Finite Group Databasebeta

Subgroups — #36475047

Order 22784.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022138	C₂	1	1	yes	maximal, characteristic
2	#26022123	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018458	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018445	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018439	C₂	1	1	yes	maximal, characteristic
6	#26018387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2264	C₂	1	1	yes	maximal, characteristic
8	#19151715	—	2	1	no	—
9	Q₁₆×Dic₈₉	—	2	1	no	—
10	#19151432	C₂²	1	1	yes	characteristic
11	#19151428	C₂²	1	1	yes	characteristic
12	#19151426	C₂²	1	1	yes	characteristic
13	#19151405	—	2	1	no	—
14	#19151402	—	2	1	no	—
15	#19151373	C₂²	1	1	yes	characteristic
16	#508×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
18	Q₆₄×C₈₉	C₄	1	2	yes	complemented
19	SD₆₄×C₈₉	C₄	1	2	yes	complemented
20	M₆₄×C₈₉	C₂²	1	1	yes	characteristic
21	#15668019	—	4	1	no	—
22	Q₈×Dic₈₉	—	4	1	no	—
23	C₂×#13083549	D₄	1	1	yes	characteristic
24	#15667948	—	2	1	no	—
25	#15667946	—	2	1	no	—
26	#15667939	—	4	1	no	—
27	#15667936	—	4	1	no	—
28	C₈×Dic₈₉	D₄	1	1	yes	characteristic
29	#15667918	—	2	1	no	—
30	#135×C₈₉	—	2	1	no	—
31	C₂×Q₁₆×C₈₉	—	2	1	no	—
32	Q₃₂×C₈₉	—	2	1	no	—
33	Q₃₂×C₈₉	C₂×C₄	1	2	yes	—
34	Q₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
35	SD₃₂×C₈₉	—	2	1	no	—
36	D₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
38	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
39	#13083558	—	4	1	no	—
40	#13083558	—	8	1	no	—
41	C₂×#10623494	—	8	1	no	—
42	C₂×#10623494	D₈	1	1	yes	characteristic
43	C₄⋊Dic₈₉	—	4	1	no	—
44	C₄⋊Dic₈₉	—	8	1	no	—
45	C₄×Dic₈₉	—	8	1	no	—
46	C₄×Dic₈₉	D₈	1	1	yes	characteristic
47	#13083549	—	2	1	no	—
48	#41×C₈₉	—	4	1	no	—
49	C₂×Q₈×C₈₉	—	4	1	no	—
50	Q₁₆×C₈₉	—	2	1	no	—
51	Q₁₆×C₈₉	—	2	2	no	—
52	Q₁₆×C₈₉	—	4	1	no	—
53	SD₁₆×C₈₉	—	4	1	no	—
54	D₈×C₈₉	—	2	1	no	—
55	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₈₉	#41	1	1	yes	characteristic
58	Q₈×C₈₉	—	4	1	no	—
59	Q₈×C₈₉	—	4	2	no	—
60	Q₈×C₈₉	—	8	1	no	—
61	D₄×C₈₉	—	4	1	no	—
62	D₄×C₈₉	—	8	1	no	—
63	C₂×C₄×C₈₉	—	8	1	no	—
64	C₂×C₄×C₈₉	—	8	1	no	—
65	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
66	C₂×Dic₈₉	—	2	1	no	—
67	C₂×Dic₈₉	—	8	1	no	—
68	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
69	C₈×C₈₉	#135	1	1	yes	characteristic
70	#10623494	—	2	1	no	—
71	#10623494	—	8	1	no	—
72	C₂²×C₈₉	—	8	1	no	—
73	C₂²×C₈₉	#509	1	1	yes	characteristic
74	C₄×C₈₉	—	8	1	no	—
75	C₄×C₈₉	—	8	2	no	—
76	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
77	C₄×C₈₉	#509	1	1	yes	characteristic
78	Dic₈₉	—	4	1	no	—
79	Dic₈₉	—	16	1	no	—
80	#13657	—	89	1	no	maximal, Sylow
81	C₂×C₈₉	—	2	1	no	—
82	C₂×C₈₉	—	16	1	no	—
83	C₂×C₈₉	#2177	1	1	yes	characteristic
84	#2264	D₈₉	1	1	yes	characteristic
85	#2179	—	89	1	no	—
86	#2176	—	89	1	no	—
87	#1426	—	89	1	no	—
88	#1420	—	89	1	no	—
89	#1418	—	89	1	no	—
90	#1398	—	89	1	no	—
91	C₈₉	#13657	1	1	yes	characteristic, complemented, Sylow
92	#508	C₂×D₈₉	1	1	yes	characteristic
93	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
94	#443	—	178	1	no	—
95	C₄×Q₁₆	—	178	1	no	—
96	Q₆₄	Dic₈₉	1	2	yes	complemented
97	SD₆₄	Dic₈₉	1	2	yes	complemented
98	M₆₄	—	89	1	no	—
99	M₆₄	C₂×D₈₉	1	1	yes	characteristic
100	#368	—	89	1	no	—
101	#367	—	89	1	no	—
102	#359	—	178	1	no	—
103	#358	—	178	1	no	—
104	#346	—	89	1	no	—
105	#135	—	2	1	no	—
106	C₂×Q₁₆	—	2	1	no	—
107	#131	—	356	1	no	—
108	C₄×Q₈	—	356	1	no	—
109	Q₃₂	—	2	1	no	—
110	Q₃₂	C₂×Dic₈₉	1	2	yes	—
111	Q₃₂	C₂×Dic₈₉	1	1	yes	characteristic
112	SD₃₂	—	2	1	no	—
113	D₁₆	C₂×Dic₈₉	1	1	yes	characteristic
114	C₂×C₁₆	—	89	1	no	—
115	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
116	#108	—	178	1	no	—
117	#107	—	178	1	no	—
118	#104	—	356	1	no	—
119	#103	—	356	1	no	—
120	C₄×C₈	—	89	1	no	—
121	C₃₂	—	178	1	no	—
122	C₃₂	C₂×Dic₈₉	1	2	yes	—
123	#41	—	4	1	no	—
124	C₂×Q₈	—	4	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	2	2	no	—
127	Q₁₆	—	4	1	no	—
128	SD₁₆	—	4	1	no	—
129	D₈	—	2	1	no	—
130	M₁₆	—	356	1	no	—
131	M₁₆	—	712	1	no	—
132	C₂×C₈	—	89	1	no	—
133	C₂×C₈	—	712	1	no	—
134	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
135	C₄⋊C₄	—	356	1	no	—
136	C₄⋊C₄	—	712	1	no	—
137	C₄²	—	89	1	no	—
138	C₄²	—	712	1	no	—
139	C₁₆	—	178	1	no	—
140	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
141	C₁₆	#13083573	1	1	yes	characteristic
142	Q₈	—	4	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	8	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	178	1	no	—
150	C₂×C₄	—	712	1	no	—
151	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
152	C₈	—	178	1	no	—
153	C₈	—	712	1	no	—
154	C₈	D₄×Dic₈₉	1	1	yes	characteristic
155	C₈	#15668031	1	1	yes	characteristic
156	C₂²	—	8	1	no	—
157	C₂²	#19152032	1	1	yes	characteristic
158	C₄	—	8	1	no	—
159	C₄	—	8	2	no	—
160	C₄	—	356	1	no	—
161	C₄	—	1424	1	no	—
162	C₄	D₈×Dic₈₉	1	1	yes	characteristic
163	C₄	#19152032	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	16	1	no	—
166	C₂	#26022128	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26022128
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#2264	C₂
socle	C₂×C₈₉	#2177
prime radical	#508×C₈₉	C₂²
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#2264	D₈₉
hypocenter	C₈₉	#13657
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. #26022123	C₂	semidirect
3. #26018458	C₂	semidirect
4. #26018445	C₂	semidirect
6. #26018387	C₂	semidirect
18. Q₆₄×C₈₉	C₄	semidirect
19. SD₆₄×C₈₉	C₄	semidirect
91. C₈₉	#13657	semidirect
96. Q₆₄	Dic₈₉	semidirect
97. SD₆₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022138	2
#26022123	2
#26018458	2
#26018445	2
#26018439	2
#26018387	2
C₈₉×#2264	2
#13657	89