Finite Group Databasebeta

← #27135841

Subgroups — #27135841

Order 12352.

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#20259894C211yesmaximal, characteristic, complemented
2#20259892C211yesmaximal, characteristic, complemented
3#20259886C211yesmaximal, characteristic, complemented
4#20259885C211yesmaximal, characteristic, complemented
5#20259882C211yesmaximal, characteristic, complemented
6#20259880C211yesmaximal, characteristic, complemented
7#20259877C211yesmaximal, characteristic, complemented
8#20259871C211yesmaximal, characteristic, complemented
9#20259870C211yesmaximal, characteristic, complemented
10#20259867C211yesmaximal, characteristic, complemented
11#20259867C211yesmaximal, characteristic, complemented
12C4×C22⋊D193C211yesmaximal, characteristic, complemented
13#20259859C211yesmaximal, characteristic, complemented
14C22⋊C4×D193C211yesmaximal, characteristic, complemented
15#121×C193C211yesmaximal, characteristic, complemented
16C23×D193C2211yescharacteristic
17C2×Dic386C2211yescharacteristic, complemented
18C2×C22⋊D193C2211yescharacteristic
19C2×C22⋊D193C2211yescharacteristic
20C2×C22⋊D193C2211yescharacteristic
21C2×C22⋊D193C2211yescharacteristic
22C22×Dic193C2211yescharacteristic, complemented
23C2×C4×D193C2211yescharacteristic
24C2×C4×D193C2211yescharacteristic
25C4⋊Dic193C2211yescharacteristic, complemented
26#15690534C2211yescharacteristic, complemented
27#15690534C2211yescharacteristic, complemented
28#15690534C2211yescharacteristic, complemented
29#15690534C2211yescharacteristic, complemented
30C22⋊Dic193C2211yescharacteristic
31C22⋊Dic193C2211yescharacteristic
32C22⋊Dic193C2211yescharacteristic
33C22⋊Dic193C2211yescharacteristic, complemented
34C22⋊Dic193C2211yescharacteristic, complemented
35C22⋊Dic193C2211yescharacteristic, complemented
36#15690532C2211yescharacteristic
37#15690532C2211yescharacteristic
38#15690532C2211yescharacteristic
39#15690532C2211yescharacteristic
40#15690532C2211yescharacteristic
41#15690532C2211yescharacteristic
42C4×Dic193C2211yescharacteristic, complemented
43C4×Dic193C2211yescharacteristic, complemented
44C2×D4×C193C2211yescharacteristic
45C2×D4×C193C2211yescharacteristic
46C2×D4×C193C2211yescharacteristic, complemented
47C22×C4×C193C2211yescharacteristic, complemented
48C4⋊C4×C193C2211yescharacteristic, complemented
49C22⋊C4×C193C2211yescharacteristic
50C22⋊C4×C193C2211yescharacteristic, complemented
51C22×D193—22no—
52C22×D193—41no—
53C22×D193—41no—
54C22×D193C2311yescharacteristic
55C22×D193C2311yescharacteristic
56Dic386—41no—
57C22⋊D193—22no—
58C22⋊D193—22no—
59C22⋊D193—41no—
60C22⋊D193—41no—
61C2×Dic193—41no—
62C2×Dic193C2311yescharacteristic
63C2×Dic193C2311yescharacteristic
64C2×Dic193C2311yescharacteristic
65C2×Dic193C2311yescharacteristic
66C2×Dic193C2311yescharacteristic
67C2×Dic193C2311yescharacteristic
68C4×D193—41no—
69C4×D193—41no—
70C23×C193C2311yescharacteristic
71C23×C193C2311yescharacteristic
72C23×C193C2311yescharacteristic
73D4×C193—41no—
74D4×C193—41no—
75D4×C193—41no—
76C2×C4×C193—41no—
77C2×C4×C193C2311yescharacteristic
78C2×C4×C193C2311yescharacteristic
79C2×C4×C193C2311yescharacteristic
80C2×C4×C193C2311yescharacteristic
81C22×C193—21no—
82C22×C193—21no—
83C22×C193—21no—
84C22×C193—21no—
85C22×C193—21no—
86C22×C193—21no—
87C22×C193—21no—
88C22×C193—21no—
89C22×C193—21no—
90C22×C193C2411yescharacteristic
91C2×D193—21no—
92C2×D193—21no—
93C2×D193—21no—
94C2×D193—21no—
95C2×D193—21no—
96C2×D193—44no—
97C2×D193#4112yes—
98C4×C193—21no—
99C4×C193—21no—
100C4×C193—21no—
101C4×C193—21no—
102Dic193—21no—
103Dic193—21no—
104Dic193—21no—
105Dic193—21no—
106Dic193—21no—
107Dic193—21no—
108C2×C193—41no—
109C2×C193—41no—
110C2×C193—41no—
111C2×C193C2×#4111yescharacteristic
112C2×C193H(5,𝔽2)11yescharacteristic
113C2×C193H(5,𝔽2)11yescharacteristic
114D193—22no—
115D193—41no—
116C193#53811yescharacteristic, complemented, Sylow
117#538—1931nomaximal, Sylow
118#126—1931no—
119#126—1931no—
120#124—1931no—
121#124—1931no—
122#123—1931no—
123#123—1931no—
124C22⋊Q8—1931no—
125#121—1931no—
126#121—1931no—
127#121D19311yescharacteristic, complemented
128#120—1931no—
129#120—1931no—
130C4×D4—1931no—
131C4×D4—1931no—
132C2×C22⋊C4—1931no—
133C24—1931no—
134C2×Q8—1931no—
135C2×D4—1931no—
136C2×D4—1931no—
137C2×D4—1931no—
138C2×D4—1931no—
139C2×D4C2×D19311yescharacteristic
140C2×D4C2×D19311yescharacteristic
141C2×D4C2×D19311yescharacteristic, complemented
142C22×C4—1931no—
143C22×C4—1931no—
144C22×C4—1931no—
145C22×C4C2×D19311yescharacteristic, complemented
146C4⋊C4—1931no—
147C4⋊C4—1931no—
148C4⋊C4—1931no—
149C4⋊C4—1931no—
150C4⋊C4—1931no—
151C4⋊C4C2×D19311yescharacteristic, complemented
152C22⋊C4—1931no—
153C22⋊C4—1931no—
154C22⋊C4—1931no—
155C22⋊C4—1931no—
156C22⋊C4—1931no—
157C22⋊C4—1931no—
158C22⋊C4—1931no—
159C22⋊C4—1931no—
160C22⋊C4—1931no—
161C22⋊C4—1931no—
162C22⋊C4—1931no—
163C22⋊C4—1931no—
164C22⋊C4C2×D19311yescharacteristic
165C22⋊C4C2×D19311yescharacteristic, complemented
166C42—1931no—
167C42—1931no—
168C23—7721no—
169C23—7721no—
170C23—3862no—
171C23—1931no—
172C23—1931no—
173C23C22×D19311yescharacteristic
174C23C22×D19311yescharacteristic
175C23C22×D19311yescharacteristic
176Q8—7721no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—7721no—
181D4—7721no—
182D4—3862no—
183D4—3862no—
184C2×C4—41no—
185C2×C4—7721no—
186C2×C4—7721no—
187C2×C4—7721no—
188C2×C4—1931no—
189C2×C4—1931no—
190C2×C4—1931no—
191C2×C4—1931no—
192C2×C4—1931no—
193C2×C4—1931no—
194C2×C4C22×D19311yescharacteristic
195C2×C4C22×D19311yescharacteristic
196C2×C4C22×D19311yescharacteristic
197C2×C4C22×D19311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—7724no—
208C22—3861no—
209C22—3861no—
210C22—3861no—
211C22—3861no—
212C22—3861no—
213C22—1932no—
214C22C23×D19311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—3861no—
220C4—3861no—
221C4—3861no—
222C4—3861no—
223C4—3861no—
224C4—3861no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—7721no—
229C2—3862no—
230C2#41×D19311yescharacteristic
231C2#2025994411yescharacteristic
232C2#2025994411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
Fitting#121×C193C2
hypercenter#121D193
hypocenterC193#538
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20259894C2semidirect
2. #20259892C2semidirect
3. #20259886C2semidirect
4. #20259885C2semidirect
5. #20259882C2semidirect
6. #20259880C2semidirect
7. #20259877C2semidirect
8. #20259871C2semidirect
9. #20259870C2semidirect
10. #20259867C2semidirect
11. #20259867C2semidirect
12. C4×C22⋊D193C2semidirect
13. #20259859C2semidirect
14. C22⋊C4×D193C2semidirect
15. #121×C193C2semidirect
17. C2×Dic386C22semidirect
22. C22×Dic193C22semidirect
25. C4⋊Dic193C22semidirect
26. #15690534C22semidirect
27. #15690534C22semidirect
28. #15690534C22semidirect
29. #15690534C22semidirect
33. C22⋊Dic193C22semidirect
34. C22⋊Dic193C22semidirect
35. C22⋊Dic193C22semidirect
42. C4×Dic193C22semidirect
43. C4×Dic193C22semidirect
46. C2×D4×C193C22semidirect
47. C22×C4×C193C22semidirect
48. C4⋊C4×C193C22semidirect
50. C22⋊C4×C193C22semidirect
116. C193#538semidirect
127. #121D193semidirect
141. C2×D4C2×D193semidirect
145. C22×C4C2×D193semidirect
151. C4⋊C4C2×D193semidirect
165. C22⋊C4C2×D193semidirect

Maximal subgroups

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

subgroupindex
#202598942
#202598922
#202598862
#202598852
#202598822
#202598802
#202598772
#202598712
#202598702
#202598672
C4×C22⋊D1932
#202598592
C22⋊C4×D1932
#121×C1932
#538193