Finite Group Databasebeta

← #11734988

Subgroups — #11734988

Order 896.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 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×#6453C211yesmaximal, characteristic, complemented
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
3C7×#397C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89731C212yesmaximal
6#89722C212yesmaximal, complemented
7#89721C211yesmaximal, characteristic, complemented
8#89449C211yesmaximal, characteristic, complemented
9#89440C212yesmaximal, complemented
10#89429C212yesmaximal, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6453—24no—
30(C22⋊D7)⋊C4—24no—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2633—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7—22no—
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C22×Dic7—22no—
48C22×Dic7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C2×C4×D7—22no—
55C2×C4×D7—42no—
56C7×C4⋊C4—22no—
57C7×C22⋊C4—24no—
58C7×C22⋊C4—22no—
59C22⋊Dic7—42no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—42no—
63C4⋊Dic7—22no—
64C4⋊Dic7—24no—
65C4⋊Dic7—42no—
66Dic7⋊C4—22no—
67Dic7⋊C4—24no—
68Dic7⋊C4—42no—
69C4×Dic7D414yes—
70C2×#125—71no—
71C2×C4×D4—71no—
72#399—71no—
73#399—71no—
74#398—72no—
75#397—71no—
76#397D711yescharacteristic, complemented
77#394—72no—
78#385—72no—
79#385—72no—
80#383—71no—
81C23×C7—21no—
82C23×C7—21no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7#4112yes—
86C23×C7C2411yescharacteristic
87C22×D7—21no—
88C22×D7—21no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7#4112yes—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7C2×D412yes—
102C22⋊D7—28no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—44no—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×D412yes—
117C4×D7—44no—
118C22×D4—71no—
119C23×C4—71no—
120C23×C4—71no—
121#125—144no—
122C4×D4—144no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—71no—
127C2×C4⋊C4—72no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4C2×D711yescharacteristic
130C2×C22⋊C4—71no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4C2×D711yescharacteristic
135C2×C22⋊C4C2×D712yes—
136C2×C42—71no—
137#95—72no—
138#95—72no—
139#95—71no—
140#95—72no—
141#95C2×D711yescharacteristic
142#95C2×D712yes—
143C22×C7—22no—
144C22×C7—24no—
145C22×C7—41no—
146C22×C7—41no—
147C22×C7—42no—
148C22×C7C22×D411yescharacteristic
149C22×C7C2×#4111yescharacteristic
150C22×C7C2×#4111yescharacteristic
151C22×C7H(5,𝔽2)12yes—
152C22×C72−1+412yes—
153C2×D7—22no—
154C2×D7—24no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—42no—
158C4×C7—22no—
159C4×C7—42no—
160C4×C7—41no—
161C4×C7—42no—
162Dic7—22no—
163Dic7—22no—
164Dic7—42no—
165Dic7—41no—
166Dic7—42no—
167C24—71no—
168C24C22×D711yescharacteristic
169C2×D4—142no—
170C2×D4—142no—
171C2×D4—142no—
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—72no—
175C22×C4—71no—
176C22×C4—72no—
177C22×C4—142no—
178C22×C4—142no—
179C22×C4—282no—
180C22×C4—282no—
181C22×C4C22×D712yes—
182C22×C4C22×D712yes—
183C22×C4C22×D711yescharacteristic
184C22×C4C22×D711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—142no—
187C4⋊C4—144no—
188C4⋊C4—144no—
189C4⋊C4—142no—
190C4⋊C4—282no—
191C4⋊C4—282no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—144no—
195C22⋊C4—142no—
196C22⋊C4—282no—
197C22⋊C4—282no—
198C42—74no—
199C2×C7—42no—
200C2×C7#52511yescharacteristic
201C2×C7#53511yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7#53912yes—
204C2×C7#53912yes—
205D7—42no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—72no—
211C23—141no—
212C23—141no—
213C23—142no—
214C23—142no—
215C23#115912yes—
216C23C23×D711yescharacteristic
217D4—148no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—72no—
228C2×C4—72no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—141no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4—142no—
240C2×C4—284no—
241C2×C4—284no—
242C2×C4D4×D712yes—
243C7#263311yescharacteristic, complemented, Sylow
244C22—22no—
245C22—24no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—144no—
250C22—142no—
251C22—281no—
252C22—282no—
253C22—281no—
254C22C2×D4×D711yescharacteristic
255C22C2×#115911yescharacteristic
256C22D7×#4111yescharacteristic
257C22#655312yes—
258C22#655412yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—41no—
263C4—142no—
264C4—142no—
265C4—282no—
266C4—281no—
267C4—282no—
268C2—42no—
269C2—282no—
270C2#9029711yescharacteristic
271C2#9033911yescharacteristic
272C2#9035612yes—
273C2#9036511yescharacteristic
274C2#9039812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC22×C22⋊D7C22
hypercenter#397D7
hypocenterC7#2633
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6453C2semidirect
3. C7×#397C2semidirect
4. #89765C2semidirect
6. #89722C2semidirect
7. #89721C2semidirect
8. #89449C2semidirect
9. #89440C2semidirect
10. #89429C2semidirect
11. #89423C2semidirect
76. #397D7semidirect
243. C7#2633semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×(C22⋊D7)⋊C42
C7×#3972
#897652
#897312
#897222
#897212
#894492
#894402
#894292
#894232
#26337