Subgroups — #27139569

Order 12416.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260788	C₂	1	2	yes	maximal
2	C₄³×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11715895	C₄	1	4	yes	complemented
4	C₂²⋊#11715895	—	2	2	no	—
5	C₄×#11715895	—	2	2	no	—
6	C₂×C₄²×C₉₇	—	2	2	no	—
7	C₂×C₄²×C₉₇	C₄	1	2	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	2	4	no	—
10	C₂²×C₄×C₉₇	—	2	2	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	D₄	1	2	yes	—
13	C₄²×C₉₇	—	2	8	no	—
14	C₄²×C₉₇	—	2	2	no	—
15	C₄²×C₉₇	C₈	1	4	yes	complemented
16	C₄²×C₉₇	C₂×C₄	1	2	yes	—
17	C₄²×C₉₇	D₄	1	1	yes	characteristic
18	C₄²×C₉₇	Q₈	1	1	yes	characteristic
19	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₉₇	—	2	8	no	—
21	C₂×C₄×C₉₇	—	2	4	no	—
22	C₂×C₄×C₉₇	—	2	4	no	—
23	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
29	#11715895	—	4	4	no	—
30	C₂²×C₉₇	—	2	2	no	—
31	C₂²×C₉₇	#95	1	1	yes	characteristic
32	C₂²×C₉₇	C₂²⋊C₈	1	2	yes	—
33	C₄×C₉₇	—	2	8	no	—
34	C₄×C₉₇	—	2	2	no	—
35	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
36	C₄×C₉₇	#97	1	1	yes	characteristic
37	C₄×C₉₇	#104	1	4	yes	—
38	C₄×C₉₇	C₄⋊C₈	1	2	yes	—
39	C₂×C₉₇	—	2	2	no	—
40	C₂×C₉₇	#336	1	1	yes	characteristic
41	C₂×C₉₇	#339	1	2	yes	—
42	#1276	—	97	1	no	maximal, Sylow
43	C₉₇	#1276	1	1	yes	characteristic, complemented, Sylow
44	#431	—	97	2	no	—
45	C₄³	D₉₇	1	1	yes	characteristic
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	Dic₉₇	1	2	yes	—
48	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
49	C₄⋊C₈	—	97	4	no	—
50	C₂²⋊C₈	—	194	2	no	—
51	C₄×C₈	—	194	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
55	C₂×C₈	—	194	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	#11715895	1	4	yes	complemented
59	C₄²	Dic₁₉₄	1	1	yes	characteristic
60	C₄²	C₄×D₉₇	1	2	yes	—
61	C₄²	D₃₈₈	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
67	C₂×C₄	#13095488	1	2	yes	—
68	C₂×C₄	#13095490	1	2	yes	—
69	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
70	C₂×C₄	C₂×#11715895	1	2	yes	—
71	C₂×C₄	#13095495	1	2	yes	—
72	C₈	—	388	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	#15690717	1	1	yes	characteristic
75	C₂²	C₂²⋊#11715895	1	2	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×#11715895	1	1	yes	characteristic
79	C₄	#15690721	1	1	yes	characteristic
80	C₄	#15690738	1	4	yes	—
81	C₄	C₄⋊#11715895	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	#20260490	1	1	yes	characteristic
84	C₂	#20260495	1	2	yes	—

	subgroup	G/H
center	C₄²	D₃₈₈
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	C₄³×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	C₄³	D₉₇
hypocenter	C₉₇	#1276
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₄⋊#11715895	C₄	semidirect
15. C₄²×C₉₇	C₈	semidirect
43. C₉₇	#1276	semidirect
58. C₄²	#11715895	semidirect

Maximal subgroups

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

subgroup	index
#20260788	2
C₄³×C₉₇	2
#1276	97