Finite Group Databasebeta

← #11733754

Subgroups — #11733754

Order 896.

512 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#6498C211yesmaximal, characteristic
2C7×#390C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89744C212yesmaximal, complemented
5#89718C212yesmaximal, complemented
6#89717C212yesmaximal, complemented
7#89716C212yesmaximal, complemented
8#89714C212yesmaximal, complemented
9#89705C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21#6498C418yescomplemented
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2376—71nomaximal, Sylow
33C24×C7C2311yescharacteristic
34C24×C7C2311yescharacteristic
35C2×C7×D4—41no—
36C2×C7×D4—41no—
37C2×C7×D4C2×C414yes—
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×Dic7—41no—
42C22×Dic7—41no—
43C22×Dic7—41no—
44C22×Dic7—42no—
45C22×Dic7C2×C414yescomplemented
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C7×C4⋊C4—41no—
51C7×C22⋊C4—42no—
52C7×C22⋊C4—42no—
53C42×C7—41no—
54C22⋊Dic7—42no—
55C22⋊Dic7—41no—
56C22⋊Dic7C2×C418yes—
57C22⋊Dic7C2×C414yescomplemented
58Dic7⋊C4—42no—
59Dic7⋊C4—42no—
60Dic7⋊C4—42no—
61Dic7⋊C4C2×C418yescomplemented
62C4×Dic7—42no—
63C2×#123—71no—
64#390D711yescharacteristic
65#389—72no—
66#388—72no—
67#388—72no—
68#387—72no—
69#386—71no—
70#385—72no—
71#380—72no—
72C23×C7—22no—
73C23×C7—21no—
74C23×C7—22no—
75C23×C7—21no—
76C23×C7—21no—
77C23×C7—21no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7C22×C412yes—
81C23×C7C22×C412yes—
82C23×C7C2411yescharacteristic
83C7×D4—44no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7C22×C412yes—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—44no—
104C2×Dic7—44no—
105C2×Dic7C22×C414yes—
106C2×Dic7C22×C414yes—
107C22×D4C2×D711yescharacteristic
108C23×C4—71no—
109C23×C4—71no—
110#123—78no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4C2×D711yescharacteristic
116C2×C22⋊C4—72no—
117C2×C22⋊C4—71no—
118C2×C22⋊C4—72no—
119C2×C22⋊C4—71no—
120C2×C22⋊C4C2×D712yes—
121C2×C22⋊C4C2×D712yes—
122C2×C42—72no—
123C2×C42C2×D711yescharacteristic
124#95—72no—
125#95—72no—
126#95—72no—
127#95—72no—
128#95—72no—
129C22×C7—22no—
130C22×C7—22no—
131C22×C7—24no—
132C22×C7—24no—
133C22×C7—41no—
134C22×C7—41no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—41no—
138C22×C7—41no—
139C22×C7C23×C411yescharacteristic
140C22×C7H(5,𝔽2)11yescharacteristic
141C22×C7H(5,𝔽2)11yescharacteristic
142C22×C7H(5,𝔽2)12yes—
143C22×C72−1+412yes—
144C4×C7—22no—
145C4×C7—42no—
146C4×C7—42no—
147Dic7—24no—
148Dic7—24no—
149Dic7—42no—
150Dic7—42no—
151C24C22×D711yescharacteristic
152C24C22×D711yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4C4×D714yes—
156C22×C4—72no—
157C22×C4—72no—
158C22×C4—74no—
159C22×C4—72no—
160C22×C4—72no—
161C22×C4—281no—
162C22×C4—281no—
163C22×C4—282no—
164C22×C4—281no—
165C22×C4C22×D712yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C4⋊C4—41no—
169C4⋊C4—78no—
170C4⋊C4—282no—
171C4⋊C4—282no—
172C4⋊C4—282no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—78no—
176C22⋊C4—74no—
177C22⋊C4—282no—
178C22⋊C4—281no—
179C42—41no—
180C42—282no—
181C2×C7—42no—
182C2×C7—42no—
183C2×C7#51811yescharacteristic
184C2×C7#52012yes—
185C2×C7#56011yescharacteristic
186C2×C7#56212yes—
187C2×C7#56311yescharacteristic
188C23—22no—
189C23—22no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23C2×C4×D712yes—
197C23C2×C4×D712yes—
198C23C23×D711yescharacteristic
199D4—44no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—74no—
209C2×C4—74no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—284no—
221C2×C4—284no—
222C2×C4C2×C4×D712yes—
223C7#237611yescharacteristic, complemented, Sylow
224C22—22no—
225C22—22no—
226C22—24no—
227C22—24no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22C22×C4×D711yescharacteristic
235C22#654811yescharacteristic
236C22#654812yes—
237C22#654811yescharacteristic
238C22#655412yes—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—144no—
243C4—144no—
244C4—282no—
245C4—282no—
246C2—42no—
247C2—42no—
248C2#9017711yescharacteristic
249C2#9023712yes—
250C2#9031212yes—
251C2#9036311yescharacteristic
252C2#9041111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#390C2
socle, cosocleC23×C7C24
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2376
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #89744C2semidirect
5. #89718C2semidirect
6. #89717C2semidirect
7. #89716C2semidirect
8. #89714C2semidirect
9. #89705C2semidirect
21. #6498C4semidirect
45. C22×Dic7C2×C4semidirect
57. C22⋊Dic7C2×C4semidirect
61. Dic7⋊C4C2×C4semidirect
223. C7#2376semidirect

Maximal subgroups

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

subgroupindex
C2×#64982
C7×#3902
#899922
#897442
#897182
#897172
#897162
#897142
#897052
#23767