Finite Group Databasebeta

← #11738763

Subgroups — #11738763

Order 896.

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

normal subgroups only

This group has 334 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#90291C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic
4#89805C211yesmaximal, characteristic
5#89742C211yesmaximal, characteristic, complemented
6#89630C212yesmaximal, complemented
7#89629C212yesmaximal, complemented
8#89540C212yesmaximal, complemented
9#89535C212yesmaximal, complemented
10#89493C211yesmaximal, characteristic, complemented
11#89490C211yesmaximal, characteristic, complemented
12C7×#121C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14#6499C2211yescharacteristic
15#6498—21no—
16D4×Dic7—21no—
17D4×Dic7C2211yescharacteristic
18C2×C4⋊Dic7—21no—
19C2×C4⋊Dic7C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×Dic7⋊C4—21no—
22C2×C4×Dic7C2211yescharacteristic
23#6453—21no—
24#6453—21no—
25#6453C2212yes—
26#6453C2211yescharacteristic
27#6453C2212yes—
28#6450C2211yescharacteristic
29#6442—21no—
30#6440—21no—
31C7×#106C2212yes—
32C7×#102C2212yes—
33C7×C22⋊C8C2211yescharacteristic
34#6406C2212yes—
35C22⋊(C7⋊C8)C2211yescharacteristic
36#6391C2212yes—
37#6390C2212yes—
38Dic7⋊C8C2212yes—
39C8×Dic7C2212yes—
40#6384C2212yes—
41#6381C2212yes—
42#6381C2212yes—
43#3232—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×Dic14C2311yescharacteristic
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2312yes—
56C7×C4⋊C4C2311yescharacteristic
57C7×C22⋊C4—21no—
58C22⋊Dic7—21no—
59C22⋊Dic7—21no—
60C22⋊Dic7—21no—
61C22⋊Dic7—21no—
62C4⋊Dic7—21no—
63C4⋊Dic7—21no—
64C4⋊Dic7—22no—
65C4⋊Dic7—22no—
66C4⋊Dic7C2312yes—
67C4⋊Dic7C2311yescharacteristic
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72Dic7⋊C4—22no—
73C4×Dic7—21no—
74C4×Dic7D411yescharacteristic
75C4×Dic7D411yescharacteristic
76C4×Dic7C2311yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79#529—71no—
80C2×#125—71no—
81#499—72no—
82#488—72no—
83#484—71no—
84#483—71no—
85#481D711yescharacteristic
86#480—71no—
87#479—72no—
88#478—72no—
89#431—71no—
90C23×C7—21no—
91C23×C7C2×D411yescharacteristic
92C7×D4—22no—
93C7×D4—42no—
94C2×C4×C7—22no—
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—22no—
103C2×Dic7—21no—
104C2×Dic7—41no—
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C2×Dic7C2×D411yescharacteristic
111Dic14—22no—
112C7×C8—22no—
113C7⋊C8—22no—
114#126—141no—
115#125—72no—
116#125—72no—
117#125—71no—
118#125—141no—
119#125—141no—
120#124—71no—
121#123—141no—
122C22⋊Q8—71no—
123#121C2×D711yescharacteristic
124C4×Q8—71no—
125C4×D4—71no—
126C4×D4—141no—
127#117—141no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—141no—
130C2×C4⋊C4—141no—
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C42—71no—
133#107—72no—
134#107—72no—
135#106—72no—
136#106C2×D712yes—
137C4⋊C8—72no—
138#103—72no—
139#103—72no—
140#102—72no—
141#102C2×D712yes—
142C22⋊C8—71no—
143C22⋊C8C2×D711yescharacteristic
144C4×C8—72no—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7#13512yes—
151C22×C7C22×D411yescharacteristic
152C4×C7—21no—
153C4×C7—41no—
154C4×C7—42no—
155C4×C72−1+412yes—
156Dic7—21no—
157Dic7—21no—
158Dic7—21no—
159Dic7—21no—
160Dic7—41no—
161Dic7—42no—
162Dic7—41no—
163C2×Q8—71no—
164C2×D4—21no—
165C2×D4C22×D711yescharacteristic
166C22×C4—21no—
167C22×C4—71no—
168C22×C4—71no—
169C22×C4—141no—
170C22×C4—141no—
171C22×C4C22×D711yescharacteristic
172C2×C8—72no—
173C2×C8C22×D712yes—
174C4⋊C4—21no—
175C4⋊C4—72no—
176C4⋊C4—71no—
177C4⋊C4—142no—
178C4⋊C4—142no—
179C4⋊C4—141no—
180C4⋊C4—141no—
181C4⋊C4—142no—
182C4⋊C4—142no—
183C4⋊C4—141no—
184C4⋊C4—141no—
185C4⋊C4—141no—
186C4⋊C4C22×D712yes—
187C4⋊C4C22×D711yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—141no—
190C22⋊C4—141no—
191C22⋊C4—141no—
192C22⋊C4—141no—
193C42—71no—
194C42—71no—
195C42—71no—
196C42—71no—
197C42—141no—
198C2×C7—22no—
199C2×C7—81no—
200C2×C7#53611yescharacteristic
201C2×C7C2×#13511yescharacteristic
202C2×C7#57511yescharacteristic
203C23—21no—
204C23D4×D711yescharacteristic
205Q8—142no—
206D4—22no—
207D4—42no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—71no—
213C2×C4—71no—
214C2×C4—71no—
215C2×C4—71no—
216C2×C4—142no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—142no—
220C2×C4—281no—
221C2×C4—282no—
222C2×C4—282no—
223C2×C4D4×D711yescharacteristic
224C2×C4C23×D711yescharacteristic
225C8—22no—
226C8—142no—
227C7#323211yescharacteristic, complemented, Sylow
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22#647912yes—
234C22C2×D4×D711yescharacteristic
235C4—21no—
236C4—41no—
237C4—42no—
238C4—141no—
239C4—141no—
240C4—141no—
241C4—141no—
242C4—282no—
243C4—281no—
244C4—281no—
245C4#655412yes—
246C2—22no—
247C2—81no—
248C2#9033911yescharacteristic
249C2C2×#647911yescharacteristic
250C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#481D7
hypocenterC7#3232
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#6453C2semidirect
5. #89742C2semidirect
6. #89630C2semidirect
7. #89629C2semidirect
8. #89540C2semidirect
9. #89535C2semidirect
10. #89493C2semidirect
11. #89490C2semidirect
227. C7#3232semidirect

Maximal subgroups

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

subgroupindex
#902912
C2×#64532
C7×#4812
#898052
#897422
#896302
#896292
#895402
#895352
#894932
#894902
#32327