Finite Group Databasebeta

← #11732283

Subgroups — #11732283

Order 896.

464 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C7×#386C211yesmaximal, characteristic
2#89993C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89984C212yesmaximal, complemented
5#89722C212yesmaximal, complemented
6#89721C212yesmaximal, complemented
7#89720C212yesmaximal, complemented
8#89433C212yesmaximal, complemented
9#89426C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C7×#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2748—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C24×C7C2311yescharacteristic
34C2×C7×D4—41no—
35C2×C7×D4—41no—
36C2×C7×D4Q814yes—
37C22×C4×C7—24no—
38C22×C4×C7—41no—
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×Dic7—41no—
43C22×Dic7—42no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C7×C22⋊C4—24no—
49C7×C22⋊C4—41no—
50C22⋊Dic7—41no—
51C22⋊Dic7—42no—
52C22⋊Dic7—42no—
53C22⋊Dic7—42no—
54C4⋊Dic7—24no—
55C4⋊Dic7—41no—
56C4⋊Dic7—42no—
57C4⋊Dic7—42no—
58Dic7⋊C4—42no—
59Dic7⋊C4—42no—
60C4×Dic7—41no—
61#399—72no—
62#398—72no—
63#396—72no—
64#390—71no—
65#387—72no—
66#386—71no—
67#386D711yescharacteristic
68#385—72no—
69#380—72no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—22no—
74C23×C7—21no—
75C23×C7—21no—
76C23×C7—21no—
77C23×C7—21no—
78C23×C7C2×Q812yes—
79C23×C7C2×Q812yes—
80C23×C7C2411yescharacteristic
81C7×D4—44no—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—44no—
90C2×C4×C7C2×Q812yes—
91C2×C4×C7#4114yes—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—44no—
105C22×D4C2×D711yescharacteristic
106C23×C4—71no—
107C23×C4C2×D711yescharacteristic
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C22⋊C4—71no—
118C2×C22⋊C4C2×D712yes—
119C2×C22⋊C4C2×D711yescharacteristic
120C2×C42—71no—
121#95—72no—
122#95—72no—
123#95—72no—
124#95—72no—
125#95C2×D712yes—
126C22×C7—22no—
127C22×C7—22no—
128C22×C7—24no—
129C22×C7—24no—
130C22×C7—42no—
131C22×C7—42no—
132C22×C7—41no—
133C22×C7—41no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7C22×Q811yescharacteristic
137C22×C7C2×#4112yes—
138C22×C7H(5,𝔽2)11yescharacteristic
139C22×C7H(5,𝔽2)11yescharacteristic
140C22×C7H(5,𝔽2)12yes—
141C4×C7—22no—
142C4×C7—24no—
143C4×C7—42no—
144Dic7—42no—
145Dic7—42no—
146Dic7—42no—
147Dic7—42no—
148C24C22×D711yescharacteristic
149C24C22×D711yescharacteristic
150C2×D4—41no—
151C2×D4—41no—
152C2×D4Dic1414yes—
153C22×C4—24no—
154C22×C4—41no—
155C22×C4—72no—
156C22×C4—72no—
157C22×C4—72no—
158C22×C4—72no—
159C22×C4—281no—
160C22×C4—282no—
161C22×C4C22×D712yes—
162C22×C4C22×D711yescharacteristic
163C22×C4C22×D712yes—
164C4⋊C4—144no—
165C4⋊C4—282no—
166C4⋊C4—282no—
167C4⋊C4—282no—
168C4⋊C4—281no—
169C4⋊C4—282no—
170C22⋊C4—24no—
171C22⋊C4—41no—
172C22⋊C4—281no—
173C22⋊C4—282no—
174C22⋊C4—282no—
175C22⋊C4—282no—
176C42—281no—
177C2×C7—42no—
178C2×C7—42no—
179C2×C7#54012yes—
180C2×C7#54311yescharacteristic
181C2×C7#55111yescharacteristic
182C2×C7#55412yes—
183C2×C7#56011yescharacteristic
184C23—22no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—22no—
192C23C2×Dic1412yes—
193C23C2×Dic1412yes—
194C23C23×D711yescharacteristic
195D4—44no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—284no—
217C2×C4C2×Dic1412yes—
218C2×C4#115914yes—
219C7#274811yescharacteristic, complemented, Sylow
220C22—22no—
221C22—22no—
222C22—24no—
223C22—24no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22C22×Dic1411yescharacteristic
231C22C2×#115912yes—
232C22#654811yescharacteristic
233C22#654811yescharacteristic
234C22#655312yes—
235C4—22no—
236C4—24no—
237C4—42no—
238C4—282no—
239C4—282no—
240C4—282no—
241C4—282no—
242C2—42no—
243C2—42no—
244C2#9017411yescharacteristic
245C2#9023412yes—
246C2#9027811yescharacteristic
247C2#9031012yes—
248C2#9035711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
prime radicalC22×C7×D4C22
hypercenter#386D7
hypocenterC7#2748
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #89984C2semidirect
5. #89722C2semidirect
6. #89721C2semidirect
7. #89720C2semidirect
8. #89433C2semidirect
9. #89426C2semidirect
219. C7#2748semidirect

Maximal subgroups

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

subgroupindex
C7×#3862
#899932
#899922
#899842
#897222
#897212
#897202
#894332
#894262
#27487