Finite Group Databasebeta

← #11734983

Subgroups — #11734983

Order 896.

528 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6451C211yesmaximal, characteristic, complemented
2C2×#6441C212yesmaximal
3C2×#6440C211yesmaximal, characteristic
4C7×#397C211yesmaximal, characteristic
5#89752C211yesmaximal, characteristic, complemented
6#89725C211yesmaximal, characteristic
7#89724C212yesmaximal
8#89433C211yesmaximal, characteristic, complemented
9#89429C212yesmaximal, complemented
10#89422C211yesmaximal, characteristic, complemented
11#89418C212yesmaximal, complemented
12C23×Dic7C2211yescharacteristic
13C22×Dic14C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28#6451—24no—
29#6441—28no—
30#6440—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2621—71nomaximal, Sylow
39C24×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×Dic7—22no—
45C22×Dic7—22no—
46C22×Dic7D414yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C2×Dic14—24no—
54C2×Dic14—24no—
55C2×Dic14—24no—
56C7×C4⋊C4—22no—
57C7×C22⋊C4—24no—
58C7×C22⋊C4—22no—
59C22⋊Dic7—24no—
60C22⋊Dic7—22no—
61C4⋊Dic7—22no—
62C4⋊Dic7—24no—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—24no—
66Dic7⋊C4—24no—
67Dic7⋊C4—24no—
68C4×Dic7—22no—
69C4×Dic7D414yes—
70C2×C4⋊Q8—71no—
71C2×C22⋊Q8—72no—
72C2×#117—71no—
73#398—71no—
74#398—72no—
75#398—71no—
76#397—72no—
77#397—71no—
78#397D711yescharacteristic
79#391—72no—
80#382—71no—
81C23×C7—21no—
82C23×C7—21no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7C2×D412yes—
86C23×C7C2411yescharacteristic
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7C2×D412yes—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—28no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7C2×D412yes—
110C2×Dic7C2×D412yes—
111C2×Dic7C2×D412yes—
112C2×Dic7C2×D412yes—
113C2×Dic7#4112yes—
114Dic14—48no—
115C22×Q8—72no—
116C23×C4—71no—
117C4⋊Q8—144no—
118C22⋊Q8—148no—
119#117—144no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—71no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C22⋊C4—71no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4C2×D711yescharacteristic
131C2×C22⋊C4C2×D712yes—
132C2×C42—71no—
133C2×C42—71no—
134#95—72no—
135#95—71no—
136#95—72no—
137#95—72no—
138#95—72no—
139#95C2×D711yescharacteristic
140#95C2×D712yes—
141C22×C7—22no—
142C22×C7—24no—
143C22×C7—41no—
144C22×C7—41no—
145C22×C7—42no—
146C22×C7C22×D411yescharacteristic
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4111yescharacteristic
149C22×C72−1+412yes—
150C22×C72−1+412yes—
151C4×C7—22no—
152C4×C7—42no—
153C4×C7—42no—
154C4×C7—41no—
155Dic7—22no—
156Dic7—22no—
157Dic7—42no—
158Dic7—41no—
159Dic7—42no—
160Dic7#12114yes—
161C24C22×D711yescharacteristic
162C2×Q8—144no—
163C2×Q8—144no—
164C2×Q8—144no—
165C22×C4—71no—
166C22×C4—74no—
167C22×C4—71no—
168C22×C4—71no—
169C22×C4—71no—
170C22×C4—72no—
171C22×C4—72no—
172C22×C4—142no—
173C22×C4—142no—
174C22×C4C22×D711yescharacteristic
175C22×C4C22×D712yes—
176C22×C4C22×D712yes—
177C22×C4C22×D711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—142no—
180C4⋊C4—144no—
181C4⋊C4—144no—
182C4⋊C4—144no—
183C4⋊C4—144no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—142no—
189C22⋊C4—144no—
190C42—74no—
191C42—142no—
192C2×C7—42no—
193C2×C7C2×#12111yescharacteristic
194C2×C7#53611yescharacteristic
195C2×C7#53612yes—
196C2×C7#53611yescharacteristic
197C2×C7#54112yes—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23D4×D712yes—
203C23C23×D711yescharacteristic
204Q8—288no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—72no—
215C2×C4—72no—
216C2×C4—72no—
217C2×C4—72no—
218C2×C4—72no—
219C2×C4—148no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4—141no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—142no—
231C2×C4D4×D712yes—
232C7#262111yescharacteristic, complemented, Sylow
233C22—22no—
234C22—24no—
235C22—42no—
236C22—41no—
237C22—41no—
238C22C2×D4×D711yescharacteristic
239C22C2×D4×D711yescharacteristic
240C22D7×#4111yescharacteristic
241C22#655412yes—
242C22#655412yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—41no—
247C4—74no—
248C4—142no—
249C4—142no—
250C4—282no—
251C4—281no—
252C4—282no—
253C2—42no—
254C2D7×#12111yescharacteristic
255C2#9033911yescharacteristic
256C2#9033912yes—
257C2#9036611yescharacteristic
258C2#9039212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2621
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
5. #89752C2semidirect
8. #89433C2semidirect
9. #89429C2semidirect
10. #89422C2semidirect
11. #89418C2semidirect
232. C7#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64412
C2×#64402
C7×#3972
#897522
#897252
#897242
#894332
#894292
#894222
#894182
#26217