Finite Group Databasebeta

← #37554863

Subgroups — #37554863

Order 23488.

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#26044256C211yesmaximal, characteristic, complemented
2#26044256C211yesmaximal, characteristic, complemented
3#26044250C211yesmaximal, characteristic, complemented
4#26044246C211yesmaximal, characteristic, complemented
5#26044246C211yesmaximal, characteristic, complemented
6#26044245C211yesmaximal, characteristic, complemented
7#26044237C211yesmaximal, characteristic, complemented
8#26044231C211yesmaximal, characteristic
9#26044223C211yesmaximal, characteristic, complemented
10C4×C22⋊D367C211yesmaximal, characteristic, complemented
11#26044219C211yesmaximal, characteristic, complemented
12#26044217C211yesmaximal, characteristic, complemented
13C2×(C2×D367)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D367C211yesmaximal, characteristic, complemented
15C22⋊Q8×C367C211yesmaximal, characteristic, complemented
16C23×D367C2211yescharacteristic
17C2×D1468C2211yescharacteristic
18C2×D1468C2211yescharacteristic
19C2×C22⋊D367C2211yescharacteristic
20C22×Dic367C2211yescharacteristic
21C2×C4×D367C2211yescharacteristic
22C2×C4×D367C2211yescharacteristic
23C2×C4×D367C2211yescharacteristic, complemented
24C4⋊Dic367C2211yescharacteristic, complemented
25C4⋊Dic367C2211yescharacteristic, complemented
26Dic367⋊C4C2211yescharacteristic, complemented
27Dic367⋊C4C2211yescharacteristic, complemented
28Dic367⋊C4C2211yescharacteristic, complemented
29C22⋊Dic367C2211yescharacteristic
30(C2×D367)⋊C4C2211yescharacteristic
31(C2×D367)⋊C4C2211yescharacteristic
32(C2×D367)⋊C4C2211yescharacteristic
33(C2×D367)⋊C4C2211yescharacteristic
34(C2×D367)⋊C4C2211yescharacteristic
35(C2×D367)⋊C4C2211yescharacteristic
36(C2×D367)⋊C4C2211yescharacteristic
37(C2×D367)⋊C4C2211yescharacteristic
38(C2×D367)⋊C4C2211yescharacteristic, complemented
39(C2×D367)⋊C4C2211yescharacteristic, complemented
40(C2×D367)⋊C4C2211yescharacteristic, complemented
41C4×Dic367C2211yescharacteristic, complemented
42C4×Dic367C2211yescharacteristic, complemented
43C4×Dic367C2211yescharacteristic, complemented
44C2×Q8×C367C2211yescharacteristic, complemented
45C22×C4×C367C2211yescharacteristic
46C4⋊C4×C367C2211yescharacteristic, complemented
47C4⋊C4×C367C2211yescharacteristic, complemented
48C4⋊C4×C367C2211yescharacteristic, complemented
49C22⋊C4×C367C2211yescharacteristic
50C22⋊C4×C367C2211yescharacteristic
51C22×D367—22no—
52C22×D367—22no—
53C22×D367—41no—
54C22×D367C2311yescharacteristic
55C22×D367C2311yescharacteristic
56C22×D367C2311yescharacteristic
57D1468—22no—
58D1468—41no—
59C22⋊D367—22no—
60C2×Dic367—22no—
61C2×Dic367C2311yescharacteristic
62C2×Dic367C2311yescharacteristic
63C2×Dic367C2311yescharacteristic
64C2×Dic367C2311yescharacteristic
65C2×Dic367C2311yescharacteristic
66C4×D367—22no—
67C4×D367—41no—
68C4×D367—41no—
69C23×C367C2311yescharacteristic
70Q8×C367—41no—
71C2×C4×C367—41no—
72C2×C4×C367C2311yescharacteristic
73C2×C4×C367C2311yescharacteristic
74C2×C4×C367C2311yescharacteristic
75C2×C4×C367C2311yescharacteristic
76C2×C4×C367C2311yescharacteristic
77C2×C4×C367C2311yescharacteristic
78C22×C367—21no—
79C22×C367—21no—
80C22×C367#4112yes—
81C22×C367C2411yescharacteristic
82C2×D367—21no—
83C2×D367—21no—
84C2×D367—21no—
85C2×D367—21no—
86C2×D367—21no—
87C2×D367—21no—
88C2×D367—21no—
89C2×D367—21no—
90C2×D367—44no—
91C2×D367#4112yes—
92C4×C367—21no—
93C4×C367—21no—
94C4×C367—21no—
95C4×C367—21no—
96C4×C367—21no—
97C4×C367—21no—
98Dic367—21no—
99Dic367—21no—
100Dic367—21no—
101Dic367—21no—
102Dic367—21no—
103C2×C367—22no—
104C2×C367C2×#4111yescharacteristic
105C2×C367C2×#4111yescharacteristic
106C2×C367H(5,𝔽2)11yescharacteristic
107D367—22no—
108D367—41no—
109D367—41no—
110C367#55111yescharacteristic, complemented, Sylow
111#551—3671nomaximal, Sylow
112#126—3671no—
113#126—3671no—
114#124—3671no—
115#123—3671no—
116#123—3671no—
117#123—3671no—
118C22⋊Q8—3671no—
119C22⋊Q8D36711yescharacteristic, complemented
120#120—3671no—
121C4×D4—3671no—
122C4×D4—3671no—
123#117—3671no—
124#117—3671no—
125C2×C22⋊C4—3671no—
126C2×C22⋊C4—3671no—
127C24—3671no—
128C2×Q8C2×D36711yescharacteristic, complemented
129C2×D4—3671no—
130C2×D4—3671no—
131C2×D4—3671no—
132C22×C4—3671no—
133C22×C4—3671no—
134C22×C4—3671no—
135C22×C4—3671no—
136C22×C4C2×D36711yescharacteristic
137C4⋊C4—3671no—
138C4⋊C4—3671no—
139C4⋊C4—3671no—
140C4⋊C4—3671no—
141C4⋊C4—3671no—
142C4⋊C4C2×D36711yescharacteristic, complemented
143C4⋊C4C2×D36711yescharacteristic, complemented
144C4⋊C4C2×D36711yescharacteristic, complemented
145C22⋊C4—3671no—
146C22⋊C4—3671no—
147C22⋊C4—3671no—
148C22⋊C4—3671no—
149C22⋊C4—3671no—
150C22⋊C4—3671no—
151C22⋊C4—3671no—
152C22⋊C4—3671no—
153C22⋊C4—3671no—
154C22⋊C4—3671no—
155C22⋊C4—3671no—
156C22⋊C4—3671no—
157C22⋊C4C2×D36711yescharacteristic
158C22⋊C4C2×D36711yescharacteristic
159C42—3671no—
160C42—3671no—
161C42—3671no—
162C23—3671no—
163C23—3671no—
164C23—3671no—
165C23—14681no—
166C23—7342no—
167C23—7342no—
168C23C22×D36711yescharacteristic
169Q8—41no—
170D4—14681no—
171D4—7342no—
172D4—7342no—
173C2×C4—41no—
174C2×C4—3671no—
175C2×C4—3671no—
176C2×C4—3671no—
177C2×C4—3671no—
178C2×C4—3671no—
179C2×C4—14681no—
180C2×C4—14681no—
181C2×C4—7342no—
182C2×C4—7342no—
183C2×C4C22×D36711yescharacteristic
184C2×C4C22×D36711yescharacteristic
185C2×C4C22×D36711yescharacteristic
186C2×C4C22×D36711yescharacteristic
187C2×C4C22×D36711yescharacteristic
188C2×C4C22×D36711yescharacteristic
189C22—21no—
190C22—21no—
191C22—3672no—
192C22—14684no—
193C22—7341no—
194C22—7341no—
195C22—7341no—
196C22—7341no—
197C22—7341no—
198C22—7341no—
199C22—7341no—
200C22—7341no—
201C22#1915457812yes—
202C22C23×D36711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—7341no—
210C4—7341no—
211C4—7341no—
212C4—7341no—
213C4—7341no—
214C2—22no—
215C2—14681no—
216C2—14681no—
217C2—7342no—
218C2#41×D36711yescharacteristic
219C2C2×#1915457811yescharacteristic
220C2#2604430611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
FittingC22⋊Q8×C367C2
prime radical#26044231C2
hypercenterC22⋊Q8D367
hypocenterC367#551
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #26044256C2semidirect
2. #26044256C2semidirect
3. #26044250C2semidirect
4. #26044246C2semidirect
5. #26044246C2semidirect
6. #26044245C2semidirect
7. #26044237C2semidirect
9. #26044223C2semidirect
10. C4×C22⋊D367C2semidirect
11. #26044219C2semidirect
12. #26044217C2semidirect
13. C2×(C2×D367)⋊C4C2semidirect
14. C22⋊C4×D367C2semidirect
15. C22⋊Q8×C367C2semidirect
23. C2×C4×D367C22semidirect
24. C4⋊Dic367C22semidirect
25. C4⋊Dic367C22semidirect
26. Dic367⋊C4C22semidirect
27. Dic367⋊C4C22semidirect
28. Dic367⋊C4C22semidirect
38. (C2×D367)⋊C4C22semidirect
39. (C2×D367)⋊C4C22semidirect
40. (C2×D367)⋊C4C22semidirect
41. C4×Dic367C22semidirect
42. C4×Dic367C22semidirect
43. C4×Dic367C22semidirect
44. C2×Q8×C367C22semidirect
46. C4⋊C4×C367C22semidirect
47. C4⋊C4×C367C22semidirect
48. C4⋊C4×C367C22semidirect
110. C367#551semidirect
119. C22⋊Q8D367semidirect
128. C2×Q8C2×D367semidirect
142. C4⋊C4C2×D367semidirect
143. C4⋊C4C2×D367semidirect
144. C4⋊C4C2×D367semidirect

Maximal subgroups

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

subgroupindex
#260442562
#260442502
#260442462
#260442452
#260442372
#260442312
#260442232
C4×C22⋊D3672
#260442192
#260442172
C2×(C2×D367)⋊C42
C22⋊C4×D3672
C22⋊Q8×C3672
#551367