Finite Group Databasebeta

← #46830759

Subgroups — #46830759

Order 34624.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#31772622C211yesmaximal, characteristic, complemented
2#31772621C211yesmaximal, characteristic, complemented
3#31772617C211yesmaximal, characteristic, complemented
4#31772612C211yesmaximal, characteristic, complemented
5#31772610C211yesmaximal, characteristic, complemented
6#31772604C211yesmaximal, characteristic, complemented
7#31772601C211yesmaximal, characteristic, complemented
8#31772601C211yesmaximal, characteristic, complemented
9#31772600C211yesmaximal, characteristic, complemented
10#31772598C211yesmaximal, characteristic, complemented
11#31772596C211yesmaximal, characteristic, complemented
12C4×D2164C211yesmaximal, characteristic, complemented
13#31772588C211yesmaximal, characteristic, complemented
14C22⋊C4×D541C211yesmaximal, characteristic, complemented
15#124×C541C211yesmaximal, characteristic, complemented
16C23×D541C2211yescharacteristic
17C2×D2164C2211yescharacteristic
18C2×D2164C2211yescharacteristic
19C2×C22⋊D541C2211yescharacteristic
20C2×C22⋊D541C2211yescharacteristic
21C2×C22⋊D541C2211yescharacteristic
22C2×C22⋊D541C2211yescharacteristic, complemented
23C22×Dic541C2211yescharacteristic, complemented
24C2×C4×D541C2211yescharacteristic
25C2×C4×D541C2211yescharacteristic
26C2×C4×D541C2211yescharacteristic, complemented
27C4⋊Dic541C2211yescharacteristic, complemented
28C4⋊Dic541C2211yescharacteristic, complemented
29#22575815C2211yescharacteristic, complemented
30#22575815C2211yescharacteristic, complemented
31#22575815C2211yescharacteristic, complemented
32#22575815C2211yescharacteristic, complemented
33C22⋊Dic541C2211yescharacteristic
34C22⋊Dic541C2211yescharacteristic
35#22575813C2211yescharacteristic
36#22575813C2211yescharacteristic
37#22575813C2211yescharacteristic
38#22575813C2211yescharacteristic
39#22575813C2211yescharacteristic
40#22575813C2211yescharacteristic
41#22575813C2211yescharacteristic, complemented
42#22575813C2211yescharacteristic, complemented
43C4×Dic541C2211yescharacteristic, complemented
44C2×Q8×C541C2211yescharacteristic, complemented
45C2×D4×C541C2211yescharacteristic
46C22⋊C4×C541C2211yescharacteristic
47C22⋊C4×C541C2211yescharacteristic
48C22⋊C4×C541C2211yescharacteristic, complemented
49C22⋊C4×C541C2211yescharacteristic, complemented
50C42×C541C2211yescharacteristic, complemented
51C22×D541—22no—
52C22×D541—41no—
53C22×D541—41no—
54C22×D541C2311yescharacteristic
55C22×D541C2311yescharacteristic
56C22×D541C2311yescharacteristic
57D2164—22no—
58D2164—41no—
59C22⋊D541—22no—
60C22⋊D541—41no—
61C22⋊D541—41no—
62C22⋊D541—41no—
63C2×Dic541—41no—
64C2×Dic541C2311yescharacteristic
65C2×Dic541C2311yescharacteristic
66C2×Dic541C2311yescharacteristic
67C2×Dic541C2311yescharacteristic
68C2×Dic541C2311yescharacteristic
69C4×D541—41no—
70C4×D541—41no—
71C4×D541—41no—
72C23×C541C2311yescharacteristic
73C23×C541C2311yescharacteristic
74Q8×C541—41no—
75D4×C541—41no—
76C2×C4×C541C2311yescharacteristic
77C2×C4×C541C2311yescharacteristic
78C2×C4×C541C2311yescharacteristic
79C2×C4×C541C2311yescharacteristic
80C2×C4×C541C2311yescharacteristic
81C22×C541—21no—
82C22×C541—21no—
83C22×C541—21no—
84C22×C541—21no—
85C22×C541—21no—
86C22×C541—21no—
87C22×C541C2411yescharacteristic
88C2×D541—21no—
89C2×D541—21no—
90C2×D541—21no—
91C2×D541—21no—
92C2×D541—21no—
93C2×D541—21no—
94C2×D541—21no—
95C2×D541—21no—
96C2×D541—44no—
97C2×D541#4112yes—
98C4×C541—21no—
99C4×C541—21no—
100C4×C541—21no—
101C4×C541—21no—
102C4×C541—21no—
103Dic541—21no—
104Dic541—21no—
105Dic541—21no—
106Dic541—21no—
107Dic541—21no—
108C2×C541—41no—
109C2×C541—41no—
110C2×C541C2×#4111yescharacteristic
111C2×C541H(5,𝔽2)11yescharacteristic
112C2×C541H(5,𝔽2)11yescharacteristic
113D541—22no—
114D541—41no—
115D541—41no—
116C541#53811yescharacteristic, complemented, Sylow
117#538—5411nomaximal, Sylow
118#126—5411no—
119#126—5411no—
120#124—5411no—
121#124D54111yescharacteristic, complemented
122#123—5411no—
123#123—5411no—
124C22⋊Q8—5411no—
125#121—5411no—
126#121—5411no—
127#121—5411no—
128#120—5411no—
129#120—5411no—
130C4×D4—5411no—
131C4×D4—5411no—
132C2×C22⋊C4—5411no—
133C24—5411no—
134C2×Q8C2×D54111yescharacteristic, complemented
135C2×D4—5411no—
136C2×D4—5411no—
137C2×D4—5411no—
138C2×D4—5411no—
139C2×D4—5411no—
140C2×D4—5411no—
141C2×D4C2×D54111yescharacteristic
142C22×C4—5411no—
143C22×C4—5411no—
144C22×C4—5411no—
145C22×C4—5411no—
146C4⋊C4—5411no—
147C4⋊C4—5411no—
148C4⋊C4—5411no—
149C4⋊C4—5411no—
150C4⋊C4—5411no—
151C4⋊C4—5411no—
152C22⋊C4—5411no—
153C22⋊C4—5411no—
154C22⋊C4—5411no—
155C22⋊C4—5411no—
156C22⋊C4—5411no—
157C22⋊C4—5411no—
158C22⋊C4—5411no—
159C22⋊C4—5411no—
160C22⋊C4—5411no—
161C22⋊C4—5411no—
162C22⋊C4C2×D54111yescharacteristic
163C22⋊C4C2×D54111yescharacteristic
164C22⋊C4C2×D54111yescharacteristic, complemented
165C22⋊C4C2×D54111yescharacteristic, complemented
166C42—5411no—
167C42C2×D54111yescharacteristic, complemented
168C23—5411no—
169C23—5411no—
170C23—5411no—
171C23—10822no—
172C23—21641no—
173C23—21641no—
174C23C22×D54111yescharacteristic
175C23C22×D54111yescharacteristic
176Q8—41no—
177D4—41no—
178D4—10822no—
179D4—10822no—
180D4—21641no—
181D4—21641no—
182D4—21641no—
183D4—21641no—
184C2×C4—5411no—
185C2×C4—5411no—
186C2×C4—5411no—
187C2×C4—5411no—
188C2×C4—5411no—
189C2×C4—21641no—
190C2×C4—21641no—
191C2×C4—21641no—
192C2×C4—21641no—
193C2×C4C22×D54111yescharacteristic
194C2×C4C22×D54111yescharacteristic
195C2×C4C22×D54111yescharacteristic
196C2×C4C22×D54111yescharacteristic
197C2×C4C22×D54111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—5412no—
205C22—10821no—
206C22—10821no—
207C22—10821no—
208C22—10821no—
209C22—10821no—
210C22—10821no—
211C22—10821no—
212C22—10821no—
213C22—21644no—
214C22C23×D54111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—10821no—
221C4—10821no—
222C4—10821no—
223C4—10821no—
224C4—10821no—
225C2—41no—
226C2—41no—
227C2—10822no—
228C2—21641no—
229C2—21641no—
230C2#41×D54111yescharacteristic
231C2#3177267311yescharacteristic
232C2#3177267411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D541
commutator, socle, cosocleC22×C541C24
Fitting#124×C541C2
hypercenter#124D541
hypocenterC541#538
LCSG ▷ C22×C541 ▷ C541
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #31772622C2semidirect
2. #31772621C2semidirect
3. #31772617C2semidirect
4. #31772612C2semidirect
5. #31772610C2semidirect
6. #31772604C2semidirect
7. #31772601C2semidirect
8. #31772601C2semidirect
9. #31772600C2semidirect
10. #31772598C2semidirect
11. #31772596C2semidirect
12. C4×D2164C2semidirect
13. #31772588C2semidirect
14. C22⋊C4×D541C2semidirect
15. #124×C541C2semidirect
22. C2×C22⋊D541C22semidirect
23. C22×Dic541C22semidirect
26. C2×C4×D541C22semidirect
27. C4⋊Dic541C22semidirect
28. C4⋊Dic541C22semidirect
29. #22575815C22semidirect
30. #22575815C22semidirect
31. #22575815C22semidirect
32. #22575815C22semidirect
41. #22575813C22semidirect
42. #22575813C22semidirect
43. C4×Dic541C22semidirect
44. C2×Q8×C541C22semidirect
48. C22⋊C4×C541C22semidirect
49. C22⋊C4×C541C22semidirect
50. C42×C541C22semidirect
116. C541#538semidirect
121. #124D541semidirect
134. C2×Q8C2×D541semidirect
164. C22⋊C4C2×D541semidirect
165. C22⋊C4C2×D541semidirect
167. C42C2×D541semidirect

Maximal subgroups

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

subgroupindex
#317726222
#317726212
#317726172
#317726122
#317726102
#317726042
#317726012
#317726002
#317725982
#317725962
C4×D21642
#317725882
C22⋊C4×D5412
#124×C5412
#538541