Finite Group Databasebeta

← #11737880

Subgroups — #11737880

Order 896.

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

normal subgroups only

This group has 386 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
1D7×C4⋊Q8C211yesmaximal, characteristic, complemented
2#90248C211yesmaximal, characteristic
3C7×#461C211yesmaximal, characteristic, complemented
4#89942C212yesmaximal
5#89939C212yesmaximal, complemented
6#89862C211yesmaximal, characteristic, complemented
7#89791C211yesmaximal, characteristic, complemented
8#89621C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic, complemented
10#89580C212yesmaximal, complemented
11#89569C212yesmaximal, complemented
12C2×Q8×D7C2212yes—
13C2×C7×SD16C2212yes—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×D4C2211yescharacteristic
16#6507C2212yes—
17C2×#1144C2212yes—
18#6501C2211yescharacteristic
19#6499C2211yescharacteristic
20C2×#1142C2212yes—
21C4×C22⋊D7—21no—
22#6489—21no—
23C2×Dic28C2212yes—
24#6460C2211yescharacteristic
25D7×C4⋊C4—21no—
26D7×C4⋊C4—21no—
27#6453C2211yescharacteristic
28#6451—21no—
29#6451—21no—
30#6446—21no—
31#6442—21no—
32C42×D7C2211yescharacteristic
33#6432C2211yescharacteristic
34C4×Dic14C2211yescharacteristic
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6409C2212yes—
38(C2×D7)⋊C8C2212yes—
39#6390C2212yes—
40#6389C2212yes—
41#6384C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3218—71nomaximal, Sylow
44C2×C7×Q8C2312yes—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48Q8×D7—42no—
49Q8×D7—42no—
50C2×C4×D7—22no—
51C2×C4×D7D411yescharacteristic
52C2×C4×D7D411yescharacteristic
53C2×C4×D7C2311yescharacteristic
54C2×Dic14—22no—
55C2×Dic14C2311yescharacteristic
56C2×Dic14C2312yes—
57C7×SD16—42no—
58C2×C7×C8C2312yes—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4C2311yescharacteristic
62C7×C22⋊C4—21no—
63C42×C7C2311yescharacteristic
64C22⋊Dic7—21no—
65C22⋊Dic7—21no—
66C22⋊Dic7—21no—
67#1144—42no—
68#1142—42no—
69(C2×D7)⋊C4—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7—21no—
72C4⋊Dic7—21no—
73C4⋊Dic7—21no—
74C4⋊Dic7C2311yescharacteristic
75Dic7⋊C4—21no—
76Dic7⋊C4—21no—
77Dic7⋊C4—22no—
78Dic7⋊C4—22no—
79C4×Dic7—21no—
80C4×Dic7D411yescharacteristic
81C4×Dic7D411yescharacteristic
82C4×Dic7C2311yescharacteristic
83C2×C7⋊C8C2312yes—
84Dic28—42no—
85C2×C4⋊Q8—71no—
86#529—71no—
87#497—72no—
88#491—72no—
89#462—71no—
90#462—71no—
91#461—71no—
92#461D711yescharacteristic, complemented
93#451—72no—
94#448—72no—
95#432—71no—
96C23×C7—21no—
97C22×D7C2×D411yescharacteristic
98C7×Q8—22no—
99C7×Q8—22no—
100C7×D4—21no—
101C7×D4—21no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—42no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2411yescharacteristic
108C22⋊D7—42no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—22no—
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C2×Dic7C2×D411yescharacteristic
115C4×D7—22no—
116C4×D7—22no—
117C4×D7—21no—
118C4×D7—21no—
119C4×D7—44no—
120Dic14—21no—
121Dic14—22no—
122Dic14—22no—
123Dic14—21no—
124Dic14—44no—
125C7×C8—22no—
126C7⋊C8—22no—
127C22×Q8—72no—
128C2×Q16—72no—
129C2×Q16—72no—
130C2×SD16—72no—
131C2×SD16C2×D712yes—
132C4⋊Q8—72no—
133C4⋊Q8—71no—
134C4⋊Q8—141no—
135C4⋊Q8—141no—
136C4⋊Q8C2×D711yescharacteristic
137#126—141no—
138#125—71no—
139#124—71no—
140#123—141no—
141C22⋊Q8—71no—
142#121—71no—
143C4×Q8—71no—
144C4×D4—141no—
145C4×D4C2×D711yescharacteristic
146#117—141no—
147C2×C4⋊C4—141no—
148C2×C4⋊C4—141no—
149C2×C42—71no—
150C4⋊C8—71no—
151C4⋊C8C2×D711yescharacteristic
152#103—72no—
153#103—72no—
154#103—72no—
155#103C2×D712yes—
156C22⋊C8—72no—
157#97—72no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7C22×D411yescharacteristic
162C2×D7—21no—
163C2×D7—21no—
164C2×D7#13712yes—
165C4×C7—21no—
166C4×C7—42no—
167C4×C7—41no—
168C4×C7#13712yes—
169C4×C7H(5,𝔽2)11yescharacteristic
170C4×C7H(5,𝔽2)11yescharacteristic
171Dic7—21no—
172Dic7—21no—
173Dic7—21no—
174Dic7—42no—
175Dic7—41no—
176Dic7—41no—
177C2×Q8—72no—
178C2×Q8—71no—
179C2×Q8—142no—
180C2×Q8—282no—
181C2×Q8—282no—
182C2×Q8C22×D712yes—
183C2×D4—141no—
184C2×D4C22×D711yescharacteristic
185C22×C4—21no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—71no—
189C22×C4—142no—
190Q16—282no—
191Q16—282no—
192SD16—42no—
193SD16—282no—
194C2×C8—72no—
195C2×C8C22×D712yes—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—71no—
199C4⋊C4—142no—
200C4⋊C4—142no—
201C4⋊C4—141no—
202C4⋊C4—141no—
203C4⋊C4—141no—
204C4⋊C4—141no—
205C4⋊C4—141no—
206C4⋊C4—141no—
207C4⋊C4C22×D711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—141no—
210C22⋊C4—141no—
211C22⋊C4—141no—
212C22⋊C4—141no—
213C42—71no—
214C42—71no—
215C42—71no—
216C42—141no—
217C42C22×D711yescharacteristic
218C2×C7—81no—
219C2×C7#53511yescharacteristic
220C2×C7C2×#13711yescharacteristic
221C2×C7C2×#13711yescharacteristic
222D7—22no—
223C23—21no—
224C23—71no—
225Q8—22no—
226Q8—22no—
227Q8—141no—
228Q8—142no—
229Q8—142no—
230Q8—141no—
231Q8—284no—
232D4—21no—
233D4—21no—
234D4—282no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—71no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—142no—
242C2×C4—142no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—284no—
249C2×C4D4×D711yescharacteristic
250C2×C4D4×D711yescharacteristic
251C2×C4C23×D711yescharacteristic
252C8—22no—
253C8—142no—
254C7#321811yescharacteristic, complemented, Sylow
255C22—41no—
256C22—41no—
257C22—41no—
258C22—72no—
259C22—141no—
260C22—141no—
261C22C2×D4×D711yescharacteristic
262C4—21no—
263C4—41no—
264C4—42no—
265C4—141no—
266C4—141no—
267C4—141no—
268C4—281no—
269C4—282no—
270C4—281no—
271C4#647812yes—
272C4#654811yescharacteristic
273C4#655311yescharacteristic
274C2—81no—
275C2—142no—
276C2#9029811yescharacteristic
277C2C2×#647811yescharacteristic
278C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenter#461D7
hypocenterC7#3218
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. D7×C4⋊Q8C2semidirect
3. C7×#461C2semidirect
5. #89939C2semidirect
6. #89862C2semidirect
7. #89791C2semidirect
8. #89621C2semidirect
9. #89607C2semidirect
10. #89580C2semidirect
11. #89569C2semidirect
92. #461D7semidirect
254. C7#3218semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
#902482
C7×#4612
#899422
#899392
#898622
#897912
#896212
#896072
#895802
#895692
#32187