Finite Group Databasebeta

← #11734989

Subgroups — #11734989

Order 896.

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

normal subgroups only

This group has 450 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×#6450C211yesmaximal, characteristic, complemented
2C2×#6442C211yesmaximal, characteristic
3C7×#397C211yesmaximal, characteristic
4#89752C211yesmaximal, characteristic, complemented
5#89720C212yesmaximal
6#89719C211yesmaximal, characteristic
7#89717C212yesmaximal
8#89429C212yesmaximal, complemented
9#89428C211yesmaximal, characteristic, complemented
10#89426C212yesmaximal, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2212yes—
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28#6450—24no—
29#6442—24no—
30C7×#95C2211yescharacteristic
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2637—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C2×Dic14—22no—
50C2×Dic14—24no—
51C7×C4⋊C4—22no—
52C7×C22⋊C4—22no—
53C7×C22⋊C4—42no—
54C22⋊Dic7—22no—
55C22⋊Dic7—22no—
56C22⋊Dic7—24no—
57C4⋊Dic7—22no—
58C4⋊Dic7—24no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—42no—
63Dic7⋊C4—42no—
64C4×Dic7—24no—
65C4×Dic7D414yes—
66C2×#126—71no—
67C2×C4×Q8—71no—
68#398—71no—
69#398—72no—
70#397D711yescharacteristic
71#396—72no—
72#395—71no—
73#393—71no—
74#388—72no—
75#387—72no—
76#382—71no—
77C23×C7—21no—
78C23×C7—22no—
79C23×C7—21no—
80C23×C7—22no—
81C23×C7—21no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
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×C7C2×D412yes—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—21no—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C2×D412yes—
107C2×Dic7C2×D412yes—
108C2×Dic7#4114yes—
109Dic14—28no—
110C22×Q8—71no—
111#126—144no—
112C4×Q8—144no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C22⋊C4—71no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127C2×C42—72no—
128#95—72no—
129#95—71no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95C2×D711yescharacteristic
134#95C2×D712yes—
135C22×C7—41no—
136C22×C7—41no—
137C22×C7—41no—
138C22×C7—42no—
139C22×C7—42no—
140C22×C7C22×D411yescharacteristic
141C22×C7C2×#4112yes—
142C22×C7H(5,𝔽2)11yescharacteristic
143C22×C72−1+412yes—
144C22×C72−1+411yescharacteristic
145C4×C7—22no—
146C4×C7—41no—
147C4×C7—42no—
148C4×C7—42no—
149Dic7—24no—
150Dic7—22no—
151Dic7—22no—
152Dic7—41no—
153Dic7—41no—
154Dic7—42no—
155C24C22×D711yescharacteristic
156C2×Q8—144no—
157C2×Q8—142no—
158C22×C4—71no—
159C22×C4—72no—
160C22×C4—71no—
161C22×C4—72no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4C22×D711yescharacteristic
165C22×C4C22×D712yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—144no—
170C4⋊C4—142no—
171C4⋊C4—144no—
172C4⋊C4—142no—
173C4⋊C4—142no—
174C4⋊C4—282no—
175C4⋊C4—282no—
176C22⋊C4—22no—
177C22⋊C4—42no—
178C22⋊C4—144no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C42—74no—
182C42—144no—
183C2×C7—81no—
184C2×C7#52511yescharacteristic
185C2×C7#53611yescharacteristic
186C2×C7#53711yescharacteristic
187C2×C7#54112yes—
188C2×C7#54212yes—
189C23—21no—
190C23—21no—
191C23—22no—
192C23—21no—
193C23—22no—
194C23C23×D711yescharacteristic
195Q8—148no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—74no—
208C2×C4—142no—
209C2×C4—141no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—142no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—141no—
220C2×C4—142no—
221C2×C4D4×D712yes—
222C7#263711yescharacteristic, complemented, Sylow
223C22—42no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—41no—
228C22C2×D4×D711yescharacteristic
229C22#654811yescharacteristic
230C22D7×#4112yes—
231C22#655411yescharacteristic
232C22#655412yes—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—41no—
237C4—142no—
238C4—144no—
239C4—142no—
240C4—281no—
241C4—281no—
242C4—282no—
243C2—81no—
244C2#9029911yescharacteristic
245C2#9034711yescharacteristic
246C2#9035312yes—
247C2#9036611yescharacteristic
248C2#9039212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2637
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6450C2semidirect
4. #89752C2semidirect
8. #89429C2semidirect
9. #89428C2semidirect
10. #89426C2semidirect
11. #89422C2semidirect
222. C7#2637semidirect

Maximal subgroups

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

subgroupindex
C2×#64502
C2×#64422
C7×#3972
#897522
#897202
#897192
#897172
#894292
#894282
#894262
#894222
#26377