Subgroups — #27158997

Order 12416.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13095525	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11715906	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095525	—	2	2	no	—
4	C₂²×#11715906	C₄	1	1	yes	characteristic
5	C₂²×#11715906	C₂²	1	1	yes	characteristic
6	C₄⋊#11715906	—	2	2	no	—
7	C₂×C₄×#87604	C₄	1	1	yes	characteristic
8	#13095525	—	4	2	no	—
9	C₂×#11715906	—	2	1	no	—
10	C₂×#11715906	—	4	1	no	—
11	C₂×#11715906	—	4	1	no	—
12	C₂×#11715906	D₄	1	2	yes	—
13	C₂²×#87604	C₈	1	1	yes	characteristic
14	C₂²×#87604	C₂×C₄	1	1	yes	characteristic
15	C₄×#87604	—	2	1	no	—
16	C₄×#87604	—	2	1	no	—
17	C₂×C₄×D₉₇	C₈	1	1	yes	characteristic
18	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
19	C₂×#87604	—	2	1	no	—
20	C₂×#87604	—	2	1	no	—
21	C₂×#87604	—	2	1	no	—
22	C₂×#87604	—	4	1	no	—
23	C₂×#87604	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×#87604	M₁₆	1	1	yes	characteristic
25	#11715906	—	2	2	no	—
26	#11715906	—	4	1	no	—
27	C₂×C₄×C₉₇	C₁₆	1	1	yes	characteristic, complemented
28	C₂×Dic₉₇	C₁₆	1	1	yes	characteristic, complemented
29	C₄×D₉₇	—	2	1	no	—
30	C₄×D₉₇	—	2	1	no	—
31	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
32	C₂×D₉₇	—	2	1	no	—
33	C₂×D₉₇	—	2	1	no	—
34	C₂×D₉₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×D₉₇	M₃₂	1	1	yes	characteristic
36	#87604	—	2	1	no	—
37	#87604	—	4	1	no	—
38	#87604	#99	1	1	yes	characteristic
39	#87604	#101	1	1	yes	characteristic
40	C₄×C₉₇	—	2	1	no	—
41	Dic₉₇	—	2	1	no	—
42	C₂×C₉₇	—	2	1	no	—
43	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	D₉₇	—	2	1	no	—
45	D₉₇	#324	1	1	yes	characteristic
46	D₉₇	#349	1	1	yes	characteristic
47	#1322	—	97	1	no	maximal, Sylow
48	C₉₇	#1322	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₄⋊C₈	—	97	1	no	—
50	C₂²⋊C₁₆	—	97	2	no	—
51	C₂²×C₈	—	97	1	no	—
52	C₂²×C₈	—	97	1	no	—
53	C₂×C₄²	—	97	1	no	—
54	C₂×C₁₆	—	194	2	no	—
55	C₄⋊C₈	—	194	2	no	—
56	C₂²×C₄	—	97	1	no	—
57	C₂²×C₄	—	97	1	no	—
58	C₂²×C₄	—	97	1	no	—
59	C₂×C₈	—	97	2	no	—
60	C₂×C₈	—	388	1	no	—
61	C₂×C₈	—	388	1	no	—
62	C₂×C₈	—	194	1	no	—
63	C₄²	—	194	1	no	—
64	C₄²	—	194	1	no	—
65	C₁₆	—	388	2	no	—
66	C₂³	—	97	1	no	—
67	C₂×C₄	—	97	1	no	—
68	C₂×C₄	—	97	1	no	—
69	C₂×C₄	—	97	1	no	—
70	C₂×C₄	—	388	1	no	—
71	C₂×C₄	—	194	1	no	—
72	C₂×C₄	—	194	1	no	—
73	C₂×C₄	—	194	1	no	—
74	C₂×C₄	—	194	1	no	—
75	C₂×C₄	—	194	1	no	—
76	C₂×C₄	#13095525	1	1	yes	characteristic, complemented
77	C₈	—	388	1	no	—
78	C₈	—	194	2	no	—
79	C₂²	—	97	1	no	—
80	C₂²	—	97	1	no	—
81	C₂²	—	194	1	no	—
82	C₂²	—	194	1	no	—
83	C₂²	C₂×#13095525	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	97	1	no	—
86	C₄	—	97	1	no	—
87	C₄	—	388	1	no	—
88	C₄	—	194	1	no	—
89	C₄	—	194	1	no	—
90	C₂	—	2	1	no	—
91	C₂	—	97	1	no	—
92	C₂	—	97	1	no	—
93	C₂	—	194	1	no	—
94	C₂	C₂²⋊#13095525	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#13095525
commutator	C₂²×C₉₇	C₂×C₁₆
Frattini	C₂²	C₂×#13095525
Fitting	C₂×C₄×C₉₇	C₁₆
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	C₂²×D₉₇	C₂×C₈
cosocle	C₂²×#11715906	C₂²
hypercenter	C₂×C₄	#13095525
hypocenter	C₉₇	#1322
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
27. C₂×C₄×C₉₇	C₁₆	semidirect
28. C₂×Dic₉₇	C₁₆	semidirect
48. C₉₇	#1322	semidirect
76. C₂×C₄	#13095525	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13095525	2
C₂×C₄⋊#11715906	2
#1322	97