Finite Group Databasebeta

← #11737883

Subgroups — #11737883

Order 896.

392 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 121 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#90419C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic
3C7×#461C211yesmaximal, characteristic, complemented
4#89943C212yesmaximal, complemented
5#89935C212yesmaximal, complemented
6#89855C211yesmaximal, characteristic, complemented
7#89780C211yesmaximal, characteristic, complemented
8#89614C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic, complemented
10#89579C212yesmaximal, complemented
11#89563C212yesmaximal, complemented
12C2×#1157—21no—
13C2×C7×SD16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×D4C2211yescharacteristic
16#6509C2212yes—
17Q8×Dic7C2212yes—
18C2×#1143C2212yes—
19#6503C2211yescharacteristic
20#6501C2211yescharacteristic
21C4×C22⋊D7—21no—
22C2×C8×D7C2212yes—
23#6460—21no—
24#6458C2211yescharacteristic
25#6455—21no—
26#6453—21no—
27#6451C2211yescharacteristic
28#6450—21no—
29#6448—21no—
30#6447—21no—
31C4×D28C2211yescharacteristic
32C42×D7C2211yescharacteristic
33#6432C2211yescharacteristic
34C7×C4⋊C8C2211yescharacteristic
35C7×#103C2212yes—
36#6409C2212yes—
37#6406C2212yes—
38#6395C2212yes—
39#6391C2212yes—
40Dic7⋊C8C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3246—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48C2×C22⋊D7—21no—
49C2×C22⋊D7—21no—
50#1157—44no—
51C2×D28—21no—
52C2×D28C2311yescharacteristic
53C2×C4×D7—21no—
54C2×C4×D7D411yescharacteristic
55C2×C4×D7D411yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14—21no—
58C2×Dic14—21no—
59C7×SD16—42no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2311yescharacteristic
64C7×C22⋊C4—21no—
65C42×C7C2311yescharacteristic
66C22⋊Dic7—21no—
67#1143—42no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—21no—
70(C2×D7)⋊C4—21no—
71C4⋊Dic7—21no—
72C4⋊Dic7—22no—
73C4⋊Dic7C2312yes—
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—21no—
76Dic7⋊C4—22no—
77C4×Dic7—22no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82C8×D7—42no—
83#533—71no—
84#532—71no—
85#476—71no—
86#476—71no—
87#475—72no—
88#465—72no—
89#465—72no—
90#461—71no—
91#461D711yescharacteristic, complemented
92#460—72no—
93C2×C4⋊C8—71no—
94C23×C7—21no—
95C22×D7—21no—
96C22×D7C2×D411yescharacteristic
97C7×Q8—22no—
98C7×Q8—22no—
99C7×D4—21no—
100C7×D4—21no—
101C2×C4×C7—22no—
102C2×C4×C7—21no—
103C2×C4×C7—42no—
104C2×C4×C7C2×D411yescharacteristic
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2411yescharacteristic
107C22⋊D7—42no—
108C22⋊D7—42no—
109C22⋊D7—42no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115D28—21no—
116D28—21no—
117D28—42no—
118C4×D7—21no—
119C4×D7—21no—
120C4×D7—41no—
121C4×D7—42no—
122C4×D7SD1614yescomplemented
123Dic14—42no—
124Dic14—42no—
125C7×C8—22no—
126C7⋊C8—22no—
127C2×#41—141no—
128C2×SD16—72no—
129C2×SD16C2×D712yes—
130C22×C8—72no—
131C4⋊Q8—71no—
132C4⋊Q8—71no—
133C4⋊Q8C2×D711yescharacteristic
134#127—71no—
135#125—141no—
136#124—141no—
137C22⋊Q8—72no—
138C22⋊Q8—141no—
139#121—71no—
140#121—71no—
141#121—141no—
142C4×Q8—72no—
143C4×Q8—141no—
144C4×D4—71no—
145C4×D4—141no—
146C4×D4C2×D711yescharacteristic
147#117—141no—
148C2×C42—71no—
149#106—72no—
150#106—72no—
151C4⋊C8—72no—
152C4⋊C8—71no—
153C4⋊C8C2×D711yescharacteristic
154#103—72no—
155#103C2×D712yes—
156#102—72no—
157#102—72no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7C22×D411yescharacteristic
162C2×D7—21no—
163C2×D7—21no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C2×D7C2×SD1612yes—
168C4×C7—21no—
169C4×C7—42no—
170C4×C7—41no—
171C4×C7C2×SD1612yes—
172C4×C7H(5,𝔽2)11yescharacteristic
173C4×C72−1+411yescharacteristic
174Dic7—21no—
175Dic7—21no—
176Dic7—41no—
177Dic7—42no—
178Dic7C2×SD1612yes—
179#41—284no—
180C2×Q8—141no—
181C2×Q8—141no—
182C2×Q8C22×D712yes—
183C2×D4—71no—
184C2×D4—141no—
185C2×D4—141no—
186C2×D4—141no—
187C2×D4—141no—
188C2×D4C22×D711yescharacteristic
189C22×C4—21no—
190C22×C4—71no—
191C22×C4—71no—
192C22×C4—71no—
193C22×C4—141no—
194SD16—42no—
195SD16—282no—
196C2×C8—72no—
197C2×C8—282no—
198C2×C8C22×D712yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—72no—
202C4⋊C4—71no—
203C4⋊C4—141no—
204C4⋊C4—142no—
205C4⋊C4—142no—
206C4⋊C4—141no—
207C4⋊C4C22×D711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—141no—
210C22⋊C4—142no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42—142no—
217C42C22×D711yescharacteristic
218C2×C7—81no—
219C2×C7#53711yescharacteristic
220C2×C7C22×SD1611yescharacteristic
221C2×C7#57511yescharacteristic
222D7—22no—
223D7—81no—
224C23—21no—
225C23—71no—
226C23—141no—
227Q8—22no—
228Q8—22no—
229Q8—282no—
230Q8—282no—
231D4—21no—
232D4—21no—
233D4—141no—
234D4—141no—
235D4—282no—
236D4—282no—
237D4—282no—
238D4—282no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—71no—
243C2×C4—74no—
244C2×C4—71no—
245C2×C4—71no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—142no—
249C2×C4—141no—
250C2×C4—281no—
251C2×C4—282no—
252C2×C4D4×D711yescharacteristic
253C2×C4D4×D711yescharacteristic
254C2×C4C23×D711yescharacteristic
255C8—22no—
256C8—142no—
257C7#324611yescharacteristic, complemented, Sylow
258C22—41no—
259C22—41no—
260C22—41no—
261C22—72no—
262C22—141no—
263C22—141no—
264C22—281no—
265C22—281no—
266C22—281no—
267C22C2×D4×D711yescharacteristic
268C4—21no—
269C4—41no—
270C4—42no—
271C4—72no—
272C4—141no—
273C4—141no—
274C4—281no—
275C4—282no—
276C4D7×SD1612yes—
277C4#654811yescharacteristic
278C4#655411yescharacteristic
279C2—81no—
280C2—142no—
281C2—561no—
282C2#9029911yescharacteristic
283C2C2×D7×SD1611yescharacteristic
284C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
prime radical#90239C2
hypercenter#461D7
hypocenterC7#3246
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#461C2semidirect
4. #89943C2semidirect
5. #89935C2semidirect
6. #89855C2semidirect
7. #89780C2semidirect
8. #89614C2semidirect
9. D7×C4⋊C8C2semidirect
10. #89579C2semidirect
11. #89563C2semidirect
91. #461D7semidirect
122. C4×D7SD16semidirect
257. C7#3246semidirect

Maximal subgroups

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

subgroupindex
#904192
#902392
C7×#4612
#899432
#899352
#898552
#897802
#896142
D7×C4⋊C82
#895792
#895632
#32467