Subgroups — #27158140

Order 12416.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261777	C₂	1	2	yes	maximal, complemented
2	#20261664	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261649	C₂	1	2	yes	maximal, complemented
4	#20261622	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261164	C₂	1	1	yes	maximal, characteristic
6	#15690897	—	2	2	no	—
7	#15690883	C₂²	1	1	yes	characteristic
8	#15690881	C₂²	1	2	yes	—
9	#15690874	—	2	2	no	—
10	C₈×#87604	C₂²	1	1	yes	characteristic
11	#15690829	C₂²	1	2	yes	—
12	Dic₇₇₆	C₄	1	1	yes	characteristic, complemented
13	#15690754	C₄	1	2	yes	complemented
14	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
15	D₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
16	#13095521	—	2	2	no	—
17	#13095520	—	4	2	no	—
18	#13095519	D₄	1	1	yes	characteristic
19	C₂×#11715897	—	4	2	no	—
20	C₄×#87604	D₄	1	1	yes	characteristic
21	#13095514	—	2	1	no	—
22	#13095510	—	2	2	no	—
23	Dic₃₈₈	C₂×C₄	1	2	yes	—
24	#13095499	—	2	2	no	—
25	C₈×D₉₇	C₂³	1	1	yes	characteristic
26	#13095486	C₂×C₄	1	1	yes	characteristic
27	D₈×C₉₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#87604	—	2	1	no	—
30	D₄×C₉₇	—	2	2	no	—
31	C₂²⋊D₉₇	—	4	2	no	—
32	C₂×Dic₉₇	—	4	2	no	—
33	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
34	Dic₁₉₄	—	2	2	no	—
35	#11715897	—	4	2	no	—
36	#11715897	D₈	1	2	yes	—
37	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	#11715895	#41	1	1	yes	characteristic
39	C₂²×C₉₇	—	4	2	no	—
40	C₂×D₉₇	#135	1	1	yes	characteristic
41	#87604	—	4	1	no	—
42	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
43	Dic₉₇	—	4	2	no	—
44	Dic₉₇	C₂×D₈	1	1	yes	characteristic
45	C₂×C₉₇	—	8	2	no	—
46	C₂×C₉₇	C₄×D₈	1	1	yes	characteristic
47	D₉₇	—	2	1	no	—
48	#2179	—	97	1	no	maximal, Sylow
49	C₉₇	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	97	1	no	—
51	#443	—	97	2	no	—
52	#368	—	97	1	no	—
53	#359	—	97	2	no	—
54	#346	—	97	1	no	—
55	#135	—	97	2	no	—
56	#131	—	194	2	no	—
57	Q₃₂	—	97	1	no	—
58	SD₃₂	—	97	2	no	—
59	D₁₆	#87604	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	97	1	no	—
61	C₂×C₁₆	—	97	1	no	—
62	#108	—	97	2	no	—
63	#104	—	194	2	no	—
64	C₄×C₈	—	97	1	no	—
65	#41	—	194	2	no	—
66	Q₁₆	—	97	2	no	—
67	SD₁₆	—	194	2	no	—
68	D₈	C₂×#87604	1	2	yes	—
69	M₁₆	—	388	2	no	—
70	M₁₆	—	194	2	no	—
71	C₂×C₈	—	97	1	no	—
72	C₂×C₈	—	97	1	no	—
73	C₂×C₈	—	388	2	no	—
74	C₄²	—	97	1	no	—
75	C₁₆	—	97	1	no	—
76	C₁₆	—	194	1	no	—
77	C₁₆	C₂×#87604	1	1	yes	characteristic
78	Q₈	—	194	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	388	2	no	—
81	C₂×C₄	—	97	1	no	—
82	C₂×C₄	—	388	2	no	—
83	C₂×C₄	—	194	1	no	—
84	C₈	—	97	2	no	—
85	C₈	—	97	1	no	—
86	C₈	—	388	2	no	—
87	C₈	C₂²×#87604	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	—	97	1	no	—
90	C₄	—	97	1	no	—
91	C₄	—	388	1	no	—
92	C₄	—	388	2	no	—
93	C₄	D₄×#87604	1	1	yes	characteristic
94	C₂	—	8	2	no	—
95	C₂	—	194	1	no	—
96	C₂	D₈×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#87604
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₈	C₂²×#87604
Fitting	D₁₆×C₉₇	C₄
socle	C₂×C₉₇	C₄×D₈
prime radical	#20261164	C₂
cosocle	C₈×D₉₇	C₂³
hypercenter	D₁₆	#87604
hypocenter	C₉₇	#2179
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #20261777	C₂	semidirect
2. #20261664	C₂	semidirect
3. #20261649	C₂	semidirect
4. #20261622	C₂	semidirect
12. Dic₇₇₆	C₄	semidirect
13. #15690754	C₄	semidirect
15. D₁₆×C₉₇	C₄	semidirect
49. C₉₇	#2179	semidirect
59. D₁₆	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261777	2
#20261664	2
#20261649	2
#20261622	2
#20261164	2
#2179	97