Subgroups — #27159005

Order 12416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690909	C₂	1	2	yes	maximal
2	C₄×#13095523	C₂	1	1	yes	maximal, characteristic
3	#15690909	C₄	1	4	yes	complemented
4	C₂×#13095523	C₄	1	1	yes	characteristic
5	C₂×#13095523	C₂²	1	1	yes	characteristic
6	C₄×#11715897	C₄	1	1	yes	characteristic
7	#13095523	C₈	1	2	yes	—
8	#13095523	C₂×C₄	1	2	yes	—
9	C₂×#11715897	C₈	1	1	yes	characteristic
10	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
11	C₄×Dic₉₇	C₈	1	1	yes	characteristic
12	C₂×C₄×C₉₇	C₁₆	1	1	yes	characteristic
13	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic
14	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
15	#11715897	C₁₆	1	2	yes	—
16	#11715897	C₄²	1	1	yes	characteristic
17	#11715897	C₂×C₈	1	1	yes	characteristic
18	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
19	C₄×C₉₇	—	2	1	no	—
20	Dic₉₇	—	2	1	no	—
21	Dic₉₇	C₄×C₈	1	1	yes	characteristic
22	Dic₉₇	C₂×C₁₆	1	1	yes	characteristic
23	C₂×C₉₇	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₉₇	M₆₄	1	1	yes	characteristic
25	C₂×C₉₇	M₆₄	1	1	yes	characteristic
26	#1397	—	97	1	no	maximal, Sylow
27	C₉₇	#1397	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	—	97	2	no	—
29	C₄×C₁₆	—	97	1	no	—
30	C₂×C₁₆	—	97	1	no	—
31	C₂×C₁₆	—	97	1	no	—
32	C₄×C₈	—	97	1	no	—
33	C₃₂	—	97	4	no	—
34	C₂×C₈	—	97	1	no	—
35	C₂×C₈	—	97	1	no	—
36	C₄²	—	97	1	no	—
37	C₁₆	—	97	2	no	—
38	C₁₆	—	97	2	no	—
39	C₂×C₄	—	97	1	no	—
40	C₂×C₄	—	97	1	no	—
41	C₂×C₄	#13095525	1	1	yes	characteristic
42	C₈	—	97	1	no	—
43	C₈	—	97	1	no	—
44	C₈	—	97	2	no	—
45	C₂²	C₂×#13095525	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	97	1	no	—
48	C₄	—	97	1	no	—
49	C₄	—	194	1	no	—
50	C₂	C₄×#13095525	1	1	yes	characteristic
51	C₂	#20261865	1	1	yes	characteristic
52	C₂	#20261866	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×#13095525
commutator	C₂×C₉₇	C₄×C₁₆
Fitting	C₂×C₄×C₉₇	C₁₆
socle, prime radical	C₂²×C₉₇	C₂×C₁₆
cosocle	C₂×#13095523	C₂²
hypercenter	C₂×C₄	#13095525
hypocenter	C₉₇	#1397
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. #15690909	C₄	semidirect
27. C₉₇	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690909	2
C₄×#13095523	2
#1397	97