Finite Group Databasebeta

← #11734244

Subgroups — #11734244

Order 896.

564 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 151 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×#6507C211yesmaximal, characteristic, complemented
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
3C7×#393C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic, complemented
5#89729C212yesmaximal
6#89719C211yesmaximal, characteristic, complemented
7#89714C212yesmaximal, complemented
8#89445C211yesmaximal, characteristic, complemented
9#89439C212yesmaximal, complemented
10#89422C212yesmaximal, complemented
11#89421C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C22×Dic14C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20#6507—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30(C22⋊D7)⋊C4—24no—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2563—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C2×C7×Q8—22no—
41C2×C7×Q8—42no—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—22no—
47C2×C22⋊D7—22no—
48C2×C22⋊D7Q814yescomplemented
49C22×Dic7—22no—
50C22×Dic7—24no—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2312yes—
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C2×C4×D7—22no—
57C2×C4×D7—24no—
58C2×Dic14—22no—
59C2×Dic14—42no—
60C7×C22⋊C4—22no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—42no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66Dic7⋊C4—22no—
67Dic7⋊C4—24no—
68Dic7⋊C4—24no—
69Dic7⋊C4—24no—
70C4×Dic7—24no—
71C4×Dic7D414yes—
72C2×C4⋊Q8—71no—
73C2×C4×D4—71no—
74#393—71no—
75#393—71no—
76#393—71no—
77#393D711yescharacteristic, complemented
78#390—72no—
79#382—72no—
80#380—72no—
81#380—72no—
82C4×C4⋊C4—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7C2×Q812yes—
88C23×C7C2411yescharacteristic
89C22×D7—21no—
90C22×D7—22no—
91C22×D7—21no—
92C22×D7—22no—
93C22×D7C2×Q812yes—
94C7×Q8—44no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7C2×D412yes—
105C22⋊D7—28no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—44no—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×D412yes—
117C2×Dic7C2×Q812yes—
118C2×Dic7#4112yes—
119C2×Dic7#4114yes—
120C4×D7—44no—
121Dic14—44no—
122C22×Q8—71no—
123C22×Q8C2×D711yescharacteristic
124C22×D4—71no—
125C23×C4—71no—
126C23×C4—71no—
127C4⋊Q8—144no—
128C4×D4—144no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—72no—
132C2×C4⋊C4—72no—
133C2×C22⋊C4—71no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4C2×D711yescharacteristic
138C2×C22⋊C4C2×D712yes—
139C2×C42—71no—
140C2×C42—72no—
141#95—71no—
142#95—72no—
143#95—72no—
144#95—72no—
145#95C2×D711yescharacteristic
146#95C2×D712yes—
147C22×C7—22no—
148C22×C7—24no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—42no—
152C22×C7C22×D411yescharacteristic
153C22×C7C22×Q811yescharacteristic
154C22×C7C2×#4111yescharacteristic
155C22×C7C2×#4112yes—
156C22×C7H(5,𝔽2)12yes—
157C2×D7—22no—
158C2×D7—24no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—42no—
162C4×C7—22no—
163C4×C7—41no—
164C4×C7—42no—
165C4×C7—42no—
166Dic7—41no—
167Dic7—42no—
168Dic7—42no—
169Dic7C22⋊Q814yes—
170Dic7#12414yes—
171C24—71no—
172C24C22×D711yescharacteristic
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—142no—
176C2×Q8—282no—
177C2×D4—74no—
178C2×D4—142no—
179C2×D4—142no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—71no—
183C22×C4—72no—
184C22×C4—72no—
185C22×C4—142no—
186C22×C4—144no—
187C22×C4—142no—
188C22×C4—144no—
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D712yes—
191C22×C4C22×D712yes—
192C22×C4C22×D711yescharacteristic
193C4⋊C4—144no—
194C4⋊C4—142no—
195C4⋊C4—144no—
196C4⋊C4—144no—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—142no—
200C22⋊C4—282no—
201C22⋊C4—282no—
202C22⋊C4—282no—
203C42—74no—
204C42—144no—
205C2×C7—42no—
206C2×C7C2×C22⋊Q811yescharacteristic
207C2×C7C2×#12411yescharacteristic
208C2×C7#53511yescharacteristic
209C2×C7#55412yes—
210C2×C7#55512yes—
211D7—42no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—72no—
217C23—141no—
218C23—142no—
219C23—142no—
220C23—141no—
221C23Q8×D712yes—
222C23C23×D711yescharacteristic
223Q8—44no—
224Q8—284no—
225D4—148no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—72no—
236C2×C4—72no—
237C2×C4—72no—
238C2×C4—72no—
239C2×C4—74no—
240C2×C4—141no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—142no—
247C2×C4—142no—
248C2×C4—284no—
249C2×C4—284no—
250C2×C4D4×D712yes—
251C7#256311yescharacteristic, complemented, Sylow
252C22—22no—
253C22—24no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—142no—
258C22—144no—
259C22—281no—
260C22—282no—
261C22—281no—
262C22C2×D4×D711yescharacteristic
263C22#654812yes—
264C22C2×Q8×D711yescharacteristic
265C22D7×#4111yescharacteristic
266C22D7×#4112yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—74no—
272C4—74no—
273C4—282no—
274C4—282no—
275C4—281no—
276C2—42no—
277C2—282no—
278C2#9028311yescharacteristic
279C2D7×C22⋊Q811yescharacteristic
280C2#9032712yes—
281C2#9036112yes—
282C2D7×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC22×C22⋊D7C22
hypercenter#393D7
hypocenterC7#2563
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#6507C2semidirect
3. C7×#393C2semidirect
4. #90003C2semidirect
6. #89719C2semidirect
7. #89714C2semidirect
8. #89445C2semidirect
9. #89439C2semidirect
10. #89422C2semidirect
11. #89421C2semidirect
48. C2×C22⋊D7Q8semidirect
77. #393D7semidirect
251. C7#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×(C22⋊D7)⋊C42
C7×#3932
#900032
#897292
#897192
#897142
#894452
#894392
#894222
#894212
#25637