Finite Group Databasebeta

← #20299277

Subgroups — #20299277

Order 7232.

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#16813617C211yesmaximal, characteristic, complemented
2#16813616C211yesmaximal, characteristic, complemented
3#16813608C211yesmaximal, characteristic, complemented
4#16813605C211yesmaximal, characteristic, complemented
5#16813604C211yesmaximal, characteristic, complemented
6#16813603C211yesmaximal, characteristic, complemented
7#16813601C211yesmaximal, characteristic, complemented
8#16813600C211yesmaximal, characteristic, complemented
9#16813590C211yesmaximal, characteristic
10C4×C22⋊D113C211yesmaximal, characteristic, complemented
11#16813579C211yesmaximal, characteristic, complemented
12#16813578C211yesmaximal, characteristic, complemented
13C2×#14268624C211yesmaximal, characteristic, complemented
14C22⋊C4×D113C211yesmaximal, characteristic, complemented
15C4×D4×C113C211yesmaximal, characteristic, complemented
16C23×D113C2211yescharacteristic
17C2×Dic226C2211yescharacteristic, complemented
18C2×C22⋊D113C2211yescharacteristic
19C2×C22⋊D113C2211yescharacteristic
20C22×Dic113C2211yescharacteristic
21C2×C4×D113C2211yescharacteristic
22C2×C4×D113C2211yescharacteristic
23C4⋊Dic113C2211yescharacteristic, complemented
24#14268626C2211yescharacteristic, complemented
25#14268626C2211yescharacteristic, complemented
26#14268626C2211yescharacteristic, complemented
27#14268626C2211yescharacteristic, complemented
28#14268626C2211yescharacteristic, complemented
29#14268626C2211yescharacteristic, complemented
30C22⋊Dic113C2211yescharacteristic
31C22⋊Dic113C2211yescharacteristic
32C22⋊Dic113C2211yescharacteristic, complemented
33C22⋊Dic113C2211yescharacteristic, complemented
34#14268624C2211yescharacteristic
35#14268624C2211yescharacteristic
36#14268624C2211yescharacteristic
37#14268624C2211yescharacteristic
38#14268624C2211yescharacteristic
39#14268624C2211yescharacteristic
40#14268624C2211yescharacteristic
41#14268624C2211yescharacteristic
42C4×Dic113C2211yescharacteristic, complemented
43C4×Dic113C2211yescharacteristic, complemented
44C2×D4×C113C2211yescharacteristic
45C22×C4×C113C2211yescharacteristic
46C22×C4×C113C2211yescharacteristic, complemented
47C4⋊C4×C113C2211yescharacteristic, complemented
48C22⋊C4×C113C2211yescharacteristic
49C22⋊C4×C113C2211yescharacteristic, complemented
50C42×C113C2211yescharacteristic, complemented
51C23×C113C2311yescharacteristic
52C23×C113C2311yescharacteristic
53C22×D113—22no—
54C22×D113—22no—
55C22×D113—41no—
56C22×D113C2311yescharacteristic
57C22×D113C2311yescharacteristic
58D4×C113—22no—
59C2×C4×C113—41no—
60C2×C4×C113—41no—
61C2×C4×C113C2311yescharacteristic
62C2×C4×C113C2311yescharacteristic
63C2×C4×C113C2311yescharacteristic
64C2×C4×C113C2311yescharacteristic
65C2×C4×C113C2311yescharacteristic
66C22⋊D113—22no—
67C22⋊D113—41no—
68C2×Dic113—22no—
69C2×Dic113C2311yescharacteristic
70C2×Dic113C2311yescharacteristic
71C2×Dic113C2311yescharacteristic
72C2×Dic113C2311yescharacteristic
73C2×Dic113C2311yescharacteristic
74C2×Dic113C2311yescharacteristic
75C4×D113—22no—
76C4×D113—41no—
77Dic226—41no—
78C22×C113—21no—
79C22×C113—21no—
80C22×C113—21no—
81C22×C113—21no—
82C22×C113—21no—
83C22×C113#4112yes—
84C22×C113C2411yescharacteristic
85C2×D113—21no—
86C2×D113—21no—
87C2×D113—21no—
88C2×D113—21no—
89C2×D113—21no—
90C2×D113—44no—
91C2×D113#4112yes—
92C4×C113—21no—
93C4×C113—21no—
94C4×C113—21no—
95C4×C113—21no—
96C4×C113—21no—
97Dic113—21no—
98Dic113—21no—
99Dic113—21no—
100Dic113—21no—
101Dic113—21no—
102Dic113—21no—
103C2×C113—22no—
104C2×C113—41no—
105C2×C113C2×#4111yescharacteristic
106C2×C113C2×#4111yescharacteristic
107C2×C113H(5,𝔽2)11yescharacteristic
108D113—22no—
109D113—41no—
110C113#55111yescharacteristic, complemented, Sylow
111#551—1131nomaximal, Sylow
112#126—1131no—
113#126—1131no—
114#124—1131no—
115#123—1131no—
116#123—1131no—
117#123—1131no—
118C22⋊Q8—1131no—
119C22⋊Q8—1131no—
120#120—1131no—
121C4×D4—1131no—
122C4×D4D11311yescharacteristic, complemented
123#117—1131no—
124#117—1131no—
125C2×C22⋊C4—1131no—
126C2×C22⋊C4—1131no—
127C24—1131no—
128C2×Q8—1131no—
129C2×D4—1131no—
130C2×D4—1131no—
131C2×D4C2×D11311yescharacteristic
132C22×C4—1131no—
133C22×C4—1131no—
134C22×C4—1131no—
135C22×C4C2×D11311yescharacteristic
136C22×C4C2×D11311yescharacteristic, complemented
137C4⋊C4—1131no—
138C4⋊C4—1131no—
139C4⋊C4—1131no—
140C4⋊C4—1131no—
141C4⋊C4—1131no—
142C4⋊C4—1131no—
143C4⋊C4—1131no—
144C4⋊C4C2×D11311yescharacteristic, complemented
145C22⋊C4—1131no—
146C22⋊C4—1131no—
147C22⋊C4—1131no—
148C22⋊C4—1131no—
149C22⋊C4—1131no—
150C22⋊C4—1131no—
151C22⋊C4—1131no—
152C22⋊C4—1131no—
153C22⋊C4—1131no—
154C22⋊C4—1131no—
155C22⋊C4—1131no—
156C22⋊C4—1131no—
157C22⋊C4C2×D11311yescharacteristic
158C22⋊C4C2×D11311yescharacteristic, complemented
159C42—1131no—
160C42—1131no—
161C42C2×D11311yescharacteristic, complemented
162C23—1131no—
163C23—1131no—
164C23—4521no—
165C23—2262no—
166C23—2262no—
167C23C22×D11311yescharacteristic
168C23C22×D11311yescharacteristic
169Q8—4521no—
170D4—22no—
171D4—4521no—
172D4—2262no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—1131no—
176C2×C4—1131no—
177C2×C4—1131no—
178C2×C4—1131no—
179C2×C4—1131no—
180C2×C4—1131no—
181C2×C4—4521no—
182C2×C4—2262no—
183C2×C4—2262no—
184C2×C4C22×D11311yescharacteristic
185C2×C4C22×D11311yescharacteristic
186C2×C4C22×D11311yescharacteristic
187C2×C4C22×D11311yescharacteristic
188C2×C4C22×D11311yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—1132no—
195C22—4524no—
196C22—2261no—
197C22—2261no—
198C22—2261no—
199C22—2261no—
200C22—2261no—
201C22#1426864712yes—
202C22C23×D11311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—2261no—
209C4—2261no—
210C4—2261no—
211C4—2261no—
212C4—2261no—
213C4—2261no—
214C2—22no—
215C2—41no—
216C2—4521no—
217C2—2262no—
218C2#41×D11311yescharacteristic
219C2C2×#1426864711yescharacteristic
220C2#1681366711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D113
commutator, socleC22×C113C24
FittingC4×D4×C113C2
hypercenterC4×D4D113
hypocenterC113#551
LCSG ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #16813617C2semidirect
2. #16813616C2semidirect
3. #16813608C2semidirect
4. #16813605C2semidirect
5. #16813604C2semidirect
6. #16813603C2semidirect
7. #16813601C2semidirect
8. #16813600C2semidirect
10. C4×C22⋊D113C2semidirect
11. #16813579C2semidirect
12. #16813578C2semidirect
13. C2×#14268624C2semidirect
14. C22⋊C4×D113C2semidirect
15. C4×D4×C113C2semidirect
17. C2×Dic226C22semidirect
23. C4⋊Dic113C22semidirect
24. #14268626C22semidirect
25. #14268626C22semidirect
26. #14268626C22semidirect
27. #14268626C22semidirect
28. #14268626C22semidirect
29. #14268626C22semidirect
32. C22⋊Dic113C22semidirect
33. C22⋊Dic113C22semidirect
42. C4×Dic113C22semidirect
43. C4×Dic113C22semidirect
46. C22×C4×C113C22semidirect
47. C4⋊C4×C113C22semidirect
49. C22⋊C4×C113C22semidirect
50. C42×C113C22semidirect
110. C113#551semidirect
122. C4×D4D113semidirect
136. C22×C4C2×D113semidirect
144. C4⋊C4C2×D113semidirect
158. C22⋊C4C2×D113semidirect
161. C42C2×D113semidirect