Finite Group Databasebeta

← #46830815

Subgroups — #46830815

Order 34624.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#22575836C211yesmaximal, characteristic, complemented
2C22×C22⋊D541C211yesmaximal, characteristic, complemented
3#31772614C211yesmaximal, characteristic, complemented
4#31772611C211yesmaximal, characteristic, complemented
5#31772610C211yesmaximal, characteristic, complemented
6#31772605C211yesmaximal, characteristic, complemented
7#31772602C211yesmaximal, characteristic, complemented
8#31772601C211yesmaximal, characteristic, complemented
9#31772601C211yesmaximal, characteristic, complemented
10#31772598C211yesmaximal, characteristic, complemented
11#31772597C211yesmaximal, characteristic, complemented
12C4×C22⋊D541C211yesmaximal, characteristic, complemented
13#31772588C211yesmaximal, characteristic, complemented
14C22⋊C4×D541C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C541C211yesmaximal, characteristic, complemented
16C23×D541C2211yescharacteristic
17#22575836—24no—
18C2×Dic1082C2211yescharacteristic, complemented
19C2×D2164C2211yescharacteristic, complemented
20C2×D2164C2211yescharacteristic, complemented
21C2×C22⋊D541—24no—
22C2×C22⋊D541C2211yescharacteristic
23C2×C22⋊D541C2211yescharacteristic
24C2×C22⋊D541C2211yescharacteristic
25C2×C22⋊D541C2211yescharacteristic
26C2×C22⋊D541C2211yescharacteristic
27C2×C22⋊D541C2211yescharacteristic, complemented
28C2×C22⋊D541C2211yescharacteristic, complemented
29C22×Dic541C2211yescharacteristic
30C2×C4×D541C2211yescharacteristic, complemented
31C2×C4×D541C2211yescharacteristic, complemented
32C2×C4×D541C2211yescharacteristic, complemented
33C4⋊Dic541C2211yescharacteristic, complemented
34#22575815C2211yescharacteristic, complemented
35#22575815C2211yescharacteristic, complemented
36#22575815C2211yescharacteristic, complemented
37C22⋊Dic541C2211yescharacteristic
38C22⋊Dic541C2211yescharacteristic, complemented
39C22⋊Dic541C2211yescharacteristic, complemented
40#22575813C2211yescharacteristic, complemented
41#22575813C2211yescharacteristic, complemented
42#22575813C2211yescharacteristic, complemented
43#22575813C2211yescharacteristic, complemented
44#22575813C2211yescharacteristic, complemented
45C4×Dic541C2211yescharacteristic, complemented
46C24×C541C2211yescharacteristic
47C22×C4×C541C2211yescharacteristic, complemented
48C22×C4×C541C2211yescharacteristic, complemented
49C22⋊C4×C541C2211yescharacteristic
50C22⋊C4×C541C2211yescharacteristic
51C22⋊C4×C541C2211yescharacteristic, complemented
52C22⋊C4×C541C2211yescharacteristic, complemented
53C22×D541—22no—
54C22×D541—22no—
55C22×D541—41no—
56C22×D541C2311yescharacteristic
57C22×D541C2311yescharacteristic
58C22×D541C2311yescharacteristic
59Dic1082—22no—
60D2164—22no—
61D2164—41no—
62C22⋊D541—22no—
63C22⋊D541—22no—
64C22⋊D541—22no—
65C22⋊D541—22no—
66C22⋊D541—22no—
67C22⋊D541—41no—
68C22⋊D541D414yescomplemented
69C2×Dic541—22no—
70C2×Dic541C2311yescharacteristic
71C2×Dic541C2311yescharacteristic
72C2×Dic541C2311yescharacteristic
73C2×Dic541C2311yescharacteristic
74C2×Dic541C2311yescharacteristic
75C4×D541—22no—
76C4×D541—22no—
77C4×D541—41no—
78C23×C541—22no—
79C23×C541—22no—
80C23×C541—41no—
81C23×C541C2311yescharacteristic
82C23×C541C2311yescharacteristic
83C23×C541C2311yescharacteristic
84C2×C4×C541—22no—
85C2×C4×C541—41no—
86C2×C4×C541C2311yescharacteristic
87C2×C4×C541C2311yescharacteristic
88C2×C4×C541C2311yescharacteristic
89C2×C4×C541C2311yescharacteristic
90C22×C541—21no—
91C22×C541—21no—
92C22×C541—21no—
93C22×C541—21no—
94C22×C541—21no—
95C22×C541—21no—
96C22×C541—21no—
97C22×C541—44no—
98C22×C541C2×D412yes—
99C22×C541#4112yescomplemented
100C22×C541C2411yescharacteristic
101C2×D541—21no—
102C2×D541—21no—
103C2×D541—21no—
104C2×D541—21no—
105C2×D541—21no—
106C2×D541—21no—
107C2×D541—21no—
108C2×D541—21no—
109C2×D541—44no—
110C2×D541C2×D412yes—
111C4×C541—21no—
112C4×C541—21no—
113C4×C541—21no—
114C4×C541—21no—
115Dic541—21no—
116Dic541—21no—
117Dic541—21no—
118Dic541—21no—
119Dic541C2×D412yes—
120C2×C541—22no—
121C2×C541—22no—
122C2×C541—41no—
123C2×C541C22×D411yescharacteristic
124C2×C541C2×#4111yescharacteristic
125C2×C541H(5,𝔽2)11yescharacteristic
126D541—22no—
127D541—41no—
128D541—41no—
129C541#54611yescharacteristic, complemented, Sylow
130#546—5411nomaximal, Sylow
131C2×#41—5411no—
132C22×D4—5411no—
133#124—5411no—
134#123—5411no—
135#123—5411no—
136C22⋊Q8—5411no—
137#121—5411no—
138#121—5411no—
139#121—5411no—
140#120—5411no—
141#120—5411no—
142C4×D4—5411no—
143C4×D4—5411no—
144C2×C22⋊C4—5411no—
145C2×C22⋊C4D54111yescharacteristic, complemented
146C24—5411no—
147C24C2×D54111yescharacteristic
148#41—10824no—
149C2×Q8—5411no—
150C2×D4—5411no—
151C2×D4—5411no—
152C2×D4—5411no—
153C2×D4—5411no—
154C2×D4—5411no—
155C2×D4—5411no—
156C2×D4—5411no—
157C2×D4—5411no—
158C2×D4—5411no—
159C2×D4—10824no—
160C22×C4—5411no—
161C22×C4—5411no—
162C22×C4—5411no—
163C22×C4—5411no—
164C22×C4C2×D54111yescharacteristic, complemented
165C22×C4C2×D54111yescharacteristic, complemented
166C4⋊C4—5411no—
167C4⋊C4—5411no—
168C4⋊C4—5411no—
169C4⋊C4—5411no—
170C22⋊C4—5411no—
171C22⋊C4—5411no—
172C22⋊C4—5411no—
173C22⋊C4—5411no—
174C22⋊C4—5411no—
175C22⋊C4—5411no—
176C22⋊C4—5411no—
177C22⋊C4—5411no—
178C22⋊C4C2×D54111yescharacteristic
179C22⋊C4C2×D54111yescharacteristic
180C22⋊C4C2×D54111yescharacteristic, complemented
181C22⋊C4C2×D54111yescharacteristic, complemented
182C42—5411no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—5411no—
187C23—5411no—
188C23—5411no—
189C23—10822no—
190C23—10822no—
191C23—21641no—
192C23C22×D54111yescharacteristic
193C23C22×D54111yescharacteristic
194C23C22×D54111yescharacteristic
195Q8—10822no—
196D4—5414no—
197D4—10822no—
198D4—10822no—
199D4—10822no—
200D4—10822no—
201D4—10822no—
202D4—10822no—
203D4—21641no—
204D4—21641no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—5411no—
208C2×C4—5411no—
209C2×C4—5411no—
210C2×C4—5411no—
211C2×C4—5411no—
212C2×C4—10822no—
213C2×C4—10822no—
214C2×C4—10822no—
215C2×C4—21641no—
216C2×C4C22×D54111yescharacteristic
217C2×C4C22×D54111yescharacteristic
218C2×C4C22×D54111yescharacteristic
219C2×C4C22×D54111yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—5412no—
229C22—10821no—
230C22—10821no—
231C22—10821no—
232C22—10821no—
233C22—10821no—
234C22—10821no—
235C22—10821no—
236C22—10821no—
237C22—21644no—
238C22D4×D54112yes—
239C22#2257583612yescomplemented
240C22C23×D54111yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—5412no—
246C4—10821no—
247C4—10821no—
248C4—10821no—
249C4—10821no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—10822no—
254C2—21641no—
255C2—21641no—
256C2C2×D4×D54111yescharacteristic
257C2C2×#2257583611yescharacteristic
258C2#3177267311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D541
commutator, socle, cosocleC22×C541C24
FittingC2×C22⋊C4×C541C2
hypercenterC2×C22⋊C4D541
hypocenterC541#546
LCSG ▷ C22×C541 ▷ C541
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#22575836C2semidirect
2. C22×C22⋊D541C2semidirect
3. #31772614C2semidirect
4. #31772611C2semidirect
5. #31772610C2semidirect
6. #31772605C2semidirect
7. #31772602C2semidirect
8. #31772601C2semidirect
9. #31772601C2semidirect
10. #31772598C2semidirect
11. #31772597C2semidirect
12. C4×C22⋊D541C2semidirect
13. #31772588C2semidirect
14. C22⋊C4×D541C2semidirect
15. C2×C22⋊C4×C541C2semidirect
18. C2×Dic1082C22semidirect
19. C2×D2164C22semidirect
20. C2×D2164C22semidirect
27. C2×C22⋊D541C22semidirect
28. C2×C22⋊D541C22semidirect
30. C2×C4×D541C22semidirect
31. C2×C4×D541C22semidirect
32. C2×C4×D541C22semidirect
33. C4⋊Dic541C22semidirect
34. #22575815C22semidirect
35. #22575815C22semidirect
36. #22575815C22semidirect
38. C22⋊Dic541C22semidirect
39. C22⋊Dic541C22semidirect
40. #22575813C22semidirect
41. #22575813C22semidirect
42. #22575813C22semidirect
43. #22575813C22semidirect
44. #22575813C22semidirect
45. C4×Dic541C22semidirect
47. C22×C4×C541C22semidirect
48. C22×C4×C541C22semidirect
51. C22⋊C4×C541C22semidirect
52. C22⋊C4×C541C22semidirect
68. C22⋊D541D4semidirect
99. C22×C541#41semidirect
129. C541#546semidirect
145. C2×C22⋊C4D541semidirect
164. C22×C4C2×D541semidirect
165. C22×C4C2×D541semidirect
180. C22⋊C4C2×D541semidirect
181. C22⋊C4C2×D541semidirect
239. C22#22575836semidirect

Maximal subgroups

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

subgroupindex
C2×#225758362
C22×C22⋊D5412
#317726142
#317726112
#317726102
#317726052
#317726022
#317726012
#317725982
#317725972
C4×C22⋊D5412
#317725882
C22⋊C4×D5412
C2×C22⋊C4×C5412
#546541