect
45. C₉₇	#2916	semidirect

Maximal subgroups

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

subgroup	index
#20261502	2
#20260784	2
C₂×#15690746	2
C₂×#131×C₉₇	2
#2916	97