Finite Group Databasebeta

← #11731524

Subgroups — #11731524

Order 896.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×Dic7⋊C4C211yesmaximal, characteristic
2C2×#6441C211yesmaximal, characteristic
3C7×#381C211yesmaximal, characteristic
4#89983C211yesmaximal, characteristic
5#89715C211yesmaximal, characteristic
6#89715C211yesmaximal, characteristic
7C22⋊C4×Dic7C211yesmaximal, characteristic
8#89432C212yesmaximal, complemented
9#89430C212yesmaximal, complemented
10#89422C212yesmaximal, complemented
11#89418C212yesmaximal, complemented
12C23×C4×C7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C23×Dic7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C22⋊Dic7C2211yescharacteristic
19C2×C22⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4—24no—
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27#6441—24no—
28C7×#95C2212yes—
29C7×#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2503—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7—22no—
40C22×C4×C7—22no—
41C22×C4×C7—22no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×Dic7—24no—
47C22×Dic7—22no—
48C22×Dic7—22no—
49C22×Dic7—22no—
50C22×Dic7D414yes—
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C2×Dic14—22no—
58C2×Dic14—24no—
59C7×C22⋊C4—22no—
60C7×C22⋊C4—22no—
61C22⋊Dic7—22no—
62C22⋊Dic7—22no—
63C4⋊Dic7—22no—
64Dic7⋊C4—22no—
65Dic7⋊C4—22no—
66Dic7⋊C4—24no—
67Dic7⋊C4—24no—
68C4×Dic7—24no—
69C2×C22⋊Q8—71no—
70C22×C4⋊C4—71no—
71#400—72no—
72#400—72no—
73#391—72no—
74#382—72no—
75#381—71no—
76#381—71no—
77#381—71no—
78#381D711yescharacteristic
79C4×C22⋊C4—71no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7C2×D412yes—
83C23×C7#4112yes—
84C23×C7#4112yes—
85C23×C7C2411yescharacteristic
86C2×C4×C7—22no—
87C2×C4×C7—21no—
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×C7—44no—
97C2×Dic7—22no—
98C2×Dic7—28no—
99C2×Dic7—21no—
100C2×Dic7—22no—
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×Dic7—44no—
110C2×Dic7C2×D412yes—
111C2×Dic7C2×D412yes—
112C2×Dic7#4114yes—
113Dic14—44no—
114C22×Q8—71no—
115C23×C4—71no—
116C23×C4—71no—
117C23×C4C2×D711yescharacteristic
118C22⋊Q8—144no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—144no—
125C2×C22⋊C4—71no—
126C2×C22⋊C4—71no—
127C2×C22⋊C4C2×D711yescharacteristic
128C2×C22⋊C4C2×D711yescharacteristic
129C2×C42—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95—72no—
135#95—72no—
136#95—72no—
137#95C2×D712yes—
138#95C2×D712yes—
139C22×C7—22no—
140C22×C7—22no—
141C22×C7—24no—
142C22×C7—24no—
143C22×C7#12414yes—
144C22×C7C22×D411yescharacteristic
145C22×C7C2×#4111yescharacteristic
146C22×C7C2×#4111yescharacteristic
147C22×C7C2×#4112yes—
148C22×C72−1+412yes—
149C4×C7—41no—
150C4×C7—42no—
151C4×C7—41no—
152C4×C7—42no—
153Dic7—22no—
154Dic7—41no—
155Dic7—42no—
156Dic7—41no—
157Dic7—42no—
158Dic7#12314yes—
159C24C22×D711yescharacteristic
160C2×Q8—142no—
161C2×Q8—144no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—71no—
166C22×C4—74no—
167C22×C4—71no—
168C22×C4—72no—
169C22×C4—71no—
170C22×C4—71no—
171C22×C4—72no—
172C22×C4—144no—
173C22×C4—142no—
174C22×C4—142no—
175C22×C4—142no—
176C22×C4C22×D712yes—
177C22×C4C22×D711yescharacteristic
178C22×C4C22×D711yescharacteristic
179C22×C4C22×D712yes—
180C4⋊C4—144no—
181C4⋊C4—144no—
182C4⋊C4—142no—
183C4⋊C4—142no—
184C4⋊C4—142no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—142no—
188C22⋊C4—142no—
189C42—144no—
190C2×C7—24no—
191C2×C7C2×#12311yescharacteristic
192C2×C7C2×#12411yescharacteristic
193C2×C7#53611yescharacteristic
194C2×C7#55212yes—
195C2×C7#55212yes—
196C23—22no—
197C23—22no—
198C23#115712yes—
199C23D4×D712yes—
200C23#115912yes—
201C23C23×D711yescharacteristic
202Q8—284no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—44no—
214C2×C4—72no—
215C2×C4—74no—
216C2×C4—72no—
217C2×C4—148no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—141no—
222C2×C4—141no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—284no—
230C7#250311yescharacteristic, complemented, Sylow
231C22—24no—
232C22—22no—
233C22—24no—
234C22—22no—
235C22#644814yes—
236C22C2×#115711yescharacteristic
237C22C2×D4×D711yescharacteristic
238C22C2×#115911yescharacteristic
239C22D7×#4112yes—
240C22#655412yes—
241C4—42no—
242C4—41no—
243C4—42no—
244C4—41no—
245C4—74no—
246C4—142no—
247C4—282no—
248C4—282no—
249C4—281no—
250C4—281no—
251C2—24no—
252C2C2×#644811yescharacteristic
253C2#9022212yes—
254C2#9033911yescharacteristic
255C2#9034112yes—
256C2D7×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC24×C7C23
hypercenter#381D7
hypocenterC7#2503
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
8. #89432C2semidirect
9. #89430C2semidirect
10. #89422C2semidirect
11. #89418C2semidirect
230. C7#2503semidirect

Maximal subgroups

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

subgroupindex
C22×Dic7⋊C42
C2×#64412
C7×#3812
#899832
#897152
C22⋊C4×Dic72
#894322
#894302
#894222
#894182
#25037