Finite Group Databasebeta

← #11734533

Subgroups — #11734533

Order 896.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6499C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic
4#89994C211yesmaximal, characteristic
5#89751C211yesmaximal, characteristic, complemented
6#89724C212yesmaximal, complemented
7#89721C212yesmaximal, complemented
8#89720C212yesmaximal, complemented
9#89720C212yesmaximal, complemented
10#89428C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6499—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2211yescharacteristic
31#6452—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2820—71nomaximal, Sylow
38C24×C7C2312yes—
39C2×C7×D4—22no—
40C2×C7×D4—42no—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—42no—
53C7×C4⋊C4—22no—
54C7×C22⋊C4—42no—
55C7×C22⋊C4—42no—
56C22⋊Dic7—24no—
57C22⋊Dic7—24no—
58C22⋊Dic7—42no—
59C22⋊Dic7—42no—
60C4⋊Dic7—22no—
61C4⋊Dic7—42no—
62Dic7⋊C4—22no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—42no—
67Dic7⋊C4—42no—
68C4×Dic7D414yes—
69C2×#125—71no—
70C2×#124—71no—
71#399—72no—
72#397—72no—
73#396—72no—
74#396—72no—
75#395—71no—
76#395—71no—
77#394—71no—
78#394D711yescharacteristic
79#382—71no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7C2411yescharacteristic
88C7×D4—44no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7C2×D412yes—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7C2×D412yes—
114C2×Dic7C2×D412yes—
115Dic14—44no—
116C22×Q8—71no—
117C22×D4C2×D711yescharacteristic
118#125—144no—
119#124—144no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4—72no—
128C2×C4⋊C4C2×D711yescharacteristic
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4C2×D712yes—
134C2×C22⋊C4C2×D712yes—
135C2×C42—71no—
136#95—71no—
137#95—72no—
138#95—72no—
139#95—72no—
140#95C2×D711yescharacteristic
141C22×C7—42no—
142C22×C7—42no—
143C22×C7—42no—
144C22×C7—42no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7C22×D411yescharacteristic
149C22×C7H(5,𝔽2)12yes—
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+412yes—
152C22×C72−1+411yescharacteristic
153C4×C7—22no—
154C4×C7—41no—
155C4×C7—41no—
156C4×C7—42no—
157Dic7—22no—
158Dic7—22no—
159Dic7—42no—
160Dic7—41no—
161Dic7—41no—
162Dic7—42no—
163C24C22×D712yes—
164C2×Q8—142no—
165C2×Q8—282no—
166C2×D4—22no—
167C2×D4—42no—
168C22×C4—71no—
169C22×C4—71no—
170C22×C4—72no—
171C22×C4—71no—
172C22×C4—72no—
173C22×C4—71no—
174C22×C4C22×D712yes—
175C22×C4C22×D711yescharacteristic
176C22×C4C22×D711yescharacteristic
177C22×C4C22×D711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—142no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—142no—
183C4⋊C4—142no—
184C4⋊C4—282no—
185C4⋊C4—282no—
186C4⋊C4—282no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—144no—
190C22⋊C4—144no—
191C22⋊C4—282no—
192C22⋊C4—282no—
193C42—74no—
194C2×C7—82no—
195C2×C7#53511yescharacteristic
196C2×C7#53611yescharacteristic
197C2×C7#53711yescharacteristic
198C2×C7#56212yes—
199C2×C7#56312yes—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23C23×D711yescharacteristic
208Q8—284no—
209D4—44no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—72no—
220C2×C4—72no—
221C2×C4—141no—
222C2×C4—141no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4—141no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4—142no—
235C2×C4D4×D712yes—
236C7#282011yescharacteristic, complemented, Sylow
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D711yescharacteristic
245C22#654812yes—
246C22#654811yescharacteristic
247C22#655411yescharacteristic
248C22#655412yes—
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—142no—
254C4—142no—
255C4—282no—
256C4—281no—
257C4—281no—
258C4—282no—
259C2—82no—
260C2#9028311yescharacteristic
261C2#9029911yescharacteristic
262C2#9031212yes—
263C2#9034212yes—
264C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2820
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
5. #89751C2semidirect
6. #89724C2semidirect
7. #89721C2semidirect
8. #89720C2semidirect
9. #89720C2semidirect
10. #89428C2semidirect
11. #89422C2semidirect
236. C7#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64522
C7×#3942
#899942
#897512
#897242
#897212
#897202
#894282
#894222
#28207