Subgroups — #27158925

Order 12416.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690907	C₂	1	2	yes	maximal
2	C₄⋊#13095523	C₂	1	4	yes	maximal, complemented
3	C₄×#13095519	C₂	1	1	yes	maximal, characteristic
4	#15690907	—	2	4	no	—
5	C₂×#13095523	C₂²	1	4	yes	—
6	C₂×#13095519	C₄	1	1	yes	characteristic
7	C₂×#13095519	C₂²	1	1	yes	characteristic
8	C₄×#11715897	C₄	1	2	yes	—
9	C₄×#11715897	C₂²	1	2	yes	—
10	C₄²×D₉₇	C₄	1	1	yes	characteristic
11	#13095523	—	2	4	no	—
12	#13095519	—	2	2	no	—
13	#13095519	D₄	1	2	yes	—
14	#13095519	Q₈	1	2	yes	—
15	C₂×#11715897	C₂×C₄	1	2	yes	—
16	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
17	C₂×#11715897	C₂³	1	1	yes	characteristic
18	C₂×C₄×D₉₇	C₈	1	2	yes	—
19	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₉₇	C₈	1	1	yes	characteristic
21	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
22	C₄²×C₉₇	C₈	1	1	yes	characteristic
23	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
26	C₂×Dic₉₇	C₂×C₈	1	2	yes	—
27	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×D₉₇	—	2	4	no	—
29	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
30	C₄×D₉₇	M₁₆	1	2	yes	—
31	#11715897	—	2	2	no	—
32	#11715897	C₄⋊C₄	1	2	yes	—
33	#11715897	C₂×D₄	1	1	yes	characteristic
34	#11715897	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
36	C₂×D₉₇	—	2	2	no	—
37	C₂×D₉₇	C₄⋊C₈	1	2	yes	—
38	C₄×C₉₇	C₄⋊C₈	1	2	yes	—
39	C₄×C₉₇	M₃₂	1	4	yes	complemented
40	Dic₉₇	—	2	2	no	—
41	Dic₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
44	C₂×C₉₇	C₂×M₃₂	1	2	yes	—
45	D₉₇	—	2	2	no	—
46	#2150	—	97	1	no	maximal, Sylow
47	C₉₇	#2150	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	97	2	no	—
49	C₂×C₄×C₈	—	97	1	no	—
50	C₄⋊C₁₆	—	97	4	no	—
51	C₂²×C₈	—	97	1	no	—
52	C₂²×C₈	—	97	1	no	—
53	C₂×C₄²	—	97	1	no	—
54	M₃₂	—	194	4	no	—
55	C₂×C₁₆	—	97	4	no	—
56	C₄×C₈	—	97	2	no	—
57	C₄×C₈	—	97	2	no	—
58	C₂²×C₄	—	97	1	no	—
59	C₂²×C₄	—	97	2	no	—
60	C₂×C₈	—	97	1	no	—
61	C₂×C₈	—	97	1	no	—
62	C₂×C₈	—	97	2	no	—
63	C₂×C₈	—	97	2	no	—
64	C₂×C₈	—	97	2	no	—
65	C₂×C₈	—	194	2	no	—
66	C₄²	—	97	1	no	—
67	C₄²	—	97	2	no	—
68	C₄²	#11715906	1	1	yes	characteristic
69	C₁₆	—	194	4	no	—
70	C₂³	—	97	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	97	2	no	—
73	C₂×C₄	—	97	2	no	—
74	C₂×C₄	—	97	2	no	—
75	C₂×C₄	—	194	4	no	—
76	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
77	C₂×C₄	C₂×#11715906	1	2	yes	—
78	C₈	—	97	1	no	—
79	C₈	—	97	2	no	—
80	C₈	—	97	1	no	—
81	C₈	—	194	2	no	—
82	C₂²	—	97	2	no	—
83	C₂²	—	194	2	no	—
84	C₂²	C₂²×#11715906	1	1	yes	characteristic
85	C₄	—	97	1	no	—
86	C₄	—	97	1	no	—
87	C₄	—	194	2	no	—
88	C₄	C₄⋊#11715906	1	2	yes	—
89	C₄	#15690907	1	4	yes	complemented
90	C₂	—	194	2	no	—
91	C₂	C₂×C₄⋊#11715906	1	1	yes	characteristic
92	C₂	C₂×#15690907	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₄²×C₉₇	C₈
prime radical	C₂²×D₉₇	C₂×C₈
cosocle	C₂×#11715897	C₂³
hypercenter	C₄²	#11715906
hypocenter	C₉₇	#2150
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊#13095523	C₂	semidirect
39. C₄×C₉₇	M₃₂	semidirect
47. C₉₇	#2150	semidirect
89. C₄	#15690907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690907	2
C₄⋊#13095523	2
C₄×#13095519	2
#2150	97