Finite Group Databasebeta

Subgroups — #22590286

Order 9088.

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	#17986434	C₂	1	2	yes	maximal, complemented
2	#17986252	C₂	1	1	yes	maximal, characteristic, complemented
3	#17985809	C₂	1	2	yes	maximal, complemented
4	#17985781	C₂	1	1	yes	maximal, characteristic, complemented
5	#575×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₆₈	C₂²	1	1	yes	characteristic, complemented
7	#14556102	—	2	1	no	—
8	#14556071	—	2	2	no	—
9	#14556069	—	2	2	no	—
10	#14556068	C₂²	1	2	yes	complemented
11	#14556060	C₂²	1	1	yes	characteristic, complemented
12	#14556050	—	2	1	no	—
13	C₂×#41×C₇₁	—	2	1	no	—
14	#137×C₇₁	—	2	2	no	—
15	AGL(1,ℤ/8)×C₇₁	—	2	2	no	—
16	#135×C₇₁	—	2	1	no	—
17	#135×C₇₁	C₂²	1	2	yes	—
18	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×D₂₈₄	—	2	1	no	—
20	C₂×C₂²⋊D₇₁	—	4	1	no	—
21	D₅₆₈	—	2	2	no	—
22	C₄⋊Dic₇₁	—	2	1	no	—
23	(C₂×D₇₁)⋊C₄	—	4	1	no	—
24	C₇₁⋊C₁₆	—	2	2	no	—
25	#41×C₇₁	—	2	2	no	—
26	#41×C₇₁	—	2	2	no	—
27	#41×C₇₁	—	4	1	no	—
28	#41×C₇₁	—	4	1	no	—
29	C₂×Q₈×C₇₁	—	2	1	no	—
30	C₂×D₄×C₇₁	—	2	1	no	—
31	C₂×D₄×C₇₁	—	4	1	no	—
32	C₂²×C₄×C₇₁	—	4	1	no	—
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	Q₁₆×C₇₁	—	2	1	no	—
35	Q₁₆×C₇₁	D₄	1	2	yes	complemented
36	SD₁₆×C₇₁	—	2	2	no	—
37	SD₁₆×C₇₁	—	2	2	no	—
38	D₈×C₇₁	—	2	1	no	—
39	D₈×C₇₁	D₄	1	2	yes	complemented
40	M₁₆×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂³×C₇₁	—	4	1	no	—
44	C₂³×C₇₁	D₈	1	1	yes	characteristic
45	C₂²×D₇₁	—	4	1	no	—
46	Q₈×C₇₁	—	2	2	no	—
47	Q₈×C₇₁	—	4	1	no	—
48	D₄×C₇₁	—	2	2	no	—
49	D₄×C₇₁	—	4	1	no	—
50	D₄×C₇₁	—	4	2	no	—
51	D₄×C₇₁	—	4	2	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	4	2	no	—
55	C₂×C₄×C₇₁	—	4	1	no	—
56	C₂×C₄×C₇₁	—	4	2	no	—
57	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²⋊D₇₁	—	8	2	no	—
60	C₂×Dic₇₁	—	4	1	no	—
61	D₂₈₄	—	4	1	no	—
62	D₂₈₄	—	4	1	no	—
63	C₈×C₇₁	—	2	1	no	—
64	C₈×C₇₁	C₂×D₄	1	2	yes	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	4	1	no	—
67	C₂²×C₇₁	—	4	2	no	—
68	C₂²×C₇₁	—	8	2	no	—
69	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
70	C₂×D₇₁	—	8	1	no	—
71	C₂×D₇₁	—	8	1	no	—
72	C₂×D₇₁	—	8	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	#120	1	1	yes	characteristic
76	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	Dic₇₁	—	8	1	no	—
78	C₂×C₇₁	—	2	1	no	—
79	C₂×C₇₁	—	4	1	no	—
80	C₂×C₇₁	—	8	2	no	—
81	C₂×C₇₁	#447	1	1	yes	characteristic
82	D₇₁	—	16	1	no	—
83	#2190	—	71	1	no	maximal, Sylow
84	C₇₁	#2190	1	1	yes	characteristic, complemented, Sylow
85	#575	D₇₁	1	1	yes	characteristic, complemented
86	#509	—	71	2	no	—
87	#469	—	71	1	no	—
88	#360	—	71	2	no	—
89	#349	—	71	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×D₇₁	1	2	yes	—
95	C₂×D₈	—	71	1	no	—
96	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
97	#121	—	142	1	no	—
98	SD₃₂	—	142	2	no	—
99	D₁₆	—	142	2	no	—
100	M₃₂	—	71	2	no	—
101	#106	—	71	1	no	—
102	#102	—	142	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	—	284	1	no	—
111	C₂×D₄	—	142	1	no	—
112	C₂²×C₄	—	4	1	no	—
113	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
114	Q₁₆	—	2	1	no	—
115	Q₁₆	C₂²⋊D₇₁	1	2	yes	complemented
116	SD₁₆	—	2	2	no	—
117	SD₁₆	—	2	2	no	—
118	D₈	—	2	1	no	—
119	D₈	—	142	2	no	—
120	D₈	C₂²⋊D₇₁	1	2	yes	complemented
121	M₁₆	—	2	2	no	—
122	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
124	C₄⋊C₄	—	142	1	no	—
125	C₂²⋊C₄	—	284	1	no	—
126	C₁₆	—	142	2	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	284	1	no	—
129	C₂³	#11765850	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	—	284	1	no	—
137	D₄	—	284	1	no	—
138	D₄	—	568	2	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	—	284	1	no	—
145	C₂×C₄	#11765850	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	C₂×C₂²⋊D₇₁	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	—	568	1	no	—
154	C₂²	—	568	1	no	—
155	C₂²	—	568	1	no	—
156	C₂²	C₂×#11765850	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	4	2	no	—
159	C₄	—	568	1	no	—
160	C₄	#14556097	1	1	yes	characteristic
161	C₄	#14556154	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	8	2	no	—
165	C₂	—	1136	1	no	—
166	C₂	#17986133	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17986133
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#575×C₇₁	C₂
socle	C₂×C₇₁	#447
hypercenter	#575	D₇₁
hypocenter	C₇₁	#2190
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
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. #17986434	C₂	semidirect
2. #17986252	C₂	semidirect
3. #17985809	C₂	semidirect
4. #17985781	C₂	semidirect
5. #575×C₇₁	C₂	semidirect
6. C₂×D₅₆₈	C₂²	semidirect
10. #14556068	C₂²	semidirect
11. #14556060	C₂²	semidirect
35. Q₁₆×C₇₁	D₄	semidirect
39. D₈×C₇₁	D₄	semidirect
84. C₇₁	#2190	semidirect
85. #575	D₇₁	semidirect
115. Q₁₆	C₂²⋊D₇₁	semidirect
120. D₈	C₂²⋊D₇₁	semidirect