Finite Group Databasebeta

← #11740331

Subgroups — #11740331

Order 896.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×C7×#123C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89982C212yesmaximal, complemented
5#89751C212yesmaximal, complemented
6#89750C212yesmaximal, complemented
7#89724C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89714C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C7×#123—24no—
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2211yescharacteristic
22#6499—24no—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33#2717—71nomaximal, Sylow
34C24×C7C2311yescharacteristic
35C24×C7C2311yescharacteristic
36C2×C7×D4—41no—
37C2×C7×D4—41no—
38C2×C7×D4D414yes—
39C22×C4×C7—22no—
40C22×C4×C7—41no—
41C22×C4×C7—41no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×Dic7—24no—
46C22×Dic7—41no—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C2×Dic14—22no—
52C2×Dic14—41no—
53C2×Dic14—41no—
54C7×C4⋊C4—24no—
55C7×C22⋊C4—22no—
56C7×C22⋊C4—24no—
57C22⋊Dic7—24no—
58C22⋊Dic7—24no—
59C22⋊Dic7—24no—
60C22⋊Dic7—41no—
61Dic7⋊C4—24no—
62Dic7⋊C4—42no—
63Dic7⋊C4—42no—
64C4×Dic7—22no—
65C2×#124—71no—
66C2×#123D711yescharacteristic
67#397—72no—
68#397—72no—
69#395—72no—
70#387—72no—
71#386—71no—
72#385—72no—
73#380—72no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—21no—
77C23×C7—21no—
78C23×C7—21no—
79C23×C7—21no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7C2×D412yes—
83C23×C7C2×D412yes—
84C23×C7C2411yescharacteristic
85C7×D4—44no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
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—44no—
95C2×C4×C7C2×D412yes—
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—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—44no—
108C2×Dic7#4114yes—
109Dic14—44no—
110C22×Q8—71no—
111C22×D4C2×D711yescharacteristic
112C23×C4—71no—
113C23×C4C2×D711yescharacteristic
114#124—144no—
115#123—24no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4C2×D712yes—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4C2×D712yes—
125C2×C22⋊C4C2×D711yescharacteristic
126C2×C42—71no—
127#95—72no—
128#95—72no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133C22×C7—22no—
134C22×C7—22no—
135C22×C7—24no—
136C22×C7—24no—
137C22×C7—42no—
138C22×C7—42no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7—41no—
142C22×C7—41no—
143C22×C7C22×D411yescharacteristic
144C22×C7C2×#4112yes—
145C22×C7H(5,𝔽2)11yescharacteristic
146C22×C7H(5,𝔽2)11yescharacteristic
147C22×C72−1+412yes—
148C4×C7—22no—
149C4×C7—42no—
150C4×C7—42no—
151Dic7—24no—
152Dic7—42no—
153Dic7—42no—
154Dic7—42no—
155C24C22×D711yescharacteristic
156C24C22×D711yescharacteristic
157C2×Q8—142no—
158C2×Q8—281no—
159C2×Q8—281no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C22⋊D714yes—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—72no—
167C22×C4—72no—
168C22×C4—72no—
169C22×C4—72no—
170C22×C4—144no—
171C22×C4—281no—
172C22×C4C22×D712yes—
173C22×C4C22×D712yes—
174C22×C4C22×D711yescharacteristic
175C4⋊C4—24no—
176C4⋊C4—144no—
177C4⋊C4—282no—
178C4⋊C4—282no—
179C22⋊C4—22no—
180C22⋊C4—24no—
181C22⋊C4—144no—
182C22⋊C4—144no—
183C22⋊C4—144no—
184C22⋊C4—281no—
185C42—142no—
186C2×C7—42no—
187C2×C7—42no—
188C2×C7#53411yescharacteristic
189C2×C7#54212yes—
190C2×C7#54712yes—
191C2×C7#55111yescharacteristic
192C2×C7#56311yescharacteristic
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23C2×C22⋊D712yes—
202C23C2×C22⋊D712yes—
203C23C23×D711yescharacteristic
204Q8—284no—
205D4—44no—
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—74no—
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—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—284no—
228C2×C4C2×C22⋊D712yes—
229C7#271711yescharacteristic, complemented, Sylow
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—42no—
239C22—42no—
240C22#654811yescharacteristic
241C22#654811yescharacteristic
242C22D7×#4112yes—
243C22#655412yes—
244C22C22×C22⋊D711yescharacteristic
245C4—22no—
246C4—42no—
247C4—42no—
248C4—144no—
249C4—282no—
250C4—282no—
251C4—282no—
252C2—42no—
253C2—42no—
254C2#9034211yescharacteristic
255C2#9035011yescharacteristic
256C2#9035312yes—
257C2#9049511yescharacteristic
258C2#9051512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC22×C7×D4C22
hypercenterC2×#123D7
hypocenterC7#2717
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. #89982C2semidirect
5. #89751C2semidirect
6. #89750C2semidirect
7. #89724C2semidirect
8. #89724C2semidirect
9. #89714C2semidirect
229. C7#2717semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#64992
#899922
#899822
#897512
#897502
#897242
#897142
#27177