Finite Group Databasebeta

← #11738903

Subgroups — #11738903

Order 896.

360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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×#6451C211yesmaximal, characteristic
3C7×#484C211yesmaximal, characteristic
4#89816C211yesmaximal, characteristic
5#89742C211yesmaximal, characteristic
6#89647C212yesmaximal, complemented
7#89646C212yesmaximal, complemented
8#89566C212yesmaximal, complemented
9#89564C212yesmaximal, complemented
10#89494C211yesmaximal, characteristic
11#89490C211yesmaximal, characteristic
12C22×Dic14—21no—
13C7×C22⋊Q8C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15Q8×Dic7C2211yescharacteristic
16#6507C2211yescharacteristic
17C2×C4⋊Dic7C2211yescharacteristic
18#6487C2211yescharacteristic
19C2×Dic7⋊C4—21no—
20C2×C4×Dic7C2211yescharacteristic
21#6452—21no—
22#6451—21no—
23#6451—21no—
24#6451C2211yescharacteristic
25#6451C2212yes—
26#6451C2212yes—
27#6450—21no—
28#6450C2211yescharacteristic
29#6441—21no—
30#6440—21no—
31C7×#107C2212yes—
32C7×#103C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6409C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6392C2212yes—
37#6390C2212yes—
38Dic7⋊C8C2212yes—
39C8×Dic7C2212yes—
40#6384C2212yes—
41#6381C2212yes—
42#6381C2212yes—
43#3230—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C22×C4×C7—21no—
46C22×C4×C7C2311yescharacteristic
47C22×Dic7—21no—
48C22×Dic7D411yescharacteristic
49C22×Dic7D411yescharacteristic
50C2×Dic14—21no—
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—44no—
54C2×Dic14C2311yescharacteristic
55C2×C7×C8C2312yes—
56C7×C4⋊C4—21no—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C7×C22⋊C4—21no—
61C22⋊Dic7—21no—
62C4⋊Dic7—21no—
63C4⋊Dic7—21no—
64C4⋊Dic7C2312yes—
65C4⋊Dic7C2311yescharacteristic
66Dic7⋊C4—21no—
67Dic7⋊C4—21no—
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73C4×Dic7—21no—
74C4×Dic7—21no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#533—71no—
81C2×C4⋊Q8—71no—
82#500—72no—
83#487—72no—
84#484—71no—
85#484—71no—
86#484—71no—
87#484D711yescharacteristic
88#477—72no—
89#477—72no—
90#431—71no—
91C23×C7C2×D411yescharacteristic
92C7×Q8—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—42no—
98C2×C4×C7C2×D411yescharacteristic
99C2×C4×C7C2411yescharacteristic
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—42no—
105C2×Dic7Q1614yes—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110Dic14—22no—
111Dic14—44no—
112Dic14—42no—
113Dic14—44no—
114C7×C8—22no—
115C7⋊C8—22no—
116C22×Q8—141no—
117C4⋊Q8—72no—
118C4⋊Q8—72no—
119C4⋊Q8—71no—
120C4⋊Q8—71no—
121C4⋊Q8—141no—
122C4⋊Q8—141no—
123#125—141no—
124C22⋊Q8—71no—
125C22⋊Q8—141no—
126C22⋊Q8C2×D711yescharacteristic
127C4×Q8—71no—
128C4×Q8—71no—
129C4×Q8—141no—
130#117—141no—
131C2×C4⋊C4—71no—
132C2×C4⋊C4—141no—
133C2×C4⋊C4C2×D711yescharacteristic
134C2×C42—71no—
135#107—72no—
136#107—72no—
137#107—72no—
138#107C2×D712yes—
139C4⋊C8—72no—
140#103—72no—
141#103—72no—
142#103—72no—
143#103C2×D712yes—
144C22⋊C8—71no—
145C22⋊C8C2×D711yescharacteristic
146C4×C8—72no—
147C22×C7—21no—
148C22×C7—21no—
149C22×C7C2×Q1612yes—
150C22×C7C22×D411yescharacteristic
151C4×C7—21no—
152C4×C7—41no—
153C4×C7—42no—
154C4×C7—41no—
155C4×C72−1+412yes—
156Dic7—21no—
157Dic7—41no—
158Dic7—42no—
159Dic7—41no—
160Dic7C2×Q1612yes—
161Dic7C2×Q1612yes—
162Dic7AGL(1,ℤ/8)12yes—
163C2×Q8—71no—
164C2×Q8—141no—
165C2×Q8—142no—
166C2×Q8—142no—
167C2×Q8—284no—
168C2×Q8C22×D711yescharacteristic
169C22×C4—21no—
170C22×C4—71no—
171C22×C4—71no—
172C22×C4—141no—
173C22×C4C22×D711yescharacteristic
174C2×C8—72no—
175C2×C8C22×D712yes—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—72no—
179C4⋊C4—71no—
180C4⋊C4—142no—
181C4⋊C4—142no—
182C4⋊C4—141no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4C22×D712yes—
190C4⋊C4C22×D711yescharacteristic
191C22⋊C4—21no—
192C22⋊C4—141no—
193C42—71no—
194C42—71no—
195C42—71no—
196C42—71no—
197C42—141no—
198C42—141no—
199C2×C7—22no—
200C2×C7#53611yescharacteristic
201C2×C7C22×Q1611yescharacteristic
202C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
203C23D4×D711yescharacteristic
204Q8—22no—
205Q8—142no—
206Q8—282no—
207Q8—284no—
208Q8—284no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—74no—
215C2×C4—71no—
216C2×C4—71no—
217C2×C4—71no—
218C2×C4—71no—
219C2×C4—142no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4—282no—
224C2×C4D4×D711yescharacteristic
225C2×C4C23×D711yescharacteristic
226C8—22no—
227C8—142no—
228C7#323011yescharacteristic, complemented, Sylow
229C22—21no—
230C22—21no—
231C22D7×Q1612yes—
232C22C2×D4×D711yescharacteristic
233C4—21no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—72no—
238C4—72no—
239C4—72no—
240C4—141no—
241C4—282no—
242C4—281no—
243C4—281no—
244C4#655412yes—
245C2—22no—
246C2#9033911yescharacteristic
247C2C2×D7×Q1611yescharacteristic
248C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#484C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#484D7
hypocenterC7#3230
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
6. #89647C2semidirect
7. #89646C2semidirect
8. #89566C2semidirect
9. #89564C2semidirect
228. C7#3230semidirect

Maximal subgroups

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

subgroupindex
#903132
C2×#64512
C7×#4842
#898162
#897422
#896472
#896462
#895662
#895642
#894942
#894902
#32307