Finite Group Databasebeta

← #11738865

Subgroups — #11738865

Order 896.

324 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 113 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#90313C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic
3C7×#483C211yesmaximal, characteristic
4#89816C211yesmaximal, characteristic
5#89743C211yesmaximal, characteristic
6#89629C212yesmaximal, complemented
7#89628C212yesmaximal, complemented
8#89569C212yesmaximal, complemented
9#89568C212yesmaximal, complemented
10#89494C211yesmaximal, characteristic
11#89491C211yesmaximal, characteristic
12C7×C22⋊Q8C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14Q8×Dic7C2211yescharacteristic
15#6507C2211yescharacteristic
16C2×C4⋊Dic7—21no—
17C2×C4⋊Dic7C2211yescharacteristic
18#6487C2211yescharacteristic
19C2×Dic7⋊C4—21no—
20C2×C4×Dic7C2211yescharacteristic
21#6453—21no—
22#6453—21no—
23#6453C2212yes—
24#6453C2212yes—
25#6452—21no—
26#6451C2211yescharacteristic
27#6450—21no—
28#6450C2211yescharacteristic
29#6441—21no—
30#6440—21no—
31C7×#106C2212yes—
32C7×#103C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6409C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6392C2212yes—
37#6390C2212yes—
38#6389C2212yes—
39Dic7⋊C8C2212yes—
40#6384C2212yes—
41#6382C2212yes—
42#6381C2212yes—
43#3236—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C22×Dic7—21no—
48C22×Dic7D411yescharacteristic
49C22×Dic7D411yescharacteristic
50C2×Dic14—21no—
51C2×Dic14C2311yescharacteristic
52C2×C7×C8C2312yes—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C7×C4⋊C4C2311yescharacteristic
57C7×C22⋊C4—21no—
58C22⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7—21no—
61C4⋊Dic7—22no—
62C4⋊Dic7—22no—
63C4⋊Dic7C2312yes—
64C4⋊Dic7C2311yescharacteristic
65Dic7⋊C4—21no—
66Dic7⋊C4—21no—
67Dic7⋊C4—21no—
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—22no—
71Dic7⋊C4—22no—
72C4×Dic7—21no—
73C4×Dic7—21no—
74C4×Dic7D411yescharacteristic
75C4×Dic7D411yescharacteristic
76C4×Dic7C2311yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79#533—71no—
80C2×#125—71no—
81#501—72no—
82#491—72no—
83#484—71no—
84#484—71no—
85#483—71no—
86#483D711yescharacteristic
87#479—72no—
88#479—72no—
89#432—71no—
90C23×C7C2×D411yescharacteristic
91C7×Q8—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—42no—
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2411yescharacteristic
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—41no—
104C2×Dic7—42no—
105C2×Dic7C2×D411yescharacteristic
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109Dic14—22no—
110Dic14—42no—
111C7×C8—22no—
112C7⋊C8—22no—
113C4⋊Q8—71no—
114C4⋊Q8—71no—
115#125—72no—
116#125—72no—
117#125—141no—
118#125—141no—
119#125—141no—
120C22⋊Q8—71no—
121C22⋊Q8—141no—
122C22⋊Q8C2×D711yescharacteristic
123C4×Q8—71no—
124C4×Q8—71no—
125C4×Q8—141no—
126#117—141no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—141no—
129C2×C4⋊C4—141no—
130C2×C4⋊C4C2×D711yescharacteristic
131C2×C42—71no—
132#107—72no—
133#107—72no—
134#106—72no—
135#106C2×D712yes—
136C4⋊C8—72no—
137#103—72no—
138#103—72no—
139#103—72no—
140#103C2×D712yes—
141C22⋊C8—71no—
142C22⋊C8C2×D711yescharacteristic
143#97—72no—
144C22×C7—21no—
145C22×C7—21no—
146C22×C7#13712yes—
147C22×C7C22×D411yescharacteristic
148C4×C7—21no—
149C4×C7—41no—
150C4×C7—42no—
151C4×C7—41no—
152C4×C72−1+412yes—
153Dic7—21no—
154Dic7—21no—
155Dic7—21no—
156Dic7—21no—
157Dic7—41no—
158Dic7—41no—
159Dic7—42no—
160C2×Q8—71no—
161C2×Q8—141no—
162C2×Q8C22×D711yescharacteristic
163C22×C4—21no—
164C22×C4—71no—
165C22×C4—71no—
166C22×C4—141no—
167C22×C4C22×D711yescharacteristic
168C2×C8—72no—
169C2×C8C22×D712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—72no—
173C4⋊C4—71no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—141no—
177C4⋊C4—141no—
178C4⋊C4—141no—
179C4⋊C4—141no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4C22×D712yes—
186C4⋊C4C22×D711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—141no—
189C42—71no—
190C42—71no—
191C42—71no—
192C42—71no—
193C42—141no—
194C42—141no—
195C2×C7—22no—
196C2×C7#53611yescharacteristic
197C2×C7C2×#13711yescharacteristic
198C2×C7#57511yescharacteristic
199C23D4×D711yescharacteristic
200Q8—22no—
201Q8—142no—
202Q8—282no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—71no—
209C2×C4—71no—
210C2×C4—71no—
211C2×C4—71no—
212C2×C4—142no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—281no—
217C2×C4—282no—
218C2×C4D4×D711yescharacteristic
219C2×C4C23×D711yescharacteristic
220C8—22no—
221C8—142no—
222C7#323611yescharacteristic, complemented, Sylow
223C22—21no—
224C22—21no—
225C22#647812yes—
226C22C2×D4×D711yescharacteristic
227C4—21no—
228C4—42no—
229C4—41no—
230C4—41no—
231C4—141no—
232C4—141no—
233C4—141no—
234C4—141no—
235C4—282no—
236C4—281no—
237C4—281no—
238C4#655412yes—
239C2—22no—
240C2#9033911yescharacteristic
241C2C2×#647811yescharacteristic
242C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#483C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#483D7
hypocenterC7#3236
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
6. #89629C2semidirect
7. #89628C2semidirect
8. #89569C2semidirect
9. #89568C2semidirect
222. C7#3236semidirect

Maximal subgroups

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

subgroupindex
#903132
C2×#64532
C7×#4832
#898162
#897432
#896292
#896282
#895692
#895682
#894942
#894912
#32367