Finite Group Databasebeta

← #45690770

Subgroups — #45690770

Order 33472.

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×#22552640C211yesmaximal, characteristic, complemented
2C22×C22⋊D523C211yesmaximal, characteristic, complemented
3#30660369C212yesmaximal, complemented
4#30660359C212yesmaximal, complemented
5#30660346C212yesmaximal, complemented
6#30660345C212yesmaximal, complemented
7#30660343C211yesmaximal, characteristic, complemented
8#30660342C211yesmaximal, characteristic, complemented
9#30660341C211yesmaximal, characteristic, complemented
10#30660329C211yesmaximal, characteristic, complemented
11#120×C523C211yesmaximal, characteristic, complemented
12C23×D523C2211yescharacteristic
13#22552640—24no—
14C2×Dic1046C2211yescharacteristic, complemented
15C2×D2092C2212yescomplemented
16C2×C22⋊D523—24no—
17C2×C22⋊D523C2212yes—
18C2×C22⋊D523C2212yes—
19C2×C22⋊D523C2211yescharacteristic
20C2×C22⋊D523C2211yescharacteristic, complemented
21C2×C22⋊D523C2212yescomplemented
22C2×C22⋊D523C2212yescomplemented
23C22×Dic523C2211yescharacteristic
24C22×Dic523C2211yescharacteristic, complemented
25C2×C4×D523C2211yescharacteristic, complemented
26Dic523⋊C4C2212yescomplemented
27C22⋊Dic523C2211yescharacteristic
28C22⋊Dic523C2212yescomplemented
29(C2×D523)⋊C4C2212yescomplemented
30(C2×D523)⋊C4C2212yescomplemented
31C4×Dic523C2212yescomplemented
32C24×C523C2211yescharacteristic
33C2×D4×C523C2211yescharacteristic
34C2×D4×C523C2212yescomplemented
35C22⋊C4×C523C2211yescharacteristic
36C22⋊C4×C523C2212yescomplemented
37C22×D523—22no—
38C22×D523—42no—
39C22×D523C2312yes—
40C22×D523C2311yescharacteristic
41Dic1046—22no—
42D2092—42no—
43C22⋊D523—22no—
44C22⋊D523—24no—
45C22⋊D523—24no—
46C22⋊D523—22no—
47C22⋊D523—42no—
48C22⋊D523—42no—
49C2×Dic523—22no—
50C2×Dic523D414yescomplemented
51C2×Dic523C2312yes—
52C2×Dic523C2311yescharacteristic
53C2×Dic523C2311yescharacteristic
54C2×Dic523C2311yescharacteristic
55C4×D523—22no—
56C23×C523—22no—
57C23×C523—42no—
58C23×C523C2312yes—
59C23×C523C2311yescharacteristic
60C23×C523C2311yescharacteristic
61D4×C523—22no—
62D4×C523—42no—
63C2×C4×C523C2312yes—
64C2×C4×C523C2311yescharacteristic
65C22×C523—22no—
66C22×C523—21no—
67C22×C523—22no—
68C22×C523—22no—
69C22×C523—22no—
70C22×C523—22no—
71C22×C523—44no—
72C22×C523C2×D412yes—
73C22×C523C2411yescharacteristic
74C2×D523—21no—
75C2×D523—22no—
76C2×D523—22no—
77C2×D523—22no—
78C2×D523—22no—
79C2×D523—44no—
80C4×C523—21no—
81C4×C523—22no—
82Dic523—21no—
83Dic523—22no—
84Dic523C2×D412yes—
85Dic523C2×D412yes—
86C2×C523—22no—
87C2×C523—41no—
88C2×C523—42no—
89C2×C523C22×D411yescharacteristic
90C2×C523H(5,𝔽2)12yes—
91D523—41no—
92D523—42no—
93C523#53511yescharacteristic, complemented, Sylow
94#535—5231nomaximal, Sylow
95C2×#41—5231no—
96C22×D4—5231no—
97#127—5232no—
98#124—5232no—
99#121—5232no—
100#121—5232no—
101#120—5231no—
102#120—5231no—
103#120—5231no—
104#120D52311yescharacteristic, complemented
105#117—5231no—
106C24—5231no—
107C24C2×D52311yescharacteristic
108#41—10464no—
109C2×Q8—5231no—
110C2×D4—5231no—
111C2×D4—5232no—
112C2×D4—5232no—
113C2×D4—5231no—
114C2×D4—5232no—
115C2×D4—5232no—
116C2×D4—5232no—
117C2×D4—10464no—
118C2×D4C2×D52311yescharacteristic
119C2×D4C2×D52312yescomplemented
120C22×C4—5231no—
121C22×C4—5231no—
122C22×C4—5231no—
123C4⋊C4—5232no—
124C22⋊C4—5231no—
125C22⋊C4—5232no—
126C22⋊C4—5232no—
127C22⋊C4—5232no—
128C22⋊C4C2×D52311yescharacteristic
129C22⋊C4C2×D52312yescomplemented
130C42—5232no—
131C23—22no—
132C23—42no—
133C23—5232no—
134C23—5231no—
135C23—10462no—
136C23—20922no—
137C23C22×D52312yes—
138C23C22×D52311yescharacteristic
139C23C22×D52311yescharacteristic
140Q8—10462no—
141D4—22no—
142D4—42no—
143D4—10464no—
144D4—10464no—
145D4—10462no—
146D4—10462no—
147D4—20922no—
148D4—20922no—
149D4—20922no—
150C2×C4—5234no—
151C2×C4—5231no—
152C2×C4—5231no—
153C2×C4—5232no—
154C2×C4—5231no—
155C2×C4—10462no—
156C2×C4—10462no—
157C2×C4C22×D52312yes—
158C2×C4C22×D52311yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—10461no—
167C22—10462no—
168C22—10462no—
169C22—10462no—
170C22—10462no—
171C22—20924no—
172C22D4×D52312yes—
173C22C23×D52311yescharacteristic
174C4—21no—
175C4—22no—
176C4—5232no—
177C4—5232no—
178C4—10461no—
179C4—10462no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—20922no—
184C2—20921no—
185C2C2×D4×D52311yescharacteristic
186C2#3066041812yes—
subgroupG/H
center, FrattiniC22C23×D523
commutator, socle, cosocleC22×C523C24
Fitting#120×C523C2
hypercenter#120D523
hypocenterC523#535
LCSG ▷ C22×C523 ▷ C523
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#22552640C2semidirect
2. C22×C22⋊D523C2semidirect
3. #30660369C2semidirect
4. #30660359C2semidirect
5. #30660346C2semidirect
6. #30660345C2semidirect
7. #30660343C2semidirect
8. #30660342C2semidirect
9. #30660341C2semidirect
10. #30660329C2semidirect
11. #120×C523C2semidirect
14. C2×Dic1046C22semidirect
15. C2×D2092C22semidirect
20. C2×C22⋊D523C22semidirect
21. C2×C22⋊D523C22semidirect
22. C2×C22⋊D523C22semidirect
24. C22×Dic523C22semidirect
25. C2×C4×D523C22semidirect
26. Dic523⋊C4C22semidirect
28. C22⋊Dic523C22semidirect
29. (C2×D523)⋊C4C22semidirect
30. (C2×D523)⋊C4C22semidirect
31. C4×Dic523C22semidirect
34. C2×D4×C523C22semidirect
36. C22⋊C4×C523C22semidirect
50. C2×Dic523D4semidirect
93. C523#535semidirect
104. #120D523semidirect
119. C2×D4C2×D523semidirect
129. C22⋊C4C2×D523semidirect

Maximal subgroups

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

subgroupindex
C2×#225526402
C22×C22⋊D5232
#306603692
#306603592
#306603462
#306603452
#306603432
#306603422
#306603412
#306603292
#120×C5232
#535523