Finite Group Databasebeta

← #11738552

Subgroups — #11738552

Order 896.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C4×#1161C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89792C211yesmaximal, characteristic, complemented
6#89663C212yesmaximal, complemented
7#89652C212yesmaximal, complemented
8#89618C211yesmaximal, characteristic, complemented
9D7×C4⋊C8C211yesmaximal, characteristic, complemented
10#89576C212yesmaximal
11#89568C212yesmaximal, complemented
12C2×#1161C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509—21no—
16Q8×Dic7—21no—
17Q8×Dic7C2211yescharacteristic
18C2×C8×D7C2212yes—
19#6460—21no—
20#6456—21no—
21#6456C2211yescharacteristic
22#6455—21no—
23#6455C2212yes—
24#6455C2211yescharacteristic
25#6453C2212yes—
26#6450—21no—
27C4×D28—21no—
28C4×D28C2211yescharacteristic
29C42×D7—21no—
30C42×D7C2211yescharacteristic
31#6432C2211yescharacteristic
32C7×#107C2212yes—
33C7×C4⋊C8C2211yescharacteristic
34C7×#103C2212yes—
35#6409C2212yes—
36#6395C2212yes—
37#6392C2212yes—
38Dic7⋊C8C2212yes—
39#6383C2212yes—
40#6382C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3076—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46#1161—22no—
47#1161Q814yescomplemented
48C2×D28—21no—
49C2×D28C2311yescharacteristic
50C2×C4×D7—21no—
51C2×C4×D7—21no—
52C2×C4×D7—21no—
53C2×C4×D7D411yescharacteristic
54C2×C4×D7D411yescharacteristic
55C2×C4×D7C2311yescharacteristic
56C2×Dic14—21no—
57C2×C7×C8C2312yes—
58C7×C4⋊C4—21no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2312yes—
61C7×C4⋊C4C2311yescharacteristic
62C42×C7—21no—
63C42×C7C2311yescharacteristic
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—22no—
66(C2×D7)⋊C4—21no—
67(C2×D7)⋊C4—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—22no—
71C4⋊Dic7C2312yes—
72C4⋊Dic7C2311yescharacteristic
73Dic7⋊C4—22no—
74C4×Dic7—21no—
75C4×Dic7—21no—
76C4×Dic7—22no—
77C4×Dic7D411yescharacteristic
78C4×Dic7D411yescharacteristic
79C4×Dic7C2311yescharacteristic
80C2×C7⋊C8C2312yes—
81C8×D7—24no—
82#533—71no—
83C4×#41—71no—
84#479—72no—
85#478—72no—
86#477D711yescharacteristic, complemented
87#476—71no—
88#475—71no—
89#474—71no—
90#427—72no—
91C2×C4⋊C8—71no—
92#416—72no—
93C22×D7—21no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—21no—
96C7×Q8—42no—
97C7×Q8C2×Q812yes—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2411yescharacteristic
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109D28—24no—
110D28—21no—
111D28C2×Q812yes—
112C4×D7—22no—
113C4×D7—24no—
114C4×D7—41no—
115C4×D7—42no—
116C4×D7—42no—
117C4×D7C2×Q812yes—
118C4×D7#4112yes—
119Dic14—42no—
120C7×C8—22no—
121C7⋊C8—22no—
122C2×#41—71no—
123C22×C8—72no—
124C4⋊Q8—71no—
125C4⋊Q8C2×D711yescharacteristic
126#125—72no—
127C22⋊Q8—141no—
128C22⋊Q8—141no—
129C4×Q8—71no—
130C4×Q8—141no—
131C4×Q8—141no—
132C4×Q8C2×D711yescharacteristic
133C4×D4—71no—
134C4×D4—71no—
135C4×D4—141no—
136C4×D4—141no—
137#117—71no—
138#117—72no—
139#117—141no—
140C2×C42—71no—
141C2×C42—141no—
142#107—72no—
143#107C2×D712yes—
144#106—72no—
145#106—72no—
146C4⋊C8—72no—
147C4⋊C8—71no—
148C4⋊C8C2×D711yescharacteristic
149#103—72no—
150#103C2×D712yes—
151#102—72no—
152#102—72no—
153C22×C7C22×D411yescharacteristic
154C2×D7—21no—
155C2×D7—21no—
156C2×D7—22no—
157C2×D7—41no—
158C2×D7—41no—
159C2×D7C22⋊Q812yes—
160C4×C7—22no—
161C4×C7—21no—
162C4×C7—41no—
163C4×C7—42no—
164C4×C7#13512yes—
165C4×C7C22×Q811yescharacteristic
166C4×C7C2×#4111yescharacteristic
167Dic7—21no—
168Dic7—21no—
169Dic7—42no—
170Dic7—41no—
171Dic7C22⋊Q812yes—
172#41—74no—
173#41—142no—
174C2×Q8—21no—
175C2×Q8—141no—
176C2×Q8C22×D711yescharacteristic
177C2×D4—71no—
178C2×D4—141no—
179C22×C4—71no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—141no—
183C22×C4—141no—
184C22×C4—141no—
185C2×C8—72no—
186C2×C8—144no—
187C2×C8C22×D712yes—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—72no—
191C4⋊C4—71no—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C4⋊C4—142no—
195C4⋊C4—141no—
196C4⋊C4C22×D712yes—
197C4⋊C4C22×D711yescharacteristic
198C22⋊C4—141no—
199C22⋊C4—141no—
200C22⋊C4—142no—
201C22⋊C4—141no—
202C42—21no—
203C42—71no—
204C42—71no—
205C42—71no—
206C42—141no—
207C42—141no—
208C42—142no—
209C42C22×D711yescharacteristic
210C2×C7C2×C22⋊Q811yescharacteristic
211C2×C7C2×#13511yescharacteristic
212C2×C7#57511yescharacteristic
213D7—22no—
214D7—42no—
215C23—71no—
216C23—141no—
217Q8—21no—
218Q8—42no—
219Q8—282no—
220Q8Q8×D712yes—
221D4—72no—
222D4—144no—
223D4—141no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—71no—
228C2×C4—72no—
229C2×C4—72no—
230C2×C4—71no—
231C2×C4—71no—
232C2×C4—144no—
233C2×C4—142no—
234C2×C4—141no—
235C2×C4—142no—
236C2×C4—281no—
237C2×C4—282no—
238C2×C4—282no—
239C2×C4D4×D711yescharacteristic
240C2×C4D4×D711yescharacteristic
241C2×C4C23×D711yescharacteristic
242C8—22no—
243C8—142no—
244C7#307611yescharacteristic, complemented, Sylow
245C22—72no—
246C22—141no—
247C22—141no—
248C22—142no—
249C22—281no—
250C22—281no—
251C22C2×D4×D711yescharacteristic
252C4—21no—
253C4—22no—
254C4—41no—
255C4—42no—
256C4—72no—
257C4—141no—
258C4—141no—
259C4—281no—
260C4—282no—
261C4#648212yes—
262C4C2×Q8×D711yescharacteristic
263C4D7×#4111yescharacteristic
264C2—142no—
265C2—282no—
266C2D7×C22⋊Q811yescharacteristic
267C2C2×#648211yescharacteristic
268C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#477D7
hypocenterC7#3076
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. #90419C2semidirect
3. C7×#477C2semidirect
4. #89856C2semidirect
5. #89792C2semidirect
6. #89663C2semidirect
7. #89652C2semidirect
8. #89618C2semidirect
9. D7×C4⋊C8C2semidirect
11. #89568C2semidirect
47. #1161Q8semidirect
86. #477D7semidirect
244. C7#3076semidirect

Maximal subgroups

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

subgroupindex
#904192
C4×#11612
C7×#4772
#898562
#897922
#896632
#896522
#896182
D7×C4⋊C82
#895762
#895682
#30767