Finite Group Databasebeta

← #11738554

Subgroups — #11738554

Order 896.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C4×#1161C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89793C211yesmaximal, characteristic, complemented
6#89662C212yesmaximal, complemented
7#89653C212yesmaximal, complemented
8#89614C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89575C212yesmaximal
11#89563C212yesmaximal, complemented
12C2×#1161C2211yescharacteristic
13C2×Q8×D7—21no—
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×Q8C2211yescharacteristic
16Q8×Dic7C2211yescharacteristic
17#6507—21no—
18C2×#1131C2212yes—
19#6456—21no—
20#6456C2211yescharacteristic
21#6455—21no—
22#6455C2211yescharacteristic
23D7×C4⋊C4—21no—
24D7×C4⋊C4C2212yes—
25#6451—21no—
26#6451C2212yes—
27C4×D28—21no—
28C4×D28C2211yescharacteristic
29C42×D7—21no—
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic
32C7×#107C2212yes—
33C7×C4⋊C8C2211yescharacteristic
34C7×#103C2212yes—
35#6409C2212yes—
36#6395C2212yes—
37#6391C2212yes—
38Dic7⋊C8C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3079—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46#1161—22no—
47#1161Q814yescomplemented
48Q8×D7—44no—
49C2×D28—21no—
50C2×D28C2311yescharacteristic
51C2×C4×D7—21no—
52C2×C4×D7—22no—
53C2×C4×D7—21no—
54C2×C4×D7—21no—
55C2×C4×D7D411yescharacteristic
56C2×C4×D7D411yescharacteristic
57C2×C4×D7C2311yescharacteristic
58C2×Dic14—21no—
59C2×Dic14—22no—
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4C2312yes—
64C7×C4⋊C4C2311yescharacteristic
65C42×C7—21no—
66C42×C7C2311yescharacteristic
67(C2×D7)⋊C4—21no—
68(C2×D7)⋊C4—21no—
69(C2×D7)⋊C4—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7—21no—
72C4⋊Dic7C2312yes—
73C4⋊Dic7C2311yescharacteristic
74Dic7⋊C4—22no—
75Dic7⋊C4—22no—
76C4×Dic7—21no—
77C4×Dic7—21no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82#1131—24no—
83C2×C4⋊Q8—71no—
84C4×#41—71no—
85#477—71no—
86#477D711yescharacteristic, complemented
87#476—71no—
88#476—71no—
89#475—72no—
90#474—72no—
91#428—72no—
92#423—71no—
93#417—72no—
94C22×D7—21no—
95C22×D7C2×D411yescharacteristic
96C7×Q8—21no—
97C7×Q8—42no—
98C7×Q8C2×Q812yescomplemented
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—22no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110D28—24no—
111D28—21no—
112D28C2×Q812yescomplemented
113C4×D7—22no—
114C4×D7—24no—
115C4×D7—22no—
116C4×D7—44no—
117C4×D7—42no—
118C4×D7—42no—
119C4×D7C2×Q812yes—
120C4×D7#4112yes—
121Dic14—42no—
122Dic14—44no—
123C7×C8—22no—
124C7⋊C8—22no—
125C2×#41—71no—
126C22×Q8—141no—
127C2×M16—72no—
128C4⋊Q8—72no—
129C4⋊Q8—71no—
130C4⋊Q8—141no—
131C4⋊Q8—141no—
132C4⋊Q8C2×D711yescharacteristic
133C4×Q8—71no—
134C4×Q8C2×D711yescharacteristic
135C4×D4—71no—
136C4×D4—71no—
137C4×D4—141no—
138C4×D4—141no—
139#117—71no—
140#117—141no—
141C2×C4⋊C4—72no—
142C2×C4⋊C4—141no—
143C2×C42—71no—
144C2×C42—141no—
145#107—72no—
146#107C2×D712yes—
147#106—72no—
148#106—72no—
149C4⋊C8—72no—
150C4⋊C8—71no—
151C4⋊C8C2×D711yescharacteristic
152#103—72no—
153#103C2×D712yes—
154#102—72no—
155#102—72no—
156C22×C7C22×D411yescharacteristic
157C2×D7—21no—
158C2×D7—21no—
159C2×D7—22no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7C22⋊Q812yes—
163C4×C7—22no—
164C4×C7—21no—
165C4×C7—42no—
166C4×C7—41no—
167C4×C7#13712yes—
168C4×C7C22×Q811yescharacteristic
169C4×C7C2×#4111yescharacteristic
170Dic7—21no—
171Dic7—42no—
172Dic7—41no—
173Dic7C22⋊Q812yes—
174Dic7AGL(1,ℤ/8)12yes—
175#41—74no—
176#41—142no—
177C2×Q8—21no—
178C2×Q8—142no—
179C2×Q8—141no—
180C2×Q8—284no—
181C2×Q8C22×D711yescharacteristic
182C2×D4—71no—
183C2×D4—141no—
184C22×C4—71no—
185C22×C4—71no—
186C22×C4—71no—
187C22×C4—141no—
188C22×C4—141no—
189C22×C4—142no—
190C22×C4—141no—
191M16—144no—
192C2×C8—72no—
193C2×C8C22×D712yes—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—72no—
197C4⋊C4—71no—
198C4⋊C4—142no—
199C4⋊C4—142no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4C22×D712yes—
203C4⋊C4C22×D711yescharacteristic
204C22⋊C4—141no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C42—21no—
208C42—71no—
209C42—71no—
210C42—71no—
211C42—141no—
212C42—141no—
213C42C22×D711yescharacteristic
214C2×C7C2×C22⋊Q811yescharacteristic
215C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C7C2×#13711yescharacteristic
217D7—22no—
218D7—42no—
219C23—71no—
220C23—141no—
221Q8—21no—
222Q8—42no—
223Q8—282no—
224Q8—284no—
225Q8Q8×D712yescomplemented
226D4—72no—
227D4—144no—
228D4—141no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—71no—
233C2×C4—71no—
234C2×C4—72no—
235C2×C4—72no—
236C2×C4—71no—
237C2×C4—142no—
238C2×C4—144no—
239C2×C4—142no—
240C2×C4—141no—
241C2×C4—142no—
242C2×C4—282no—
243C2×C4—284no—
244C2×C4—282no—
245C2×C4D4×D711yescharacteristic
246C2×C4D4×D711yescharacteristic
247C2×C4C23×D711yescharacteristic
248C8—22no—
249C8—142no—
250C7#307911yescharacteristic, complemented, Sylow
251C22—72no—
252C22—141no—
253C22—141no—
254C22—142no—
255C22—281no—
256C22—281no—
257C22C2×D4×D711yescharacteristic
258C4—21no—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—72no—
263C4—72no—
264C4—141no—
265C4—281no—
266C4—282no—
267C4#648112yes—
268C4C2×Q8×D711yescharacteristic
269C4D7×#4111yescharacteristic
270C2—142no—
271C2—282no—
272C2D7×C22⋊Q811yescharacteristic
273C2C2×#648111yescharacteristic
274C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#477D7
hypocenterC7#3079
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#477C2semidirect
4. #89856C2semidirect
5. #89793C2semidirect
6. #89662C2semidirect
7. #89653C2semidirect
8. #89614C2semidirect
9. #89609C2semidirect
11. #89563C2semidirect
47. #1161Q8semidirect
86. #477D7semidirect
98. C7×Q8C2×Q8semidirect
112. D28C2×Q8semidirect
225. Q8Q8×D7semidirect
250. C7#3079semidirect

Maximal subgroups

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

subgroupindex
D7×C4⋊Q82
C4×#11612
C7×#4772
#898562
#897932
#896622
#896532
#896142
#896092
#895752
#895632
#30797