Finite Group Databasebeta

← #11735462

Subgroups — #11735462

Order 896.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#6455C211yesmaximal, characteristic, complemented
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic, complemented
4#89765C212yesmaximal, complemented
5#89754C211yesmaximal, characteristic, complemented
6C4⋊C4×Dic7C212yesmaximal, complemented
7#89733C212yesmaximal
8#89725C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic7C212yesmaximal, complemented
10#89438C211yesmaximal, characteristic, complemented
11#89433C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2211yescharacteristic
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2212yes—
29#6455—24no—
30(C22⋊D7)⋊C4—24no—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2395—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C2×C22⋊D7—22no—
45C2×C22⋊D7—22no—
46C2×C22⋊D7—22no—
47C22×Dic7—22no—
48C22×Dic7—24no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2312yes—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C2×C4×D7—24no—
55C2×C4×D7D414yes—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—24no—
58C7×C22⋊C4—22no—
59C7×C22⋊C4—24no—
60C22⋊Dic7—24no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64C4⋊Dic7—22no—
65C4⋊Dic7—24no—
66Dic7⋊C4—22no—
67C4×Dic7—22no—
68C4×Dic7—24no—
69C4×Dic7—24no—
70C2×C4×D4—71no—
71C2×#117—71no—
72#400—71no—
73#399D711yescharacteristic, complemented
74#398—71no—
75#397—72no—
76#397—71no—
77#394—72no—
78C4×C4⋊C4—72no—
79C4×C22⋊C4—72no—
80C4×C22⋊C4—71no—
81C23×C7—21no—
82C23×C7—22no—
83C23×C7—22no—
84C23×C7—21no—
85C23×C7#4112yes—
86C23×C7C2411yescharacteristic
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—21no—
90C22×D7—21no—
91C22×D7C2×D412yes—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7C2×D412yes—
101C2×C4×C7#4114yes—
102C22⋊D7—28no—
103C2×Dic7—21no—
104C2×Dic7—21no—
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—44no—
114C2×Dic7C2×D412yes—
115C2×Dic7#4112yes—
116C2×Dic7#4114yes—
117C4×D7—28no—
118C22×D4—71no—
119C23×C4—71no—
120C23×C4—71no—
121C4×D4—144no—
122#117—144no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4C2×D711yescharacteristic
127C2×C4⋊C4C2×D712yes—
128C2×C22⋊C4—72no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—71no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4C2×D711yescharacteristic
133C2×C22⋊C4C2×D712yes—
134C2×C42—72no—
135C2×C42—71no—
136C2×C42—72no—
137#95—72no—
138#95—71no—
139#95—72no—
140#95—72no—
141#95—72no—
142#95C2×D711yescharacteristic
143C22×C7—22no—
144C22×C7—24no—
145C22×C7—41no—
146C22×C7—42no—
147C22×C7—41no—
148C22×C7C22×D411yescharacteristic
149C22×C7C2×#4112yes—
150C22×C7C2×#4111yescharacteristic
151C22×C7C2×#4112yes—
152C22×C7C2×#4111yescharacteristic
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—22no—
156C2×D7—42no—
157C2×D7—42no—
158C4×C7—22no—
159C4×C7—24no—
160C4×C7—41no—
161C4×C7—42no—
162Dic7—22no—
163Dic7—41no—
164Dic7—42no—
165Dic7—42no—
166Dic7#12314yes—
167C24—71no—
168C24C22×D711yescharacteristic
169C2×D4—142no—
170C2×D4—142no—
171C2×D4—142no—
172C22×C4—71no—
173C22×C4—71no—
174C22×C4—74no—
175C22×C4—72no—
176C22×C4—71no—
177C22×C4—72no—
178C22×C4—144no—
179C22×C4—142no—
180C22×C4—144no—
181C22×C4C22×D711yescharacteristic
182C22×C4C22×D712yes—
183C22×C4C22×D711yescharacteristic
184C22×C4C22×D712yes—
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—144no—
188C4⋊C4—142no—
189C4⋊C4—142no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—144no—
193C22⋊C4—142no—
194C22⋊C4—144no—
195C22⋊C4—144no—
196C42—144no—
197C42—144no—
198C42—142no—
199C2×C7—42no—
200C2×C7C2×#12311yescharacteristic
201C2×C7#52511yescharacteristic
202C2×C7#52912yes—
203C2×C7#52912yes—
204C2×C7#53211yescharacteristic
205D7—42no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—72no—
211C23—142no—
212C23—141no—
213C23—142no—
214C23—141no—
215C23#116112yes—
216C23C23×D711yescharacteristic
217D4—148no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—72no—
227C2×C4—74no—
228C2×C4—72no—
229C2×C4—148no—
230C2×C4—141no—
231C2×C4—141no—
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—284no—
241C2×C4D4×D712yes—
242C2×C4#115914yes—
243C7#239511yescharacteristic, complemented, Sylow
244C22—24no—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—142no—
250C22—142no—
251C22—142no—
252C22—282no—
253C22—282no—
254C22C2×D4×D711yescharacteristic
255C22C2×#115912yes—
256C22C2×#116111yescharacteristic
257C22D7×#4112yes—
258C22D7×#4111yescharacteristic
259C4—24no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—74no—
264C4—142no—
265C4—282no—
266C4—282no—
267C4—281no—
268C2—42no—
269C2—282no—
270C2#9031512yes—
271C2#9031811yescharacteristic
272C2#9034012yes—
273C2D7×#12311yescharacteristic
274C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2395
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6455C2semidirect
3. C7×#399C2semidirect
4. #89765C2semidirect
5. #89754C2semidirect
6. C4⋊C4×Dic7C2semidirect
8. #89725C2semidirect
9. C22⋊C4×Dic7C2semidirect
10. #89438C2semidirect
11. #89433C2semidirect
73. #399D7semidirect
243. C7#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#64552
C2×(C22⋊D7)⋊C42
C7×#3992
#897652
#897542
C4⋊C4×Dic72
#897332
#897252
C22⋊C4×Dic72
#894382
#894332
#23957