Finite Group Databasebeta

Subgroups — #27154875

Order 12416.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261540	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261171	C₂	1	3	yes	maximal, complemented
3	#20261170	C₂	1	3	yes	maximal, complemented
4	#20261165	C₂	1	3	yes	maximal, complemented
5	C₂×#15690751	C₂	1	3	yes	maximal, complemented
6	#20261152	C₂	1	1	yes	maximal, characteristic, complemented
7	#576×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690856	—	2	1	no	—
9	#15690838	—	2	3	no	—
10	#15690832	C₂²	1	3	yes	complemented
11	C₂×D₇₇₆	C₂²	1	3	yes	complemented
12	#15690756	C₂²	1	1	yes	characteristic, complemented
13	#15690754	C₂²	1	3	yes	complemented
14	#15690753	C₂²	1	3	yes	complemented
15	#15690751	C₂²	1	3	yes	complemented
16	#15690751	C₂²	1	6	yes	complemented
17	#15690750	C₂²	1	3	yes	complemented
18	C₂×#13095486	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₉₇	—	2	1	no	—
20	AGL(1,ℤ/8)×C₉₇	—	2	3	no	—
21	#135×C₉₇	C₂²	1	3	yes	complemented
22	C₂×D₈×C₉₇	C₂²	1	3	yes	complemented
23	#131×C₉₇	C₂²	1	1	yes	characteristic
24	#13095512	—	4	1	no	—
25	#13095511	—	2	3	no	—
26	C₂×D₃₈₈	—	2	3	no	—
27	D₄×D₉₇	—	4	3	no	—
28	Dic₃₈₈	C₂³	1	1	yes	characteristic
29	#13095500	—	2	3	no	—
30	D₇₇₆	—	2	3	no	—
31	D₇₇₆	C₂³	1	3	yes	—
32	#13095486	C₂³	1	3	yes	—
33	#13095486	C₂³	1	1	yes	characteristic
34	#41×C₉₇	—	2	3	no	—
35	#41×C₉₇	—	4	1	no	—
36	#41×C₉₇	D₄	1	1	yes	characteristic
37	C₂×D₄×C₉₇	—	2	3	no	—
38	C₂×D₄×C₉₇	—	4	3	no	—
39	Q₁₆×C₉₇	C₂³	1	1	yes	characteristic
40	SD₁₆×C₉₇	—	2	3	no	—
41	D₈×C₉₇	—	2	3	no	—
42	D₈×C₉₇	C₂³	1	3	yes	—
43	M₁₆×C₉₇	D₄	1	3	yes	—
44	C₂×C₈×C₉₇	C₂³	1	3	yes	—
45	C₂³×C₉₇	—	4	3	no	—
46	C₂²×D₉₇	—	4	3	no	—
47	Q₈×C₉₇	—	2	1	no	—
48	Q₈×C₉₇	D₈	1	1	yes	characteristic
49	D₄×C₉₇	—	2	3	no	—
50	D₄×C₉₇	—	4	3	no	—
51	D₄×C₉₇	—	4	3	no	—
52	D₄×C₉₇	D₈	1	3	yes	—
53	C₂×C₄×C₉₇	—	4	3	no	—
54	C₂×C₄×C₉₇	C₂×D₄	1	3	yes	—
55	C₂²⋊D₉₇	—	4	3	no	—
56	D₃₈₈	—	2	3	no	—
57	D₃₈₈	—	4	3	no	—
58	C₄×D₉₇	—	4	3	no	—
59	Dic₁₉₄	—	2	1	no	—
60	C₈×C₉₇	C₂×D₄	1	3	yes	—
61	C₈×C₉₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₉₇	—	4	3	no	—
63	C₂²×C₉₇	—	16	3	no	—
64	C₂²×C₉₇	C₂×D₈	1	3	yes	—
65	C₂×D₉₇	—	4	3	no	—
66	C₂×D₉₇	—	16	3	no	—
67	C₄×C₉₇	—	4	1	no	—
68	C₄×C₉₇	C₂×D₈	1	3	yes	—
69	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₉₇	—	4	1	no	—
71	C₂×C₉₇	—	2	3	no	—
72	C₂×C₉₇	—	8	3	no	—
73	C₂×C₉₇	C₂²×D₈	1	1	yes	characteristic
74	D₉₇	—	8	3	no	—
75	#3415	—	97	1	no	maximal, Sylow
76	C₉₇	#3415	1	1	yes	characteristic, complemented, Sylow
77	#576	—	97	1	no	—
78	#576	D₉₇	1	1	yes	characteristic, complemented
79	#509	—	97	3	no	—
80	#509	—	97	3	no	—
81	#508	—	97	3	no	—
82	C₂×D₁₆	—	97	3	no	—
83	#504	—	97	1	no	—
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	194	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	194	3	no	—
88	#135	—	97	3	no	—
89	#135	C₂×D₉₇	1	3	yes	complemented
90	C₂×D₈	—	97	3	no	—
91	C₂×D₈	C₂×D₉₇	1	3	yes	complemented
92	#131	C₂×D₉₇	1	1	yes	characteristic
93	Q₃₂	—	97	1	no	—
94	SD₃₂	—	97	3	no	—
95	SD₃₂	—	97	3	no	—
96	D₁₆	—	97	3	no	—
97	D₁₆	—	97	6	no	—
98	M₃₂	—	97	3	no	—
99	C₂×C₁₆	—	97	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	388	1	no	—
103	#41	—	194	3	no	—
104	#41	C₂²⋊D₉₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	388	3	no	—
108	C₂×D₄	—	194	3	no	—
109	Q₁₆	—	97	1	no	—
110	Q₁₆	C₂²×D₉₇	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	194	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	97	3	no	—
115	D₈	—	194	3	no	—
116	D₈	C₂²×D₉₇	1	3	yes	—
117	M₁₆	C₂²⋊D₉₇	1	3	yes	—
118	C₂×C₈	C₂²×D₉₇	1	3	yes	—
119	C₁₆	—	97	1	no	—
120	C₁₆	—	97	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	388	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	194	1	no	—
125	Q₈	#13095496	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	388	3	no	—
130	D₄	—	388	3	no	—
131	D₄	—	194	3	no	—
132	D₄	#13095496	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	388	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₉₇	1	3	yes	—
136	C₈	C₂×C₂²⋊D₉₇	1	3	yes	—
137	C₈	C₂³×D₉₇	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	1552	3	no	—
141	C₂²	—	388	3	no	—
142	C₂²	C₂×#13095496	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	388	1	no	—
145	C₄	C₂×#13095496	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	776	3	no	—
150	C₂	C₂²×#13095496	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#13095496
commutator, cosocle	C₈×C₉₇	C₂⁴
Frattini	C₈	C₂³×D₉₇
Fitting	#576×C₉₇	C₂
socle	C₂×C₉₇	C₂²×D₈
hypercenter	#576	D₉₇
hypocenter	C₉₇	#3415
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

Complements

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

N	H ≅ G/N	type
1. #20261540	C₂	semidirect
2. #20261171	C₂	semidirect
3. #20261170	C₂	semidirect
4. #20261165	C₂	semidirect
5. C₂×#15690751	C₂	semidirect
6. #20261152	C₂	semidirect
7. #576×C₉₇	C₂	semidirect
10. #15690832	C₂²	semidirect
11. C₂×D₇₇₆	C₂²	semidirect
12. #15690756	C₂²	semidirect
13. #15690754	C₂²	semidirect
14. #15690753	C₂²	semidirect
15. #15690751	C₂²	semidirect
16. #15690751	C₂²	semidirect
17. #15690750	C₂²	semidirect
18. C₂×#13095486	C₂²	semidirect
21. #135×C₉₇	C₂²	semidirect
22. C₂×D₈×C₉₇	C₂²	semidirect
76. C₉₇	#3415	semidirect
78. #576	D₉₇	semidirect
89. #135	C₂×D₉₇	semidirect
91. C₂×D₈	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261540	2
#20261171	2
#20261170	2
#20261165	2
C₂×#15690751	2
#20261152	2
#576×C₉₇	2
#3415	97