Subgroups — #27141642

Order 12416.

220 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	C₄×#13095495	C₂	1	2	yes	maximal
3	#20260500	C₂	1	2	yes	maximal
4	C₂×#97×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	4	yes	—
6	#15690746	C₄	1	8	yes	complemented
7	#15690721	—	2	2	no	—
8	C₄×#11715895	—	2	2	no	—
9	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	#97×C₉₇	C₄	1	4	yes	—
13	#13095495	—	2	4	no	—
14	#13095495	C₂×C₄	1	8	yes	—
15	C₂×#11715895	—	2	4	no	—
16	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
17	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₉₇	—	2	2	no	—
19	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
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	D₄	1	2	yes	—
24	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
27	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	#41	1	2	yes	—
31	C₈×C₉₇	—	2	2	no	—
32	C₈×C₉₇	C₄²	1	4	yes	—
33	#11715895	—	2	8	no	—
34	C₂²×C₉₇	—	2	2	no	—
35	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
37	C₄×C₉₇	—	2	4	no	—
38	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₉₇	#117	1	1	yes	characteristic
40	C₄×C₉₇	C₄×D₄	1	2	yes	—
41	C₂×C₉₇	—	2	2	no	—
42	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₉₇	#430	1	2	yes	—
44	#1778	—	97	1	no	maximal, Sylow
45	C₉₇	#1778	1	1	yes	characteristic, complemented, Sylow
46	C₂×#108	—	97	2	no	—
47	C₄×M₁₆	—	97	2	no	—
48	C₂×#97	D₉₇	1	1	yes	characteristic
49	#341	—	97	2	no	—
50	C₂×M₁₆	—	97	4	no	—
51	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
53	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
54	#108	—	97	8	no	—
55	#97	—	194	2	no	—
56	#97	Dic₉₇	1	4	yes	—
57	C₄×C₈	—	194	2	no	—
58	C₂²×C₄	Dic₁₉₄	1	2	yes	—
59	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
60	M₁₆	—	97	8	no	—
61	M₁₆	—	194	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	194	4	no	—
64	C₂×C₈	C₄×D₉₇	1	4	yes	—
65	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
66	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
67	C₄²	D₃₈₈	1	2	yes	—
68	C₄²	C₂×Dic₉₇	1	2	yes	—
69	C₂³	C₂×Dic₁₉₄	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
73	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
74	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
75	C₂×C₄	#13095511	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	194	8	no	—
78	C₈	C₄×Dic₉₇	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
81	C₂²	C₄×Dic₁₉₄	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₂×C₄×Dic₉₇	1	1	yes	characteristic
84	C₄	#15690769	1	1	yes	characteristic
85	C₄	C₄×D₃₈₈	1	2	yes	—
86	C₂	—	2	2	no	—
87	C₂	C₄×C₄⋊Dic₉₇	1	1	yes	characteristic
88	C₂	#20260784	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#97×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	#1778
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
6. #15690746	C₄	semidirect
45. C₉₇	#1778	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
C₄×#13095495	2
#20260500	2
C₂×#97×C₉₇	2
#1778	97