Subgroups — #27141856

Order 12416.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11715895	C₂	1	2	yes	maximal
2	#20260490	C₂	1	4	yes	maximal
3	#376×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₂²	1	4	yes	—
5	C₄⋊#11715895	—	2	4	no	—
6	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	#95×C₉₇	C₄	1	4	yes	—
9	C₂×#11715895	—	2	4	no	—
10	C₂×#11715895	—	2	8	no	—
11	C₂²×C₄×C₉₇	C₂×C₄	1	4	yes	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
14	C₄²×C₉₇	—	2	4	no	—
15	C₄²×C₉₇	D₄	1	2	yes	—
16	C₄²×C₉₇	Q₈	1	2	yes	—
17	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	—	2	4	no	—
19	C₂×C₄×C₉₇	—	2	4	no	—
20	C₂×C₄×C₉₇	—	2	8	no	—
21	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	#41	1	4	yes	—
25	#11715895	—	4	4	no	—
26	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₉₇	#117	1	2	yes	—
28	C₂²×C₉₇	#131	1	4	yes	—
29	C₄×C₉₇	—	2	4	no	—
30	C₄×C₉₇	—	4	4	no	—
31	C₄×C₉₇	#124	1	2	yes	—
32	C₄×C₉₇	#125	1	2	yes	—
33	C₂×C₉₇	#382	1	1	yes	characteristic
34	C₂×C₉₇	#424	1	2	yes	—
35	C₂×C₉₇	#433	1	4	yes	—
36	#1832	—	97	1	no	maximal, Sylow
37	C₉₇	#1832	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄⋊C₈	—	97	2	no	—
39	#376	D₉₇	1	1	yes	characteristic
40	#336	—	97	4	no	—
41	C₂²×C₈	—	97	4	no	—
42	C₂×C₄²	C₂×D₉₇	1	2	yes	—
43	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
44	C₄⋊C₈	—	194	4	no	—
45	#95	Dic₉₇	1	4	yes	—
46	C₂²×C₄	C₂×Dic₉₇	1	4	yes	—
47	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
48	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
49	C₂×C₈	—	194	4	no	—
50	C₂×C₈	—	194	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	Dic₁₉₄	1	2	yes	—
53	C₄²	D₃₈₈	1	2	yes	—
54	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	8	no	—
58	C₂×C₄	C₄⋊Dic₉₇	1	4	yes	—
59	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
61	C₂×C₄	#13095511	1	4	yes	—
62	C₈	—	388	4	no	—
63	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
64	C₂²	#15690769	1	2	yes	—
65	C₂²	#15690817	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	4	4	no	—
68	C₄	#15690798	1	2	yes	—
69	C₄	#15690802	1	2	yes	—
70	C₂	#20260601	1	1	yes	characteristic
71	C₂	#20260762	1	2	yes	—
72	C₂	#20260796	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	#376×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#376	D₉₇
hypocenter	C₉₇	#1832
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
37. C₉₇	#1832	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715895	2
#20260490	2
#376×C₉₇	2
#1832	97