Finite Group Databasebeta

← #11735469

Subgroups — #11735469

Order 896.

660 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 145 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×#6457C212yesmaximal, complemented
2C2×(C22⋊D7)⋊C4C212yesmaximal, complemented
3C7×#399C211yesmaximal, characteristic, complemented
4#89761C212yesmaximal, complemented
5#89745C212yesmaximal, complemented
6#89733C212yesmaximal, complemented
7#89732C211yesmaximal, characteristic
8#89718C211yesmaximal, characteristic, complemented
9#89442C212yesmaximal, complemented
10C22×C22⋊D7C2212yes—
11C23×Dic7C2211yescharacteristic
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C2×C7×C4⋊C4C2212yes—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C22⋊Dic7C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27#6457—28no—
28(C22⋊D7)⋊C4—28no—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33#2455—71nomaximal, Sylow
34C24×C7C2311yescharacteristic
35C23×D7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C2×C22⋊D7—24no—
41C2×C22⋊D7—24no—
42C2×C22⋊D7—24no—
43C22×Dic7—24no—
44C22×Dic7—41no—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C2×D28—24no—
49C2×D28—41no—
50C2×C4×D7—24no—
51C2×C4×D7—24no—
52C2×C4×D7—42no—
53C7×C4⋊C4—24no—
54C7×C4⋊C4—41no—
55C7×C22⋊C4—24no—
56C7×C22⋊C4—41no—
57C22⋊Dic7—22no—
58(C2×D7)⋊C4—24no—
59(C2×D7)⋊C4—24no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4D418yescomplemented
62C4⋊Dic7—24no—
63Dic7⋊C4—24no—
64Dic7⋊C4—24no—
65C4×Dic7—24no—
66C2×#123—72no—
67C2×C4×D4—72no—
68#399D711yescharacteristic, complemented
69#394—72no—
70#392—71no—
71#389—72no—
72#388—72no—
73#385—72no—
74#385—71no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—21no—
78C23×C7—21no—
79C23×C7#4112yes—
80C23×C7C2411yescharacteristic
81C22×D7—22no—
82C22×D7—22no—
83C22×D7—22no—
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7C2×D414yes—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7C2×D414yes—
98C22⋊D7—216no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—44no—
107C2×Dic7C2×D414yes—
108C2×Dic7#4114yes—
109D28—44no—
110C4×D7—48no—
111C22×D4—72no—
112C22×D4—71no—
113C23×C4—71no—
114C23×C4—72no—
115#123—148no—
116C4×D4—148no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4C2×D712yes—
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C22⋊C4—72no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C22⋊C4—72no—
127C2×C22⋊C4C2×D712yes—
128C2×C22⋊C4C2×D711yescharacteristic
129C2×C42—72no—
130#95—72no—
131#95—72no—
132#95—71no—
133#95C2×D711yescharacteristic
134C22×C7—24no—
135C22×C7—22no—
136C22×C7—41no—
137C22×C7—42no—
138C22×C7—41no—
139C22×C7C22×D412yes—
140C22×C7C2×#4111yescharacteristic
141C22×C7C2×#4112yes—
142C22×C7H(5,𝔽2)11yescharacteristic
143C22×C72−1+411yescharacteristic
144C2×D7—24no—
145C2×D7—24no—
146C2×D7—24no—
147C2×D7—42no—
148C2×D7—42no—
149C2×D7—42no—
150C2×D7—42no—
151C4×C7—24no—
152C4×C7—42no—
153C4×C7—41no—
154C4×C7—41no—
155Dic7—24no—
156Dic7—24no—
157Dic7—42no—
158C24—72no—
159C24C22×D711yescharacteristic
160C2×D4—144no—
161C2×D4—144no—
162C2×D4—144no—
163C2×D4—144no—
164C2×D4—281no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—72no—
168C22×C4—144no—
169C22×C4—144no—
170C22×C4—144no—
171C22×C4—282no—
172C22×C4—281no—
173C22×C4C22×D711yescharacteristic
174C22×C4C22×D712yes—
175C22×C4C22×D711yescharacteristic
176C22×C4C22×D712yes—
177C4⋊C4—24no—
178C4⋊C4—41no—
179C4⋊C4—144no—
180C4⋊C4—144no—
181C4⋊C4—144no—
182C22⋊C4—24no—
183C22⋊C4—41no—
184C22⋊C4—78no—
185C22⋊C4—144no—
186C22⋊C4—144no—
187C22⋊C4—142no—
188C22⋊C4—144no—
189C42—144no—
190C2×C7—42no—
191C2×C7#52512yes—
192C2×C7#53911yescharacteristic
193C2×C7D4211yescharacteristic
194C2×C7#54712yes—
195C2×C7#55911yescharacteristic
196D7—44no—
197C23—22no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—74no—
202C23—142no—
203C23—142no—
204C23—142no—
205C23—142no—
206C23—142no—
207C23—142no—
208C23#116112yes—
209C23C23×D711yescharacteristic
210D4—1416no—
211D4—284no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—74no—
222C2×C4—74no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—288no—
231C2×C4—284no—
232C2×C4D4×D714yes—
233C7#245511yescharacteristic, complemented, Sylow
234C22—24no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—144no—
240C22—144no—
241C22—144no—
242C22—282no—
243C22—282no—
244C22—282no—
245C22—282no—
246C22C2×D4×D712yes—
247C22#654811yescharacteristic
248C22C2×#116111yescharacteristic
249C22D7×#4112yes—
250C22#655411yescharacteristic
251C4—24no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—144no—
256C4—144no—
257C4—282no—
258C2—42no—
259C2—284no—
260C2#9031812yes—
261C2#9033611yescharacteristic
262C2#9034612yes—
263C2#9035511yescharacteristic
264C2#9040811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radical#89732C2
hypercenter#399D7
hypocenterC7#2455
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6457C2semidirect
2. C2×(C22⋊D7)⋊C4C2semidirect
3. C7×#399C2semidirect
4. #89761C2semidirect
5. #89745C2semidirect
6. #89733C2semidirect
8. #89718C2semidirect
9. #89442C2semidirect
61. (C2×D7)⋊C4D4semidirect
68. #399D7semidirect
233. C7#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#64572
C2×(C22⋊D7)⋊C42
C7×#3992
#897612
#897452
#897332
#897322
#897182
#894422
#24557