Finite Group Databasebeta

← #46830735

Subgroups — #46830735

Order 34624.

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

normal subgroups only

This group has 346 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
1C22×C22⋊D541C211yesmaximal, characteristic, complemented
2C2×D4×D541C211yesmaximal, characteristic, complemented
3#31772612C211yesmaximal, characteristic, complemented
4#31772610C212yesmaximal, complemented
5#31772609C211yesmaximal, characteristic, complemented
6#31772601C212yesmaximal, complemented
7#31772599C212yesmaximal, complemented
8#31772597C211yesmaximal, characteristic, complemented
9#31772596C212yesmaximal, complemented
10C22⋊C4×D541C211yesmaximal, characteristic, complemented
11#120×C541C211yesmaximal, characteristic, complemented
12C23×D541C2211yescharacteristic
13C23×D541C2211yescharacteristic, complemented
14C2×D2164C2211yescharacteristic, complemented
15C2×C22⋊D541—24no—
16C2×C22⋊D541C2212yes—
17C2×C22⋊D541C2211yescharacteristic
18C2×C22⋊D541C2211yescharacteristic, complemented
19C2×C22⋊D541C2212yescomplemented
20C2×C22⋊D541C2212yescomplemented
21D4×D541—24no—
22C22×Dic541C2211yescharacteristic, complemented
23C2×C4×D541C2211yescharacteristic, complemented
24C2×C4×D541C2212yescomplemented
25C4⋊Dic541C2212yescomplemented
26#22575815C2212yescomplemented
27C22⋊Dic541C2211yescharacteristic, complemented
28C22⋊Dic541C2212yescomplemented
29C22⋊Dic541C2212yescomplemented
30#22575813C2212yescomplemented
31#22575813C2212yescomplemented
32C24×C541C2211yescharacteristic, complemented
33C2×D4×C541C2211yescharacteristic
34C2×D4×C541C2212yescomplemented
35C22⋊C4×C541C2211yescharacteristic, complemented
36C22⋊C4×C541C2212yescomplemented
37C22×D541—22no—
38C22×D541—42no—
39C22×D541—42no—
40C22×D541D414yescomplemented
41C22×D541C2311yescharacteristic
42C22×D541C2311yescharacteristic
43C22×D541C2311yescharacteristic
44D2164—22no—
45C22⋊D541—22no—
46C22⋊D541—22no—
47C22⋊D541—24no—
48C22⋊D541—24no—
49C22⋊D541—42no—
50C2×Dic541—22no—
51C2×Dic541C2312yes—
52C2×Dic541C2312yes—
53C2×Dic541C2311yescharacteristic
54C4×D541—22no—
55C4×D541—42no—
56C23×C541—22no—
57C23×C541—42no—
58C23×C541C2312yes—
59C23×C541C2311yescharacteristic
60C23×C541C2311yescharacteristic
61D4×C541—22no—
62D4×C541—42no—
63C2×C4×C541C2311yescharacteristic
64C2×C4×C541C2312yes—
65C22×C541—22no—
66C22×C541—22no—
67C22×C541—21no—
68C22×C541—22no—
69C22×C541—22no—
70C22×C541—22no—
71C22×C541—44no—
72C22×C541C2×D412yes—
73C22×C541C2411yescharacteristic
74C2×D541—22no—
75C2×D541—22no—
76C2×D541—21no—
77C2×D541—22no—
78C2×D541—44no—
79C2×D541—44no—
80C2×D541C2×D412yes—
81C2×D541C2×D412yes—
82C4×C541—21no—
83C4×C541—22no—
84Dic541—21no—
85Dic541—22no—
86Dic541—22no—
87C2×C541—22no—
88C2×C541—41no—
89C2×C541—42no—
90C2×C541C22×D411yescharacteristic
91C2×C541H(5,𝔽2)12yes—
92D541—22no—
93D541—22no—
94D541—41no—
95C541#53411yescharacteristic, complemented, Sylow
96#534—5411nomaximal, Sylow
97C22×D4—5411no—
98C22×D4—5411no—
99#123—5411no—
100#123—5412no—
101#123—5411no—
102#121—5412no—
103#121—5412no—
104#120—5411no—
105#120—5412no—
106#120D54111yescharacteristic, complemented
107C2×C22⋊C4—5411no—
108C24—5411no—
109C24—5411no—
110C24C2×D54111yescharacteristic, complemented
111C2×D4—5412no—
112C2×D4—5411no—
113C2×D4—5411no—
114C2×D4—5412no—
115C2×D4—5412no—
116C2×D4—5411no—
117C2×D4—10824no—
118C2×D4—10824no—
119C2×D4C2×D54111yescharacteristic
120C2×D4C2×D54112yescomplemented
121C22×C4—5411no—
122C22×C4—5411no—
123C22×C4—5412no—
124C4⋊C4—5412no—
125C4⋊C4—5412no—
126C22⋊C4—5411no—
127C22⋊C4—5412no—
128C22⋊C4—5412no—
129C22⋊C4—5412no—
130C22⋊C4—5412no—
131C22⋊C4C2×D54111yescharacteristic, complemented
132C22⋊C4C2×D54112yescomplemented
133C23—22no—
134C23—42no—
135C23—5411no—
136C23—5411no—
137C23—5414no—
138C23—5411no—
139C23—10822no—
140C23—21642no—
141C23—21642no—
142C23C22×D54111yescharacteristic
143C23C22×D54112yes—
144C23C22×D54111yescharacteristic
145D4—22no—
146D4—42no—
147D4—10822no—
148D4—10822no—
149D4—10824no—
150D4—10824no—
151D4—10822no—
152D4—21642no—
153C2×C4—5412no—
154C2×C4—5412no—
155C2×C4—5411no—
156C2×C4—10822no—
157C2×C4—10822no—
158C2×C4—21642no—
159C2×C4C22×D54112yes—
160C2×C4C22×D54111yescharacteristic
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—21no—
166C22—22no—
167C22—44no—
168C22—5412no—
169C22—5412no—
170C22—10822no—
171C22—10822no—
172C22—10822no—
173C22—10821no—
174C22—21644no—
175C22—21644no—
176C22D4×D54112yes—
177C22C23×D54111yescharacteristic
178C4—22no—
179C4—21no—
180C4—10822no—
181C4—10821no—
182C4—10822no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—10822no—
187C2—10822no—
188C2—21641no—
189C2C2×D4×D54111yescharacteristic
190C2#3177267312yes—
subgroupG/H
center, FrattiniC22C23×D541
commutator, socle, cosocleC22×C541C24
Fitting#120×C541C2
hypercenter#120D541
hypocenterC541#534
LCSG ▷ C22×C541 ▷ C541
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D541C2semidirect
2. C2×D4×D541C2semidirect
3. #31772612C2semidirect
4. #31772610C2semidirect
5. #31772609C2semidirect
6. #31772601C2semidirect
7. #31772599C2semidirect
8. #31772597C2semidirect
9. #31772596C2semidirect
10. C22⋊C4×D541C2semidirect
11. #120×C541C2semidirect
13. C23×D541C22semidirect
14. C2×D2164C22semidirect
18. C2×C22⋊D541C22semidirect
19. C2×C22⋊D541C22semidirect
20. C2×C22⋊D541C22semidirect
22. C22×Dic541C22semidirect
23. C2×C4×D541C22semidirect
24. C2×C4×D541C22semidirect
25. C4⋊Dic541C22semidirect
26. #22575815C22semidirect
27. C22⋊Dic541C22semidirect
28. C22⋊Dic541C22semidirect
29. C22⋊Dic541C22semidirect
30. #22575813C22semidirect
31. #22575813C22semidirect
32. C24×C541C22semidirect
34. C2×D4×C541C22semidirect
35. C22⋊C4×C541C22semidirect
36. C22⋊C4×C541C22semidirect
40. C22×D541D4semidirect
95. C541#534semidirect
106. #120D541semidirect
110. C24C2×D541semidirect
120. C2×D4C2×D541semidirect
131. C22⋊C4C2×D541semidirect
132. C22⋊C4C2×D541semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D5412
C2×D4×D5412
#317726122
#317726102
#317726092
#317726012
#317725992
#317725972
#317725962
C22⋊C4×D5412
#120×C5412
#534541