Subgroups — #27139799

Order 12416.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095486	C₂	1	1	yes	maximal, characteristic
2	C₈×#11715895	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095486	—	2	1	no	—
5	C₄×#11715895	C₄	1	1	yes	characteristic
6	C₄×#11715895	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₉₇	—	2	1	no	—
8	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	2	1	no	—
10	C₂×#11715895	C₈	1	1	yes	characteristic
11	C₂×#11715895	C₈	1	1	yes	characteristic
12	#13095486	—	4	1	no	—
13	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
15	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₉₇	—	4	1	no	—
17	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
20	C₈×C₉₇	—	2	1	no	—
21	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
22	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
23	#11715895	—	2	2	no	—
24	#11715895	—	2	1	no	—
25	#11715895	C₁₆	1	2	yes	complemented
26	C₂²×C₉₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₉₇	—	2	1	no	—
28	C₄×C₉₇	#106	1	1	yes	characteristic
29	C₄×C₉₇	#108	1	1	yes	characteristic
30	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₉₇	M₃₂	1	1	yes	characteristic
32	C₂×C₉₇	#334	1	1	yes	characteristic
33	C₂×C₉₇	C₄⋊C₁₆	1	1	yes	characteristic
34	C₂×C₉₇	#364	1	1	yes	characteristic
35	#1367	—	97	1	no	maximal, Sylow
36	C₉₇	#1367	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊C₁₆	—	97	1	no	—
38	C₄⋊C₁₆	D₉₇	1	1	yes	characteristic
39	C₈²	—	97	1	no	—
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	194	1	no	—
42	C₄×C₈	—	97	1	no	—
43	C₄×C₈	—	97	1	no	—
44	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
45	C₂×C₈	—	97	1	no	—
46	C₂×C₈	—	97	1	no	—
47	C₂×C₈	—	194	1	no	—
48	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
50	C₄²	C₄×D₉₇	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	388	1	no	—
53	C₂×C₄	#13095490	1	1	yes	characteristic
54	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
55	C₂×C₄	#13095494	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	97	2	no	—
58	C₈	—	194	2	no	—
59	C₈	—	194	1	no	—
60	C₈	#13095498	1	1	yes	characteristic
61	C₈	#13095499	1	1	yes	characteristic
62	C₂²	#15690739	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#15690741	1	1	yes	characteristic
65	C₄	#15690745	1	1	yes	characteristic
66	C₄	C₁₆×D₉₇	1	1	yes	characteristic
67	C₄	#15690749	1	1	yes	characteristic
68	C₂	#20260485	1	1	yes	characteristic
69	C₂	#20260553	1	1	yes	characteristic
70	C₂	#20260555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₉₇
Fitting	C₄⋊C₁₆×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₄⋊C₈
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	C₄⋊C₁₆	D₉₇
hypocenter	C₉₇	#1367
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. #11715895	C₁₆	semidirect
36. C₉₇	#1367	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095486	2
C₈×#11715895	2
C₄⋊C₁₆×C₉₇	2
#1367	97