Subgroups — #27141875

Order 12416.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260758	C₂	1	2	yes	maximal
2	C₂×#15690720	C₂	1	1	yes	maximal, characteristic
3	#20260500	C₂	1	1	yes	maximal, characteristic
4	#20260499	C₂	1	2	yes	maximal
5	#341×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690739	—	2	4	no	—
10	#15690720	C₄	1	4	yes	—
11	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₉₇	Q₈	1	2	yes	—
14	#13095495	—	2	2	no	—
15	#13095495	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	C₂×C₄	1	2	yes	—
19	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
20	C₄×Dic₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₉₇	—	2	2	no	—
23	M₁₆×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
27	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	#41	1	2	yes	—
31	C₂×C₄×C₉₇	#41	1	2	yes	—
32	C₂×Dic₉₇	—	4	2	no	—
33	C₂×Dic₉₇	C₄⋊C₄	1	4	yes	—
34	C₈×C₉₇	—	2	2	no	—
35	C₈×C₉₇	—	4	2	no	—
36	#11715895	—	2	2	no	—
37	#11715895	—	4	2	no	—
38	C₂²×C₉₇	—	2	2	no	—
39	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₉₇	#125	1	2	yes	—
41	C₄×C₉₇	#117	1	1	yes	characteristic
42	C₄×C₉₇	#117	1	1	yes	characteristic
43	C₄×C₉₇	#124	1	2	yes	—
44	Dic₉₇	—	2	4	no	—
45	C₂×C₉₇	—	2	2	no	—
46	C₂×C₉₇	#382	1	1	yes	characteristic
47	C₂×C₉₇	#430	1	2	yes	—
48	#1838	—	97	1	no	maximal, Sylow
49	C₉₇	#1838	1	1	yes	characteristic, complemented, Sylow
50	#423	—	97	2	no	—
51	C₂×#97	—	97	1	no	—
52	#341	—	97	2	no	—
53	#341	—	97	1	no	—
54	#341	D₉₇	1	1	yes	characteristic
55	C₂×M₁₆	—	97	2	no	—
56	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
57	C₂²×C₈	—	97	1	no	—
58	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
59	C₂×C₄²	—	97	1	no	—
60	C₄⋊C₈	—	194	4	no	—
61	#97	—	97	4	no	—
62	C₂²×C₄	—	97	2	no	—
63	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	194	2	no	—
67	M₁₆	—	194	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	97	2	no	—
71	C₂×C₈	—	194	2	no	—
72	C₂×C₈	—	194	2	no	—
73	C₂×C₈	C₄×D₉₇	1	2	yes	—
74	C₄²	—	97	2	no	—
75	C₄²	—	97	2	no	—
76	C₂³	Q₈×D₉₇	1	1	yes	characteristic
77	C₂×C₄	—	97	4	no	—
78	C₂×C₄	—	388	2	no	—
79	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
80	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
81	C₂×C₄	#13095510	1	2	yes	—
82	C₂×C₄	#13095511	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	—	388	2	no	—
86	C₈	—	194	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
89	C₂²	#15690800	1	2	yes	—
90	C₄	—	194	4	no	—
91	C₄	#15690766	1	1	yes	characteristic
92	C₄	#15690767	1	1	yes	characteristic
93	C₄	#15690796	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#20260599	1	1	yes	characteristic
96	C₂	#20260782	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#341×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#341	D₉₇
hypocenter	C₉₇	#1838
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
49. C₉₇	#1838	semidirect

Maximal subgroups

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

subgroup	index
#20260758	2
C₂×#15690720	2
#20260500	2
#20260499	2
#341×C₉₇	2
#1838	97