Finite Group Databasebeta

← #11734422

Subgroups — #11734422

Order 896.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#6450C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic
4#89994C211yesmaximal, characteristic
5#89751C211yesmaximal, characteristic, complemented
6#89723C212yesmaximal, complemented
7#89721C212yesmaximal, complemented
8#89719C212yesmaximal, complemented
9#89717C212yesmaximal, complemented
10#89428C211yesmaximal, characteristic, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22#6499—24no—
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30#6450—24no—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2632—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×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—22no—
54C2×Dic14—22no—
55C2×Dic14—42no—
56C7×C4⋊C4—22no—
57C7×C22⋊C4—24no—
58C7×C22⋊C4—42no—
59C22⋊Dic7—24no—
60C22⋊Dic7—24no—
61C22⋊Dic7—24no—
62C22⋊Dic7—42no—
63C4⋊Dic7—42no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—42no—
67C4×Dic7—22no—
68C4×Dic7—22no—
69C4×Dic7D414yes—
70C2×#124—71no—
71C2×C4×Q8—71no—
72#399—72no—
73#395—71no—
74#395—71no—
75#394—71no—
76#394D711yescharacteristic
77#393—72no—
78#388—72no—
79#388—72no—
80#383—71no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7C2411yescharacteristic
89C7×D4—44no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7C2×D412yes—
99C2×C4×C7#4112yes—
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7—21no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7C2×D412yes—
114C2×Dic7C2×D412yes—
115C2×Dic7#4112yes—
116Dic14—28no—
117Dic14—44no—
118C22×Q8—71no—
119C22×Q8—71no—
120C22×D4C2×D711yescharacteristic
121#124—144no—
122C4×Q8—144no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C22⋊C4—72no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4C2×D712yes—
133C2×C22⋊C4C2×D712yes—
134C2×C42—71no—
135C2×C42—71no—
136C2×C42—71no—
137#95—72no—
138#95—72no—
139#95—71no—
140#95—72no—
141#95—72no—
142#95C2×D711yescharacteristic
143C22×C7—42no—
144C22×C7—42no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—42no—
150C22×C7C22×D411yescharacteristic
151C22×C7C2×#4111yescharacteristic
152C22×C7C2×#4111yescharacteristic
153C22×C7H(5,𝔽2)12yes—
154C22×C72−1+412yes—
155C4×C7—22no—
156C4×C7—22no—
157C4×C7—41no—
158C4×C7—42no—
159Dic7—22no—
160Dic7—22no—
161Dic7—22no—
162Dic7—42no—
163Dic7—41no—
164Dic7—42no—
165C24C22×D712yes—
166C2×Q8—142no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—142no—
170C2×Q8—282no—
171C2×D4—22no—
172C2×D4—42no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—71no—
176C22×C4—72no—
177C22×C4—72no—
178C22×C4—71no—
179C22×C4C22×D712yes—
180C22×C4C22×D711yescharacteristic
181C22×C4C22×D711yescharacteristic
182C22×C4C22×D711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—282no—
187C4⋊C4—282no—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—144no—
191C22⋊C4—144no—
192C22⋊C4—144no—
193C22⋊C4—282no—
194C42—74no—
195C42—142no—
196C42—142no—
197C2×C7—82no—
198C2×C7#52511yescharacteristic
199C2×C7#53511yescharacteristic
200C2×C7#53611yescharacteristic
201C2×C7#54212yes—
202C2×C7#54212yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23C23×D711yescharacteristic
211Q8—148no—
212Q8—284no—
213D4—44no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—72no—
223C2×C4—72no—
224C2×C4—72no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4—141no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4D4×D712yes—
239C2×C4#115912yes—
240C7#263211yescharacteristic, complemented, Sylow
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D711yescharacteristic
249C22C2×#115911yescharacteristic
250C22#654812yes—
251C22D7×#4111yescharacteristic
252C22#655412yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—142no—
258C4—142no—
259C4—142no—
260C4—282no—
261C4—281no—
262C4—282no—
263C2—82no—
264C2#9028311yescharacteristic
265C2#9029412yes—
266C2#9029711yescharacteristic
267C2#9035312yes—
268C2#9036611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2632
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6450C2semidirect
5. #89751C2semidirect
6. #89723C2semidirect
7. #89721C2semidirect
8. #89719C2semidirect
9. #89717C2semidirect
10. #89428C2semidirect
11. #89423C2semidirect
240. C7#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64502
C7×#3942
#899942
#897512
#897232
#897212
#897192
#897172
#894282
#894232
#26327