Finite Group Databasebeta

← #11735019

Subgroups — #11735019

Order 896.

568 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×#6446C212yesmaximal, complemented
2C7×#397C211yesmaximal, characteristic, complemented
3#89766C212yesmaximal, complemented
4#89732C211yesmaximal, characteristic
5#89730C212yesmaximal, complemented
6#89718C211yesmaximal, characteristic, complemented
7#89439C212yesmaximal, complemented
8#89437C212yesmaximal, complemented
9#89432C212yesmaximal, complemented
10C22×C22⋊D7C2212yes—
11C23×Dic7C2211yescharacteristic
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C22⋊Dic7C2211yescharacteristic
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25#6446—28no—
26C7×#95C2212yes—
27C7×#95C2211yescharacteristic
28C7⋊#95C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2693—71nomaximal, Sylow
33C24×C7C2311yescharacteristic
34C23×D7C2312yes—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—24no—
40C2×C22⋊D7—24no—
41C2×C22⋊D7—42no—
42C22×Dic7—42no—
43C22×Dic7—41no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C2×D28—24no—
48C2×D28—41no—
49C2×C4×D7—24no—
50C2×C4×D7—24no—
51C2×C4×D7—42no—
52C7×C4⋊C4—41no—
53C7×C22⋊C4—24no—
54C7×C22⋊C4—41no—
55C22⋊Dic7D414yescomplemented
56(C2×D7)⋊C4—24no—
57(C2×D7)⋊C4—42no—
58(C2×D7)⋊C4—42no—
59(C2×D7)⋊C4—42no—
60C4⋊Dic7—24no—
61Dic7⋊C4—24no—
62Dic7⋊C4—42no—
63C2×#123—72no—
64#400—72no—
65#397D711yescharacteristic, complemented
66#396—72no—
67#392—71no—
68#386—72no—
69#386—72no—
70#385—71no—
71#380—72no—
72C23×C7—22no—
73C23×C7—21no—
74C23×C7—22no—
75C23×C7—21no—
76C23×C7C2×D412yes—
77C23×C7C2411yescharacteristic
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C22×D7—22no—
83C22×D7—22no—
84C22×D7#4114yes—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C22⋊D7—48no—
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—
105C2×Dic7C2×D414yes—
106D28—44no—
107C4×D7—48no—
108C22×D4—72no—
109C22×D4—71no—
110C23×C4—71no—
111C23×C4—72no—
112#123—148no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4C2×D711yescharacteristic
117C2×C22⋊C4—71no—
118C2×C22⋊C4—72no—
119C2×C22⋊C4—72no—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4C2×D712yes—
123C2×C22⋊C4C2×D711yescharacteristic
124#95—72no—
125#95—72no—
126#95—72no—
127#95—71no—
128#95C2×D711yescharacteristic
129#95C2×D712yes—
130C22×C7—22no—
131C22×C7—24no—
132C22×C7—42no—
133C22×C7—41no—
134C22×C7—41no—
135C22×C7C22×D411yescharacteristic
136C22×C7C2×#4112yes—
137C22×C7H(5,𝔽2)11yescharacteristic
138C22×C7H(5,𝔽2)12yes—
139C22×C72−1+411yescharacteristic
140C2×D7—24no—
141C2×D7—24no—
142C2×D7—24no—
143C2×D7—42no—
144C2×D7—42no—
145C2×D7—42no—
146C2×D7—42no—
147C4×C7—42no—
148C4×C7—42no—
149C4×C7—41no—
150C4×C7—41no—
151Dic7—24no—
152Dic7—42no—
153Dic7—42no—
154C24—72no—
155C24C22×D711yescharacteristic
156C2×D4—144no—
157C2×D4—144no—
158C2×D4—144no—
159C2×D4—281no—
160C2×D4—282no—
161C22×C4—72no—
162C22×C4—72no—
163C22×C4—72no—
164C22×C4—144no—
165C22×C4—144no—
166C22×C4—282no—
167C22×C4—281no—
168C22×C4—282no—
169C22×C4C22×D711yescharacteristic
170C22×C4C22×D712yes—
171C22×C4C22×D712yes—
172C22×C4C22×D711yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—144no—
175C4⋊C4—144no—
176C4⋊C4—282no—
177C22⋊C4—24no—
178C22⋊C4—41no—
179C22⋊C4—74no—
180C22⋊C4—144no—
181C22⋊C4—282no—
182C22⋊C4—282no—
183C22⋊C4—282no—
184C2×C7—42no—
185C2×C7#53711yescharacteristic
186C2×C7#53912yes—
187C2×C7#54612yes—
188C2×C7#55111yescharacteristic
189C2×C7#56011yescharacteristic
190D7—44no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—21no—
195C23—74no—
196C23—142no—
197C23—142no—
198C23—142no—
199C23—142no—
200C23—142no—
201C23—142no—
202C23D4×D712yes—
203C23C23×D711yescharacteristic
204D4—284no—
205D4—288no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—74no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—288no—
226C2×C4—284no—
227C7#269311yescharacteristic, complemented, Sylow
228C22—22no—
229C22—24no—
230C22—41no—
231C22—42no—
232C22—41no—
233C22—144no—
234C22—144no—
235C22—144no—
236C22—282no—
237C22—282no—
238C22—282no—
239C22—282no—
240C22C2×D4×D711yescharacteristic
241C22#654811yescharacteristic
242C22D7×#4112yes—
243C22#655312yes—
244C22#655411yescharacteristic
245C4—41no—
246C4—42no—
247C4—42no—
248C4—41no—
249C4—144no—
250C4—282no—
251C4—282no—
252C2—42no—
253C2—284no—
254C2#9028811yescharacteristic
255C2#9034512yes—
256C2#9035711yescharacteristic
257C2#9037011yescharacteristic
258C2#9039812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radical#89732C2
hypercenter#397D7
hypocenterC7#2693
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6446C2semidirect
2. C7×#397C2semidirect
3. #89766C2semidirect
5. #89730C2semidirect
6. #89718C2semidirect
7. #89439C2semidirect
8. #89437C2semidirect
9. #89432C2semidirect
55. C22⋊Dic7D4semidirect
65. #397D7semidirect
227. C7#2693semidirect

Maximal subgroups

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

subgroupindex
C2×#64462
C7×#3972
#897662
#897322
#897302
#897182
#894392
#894372
#894322
#26937