Finite Group Databasebeta

← #11738722

Subgroups — #11738722

Order 896.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#90291C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic
4#89805C211yesmaximal, characteristic
5#89743C211yesmaximal, characteristic, complemented
6#89648C212yesmaximal, complemented
7#89647C212yesmaximal, complemented
8#89537C212yesmaximal, complemented
9#89533C212yesmaximal, complemented
10#89493C211yesmaximal, characteristic, complemented
11#89491C211yesmaximal, characteristic, complemented
12C22×Dic14—21no—
13C7×#121C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15#6499C2211yescharacteristic
16#6498—21no—
17D4×Dic7—21no—
18D4×Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×Dic7⋊C4—21no—
22C2×C4×Dic7C2211yescharacteristic
23#6453C2211yescharacteristic
24#6451—21no—
25#6451—21no—
26#6451C2212yes—
27#6451C2212yes—
28#6450C2211yescharacteristic
29#6442—21no—
30#6440—21no—
31C7×#107C2212yes—
32C7×#102C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6406C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6391C2212yes—
37#6390C2212yes—
38#6389C2212yes—
39Dic7⋊C8C2212yes—
40#6384C2212yes—
41#6382C2212yes—
42#6381C2212yes—
43#3238—71nomaximal, Sylow
44C2×C7×D4—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C22×C4×C7C2311yescharacteristic
48C22×Dic7—21no—
49C22×Dic7—21no—
50C22×Dic7D411yescharacteristic
51C22×Dic7D411yescharacteristic
52C2×Dic14—22no—
53C2×Dic14—22no—
54C2×Dic14—44no—
55C2×Dic14C2311yescharacteristic
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4C2312yes—
59C7×C4⋊C4C2311yescharacteristic
60C7×C22⋊C4—21no—
61C22⋊Dic7—21no—
62C22⋊Dic7—21no—
63C22⋊Dic7—21no—
64C22⋊Dic7—21no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7C2312yes—
68C4⋊Dic7C2311yescharacteristic
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—22no—
73Dic7⋊C4—22no—
74C4×Dic7—21no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80C2×C4⋊Q8—71no—
81#529—71no—
82#501—72no—
83#489—72no—
84#483—71no—
85#483—71no—
86#480—71no—
87#480D711yescharacteristic
88#477—72no—
89#476—72no—
90#432—71no—
91C23×C7—21no—
92C23×C7C2×D411yescharacteristic
93C7×D4—22no—
94C7×D4—42no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—42no—
99C2×C4×C7C2×D411yescharacteristic
100C2×C4×C7C2411yescharacteristic
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111C2×Dic7C2×D411yescharacteristic
112Dic14—22no—
113Dic14—44no—
114Dic14—44no—
115C7×C8—22no—
116C7⋊C8—22no—
117C22×Q8—141no—
118C4⋊Q8—72no—
119C4⋊Q8—72no—
120C4⋊Q8—141no—
121C4⋊Q8—141no—
122#126—141no—
123#125—71no—
124#124—71no—
125#123—141no—
126C22⋊Q8—71no—
127#121C2×D711yescharacteristic
128C4×Q8—71no—
129C4×D4—71no—
130C4×D4—141no—
131#117—141no—
132C2×C4⋊C4—71no—
133C2×C4⋊C4—141no—
134C2×C4⋊C4C2×D711yescharacteristic
135C2×C42—71no—
136#107—72no—
137#107C2×D712yes—
138#106—72no—
139#106—72no—
140C4⋊C8—72no—
141#103—72no—
142#103—72no—
143#102—72no—
144#102C2×D712yes—
145C22⋊C8—71no—
146C22⋊C8C2×D711yescharacteristic
147#97—72no—
148C22×C7—21no—
149C22×C7—21no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7AGL(1,ℤ/8)12yes—
154C22×C7C22×D411yescharacteristic
155C4×C7—21no—
156C4×C7—42no—
157C4×C7—41no—
158C4×C72−1+412yes—
159Dic7—21no—
160Dic7—21no—
161Dic7—21no—
162Dic7—41no—
163Dic7—42no—
164Dic7—41no—
165Dic7#13712yes—
166C2×Q8—71no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—284no—
170C2×D4—21no—
171C2×D4C22×D711yescharacteristic
172C22×C4—21no—
173C22×C4—71no—
174C22×C4—71no—
175C22×C4—141no—
176C22×C4—141no—
177C22×C4C22×D711yescharacteristic
178C2×C8—72no—
179C2×C8C22×D712yes—
180C4⋊C4—21no—
181C4⋊C4—72no—
182C4⋊C4—71no—
183C4⋊C4—142no—
184C4⋊C4—142no—
185C4⋊C4—141no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4C22×D712yes—
191C4⋊C4C22×D711yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—141no—
194C22⋊C4—141no—
195C22⋊C4—141no—
196C22⋊C4—141no—
197C42—71no—
198C42—71no—
199C42—71no—
200C42—71no—
201C42—141no—
202C2×C7—22no—
203C2×C7—81no—
204C2×C7#53611yescharacteristic
205C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
206C2×C7C2×#13711yescharacteristic
207C23—21no—
208C23D4×D711yescharacteristic
209Q8—142no—
210Q8—284no—
211Q8—284no—
212D4—22no—
213D4—42no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—71no—
219C2×C4—71no—
220C2×C4—71no—
221C2×C4—71no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—142no—
227C2×C4—282no—
228C2×C4—282no—
229C2×C4D4×D711yescharacteristic
230C2×C4C23×D711yescharacteristic
231C8—22no—
232C8—142no—
233C7#323811yescharacteristic, complemented, Sylow
234C22—21no—
235C22—21no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22#647412yes—
240C22C2×D4×D711yescharacteristic
241C4—21no—
242C4—42no—
243C4—41no—
244C4—72no—
245C4—141no—
246C4—141no—
247C4—141no—
248C4—282no—
249C4—281no—
250C4—281no—
251C4#655412yes—
252C2—22no—
253C2—81no—
254C2#9033911yescharacteristic
255C2C2×#647411yescharacteristic
256C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#480D7
hypocenterC7#3238
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
5. #89743C2semidirect
6. #89648C2semidirect
7. #89647C2semidirect
8. #89537C2semidirect
9. #89533C2semidirect
10. #89493C2semidirect
11. #89491C2semidirect
233. C7#3238semidirect

Maximal subgroups

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

subgroupindex
#902912
C2×#64512
C7×#4802
#898052
#897432
#896482
#896472
#895372
#895332
#894932
#894912
#32387