Finite Group Databasebeta

← #11738443

Subgroups — #11738443

Order 896.

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

normal subgroups only

This group has 380 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
2C4×Q8×D7C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic, complemented
4#89866C211yesmaximal, characteristic, complemented
5#89803C211yesmaximal, characteristic, complemented
6#89634C212yesmaximal
7#89625C212yesmaximal, complemented
8#89621C211yesmaximal, characteristic, complemented
9#89607C211yesmaximal, characteristic
10#89580C212yesmaximal
11#89561C212yesmaximal, complemented
12C2×Q8×D7C2211yescharacteristic
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×Q8C2211yescharacteristic
15#6509—21no—
16Q8×Dic7—21no—
17Q8×Dic7C2211yescharacteristic
18C2×#1144C2212yes—
19C2×Dic28C2212yes—
20#6460C2212yes—
21#6455—21no—
22D7×C4⋊C4—21no—
23D7×C4⋊C4C2211yescharacteristic
24#6453—21no—
25#6450—21no—
26#6450C2211yescharacteristic
27#6450C2212yes—
28C42×D7—21no—
29C42×D7C2211yescharacteristic
30#6432C2211yescharacteristic
31C4×Dic14—21no—
32C4×Dic14C2211yescharacteristic
33C7×#106C2212yes—
34C7×C4⋊C8C2211yescharacteristic
35C7×#103C2212yes—
36(C2×D7)⋊C8C2212yes—
37#6390C2212yes—
38#6389C2212yes—
39#6384C2212yes—
40#6384C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3113—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46Q8×D7—22no—
47Q8×D7—22no—
48C2×C4×D7—21no—
49C2×C4×D7—21no—
50C2×C4×D7D411yescharacteristic
51C2×C4×D7D411yescharacteristic
52C2×C4×D7C2311yescharacteristic
53C2×Dic14—21no—
54C2×Dic14C2312yes—
55C2×Dic14C2311yescharacteristic
56C2×C7×C8C2312yes—
57C7×C4⋊C4—21no—
58C7×C4⋊C4—21no—
59C7×C4⋊C4C2312yes—
60C7×C4⋊C4C2311yescharacteristic
61C42×C7—21no—
62C42×C7C2311yescharacteristic
63#1144—24no—
64(C2×D7)⋊C4—22no—
65C4⋊Dic7—21no—
66C4⋊Dic7—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—22no—
69C4⋊Dic7C2311yescharacteristic
70Dic7⋊C4—21no—
71Dic7⋊C4—21no—
72Dic7⋊C4—21no—
73Dic7⋊C4—22no—
74C4×Dic7—21no—
75C4×Dic7—21no—
76C4×Dic7—21no—
77C4×Dic7—22no—
78C4×Dic7D411yescharacteristic
79C4×Dic7D411yescharacteristic
80C4×Dic7C2311yescharacteristic
81C2×C7⋊C8C2312yes—
82Dic28—24no—
83#533—71no—
84C2×C4×Q8—71no—
85#483—72no—
86#475—71no—
87#475D711yescharacteristic, complemented
88#462—71no—
89#462—71no—
90#451—72no—
91#441—72no—
92#441—72no—
93#432—71no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—21no—
96C7×Q8—42no—
97C7×Q8#4112yes—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—22no—
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2411yescharacteristic
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109C2×Dic7C2×D411yescharacteristic
110C4×D7—24no—
111C4×D7—22no—
112C4×D7—22no—
113C4×D7—21no—
114C4×D7—21no—
115C4×D7—42no—
116Dic14—24no—
117Dic14—21no—
118Dic14—22no—
119Dic14—22no—
120Dic14#4112yes—
121C7×C8—22no—
122C7⋊C8—22no—
123C22×Q8—71no—
124C2×Q16—72no—
125C2×Q16—72no—
126C4⋊Q8—71no—
127C4⋊Q8C2×D711yescharacteristic
128#125—141no—
129C22⋊Q8—72no—
130C22⋊Q8—141no—
131C4×Q8—71no—
132C4×Q8—71no—
133C4×Q8—72no—
134C4×Q8—71no—
135C4×Q8—141no—
136C4×Q8—141no—
137C4×Q8—141no—
138C4×Q8C2×D711yescharacteristic
139#117—141no—
140C2×C4⋊C4—71no—
141C2×C4⋊C4—141no—
142C2×C42—71no—
143C2×C42—141no—
144#106—72no—
145#106C2×D712yes—
146C4⋊C8—71no—
147C4⋊C8C2×D711yescharacteristic
148#103—72no—
149#103—72no—
150#103—72no—
151#103C2×D712yes—
152C22⋊C8—72no—
153#97—72no—
154C22×C7C22×D411yescharacteristic
155C2×D7—21no—
156C2×D7—21no—
157C2×D7#13712yes—
158C4×C7—22no—
159C4×C7—21no—
160C4×C7—41no—
161C4×C7—42no—
162C4×C7#13712yes—
163C4×C7C2×#4111yescharacteristic
164C4×C7C2×#4111yescharacteristic
165Dic7—21no—
166Dic7—22no—
167Dic7—21no—
168Dic7—21no—
169Dic7—42no—
170Dic7—41no—
171C2×Q8—21no—
172C2×Q8—71no—
173C2×Q8—72no—
174C2×Q8—141no—
175C2×Q8—142no—
176C2×Q8—142no—
177C2×Q8C22×D711yescharacteristic
178C22×C4—71no—
179C22×C4—71no—
180C22×C4—71no—
181C22×C4—141no—
182C22×C4—141no—
183Q16—144no—
184Q16—144no—
185C2×C8—72no—
186C2×C8C22×D712yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—71no—
190C4⋊C4—141no—
191C4⋊C4—141no—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C4⋊C4—142no—
195C4⋊C4—141no—
196C4⋊C4—141no—
197C4⋊C4—141no—
198C4⋊C4C22×D711yescharacteristic
199C4⋊C4C22×D712yes—
200C22⋊C4—142no—
201C42—21no—
202C42—71no—
203C42—71no—
204C42—71no—
205C42—141no—
206C42—141no—
207C42—142no—
208C42—141no—
209C42C22×D711yescharacteristic
210C2×C7#52511yescharacteristic
211C2×C7C2×#13711yescharacteristic
212C2×C7C2×#13711yescharacteristic
213D7—22no—
214C23—71no—
215Q8—21no—
216Q8—42no—
217Q8—72no—
218Q8—144no—
219Q8—141no—
220Q8—142no—
221Q8—142no—
222Q8#116112yes—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—71no—
227C2×C4—71no—
228C2×C4—71no—
229C2×C4—142no—
230C2×C4—144no—
231C2×C4—141no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4—141no—
235C2×C4—141no—
236C2×C4—142no—
237C2×C4—282no—
238C2×C4D4×D711yescharacteristic
239C2×C4D4×D711yescharacteristic
240C2×C4C23×D711yescharacteristic
241C8—22no—
242C8—142no—
243C7#311311yescharacteristic, complemented, Sylow
244C22—72no—
245C22—141no—
246C22—141no—
247C22C2×D4×D711yescharacteristic
248C4—21no—
249C4—22no—
250C4—42no—
251C4—41no—
252C4—141no—
253C4—141no—
254C4—141no—
255C4—142no—
256C4—281no—
257C4—282no—
258C4#647812yes—
259C4C2×#116111yescharacteristic
260C4D7×#4111yescharacteristic
261C2—142no—
262C2#9031811yescharacteristic
263C2C2×#647811yescharacteristic
264C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#475D7
hypocenterC7#3113
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. C7×#475C2semidirect
4. #89866C2semidirect
5. #89803C2semidirect
7. #89625C2semidirect
8. #89621C2semidirect
11. #89561C2semidirect
87. #475D7semidirect
243. C7#3113semidirect

Maximal subgroups

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

subgroupindex
#904192
C4×Q8×D72
C7×#4752
#898662
#898032
#896342
#896252
#896212
#896072
#895802
#895612
#31137