Finite Group Databasebeta

Subgroups — #27143682

Order 12416.

226 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20261170	C₂	1	2	yes	maximal, complemented
2	#20260988	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260545	C₂	1	2	yes	maximal, complemented
4	#20260517	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₇₆	C₂²	1	1	yes	characteristic, complemented
7	#15690784	—	2	1	no	—
8	#15690753	—	2	2	no	—
9	#15690751	—	2	2	no	—
10	#15690750	C₂²	1	2	yes	complemented
11	#15690742	C₂²	1	1	yes	characteristic, complemented
12	#15690732	—	2	1	no	—
13	C₂×#41×C₉₇	—	2	1	no	—
14	#137×C₉₇	—	2	2	no	—
15	AGL(1,ℤ/8)×C₉₇	—	2	2	no	—
16	#135×C₉₇	—	2	1	no	—
17	#135×C₉₇	C₂²	1	2	yes	—
18	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×D₃₈₈	—	2	1	no	—
20	C₂×C₂²⋊D₉₇	—	4	1	no	—
21	D₇₇₆	—	2	2	no	—
22	C₄⋊Dic₉₇	—	2	1	no	—
23	#13095488	—	4	1	no	—
24	#13095486	—	2	2	no	—
25	#41×C₉₇	—	2	2	no	—
26	#41×C₉₇	—	2	2	no	—
27	#41×C₉₇	—	4	1	no	—
28	#41×C₉₇	—	4	1	no	—
29	C₂×Q₈×C₉₇	—	2	1	no	—
30	C₂×D₄×C₉₇	—	2	1	no	—
31	C₂×D₄×C₉₇	—	4	1	no	—
32	C₂²×C₄×C₉₇	—	4	1	no	—
33	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
34	Q₁₆×C₉₇	—	2	1	no	—
35	Q₁₆×C₉₇	D₄	1	2	yes	complemented
36	SD₁₆×C₉₇	—	2	2	no	—
37	SD₁₆×C₉₇	—	2	2	no	—
38	D₈×C₉₇	—	2	1	no	—
39	D₈×C₉₇	D₄	1	2	yes	complemented
40	M₁₆×C₉₇	—	2	2	no	—
41	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
42	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
43	C₂³×C₉₇	—	4	1	no	—
44	C₂³×C₉₇	D₈	1	1	yes	characteristic
45	C₂²×D₉₇	—	4	1	no	—
46	Q₈×C₉₇	—	2	2	no	—
47	Q₈×C₉₇	—	4	1	no	—
48	D₄×C₉₇	—	2	2	no	—
49	D₄×C₉₇	—	4	1	no	—
50	D₄×C₉₇	—	4	2	no	—
51	D₄×C₉₇	—	4	2	no	—
52	C₂×C₄×C₉₇	—	2	1	no	—
53	C₂×C₄×C₉₇	—	2	1	no	—
54	C₂×C₄×C₉₇	—	4	2	no	—
55	C₂×C₄×C₉₇	—	4	1	no	—
56	C₂×C₄×C₉₇	—	4	2	no	—
57	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
58	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₉₇	—	8	2	no	—
60	C₂×Dic₉₇	—	4	1	no	—
61	D₃₈₈	—	4	1	no	—
62	D₃₈₈	—	4	1	no	—
63	C₈×C₉₇	—	2	1	no	—
64	C₈×C₉₇	C₂×D₄	1	2	yes	—
65	C₂²×C₉₇	—	2	1	no	—
66	C₂²×C₉₇	—	4	1	no	—
67	C₂²×C₉₇	—	4	2	no	—
68	C₂²×C₉₇	—	8	2	no	—
69	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
70	C₂×D₉₇	—	8	1	no	—
71	C₂×D₉₇	—	8	1	no	—
72	C₂×D₉₇	—	8	1	no	—
73	C₄×C₉₇	—	2	1	no	—
74	C₄×C₉₇	—	4	2	no	—
75	C₄×C₉₇	#120	1	1	yes	characteristic
76	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₉₇	—	8	1	no	—
78	C₂×C₉₇	—	2	1	no	—
79	C₂×C₉₇	—	4	1	no	—
80	C₂×C₉₇	—	8	2	no	—
81	C₂×C₉₇	#447	1	1	yes	characteristic
82	D₉₇	—	16	1	no	—
83	#2190	—	97	1	no	maximal, Sylow
84	C₉₇	#2190	1	1	yes	characteristic, complemented, Sylow
85	#575	D₉₇	1	1	yes	characteristic, complemented
86	#509	—	97	2	no	—
87	#469	—	97	1	no	—
88	#360	—	97	2	no	—
89	#349	—	97	1	no	—
90	C₂×#41	—	2	1	no	—
91	#137	—	2	2	no	—
92	AGL(1,ℤ/8)	—	2	2	no	—
93	#135	—	2	1	no	—
94	#135	C₂×D₉₇	1	2	yes	—
95	C₂×D₈	—	97	1	no	—
96	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
97	#121	—	194	1	no	—
98	SD₃₂	—	194	2	no	—
99	D₁₆	—	194	2	no	—
100	M₃₂	—	97	2	no	—
101	#106	—	97	1	no	—
102	#102	—	194	1	no	—
103	#41	—	2	2	no	—
104	#41	—	2	2	no	—
105	#41	—	4	1	no	—
106	#41	—	4	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	4	1	no	—
110	C₂×D₄	—	388	1	no	—
111	C₂×D₄	—	194	1	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
114	Q₁₆	—	2	1	no	—
115	Q₁₆	C₂²⋊D₉₇	1	2	yes	complemented
116	SD₁₆	—	2	2	no	—
117	SD₁₆	—	2	2	no	—
118	D₈	—	2	1	no	—
119	D₈	—	194	2	no	—
120	D₈	C₂²⋊D₉₇	1	2	yes	complemented
121	M₁₆	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
124	C₄⋊C₄	—	194	1	no	—
125	C₂²⋊C₄	—	388	1	no	—
126	C₁₆	—	194	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	388	1	no	—
129	C₂³	#13095496	1	1	yes	characteristic
130	Q₈	—	2	2	no	—
131	Q₈	—	4	1	no	—
132	D₄	—	2	2	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	776	2	no	—
137	D₄	—	388	1	no	—
138	D₄	—	388	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	388	1	no	—
145	C₂×C₄	#13095496	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	C₂×C₂²⋊D₉₇	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	776	1	no	—
154	C₂²	—	776	1	no	—
155	C₂²	—	776	1	no	—
156	C₂²	C₂×#13095496	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	776	1	no	—
160	C₄	#15690779	1	1	yes	characteristic
161	C₄	#15690836	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	1552	1	no	—
166	C₂	#20260869	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260869
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#575×C₉₇	C₂
socle	C₂×C₉₇	#447
hypercenter	#575	D₉₇
hypocenter	C₉₇	#2190
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #575

Complements

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

N	H ≅ G/N	type
1. #20261170	C₂	semidirect
2. #20260988	C₂	semidirect
3. #20260545	C₂	semidirect
4. #20260517	C₂	semidirect
5. #575×C₉₇	C₂	semidirect
6. C₂×D₇₇₆	C₂²	semidirect
10. #15690750	C₂²	semidirect
11. #15690742	C₂²	semidirect
35. Q₁₆×C₉₇	D₄	semidirect
39. D₈×C₉₇	D₄	semidirect
84. C₉₇	#2190	semidirect
85. #575	D₉₇	semidirect
115. Q₁₆	C₂²⋊D₉₇	semidirect
120. D₈	C₂²⋊D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261170	2
#20260988	2
#20260545	2
#20260517	2
#575×C₉₇	2
#2190	97