Finite Group Databasebeta

← #11735271

Subgroups — #11735271

Order 896.

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

normal subgroups only

This group has 520 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×#6460C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic
3C7×#398C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89763C212yesmaximal
6#89753C212yesmaximal, complemented
7#89751C211yesmaximal, characteristic, complemented
8#89447C212yesmaximal, complemented
9#89445C211yesmaximal, characteristic, complemented
10#89440C212yesmaximal, complemented
11#89436C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×C4⋊Dic7C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28#6460—24no—
29#6456—24no—
30C7×#95C2211yescharacteristic
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2638—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C2×D28—22no—
50C2×D28—22no—
51C2×D28—22no—
52C2×C4×D7—22no—
53C2×C4×D7—42no—
54C2×C4×D7—42no—
55C2×C4×D7D414yes—
56C2×Dic14—22no—
57C2×Dic14—42no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—22no—
60C7×C4⋊C4—24no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—42no—
66(C2×D7)⋊C4—42no—
67C4⋊Dic7—24no—
68Dic7⋊C4—42no—
69C4×Dic7—22no—
70C2×C22⋊Q8—71no—
71C2×C4×D4—71no—
72#398D711yescharacteristic, complemented
73#397—71no—
74#397—72no—
75#395—71no—
76#394—72no—
77#393—71no—
78#387—72no—
79#385—72no—
80#381—71no—
81C23×C7C2411yescharacteristic
82C22×D7—21no—
83C22×D7—21no—
84C22×D7—21no—
85C22×D7—21no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7C2×D412yes—
91C22×D7#4112yes—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7C2×D412yes—
104C2×C4×C7#4112yes—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7C2×D412yes—
115D28—28no—
116C4×D7—44no—
117C4×D7—44no—
118Dic14—44no—
119C22×Q8—71no—
120C22×D4—71no—
121C23×C4—71no—
122C23×C4—71no—
123C22⋊Q8—144no—
124C4×D4—144no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C4⋊C4C2×D711yescharacteristic
129C2×C4⋊C4C2×D712yes—
130C2×C22⋊C4—71no—
131C2×C22⋊C4—71no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C42—71no—
137#95—72no—
138#95—71no—
139#95—72no—
140#95—72no—
141#95—72no—
142#95C2×D711yescharacteristic
143#95C2×D712yes—
144C22×C7C22×D411yescharacteristic
145C22×C7C2×#4111yescharacteristic
146C22×C7C2×#4111yescharacteristic
147C22×C7H(5,𝔽2)12yes—
148C22×C72−1+412yes—
149C2×D7—22no—
150C2×D7—22no—
151C2×D7—22no—
152C2×D7—22no—
153C2×D7—24no—
154C2×D7—41no—
155C2×D7—41no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C4×C7—22no—
160C4×C7—22no—
161C4×C7—42no—
162C4×C7—41no—
163C4×C7—42no—
164Dic7—22no—
165Dic7—42no—
166Dic7—41no—
167Dic7—42no—
168C24—71no—
169C24—71no—
170C2×Q8—142no—
171C2×Q8—282no—
172C2×D4—142no—
173C2×D4—142no—
174C2×D4—142no—
175C22×C4—71no—
176C22×C4—74no—
177C22×C4—71no—
178C22×C4—72no—
179C22×C4—72no—
180C22×C4—142no—
181C22×C4—282no—
182C22×C4—282no—
183C22×C4C22×D712yes—
184C22×C4C22×D712yes—
185C22×C4C22×D711yescharacteristic
186C22×C4C22×D711yescharacteristic
187C22×C4C22×D711yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—24no—
191C4⋊C4—144no—
192C4⋊C4—282no—
193C22⋊C4—144no—
194C22⋊C4—142no—
195C22⋊C4—144no—
196C22⋊C4—142no—
197C22⋊C4—282no—
198C22⋊C4—282no—
199C42—142no—
200C2×C7#52511yescharacteristic
201C2×C7#53411yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7#53912yes—
204C2×C7#54212yes—
205D7—42no—
206D7—42no—
207C23—72no—
208C23—72no—
209C23—141no—
210C23—141no—
211C23—141no—
212C23—142no—
213C23—142no—
214C23—141no—
215C23—142no—
216C23—142no—
217C23C23×D711yescharacteristic
218Q8—284no—
219D4—148no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—72no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—284no—
242C2×C4—284no—
243C2×C4D4×D712yes—
244C2×C4#115912yes—
245C7#263811yescharacteristic, complemented, Sylow
246C22—144no—
247C22—142no—
248C22—142no—
249C22—142no—
250C22—142no—
251C22—282no—
252C22—281no—
253C22—282no—
254C22—282no—
255C22—281no—
256C22C2×D4×D711yescharacteristic
257C22C2×#115911yescharacteristic
258C22#655112yes—
259C22D7×#4111yescharacteristic
260C22#655312yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—41no—
266C4—142no—
267C4—282no—
268C4—281no—
269C4—282no—
270C2—282no—
271C2—282no—
272C2#9029711yescharacteristic
273C2#9033412yes—
274C2#9034411yescharacteristic
275C2#9038312yes—
276C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC22×D28C22
hypercenter#398D7
hypocenterC7#2638
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
3. C7×#398C2semidirect
4. #89765C2semidirect
6. #89753C2semidirect
7. #89751C2semidirect
8. #89447C2semidirect
9. #89445C2semidirect
10. #89440C2semidirect
11. #89436C2semidirect
72. #398D7semidirect
245. C7#2638semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64562
C7×#3982
#897652
#897632
#897532
#897512
#894472
#894452
#894402
#894362
#26387