Finite Group Databasebeta

Subgroups — #36475038

Order 22784.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#26022559	C₂	1	1	yes	maximal, characteristic
2	#26022133	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022132	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018455	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018437	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018385	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1417	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152030	C₂²	1	1	yes	characteristic
9	#19151715	—	2	1	no	—
10	#19151714	—	2	1	no	—
11	#19151438	C₄	1	1	yes	characteristic, complemented
12	#19151436	C₄	1	1	yes	characteristic, complemented
13	#19151435	C₄	1	1	yes	characteristic, complemented
14	#19151433	C₄	1	1	yes	characteristic, complemented
15	C₂×#15667918	C₂²	1	1	yes	characteristic
16	#19151428	C₂²	1	1	yes	characteristic
17	#19151405	—	2	1	no	—
18	#19151404	—	2	1	no	—
19	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
20	#508×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
22	#368×C₈₉	C₂²	1	1	yes	characteristic
23	#15668034	—	2	1	no	—
24	#15668020	—	4	1	no	—
25	#15668019	—	4	1	no	—
26	Dic₇₁₂	C₂×C₄	1	1	yes	characteristic
27	#15667957	—	2	1	no	—
28	D₁₄₂₄	C₂×C₄	1	1	yes	characteristic
29	C₂×#13083549	D₄	1	1	yes	characteristic
30	#15667948	—	2	1	no	—
31	#15667939	—	4	1	no	—
32	#15667938	—	4	1	no	—
33	C₈×Dic₈₉	D₄	1	1	yes	characteristic
34	#15667918	C₂×C₄	1	1	yes	characteristic
35	#15667918	C₂×C₄	1	1	yes	characteristic
36	#135×C₈₉	—	2	1	no	—
37	Q₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
38	SD₃₂×C₈₉	—	2	1	no	—
39	D₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
41	#108×C₈₉	—	2	1	no	—
42	C₃₂×C₈₉	—	2	1	no	—
43	#13083574	—	4	1	no	—
44	Dic₃₅₆	—	2	1	no	—
45	#13083563	—	4	1	no	—
46	D₇₁₂	—	2	1	no	—
47	#13083558	—	4	1	no	—
48	#13083558	—	8	1	no	—
49	#13083557	—	8	1	no	—
50	C₂×#10623494	—	8	1	no	—
51	C₂×#10623494	D₈	1	1	yes	characteristic
52	C₈×D₈₉	—	8	1	no	—
53	C₄×Dic₈₉	D₈	1	1	yes	characteristic
54	#13083549	—	2	1	no	—
55	#41×C₈₉	—	4	1	no	—
56	Q₁₆×C₈₉	—	2	1	no	—
57	SD₁₆×C₈₉	—	4	1	no	—
58	D₈×C₈₉	—	2	1	no	—
59	M₁₆×C₈₉	—	4	1	no	—
60	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₁₆×C₈₉	#41	1	1	yes	characteristic
63	Q₈×C₈₉	—	4	1	no	—
64	D₄×C₈₉	—	4	1	no	—
65	D₄×C₈₉	—	8	1	no	—
66	C₂×C₄×C₈₉	—	8	1	no	—
67	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₂²⋊D₈₉	—	8	1	no	—
69	C₂×Dic₈₉	—	2	1	no	—
70	D₃₅₆	—	4	1	no	—
71	C₄×D₈₉	—	8	1	no	—
72	Dic₁₇₈	—	4	1	no	—
73	C₈×C₈₉	—	8	1	no	—
74	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
75	C₈×C₈₉	#135	1	1	yes	characteristic
76	#10623494	—	8	1	no	—
77	#10623494	D₁₆	1	1	yes	characteristic
78	#10623494	D₁₆	1	1	yes	characteristic
79	C₂²×C₈₉	—	8	1	no	—
80	C₂²×C₈₉	#508	1	1	yes	characteristic
81	C₂×D₈₉	—	8	1	no	—
82	C₄×C₈₉	—	8	1	no	—
83	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
84	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
85	Dic₈₉	—	4	1	no	—
86	Dic₈₉	—	8	1	no	—
87	#13656	—	89	1	no	maximal, Sylow
88	C₂×C₈₉	—	2	1	no	—
89	C₂×C₈₉	—	16	1	no	—
90	C₂×C₈₉	C₄×D₁₆	1	1	yes	characteristic
91	D₈₉	—	16	1	no	—
92	#2262	—	89	1	no	—
93	#2178	—	89	1	no	—
94	#2178	—	89	1	no	—
95	#1425	—	89	1	no	—
96	#1417	—	89	1	no	—
97	#1417	D₈₉	1	1	yes	characteristic, complemented
98	#1397	—	89	1	no	—
99	C₈₉	#13656	1	1	yes	characteristic, complemented, Sylow
100	#508	—	89	1	no	—
101	#508	C₂×D₈₉	1	1	yes	characteristic
102	#443	—	178	1	no	—
103	#443	—	178	1	no	—
104	Q₆₄	—	89	1	no	—
105	SD₆₄	—	89	1	no	—
106	SD₆₄	—	89	1	no	—
107	D₃₂	—	89	1	no	—
108	C₂×C₃₂	—	89	1	no	—
109	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
110	#368	—	89	1	no	—
111	#368	C₂×D₈₉	1	1	yes	characteristic
112	#359	—	178	1	no	—
113	#359	—	178	1	no	—
114	C₄×C₁₆	—	89	1	no	—
115	#135	—	2	1	no	—
116	#135	—	178	1	no	—
117	#131	—	356	1	no	—
118	#131	—	356	1	no	—
119	Q₃₂	—	89	1	no	—
120	Q₃₂	C₄×D₈₉	1	1	yes	characteristic
121	SD₃₂	—	2	1	no	—
122	SD₃₂	—	178	1	no	—
123	D₁₆	—	89	1	no	—
124	D₁₆	C₄×D₈₉	1	1	yes	characteristic
125	C₂×C₁₆	—	89	1	no	—
126	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
127	#108	—	2	1	no	—
128	#108	—	178	1	no	—
129	#104	—	356	1	no	—
130	#104	—	356	1	no	—
131	C₄×C₈	—	89	1	no	—
132	C₃₂	—	2	1	no	—
133	C₃₂	—	89	1	no	—
134	C₃₂	—	89	1	no	—
135	#41	—	4	1	no	—
136	#41	—	356	1	no	—
137	Q₁₆	—	2	1	no	—
138	Q₁₆	—	178	1	no	—
139	SD₁₆	—	4	1	no	—
140	SD₁₆	—	356	1	no	—
141	D₈	—	2	1	no	—
142	D₈	—	178	1	no	—
143	M₁₆	—	4	1	no	—
144	M₁₆	—	356	1	no	—
145	M₁₆	—	712	1	no	—
146	M₁₆	—	712	1	no	—
147	C₂×C₈	—	89	1	no	—
148	C₂×C₈	—	712	1	no	—
149	C₂×C₈	—	712	1	no	—
150	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
151	C₄²	—	89	1	no	—
152	C₁₆	—	178	1	no	—
153	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
154	C₁₆	#13083573	1	1	yes	characteristic
155	Q₈	—	4	1	no	—
156	Q₈	—	356	1	no	—
157	D₄	—	4	1	no	—
158	D₄	—	8	1	no	—
159	D₄	—	356	1	no	—
160	D₄	—	712	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	178	1	no	—
163	C₂×C₄	—	712	1	no	—
164	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
165	C₈	—	8	1	no	—
166	C₈	—	89	1	no	—
167	C₈	—	89	1	no	—
168	C₈	—	712	1	no	—
169	C₈	#15667972	1	1	yes	characteristic
170	C₈	#15668033	1	1	yes	characteristic
171	C₂²	—	8	1	no	—
172	C₂²	—	712	1	no	—
173	C₂²	#19152029	1	1	yes	characteristic
174	C₄	—	8	1	no	—
175	C₄	—	356	1	no	—
176	C₄	—	712	1	no	—
177	C₄	#19151678	1	1	yes	characteristic
178	C₄	D₁₆×D₈₉	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	16	1	no	—
181	C₂	—	1424	1	no	—
182	C₂	#26022096	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆×D₈₉
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1417	C₂
socle	C₂×C₈₉	C₄×D₁₆
prime radical	#26022559	C₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1417	D₈₉
hypocenter	C₈₉	#13656
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1417

Complements

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

N	H ≅ G/N	type
2. #26022133	C₂	semidirect
3. #26022132	C₂	semidirect
4. #26018455	C₂	semidirect
5. #26018437	C₂	semidirect
6. #26018385	C₂	semidirect
7. C₈₉×#1417	C₂	semidirect
11. #19151438	C₄	semidirect
12. #19151436	C₄	semidirect
13. #19151435	C₄	semidirect
14. #19151433	C₄	semidirect
97. #1417	D₈₉	semidirect
99. C₈₉	#13656	semidirect

Maximal subgroups

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

subgroup	index
#26022559	2
#26022133	2
#26022132	2
#26018455	2
#26018437	2
#26018385	2
C₈₉×#1417	2
#13656	89