Finite Group Databasebeta

Subgroups — #36475050

Order 22784.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26022564	C₂	1	1	yes	maximal, characteristic
2	#26022138	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022127	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018457	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018443	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
9	#19151715	—	2	1	no	—
10	#19151680	—	2	1	no	—
11	#19151435	C₄	1	2	yes	complemented
12	#19151433	C₄	1	2	yes	complemented
13	#19151432	C₂²	1	1	yes	characteristic
14	#19151428	C₂²	1	1	yes	characteristic
15	#19151405	—	2	1	no	—
16	#19151398	—	2	1	no	—
17	#19151373	C₂²	1	1	yes	characteristic
18	#508×C₈₉	C₂²	1	1	yes	characteristic
19	M₆₄×C₈₉	C₂²	1	1	yes	characteristic
20	#367×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×D₇₁₂	—	2	1	no	—
22	#15668019	—	4	1	no	—
23	#15667974	—	4	1	no	—
24	D₁₄₂₄	—	2	1	no	—
25	D₁₄₂₄	C₂×C₄	1	2	yes	—
26	C₂×#13083549	D₄	1	1	yes	characteristic
27	#15667948	—	2	1	no	—
28	#15667939	—	4	1	no	—
29	#15667932	—	4	1	no	—
30	C₈×Dic₈₉	D₄	1	1	yes	characteristic
31	#15667918	C₂×C₄	1	2	yes	complemented
32	#135×C₈₉	—	2	1	no	—
33	Q₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
34	SD₃₂×C₈₉	—	2	1	no	—
35	D₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
37	#107×C₈₉	—	2	1	no	—
38	C₃₂×C₈₉	—	2	1	no	—
39	C₂×D₃₅₆	—	4	1	no	—
40	C₂×C₄×D₈₉	—	8	1	no	—
41	D₇₁₂	—	2	2	no	—
42	D₇₁₂	—	4	1	no	—
43	#13083558	—	4	1	no	—
44	#13083558	—	8	1	no	—
45	C₂×#10623494	—	8	1	no	—
46	C₂×#10623494	D₈	1	1	yes	characteristic
47	#13083551	—	8	1	no	—
48	C₄×Dic₈₉	D₈	1	1	yes	characteristic
49	#13083549	—	2	1	no	—
50	#41×C₈₉	—	4	1	no	—
51	Q₁₆×C₈₉	—	2	1	no	—
52	SD₁₆×C₈₉	—	4	1	no	—
53	D₈×C₈₉	—	2	1	no	—
54	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₈₉	—	4	1	no	—
56	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₈₉	#41	1	1	yes	characteristic
58	C₂²×D₈₉	—	8	1	no	—
59	Q₈×C₈₉	—	4	1	no	—
60	D₄×C₈₉	—	4	1	no	—
61	D₄×C₈₉	—	8	1	no	—
62	C₂×C₄×C₈₉	—	8	1	no	—
63	C₂×C₄×C₈₉	—	8	1	no	—
64	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
65	C₂×Dic₈₉	—	2	1	no	—
66	D₃₅₆	—	4	2	no	—
67	D₃₅₆	—	8	1	no	—
68	C₄×D₈₉	—	16	2	no	—
69	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
70	C₈×C₈₉	#135	1	1	yes	characteristic
71	#10623494	—	2	1	no	—
72	#10623494	—	8	1	no	—
73	C₂²×C₈₉	—	8	1	no	—
74	C₂²×C₈₉	#509	1	1	yes	characteristic
75	C₂×D₈₉	—	8	2	no	—
76	C₂×D₈₉	—	16	1	no	—
77	C₂×D₈₉	—	16	1	no	—
78	C₄×C₈₉	—	8	1	no	—
79	C₄×C₈₉	—	16	1	no	—
80	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
81	C₄×C₈₉	#509	1	1	yes	characteristic
82	Dic₈₉	—	4	1	no	—
83	#13658	—	89	1	no	maximal, Sylow
84	C₂×C₈₉	—	2	1	no	—
85	C₂×C₈₉	—	16	1	no	—
86	C₂×C₈₉	#2177	1	1	yes	characteristic
87	D₈₉	—	16	2	no	—
88	#2263	—	89	1	no	—
89	#2179	—	89	1	no	—
90	#2177	—	89	1	no	—
91	#1426	—	89	1	no	—
92	#1419	—	89	1	no	—
93	#1418	D₈₉	1	1	yes	characteristic, complemented
94	#1398	—	89	1	no	—
95	C₈₉	#13658	1	1	yes	characteristic, complemented, Sylow
96	#508	C₂×D₈₉	1	1	yes	characteristic
97	C₂×D₁₆	—	89	1	no	—
98	#443	—	178	1	no	—
99	C₄×D₈	—	178	1	no	—
100	SD₆₄	—	89	2	no	—
101	D₃₂	—	89	2	no	—
102	M₆₄	—	89	1	no	—
103	M₆₄	C₂×D₈₉	1	1	yes	characteristic
104	#368	—	89	1	no	—
105	#367	C₂×D₈₉	1	1	yes	characteristic
106	#359	—	178	1	no	—
107	#357	—	178	1	no	—
108	#346	—	89	1	no	—
109	#135	—	2	1	no	—
110	C₂×D₈	—	178	1	no	—
111	#131	—	356	1	no	—
112	C₄×D₄	—	356	1	no	—
113	Q₃₂	C₄×D₈₉	1	1	yes	characteristic, complemented
114	SD₃₂	—	2	1	no	—
115	D₁₆	—	89	2	no	—
116	D₁₆	—	178	1	no	—
117	D₁₆	C₄×D₈₉	1	1	yes	characteristic, complemented
118	C₂×C₁₆	—	89	1	no	—
119	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
120	#108	—	178	1	no	—
121	#107	—	2	1	no	—
122	#104	—	356	1	no	—
123	#102	—	356	1	no	—
124	C₄×C₈	—	89	1	no	—
125	C₃₂	—	2	1	no	—
126	C₃₂	—	89	2	no	—
127	#41	—	4	1	no	—
128	C₂×D₄	—	356	1	no	—
129	C₂²×C₄	—	712	1	no	—
130	Q₁₆	—	2	1	no	—
131	SD₁₆	—	4	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	356	1	no	—
134	D₈	—	178	2	no	—
135	M₁₆	—	356	1	no	—
136	M₁₆	—	712	1	no	—
137	C₂×C₈	—	89	1	no	—
138	C₂×C₈	—	712	1	no	—
139	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
140	C₄⋊C₄	—	4	1	no	—
141	C₂²⋊C₄	—	712	1	no	—
142	C₄²	—	89	1	no	—
143	C₁₆	—	178	1	no	—
144	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
145	C₁₆	#13083573	1	1	yes	characteristic
146	C₂³	—	712	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	D₄	—	8	1	no	—
150	D₄	—	356	2	no	—
151	D₄	—	712	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	8	1	no	—
154	C₂×C₄	—	1424	2	no	—
155	C₂×C₄	—	178	1	no	—
156	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
157	C₈	—	178	1	no	—
158	C₈	—	712	1	no	—
159	C₈	#15667972	1	1	yes	characteristic
160	C₈	#15668033	1	1	yes	characteristic
161	C₂²	—	8	1	no	—
162	C₂²	—	1424	1	no	—
163	C₂²	—	1424	1	no	—
164	C₂²	—	712	2	no	—
165	C₂²	#19152032	1	1	yes	characteristic
166	C₄	—	8	1	no	—
167	C₄	—	16	1	no	—
168	C₄	—	356	1	no	—
169	C₄	#19151678	1	1	yes	characteristic
170	C₄	#19152035	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	16	1	no	—
173	C₂	—	1424	2	no	—
174	C₂	#26022125	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26022125
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1418	C₂
socle	C₂×C₈₉	#2177
prime radical	#26022564	C₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1418	D₈₉
hypocenter	C₈₉	#13658
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1418

Complements

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

N	H ≅ G/N	type
2. #26022138	C₂	semidirect
3. #26022127	C₂	semidirect
4. #26018457	C₂	semidirect
5. #26018443	C₂	semidirect
6. #26018387	C₂	semidirect
7. C₈₉×#1418	C₂	semidirect
11. #19151435	C₄	semidirect
12. #19151433	C₄	semidirect
31. #15667918	C₂×C₄	semidirect
33. Q₃₂×C₈₉	C₂×C₄	semidirect
35. D₁₆×C₈₉	C₂×C₄	semidirect
93. #1418	D₈₉	semidirect
95. C₈₉	#13658	semidirect
113. Q₃₂	C₄×D₈₉	semidirect
117. D₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022564	2
#26022138	2
#26022127	2
#26018457	2
#26018443	2
#26018387	2
C₈₉×#1418	2
#13658	89