Finite Group Databasebeta

← #11738506

Subgroups — #11738506

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×#1159C211yesmaximal, characteristic
3C7×#476C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89780C211yesmaximal, characteristic, complemented
6#89633C212yesmaximal, complemented
7#89626C212yesmaximal, complemented
8#89622C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89543C212yesmaximal
11#89531C212yesmaximal, complemented
12C2×Q8×D7—21no—
13C2×#1159C2211yescharacteristic
14C7×C4⋊Q8C2211yescharacteristic
15C4×C7×D4C2211yescharacteristic
16#6507—21no—
17D4×Dic7C2211yescharacteristic
18C4×C22⋊D7—21no—
19C2×C4×Dic7—21no—
20C2×#1131C2212yes—
21#6455C2211yescharacteristic
22D7×C4⋊C4—21no—
23D7×C4⋊C4C2212yes—
24#6451—21no—
25#6451C2212yes—
26#6450C2211yescharacteristic
27(C22⋊D7)⋊C4—21no—
28#6440—21no—
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C4×Dic14C2211yescharacteristic
32C7×#106C2212yes—
33C7×C4⋊C8C2211yescharacteristic
34C7×#102C2212yes—
35#6406C2212yes—
36#6392C2212yes—
37#6390C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3079—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×D4C2311yescharacteristic
46C22×C4×C7—21no—
47C2×C22⋊D7—21no—
48C22×Dic7—21no—
49C22×Dic7—21no—
50Q8×D7—44no—
51#1159—22no—
52#1159Q814yescomplemented
53C2×C4×D7—22no—
54C2×C4×D7D411yescharacteristic
55C2×C4×D7D411yescharacteristic
56C2×C4×D7C2311yescharacteristic
57C2×Dic14—21no—
58C2×Dic14—22no—
59C2×Dic14C2311yescharacteristic
60C2×C7×C8C2312yes—
61C7×C4⋊C4—21no—
62C7×C4⋊C4C2312yes—
63C7×C4⋊C4C2311yescharacteristic
64C7×C22⋊C4—21no—
65C42×C7C2311yescharacteristic
66C22⋊Dic7—21no—
67(C2×D7)⋊C4—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7C2312yes—
70C4⋊Dic7C2311yescharacteristic
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74Dic7⋊C4—22no—
75C4×Dic7—21no—
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#477—71no—
87#476—71no—
88#476D711yescharacteristic, complemented
89#475—72no—
90#474—72no—
91#428—72no—
92#423—71no—
93#417—72no—
94C23×C7—21no—
95C22×D7C2×D411yescharacteristic
96C7×Q8—42no—
97C7×D4—21no—
98C7×D4C2×Q812yescomplemented
99C2×C4×C7—21no—
100C2×C4×C7—22no—
101C2×C4×C7—42no—
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2×D411yescharacteristic
104C2×C4×C7C2411yescharacteristic
105C22⋊D7—24no—
106C2×Dic7—21no—
107C2×Dic7—24no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—42no—
111C2×Dic7C2×D411yescharacteristic
112C2×Dic7C2×D411yescharacteristic
113C2×Dic7C2×D411yescharacteristic
114C4×D7—22no—
115C4×D7—22no—
116C4×D7—44no—
117C4×D7C2×Q812yes—
118C4×D7#4112yes—
119Dic14—21no—
120Dic14—42no—
121Dic14—44no—
122Dic14C2×Q812yescomplemented
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×Q8—71no—
135C4×D4—71no—
136C4×D4—141no—
137C4×D4—141no—
138C4×D4C2×D711yescharacteristic
139#117—71no—
140#117—141no—
141C2×C4⋊C4—72no—
142C2×C4⋊C4—141no—
143C2×C42—71no—
144C2×C42—141no—
145#107—72no—
146#107—72no—
147#106—72no—
148#106C2×D712yes—
149C4⋊C8—72no—
150C4⋊C8—71no—
151C4⋊C8C2×D711yescharacteristic
152#103—72no—
153#103—72no—
154#102—72no—
155#102C2×D712yes—
156C22×C7—22no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7C22×D411yescharacteristic
160C2×D7—21no—
161C2×D7—21no—
162C2×D7C22⋊Q812yes—
163C4×C7—21no—
164C4×C7—42no—
165C4×C7—41no—
166C4×C7#13712yes—
167C4×C7C22×Q811yescharacteristic
168C4×C7C2×#4111yescharacteristic
169Dic7—22no—
170Dic7—21no—
171Dic7—42no—
172Dic7—41no—
173Dic7C22⋊Q812yes—
174Dic7AGL(1,ℤ/8)12yes—
175#41—74no—
176#41—142no—
177C2×Q8—21no—
178C2×Q8—71no—
179C2×Q8—142no—
180C2×Q8—141no—
181C2×Q8—284no—
182C2×D4—141no—
183C2×D4C22×D711yescharacteristic
184C22×C4—21no—
185C22×C4—71no—
186C22×C4—71no—
187C22×C4—71no—
188C22×C4—142no—
189C22×C4—141no—
190C22×C4—141no—
191M16—144no—
192C2×C8—72no—
193C2×C8C22×D712yes—
194C4⋊C4—21no—
195C4⋊C4—72no—
196C4⋊C4—71no—
197C4⋊C4—142no—
198C4⋊C4—142no—
199C4⋊C4—141no—
200C4⋊C4—141no—
201C4⋊C4—141no—
202C4⋊C4C22×D712yes—
203C4⋊C4C22×D711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—141no—
206C22⋊C4—141no—
207C42—71no—
208C42—71no—
209C42—71no—
210C42—141no—
211C42—141no—
212C42—141no—
213C42C22×D711yescharacteristic
214C2×C7—42no—
215C2×C7C2×C22⋊Q811yescharacteristic
216C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
217C2×C7C2×#13711yescharacteristic
218D7—22no—
219C23—21no—
220C23—71no—
221Q8—42no—
222Q8—72no—
223Q8—141no—
224Q8—282no—
225Q8—284no—
226D4—21no—
227D4—144no—
228D4Q8×D712yescomplemented
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—71no—
233C2×C4—71no—
234C2×C4—72no—
235C2×C4—72no—
236C2×C4—71no—
237C2×C4—142no—
238C2×C4—144no—
239C2×C4—141no—
240C2×C4—142no—
241C2×C4—141no—
242C2×C4—142no—
243C2×C4—284no—
244C2×C4—282no—
245C2×C4D4×D711yescharacteristic
246C2×C4D4×D711yescharacteristic
247C2×C4C23×D711yescharacteristic
248C8—22no—
249C8—142no—
250C7#307911yescharacteristic, complemented, Sylow
251C22—22no—
252C22—41no—
253C22—41no—
254C22—72no—
255C22—141no—
256C22—141no—
257C22C2×D4×D711yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—72no—
262C4—72no—
263C4—141no—
264C4—142no—
265C4—282no—
266C4—281no—
267C4#647812yes—
268C4C2×Q8×D711yescharacteristic
269C4D7×#4111yescharacteristic
270C2—42no—
271C2—142no—
272C2D7×C22⋊Q811yescharacteristic
273C2C2×#647811yescharacteristic
274C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#476C2
socleC22×C7C22×D4
prime radicalC2×#1159C22
hypercenter#476D7
hypocenterC7#3079
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. D7×C4⋊Q8C2semidirect
3. C7×#476C2semidirect
4. #89863C2semidirect
5. #89780C2semidirect
6. #89633C2semidirect
7. #89626C2semidirect
8. #89622C2semidirect
9. #89609C2semidirect
11. #89531C2semidirect
52. #1159Q8semidirect
88. #476D7semidirect
98. C7×D4C2×Q8semidirect
122. Dic14C2×Q8semidirect
228. D4Q8×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×#11592
C7×#4762
#898632
#897802
#896332
#896262
#896222
#896092
#895432
#895312
#30797