Finite Group Databasebeta

← #11731588

Subgroups — #11731588

Order 896.

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

normal subgroups only

This group has 520 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
1C7×#381C211yesmaximal, characteristic, complemented
2#89985C212yesmaximal, complemented
3#89732C211yesmaximal, characteristic
4#89728C212yesmaximal, complemented
5#89719C211yesmaximal, characteristic, complemented
6#89445C212yesmaximal, complemented
7#89437C212yesmaximal, complemented
8#89436C212yesmaximal, complemented
9C7⋊#401C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×Dic14C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C7×#95C2212yes—
23C7×#95C2212yes—
24C7⋊#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2760—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7C2312yes—
34C22×C4×C7—22no—
35C22×C4×C7—41no—
36C22×C4×C7—41no—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C2×C22⋊D7—24no—
42C2×C22⋊D7—24no—
43C2×C22⋊D7—42no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C2×D28—24no—
48C2×D28—41no—
49C2×C4×D7—24no—
50C2×C4×D7—42no—
51C2×C4×D7—42no—
52C2×Dic14—41no—
53C2×Dic14—42no—
54C7×C22⋊C4—41no—
55C7×C22⋊C4—41no—
56C22⋊Dic7—42no—
57(C2×D7)⋊C4—42no—
58(C2×D7)⋊C4—42no—
59(C2×D7)⋊C4—42no—
60(C2×D7)⋊C4—42no—
61#401—72no—
62#393—71no—
63#393—72no—
64#392—71no—
65#386—72no—
66#386—72no—
67#386—72no—
68#381—72no—
69#381D711yescharacteristic, complemented
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—21no—
73C23×C7—21no—
74C23×C7#4112yes—
75C23×C7C2411yescharacteristic
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C22×D7#4114yes—
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—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—44no—
94C22⋊D7—48no—
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—
104D28—44no—
105C4×D7—48no—
106Dic14—44no—
107C22×Q8—71no—
108C22×D4—72no—
109C22×D4—71no—
110C23×C4—72no—
111C23×C4C2×D711yescharacteristic
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×D711yescharacteristic
119#95—71no—
120#95—72no—
121#95—72no—
122#95—72no—
123#95—72no—
124#95—72no—
125#95—71no—
126#95C2×D712yes—
127#95C2×D712yes—
128C22×C7—22no—
129C22×C7—24no—
130C22×C7—42no—
131C22×C7—41no—
132C22×C7—41no—
133C22×C7C2×#4111yescharacteristic
134C22×C7C2×#4112yes—
135C22×C7H(5,𝔽2)11yescharacteristic
136C22×C7H(5,𝔽2)11yescharacteristic
137C22×C7H(5,𝔽2)12yes—
138C2×D7—24no—
139C2×D7—24no—
140C2×D7—24no—
141C2×D7—42no—
142C2×D7—42no—
143C2×D7—42no—
144C2×D7—42no—
145C4×C7—42no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7—41no—
149Dic7—42no—
150Dic7—42no—
151Dic7—42no—
152C24—72no—
153C24C22×D711yescharacteristic
154C2×Q8—281no—
155C2×Q8—282no—
156C2×D4—144no—
157C2×D4—144no—
158C2×D4—144no—
159C2×D4—281no—
160C2×D4—282no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—144no—
168C22×C4—282no—
169C22×C4—282no—
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D712yes—
172C22×C4C22×D712yes—
173C22×C4C22×D711yescharacteristic
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—282no—
177C22⋊C4—282no—
178C22⋊C4—282no—
179C22⋊C4—282no—
180C22⋊C4—282no—
181C2×C7—42no—
182C2×C7#53811yescharacteristic
183C2×C7#53812yes—
184C2×C7#55112yes—
185C2×C7#55111yescharacteristic
186C2×C7#56011yescharacteristic
187D7—44no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—22no—
192C23—74no—
193C23—142no—
194C23—142no—
195C23—142no—
196C23—142no—
197C23—142no—
198C23—142no—
199C23#115712yes—
200C23C23×D711yescharacteristic
201Q8—284no—
202D4—284no—
203D4—288no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—288no—
225C7#276011yescharacteristic, complemented, Sylow
226C22—22no—
227C22—24no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—144no—
232C22—144no—
233C22—144no—
234C22—282no—
235C22—282no—
236C22—282no—
237C22—282no—
238C22C2×#115711yescharacteristic
239C22#654811yescharacteristic
240C22#654811yescharacteristic
241C22D7×#4112yes—
242C22#655312yes—
243C4—42no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—282no—
248C4—282no—
249C4—282no—
250C2—42no—
251C2—284no—
252C2#9018511yescharacteristic
253C2#9021812yes—
254C2#9034812yes—
255C2#9035011yescharacteristic
256C2#9035711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radical#89732C2
hypercenter#381D7
hypocenterC7#2760
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C7×#381C2semidirect
2. #89985C2semidirect
4. #89728C2semidirect
5. #89719C2semidirect
6. #89445C2semidirect
7. #89437C2semidirect
8. #89436C2semidirect
9. C7⋊#401C2semidirect
69. #381D7semidirect
225. C7#2760semidirect

Maximal subgroups

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

subgroupindex
C7×#3812
#899852
#897322
#897282
#897192
#894452
#894372
#894362
C7⋊#4012
#27607