Finite Group Databasebeta

← #38786679

Subgroups — #38786679

Order 25664.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20264365C211yesmaximal, characteristic, complemented
2C22×C22⋊D401C211yesmaximal, characteristic, complemented
3#27165733C212yesmaximal, complemented
4#27165723C212yesmaximal, complemented
5#27165710C212yesmaximal, complemented
6#27165709C212yesmaximal, complemented
7#27165707C211yesmaximal, characteristic, complemented
8#27165706C211yesmaximal, characteristic, complemented
9#27165705C211yesmaximal, characteristic, complemented
10#27165693C211yesmaximal, characteristic, complemented
11#120×C401C211yesmaximal, characteristic, complemented
12C23×D401C2211yescharacteristic
13#20264365—24no—
14C2×Dic802C2211yescharacteristic, complemented
15C2×D1604C2212yescomplemented
16C2×C22⋊D401—24no—
17C2×C22⋊D401C2211yescharacteristic
18C2×C22⋊D401C2212yes—
19C2×C22⋊D401C2212yes—
20C2×C22⋊D401C2211yescharacteristic, complemented
21C2×C22⋊D401C2212yescomplemented
22C2×C22⋊D401C2212yescomplemented
23C22×Dic401C2211yescharacteristic
24C22×Dic401C2211yescharacteristic, complemented
25C2×C4×D401C2211yescharacteristic, complemented
26#20264345C2212yescomplemented
27C22⋊Dic401C2211yescharacteristic
28C22⋊Dic401C2212yescomplemented
29#20264343C2212yescomplemented
30#20264343C2212yescomplemented
31C4×Dic401C2212yescomplemented
32C24×C401C2211yescharacteristic
33C2×D4×C401C2211yescharacteristic
34C2×D4×C401C2212yescomplemented
35C22⋊C4×C401C2211yescharacteristic
36C22⋊C4×C401C2212yescomplemented
37C22×D401—22no—
38C22×D401—42no—
39C22×D401C2312yes—
40C22×D401C2311yescharacteristic
41Dic802—22no—
42D1604—42no—
43C22⋊D401—22no—
44C22⋊D401—24no—
45C22⋊D401—24no—
46C22⋊D401—22no—
47C22⋊D401—42no—
48C22⋊D401—42no—
49C2×Dic401—22no—
50C2×Dic401D414yescomplemented
51C2×Dic401C2312yes—
52C2×Dic401C2311yescharacteristic
53C2×Dic401C2311yescharacteristic
54C2×Dic401C2311yescharacteristic
55C4×D401—22no—
56C23×C401—22no—
57C23×C401—42no—
58C23×C401C2312yes—
59C23×C401C2311yescharacteristic
60C23×C401C2311yescharacteristic
61D4×C401—22no—
62D4×C401—42no—
63C2×C4×C401C2312yes—
64C2×C4×C401C2311yescharacteristic
65C22×C401—22no—
66C22×C401—21no—
67C22×C401—22no—
68C22×C401—22no—
69C22×C401—22no—
70C22×C401—22no—
71C22×C401—44no—
72C22×C401C2×D412yes—
73C22×C401C2411yescharacteristic
74C2×D401—21no—
75C2×D401—22no—
76C2×D401—22no—
77C2×D401—22no—
78C2×D401—22no—
79C2×D401—44no—
80C4×C401—21no—
81C4×C401—22no—
82Dic401—21no—
83Dic401—22no—
84Dic401C2×D412yes—
85Dic401C2×D412yes—
86C2×C401—22no—
87C2×C401—41no—
88C2×C401—42no—
89C2×C401C22×D411yescharacteristic
90C2×C401H(5,𝔽2)12yes—
91D401—41no—
92D401—42no—
93C401#53511yescharacteristic, complemented, Sylow
94#535—4011nomaximal, Sylow
95C2×#41—4011no—
96C22×D4—4011no—
97#127—4012no—
98#124—4012no—
99#121—4012no—
100#121—4012no—
101#120—4011no—
102#120—4011no—
103#120—4011no—
104#120D40111yescharacteristic, complemented
105#117—4011no—
106C24—4011no—
107C24C2×D40111yescharacteristic
108#41—8024no—
109C2×Q8—4011no—
110C2×D4—8024no—
111C2×D4—4011no—
112C2×D4—4012no—
113C2×D4—4012no—
114C2×D4—4011no—
115C2×D4—4012no—
116C2×D4—4012no—
117C2×D4—4012no—
118C2×D4C2×D40111yescharacteristic
119C2×D4C2×D40112yescomplemented
120C22×C4—4011no—
121C22×C4—4011no—
122C22×C4—4011no—
123C4⋊C4—4012no—
124C22⋊C4—4011no—
125C22⋊C4—4012no—
126C22⋊C4—4012no—
127C22⋊C4—4012no—
128C22⋊C4C2×D40111yescharacteristic
129C22⋊C4C2×D40112yescomplemented
130C42—4012no—
131C23—22no—
132C23—42no—
133C23—8022no—
134C23—16042no—
135C23—4012no—
136C23—4011no—
137C23C22×D40112yes—
138C23C22×D40111yescharacteristic
139C23C22×D40111yescharacteristic
140Q8—8022no—
141D4—22no—
142D4—42no—
143D4—8022no—
144D4—8024no—
145D4—8024no—
146D4—8022no—
147D4—16042no—
148D4—16042no—
149D4—16042no—
150C2×C4—8022no—
151C2×C4—8022no—
152C2×C4—4012no—
153C2×C4—4011no—
154C2×C4—4014no—
155C2×C4—4011no—
156C2×C4—4011no—
157C2×C4C22×D40112yes—
158C2×C4C22×D40111yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—8021no—
167C22—8022no—
168C22—8022no—
169C22—8022no—
170C22—8022no—
171C22—16044no—
172C22D4×D40112yes—
173C22C23×D40111yescharacteristic
174C4—21no—
175C4—22no—
176C4—8021no—
177C4—8022no—
178C4—4012no—
179C4—4012no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—16042no—
184C2—16041no—
185C2C2×D4×D40111yescharacteristic
186C2#2716578212yes—
subgroupG/H
center, FrattiniC22C23×D401
commutator, socle, cosocleC22×C401C24
Fitting#120×C401C2
hypercenter#120D401
hypocenterC401#535
LCSG ▷ C22×C401 ▷ C401
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20264365C2semidirect
2. C22×C22⋊D401C2semidirect
3. #27165733C2semidirect
4. #27165723C2semidirect
5. #27165710C2semidirect
6. #27165709C2semidirect
7. #27165707C2semidirect
8. #27165706C2semidirect
9. #27165705C2semidirect
10. #27165693C2semidirect
11. #120×C401C2semidirect
14. C2×Dic802C22semidirect
15. C2×D1604C22semidirect
20. C2×C22⋊D401C22semidirect
21. C2×C22⋊D401C22semidirect
22. C2×C22⋊D401C22semidirect
24. C22×Dic401C22semidirect
25. C2×C4×D401C22semidirect
26. #20264345C22semidirect
28. C22⋊Dic401C22semidirect
29. #20264343C22semidirect
30. #20264343C22semidirect
31. C4×Dic401C22semidirect
34. C2×D4×C401C22semidirect
36. C22⋊C4×C401C22semidirect
50. C2×Dic401D4semidirect
93. C401#535semidirect
104. #120D401semidirect
119. C2×D4C2×D401semidirect
129. C22⋊C4C2×D401semidirect

Maximal subgroups

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

subgroupindex
C2×#202643652
C22×C22⋊D4012
#271657332
#271657232
#271657102
#271657092
#271657072
#271657062
#271657052
#271656932
#120×C4012
#535401