Finite Group Databasebeta

← #37560099

Subgroups — #37560099

Order 23872.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#26045787C211yesmaximal, characteristic, complemented
2#26045787C211yesmaximal, characteristic, complemented
3#26045781C211yesmaximal, characteristic, complemented
4#26045777C211yesmaximal, characteristic, complemented
5#26045777C211yesmaximal, characteristic, complemented
6#26045776C211yesmaximal, characteristic, complemented
7#26045768C211yesmaximal, characteristic, complemented
8#26045762C211yesmaximal, characteristic
9#26045754C211yesmaximal, characteristic, complemented
10C4×C22⋊D373C211yesmaximal, characteristic, complemented
11#26045750C211yesmaximal, characteristic, complemented
12#26045748C211yesmaximal, characteristic, complemented
13C2×#20238587C211yesmaximal, characteristic, complemented
14C22⋊C4×D373C211yesmaximal, characteristic, complemented
15C22⋊Q8×C373C211yesmaximal, characteristic, complemented
16C23×D373C2211yescharacteristic
17C2×D1492C2211yescharacteristic
18C2×D1492C2211yescharacteristic
19C2×C22⋊D373C2211yescharacteristic
20C22×Dic373C2211yescharacteristic
21C2×C4×D373C2211yescharacteristic
22C2×C4×D373C2211yescharacteristic
23C2×C4×D373C2211yescharacteristic, complemented
24C4⋊Dic373C2211yescharacteristic, complemented
25C4⋊Dic373C2211yescharacteristic, complemented
26#20238589C2211yescharacteristic, complemented
27#20238589C2211yescharacteristic, complemented
28#20238589C2211yescharacteristic, complemented
29C22⋊Dic373C2211yescharacteristic
30#20238587C2211yescharacteristic
31#20238587C2211yescharacteristic
32#20238587C2211yescharacteristic
33#20238587C2211yescharacteristic
34#20238587C2211yescharacteristic
35#20238587C2211yescharacteristic
36#20238587C2211yescharacteristic
37#20238587C2211yescharacteristic
38#20238587C2211yescharacteristic, complemented
39#20238587C2211yescharacteristic, complemented
40#20238587C2211yescharacteristic, complemented
41C4×Dic373C2211yescharacteristic, complemented
42C4×Dic373C2211yescharacteristic, complemented
43C4×Dic373C2211yescharacteristic, complemented
44C2×Q8×C373C2211yescharacteristic, complemented
45C22×C4×C373C2211yescharacteristic
46C4⋊C4×C373C2211yescharacteristic, complemented
47C4⋊C4×C373C2211yescharacteristic, complemented
48C4⋊C4×C373C2211yescharacteristic, complemented
49C22⋊C4×C373C2211yescharacteristic
50C22⋊C4×C373C2211yescharacteristic
51C22×D373—22no—
52C22×D373—22no—
53C22×D373—41no—
54C22×D373C2311yescharacteristic
55C22×D373C2311yescharacteristic
56C22×D373C2311yescharacteristic
57D1492—22no—
58D1492—41no—
59C22⋊D373—22no—
60C2×Dic373—22no—
61C2×Dic373C2311yescharacteristic
62C2×Dic373C2311yescharacteristic
63C2×Dic373C2311yescharacteristic
64C2×Dic373C2311yescharacteristic
65C2×Dic373C2311yescharacteristic
66C4×D373—22no—
67C4×D373—41no—
68C4×D373—41no—
69C23×C373C2311yescharacteristic
70Q8×C373—41no—
71C2×C4×C373—41no—
72C2×C4×C373C2311yescharacteristic
73C2×C4×C373C2311yescharacteristic
74C2×C4×C373C2311yescharacteristic
75C2×C4×C373C2311yescharacteristic
76C2×C4×C373C2311yescharacteristic
77C2×C4×C373C2311yescharacteristic
78C22×C373—21no—
79C22×C373—21no—
80C22×C373#4112yes—
81C22×C373C2411yescharacteristic
82C2×D373—21no—
83C2×D373—21no—
84C2×D373—21no—
85C2×D373—21no—
86C2×D373—21no—
87C2×D373—21no—
88C2×D373—21no—
89C2×D373—21no—
90C2×D373—44no—
91C2×D373#4112yes—
92C4×C373—21no—
93C4×C373—21no—
94C4×C373—21no—
95C4×C373—21no—
96C4×C373—21no—
97C4×C373—21no—
98Dic373—21no—
99Dic373—21no—
100Dic373—21no—
101Dic373—21no—
102Dic373—21no—
103C2×C373—22no—
104C2×C373C2×#4111yescharacteristic
105C2×C373C2×#4111yescharacteristic
106C2×C373H(5,𝔽2)11yescharacteristic
107D373—22no—
108D373—41no—
109D373—41no—
110C373#55111yescharacteristic, complemented, Sylow
111#551—3731nomaximal, Sylow
112#126—3731no—
113#126—3731no—
114#124—3731no—
115#123—3731no—
116#123—3731no—
117#123—3731no—
118C22⋊Q8—3731no—
119C22⋊Q8D37311yescharacteristic, complemented
120#120—3731no—
121C4×D4—3731no—
122C4×D4—3731no—
123#117—3731no—
124#117—3731no—
125C2×C22⋊C4—3731no—
126C2×C22⋊C4—3731no—
127C24—3731no—
128C2×Q8C2×D37311yescharacteristic, complemented
129C2×D4—3731no—
130C2×D4—3731no—
131C2×D4—3731no—
132C22×C4—3731no—
133C22×C4—3731no—
134C22×C4—3731no—
135C22×C4—3731no—
136C22×C4C2×D37311yescharacteristic
137C4⋊C4—3731no—
138C4⋊C4—3731no—
139C4⋊C4—3731no—
140C4⋊C4—3731no—
141C4⋊C4—3731no—
142C4⋊C4C2×D37311yescharacteristic, complemented
143C4⋊C4C2×D37311yescharacteristic, complemented
144C4⋊C4C2×D37311yescharacteristic, complemented
145C22⋊C4—3731no—
146C22⋊C4—3731no—
147C22⋊C4—3731no—
148C22⋊C4—3731no—
149C22⋊C4—3731no—
150C22⋊C4—3731no—
151C22⋊C4—3731no—
152C22⋊C4—3731no—
153C22⋊C4—3731no—
154C22⋊C4—3731no—
155C22⋊C4—3731no—
156C22⋊C4—3731no—
157C22⋊C4C2×D37311yescharacteristic
158C22⋊C4C2×D37311yescharacteristic
159C42—3731no—
160C42—3731no—
161C42—3731no—
162C23—3731no—
163C23—3731no—
164C23—3731no—
165C23—14921no—
166C23—7462no—
167C23—7462no—
168C23C22×D37311yescharacteristic
169Q8—41no—
170D4—14921no—
171D4—7462no—
172D4—7462no—
173C2×C4—41no—
174C2×C4—3731no—
175C2×C4—3731no—
176C2×C4—3731no—
177C2×C4—3731no—
178C2×C4—3731no—
179C2×C4—14921no—
180C2×C4—14921no—
181C2×C4—7462no—
182C2×C4—7462no—
183C2×C4C22×D37311yescharacteristic
184C2×C4C22×D37311yescharacteristic
185C2×C4C22×D37311yescharacteristic
186C2×C4C22×D37311yescharacteristic
187C2×C4C22×D37311yescharacteristic
188C2×C4C22×D37311yescharacteristic
189C22—21no—
190C22—21no—
191C22—3732no—
192C22—14924no—
193C22—7461no—
194C22—7461no—
195C22—7461no—
196C22—7461no—
197C22—7461no—
198C22—7461no—
199C22—7461no—
200C22—7461no—
201C22#2023861112yes—
202C22C23×D37311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—7461no—
210C4—7461no—
211C4—7461no—
212C4—7461no—
213C4—7461no—
214C2—22no—
215C2—14921no—
216C2—14921no—
217C2—7462no—
218C2#41×D37311yescharacteristic
219C2C2×#2023861111yescharacteristic
220C2#2604583711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
FittingC22⋊Q8×C373C2
prime radical#26045762C2
hypercenterC22⋊Q8D373
hypocenterC373#551
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26045787C2semidirect
2. #26045787C2semidirect
3. #26045781C2semidirect
4. #26045777C2semidirect
5. #26045777C2semidirect
6. #26045776C2semidirect
7. #26045768C2semidirect
9. #26045754C2semidirect
10. C4×C22⋊D373C2semidirect
11. #26045750C2semidirect
12. #26045748C2semidirect
13. C2×#20238587C2semidirect
14. C22⋊C4×D373C2semidirect
15. C22⋊Q8×C373C2semidirect
23. C2×C4×D373C22semidirect
24. C4⋊Dic373C22semidirect
25. C4⋊Dic373C22semidirect
26. #20238589C22semidirect
27. #20238589C22semidirect
28. #20238589C22semidirect
38. #20238587C22semidirect
39. #20238587C22semidirect
40. #20238587C22semidirect
41. C4×Dic373C22semidirect
42. C4×Dic373C22semidirect
43. C4×Dic373C22semidirect
44. C2×Q8×C373C22semidirect
46. C4⋊C4×C373C22semidirect
47. C4⋊C4×C373C22semidirect
48. C4⋊C4×C373C22semidirect
110. C373#551semidirect
119. C22⋊Q8D373semidirect
128. C2×Q8C2×D373semidirect
142. C4⋊C4C2×D373semidirect
143. C4⋊C4C2×D373semidirect
144. C4⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
#260457872
#260457812
#260457772
#260457762
#260457682
#260457622
#260457542
C4×C22⋊D3732
#260457502
#260457482
C2×#202385872
C22⋊C4×D3732
C22⋊Q8×C3732
#551373