Finite Group Databasebeta

← #11734331

Subgroups — #11734331

Order 896.

472 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 117 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×#393C211yesmaximal, characteristic, complemented
2#90003C212yesmaximal, complemented
3#89733C212yesmaximal, complemented
4#89732C211yesmaximal, characteristic
5#89730C212yesmaximal, complemented
6#89718C211yesmaximal, characteristic, complemented
7#89446C212yesmaximal, complemented
8#89444C212yesmaximal, complemented
9#89432C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C23×Dic7C2211yescharacteristic
13C22×D28C2211yescharacteristic
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C7×#95C2212yes—
25C7×#95C2211yescharacteristic
26C7⋊#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2824—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7C2312yes—
34C2×C7×Q8—42no—
35C2×C7×Q8—41no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C2×C22⋊D7—24no—
41C2×C22⋊D7—42no—
42C2×C22⋊D7—42no—
43C22×Dic7—42no—
44C22×Dic7—41no—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C2×D28—42no—
49C2×D28—41no—
50C7×C22⋊C4—42no—
51C7×C22⋊C4—41no—
52C22⋊Dic7—22no—
53(C2×D7)⋊C4—42no—
54(C2×D7)⋊C4—42no—
55(C2×D7)⋊C4—42no—
56(C2×D7)⋊C4—42no—
57(C2×D7)⋊C4—42no—
58C4⋊Dic7—42no—
59Dic7⋊C4—42no—
60#400—72no—
61#397—72no—
62#394—72no—
63#394—72no—
64#393—72no—
65#393D711yescharacteristic, complemented
66#392—71no—
67#386—72no—
68#385—71no—
69C23×C7—21no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—21no—
73C23×C7#4112yes—
74C23×C7C2411yescharacteristic
75C22×D7—22no—
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C7×Q8—44no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C22⋊D7—48no—
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—44no—
104D28—44no—
105C22×Q8C2×D711yescharacteristic
106C22×D4—72no—
107C22×D4—71no—
108C23×C4—71no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—72no—
111C2×C22⋊C4—71no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C22⋊C4C2×D711yescharacteristic
118C2×C22⋊C4C2×D712yes—
119#95—72no—
120#95—72no—
121#95—72no—
122#95—71no—
123#95—72no—
124#95C2×D712yes—
125#95C2×D711yescharacteristic
126C22×C7—24no—
127C22×C7—22no—
128C22×C7—41no—
129C22×C7—41no—
130C22×C7—42no—
131C22×C7C2×#4111yescharacteristic
132C22×C7H(5,𝔽2)12yes—
133C22×C7H(5,𝔽2)11yescharacteristic
134C22×C7H(5,𝔽2)12yes—
135C22×C72−1+411yescharacteristic
136C2×D7—42no—
137C2×D7—42no—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7—42no—
141C2×D7—42no—
142C2×D7—42no—
143C4×C7—41no—
144C4×C7—42no—
145C4×C7—41no—
146C4×C7—42no—
147Dic7—42no—
148Dic7—42no—
149Dic7—42no—
150C24—72no—
151C24C22×D711yescharacteristic
152C2×Q8—41no—
153C2×Q8—42no—
154C2×D4—144no—
155C2×D4—281no—
156C2×D4—282no—
157C2×D4—282no—
158C2×D4—282no—
159C22×C4—72no—
160C22×C4—72no—
161C22×C4—72no—
162C22×C4—281no—
163C22×C4—282no—
164C22×C4C22×D711yescharacteristic
165C22×C4C22×D712yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C4⋊C4—282no—
169C4⋊C4—282no—
170C22⋊C4—42no—
171C22⋊C4—41no—
172C22⋊C4—142no—
173C22⋊C4—282no—
174C22⋊C4—282no—
175C22⋊C4—282no—
176C22⋊C4—282no—
177C22⋊C4—282no—
178C2×C7—42no—
179C2×C7#53812yes—
180C2×C7#53911yescharacteristic
181C2×C7#56011yescharacteristic
182C2×C7H(3,𝔽4)11yescharacteristic
183C2×C7#56212yes—
184D7—82no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—142no—
190C23—142no—
191C23—142no—
192C23—142no—
193C23—142no—
194C23—142no—
195C23—142no—
196C23#116112yes—
197C23C23×D711yescharacteristic
198Q8—44no—
199D4—284no—
200D4—288no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—284no—
221C7#282411yescharacteristic, complemented, Sylow
222C22—24no—
223C22—22no—
224C22—41no—
225C22—42no—
226C22—41no—
227C22—282no—
228C22—282no—
229C22—282no—
230C22—282no—
231C22—282no—
232C22—282no—
233C22—282no—
234C22#654811yescharacteristic
235C22#654812yes—
236C22C2×#116111yescharacteristic
237C22#655312yes—
238C22#655411yescharacteristic
239C4—41no—
240C4—42no—
241C4—41no—
242C4—42no—
243C4—282no—
244C4—282no—
245C4—282no—
246C2—42no—
247C2—562no—
248C2#9028611yescharacteristic
249C2#9033512yes—
250C2#9033611yescharacteristic
251C2#9037411yescharacteristic
252C2#9037512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radical#89732C2
hypercenter#393D7
hypocenterC7#2824
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C7×#393C2semidirect
2. #90003C2semidirect
3. #89733C2semidirect
5. #89730C2semidirect
6. #89718C2semidirect
7. #89446C2semidirect
8. #89444C2semidirect
9. #89432C2semidirect
65. #393D7semidirect
221. C7#2824semidirect

Maximal subgroups

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

subgroupindex
C7×#3932
#900032
#897332
#897322
#897302
#897182
#894462
#894442
#894322
#28247