Finite Group Databasebeta

← #11735503

Subgroups — #11735503

Order 896.

498 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6446C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic, complemented
4#89765C212yesmaximal, complemented
5#89757C212yesmaximal, complemented
6#89754C211yesmaximal, characteristic, complemented
7#89733C212yesmaximal
8#89723C212yesmaximal, complemented
9#89719C211yesmaximal, characteristic, complemented
10#89438C211yesmaximal, characteristic, complemented
11#89428C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2212yes—
26C2×C4⋊Dic7C2212yes—
27C2×Dic7⋊C4C2211yescharacteristic
28C2×C4×Dic7C2212yes—
29#6460—24no—
30#6446—24no—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2663—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—42no—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C2×C4×D7—42no—
52C2×C4×D7D414yes—
53C2×Dic14—22no—
54C2×Dic14—42no—
55C7×C4⋊C4—22no—
56C7×C4⋊C4—24no—
57C7×C22⋊C4—22no—
58C7×C22⋊C4—24no—
59C22⋊Dic7—22no—
60C22⋊Dic7—42no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64C4⋊Dic7—22no—
65C4⋊Dic7—24no—
66C4⋊Dic7—42no—
67Dic7⋊C4—22no—
68C4×Dic7—42no—
69C2×#123—71no—
70C2×C22⋊Q8—71no—
71#400—71no—
72#399D711yescharacteristic, complemented
73#397—72no—
74#395—71no—
75#394—72no—
76#393—71no—
77#389—72no—
78#388—72no—
79C4×C22⋊C4—71no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7C2411yescharacteristic
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—21no—
89C22×D7—21no—
90C22×D7C2×D412yes—
91C2×C4×C7—22no—
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×C7C2×D412yes—
100C2×C4×C7#4114yes—
101C22⋊D7—44no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—22no—
113C2×Dic7—21no—
114C2×Dic7C2×D412yes—
115C4×D7—44no—
116Dic14—44no—
117C22×Q8—71no—
118C22×D4—71no—
119C23×C4—71no—
120#123—144no—
121C22⋊Q8—144no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4C2×D711yescharacteristic
127C2×C4⋊C4C2×D712yes—
128C2×C22⋊C4—71no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—71no—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4C2×D711yescharacteristic
134C2×C22⋊C4C2×D712yes—
135C2×C42—72no—
136#95—71no—
137#95—72no—
138#95—72no—
139#95—72no—
140#95—72no—
141#95C2×D711yescharacteristic
142C22×C7—41no—
143C22×C7—41no—
144C22×C7—41no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7H(5,𝔽2)12yes—
150C22×C72−1+411yescharacteristic
151C22×C72−1+411yescharacteristic
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—42no—
156C2×D7—42no—
157C4×C7—22no—
158C4×C7—24no—
159C4×C7—41no—
160C4×C7—42no—
161Dic7—22no—
162Dic7—42no—
163Dic7—41no—
164Dic7—41no—
165Dic7—42no—
166C24—71no—
167C24C22×D711yescharacteristic
168C2×Q8—142no—
169C2×Q8—282no—
170C2×D4—142no—
171C2×D4—282no—
172C22×C4—71no—
173C22×C4—74no—
174C22×C4—72no—
175C22×C4—71no—
176C22×C4—72no—
177C22×C4—71no—
178C22×C4—282no—
179C22×C4C22×D711yescharacteristic
180C22×C4C22×D712yes—
181C22×C4C22×D711yescharacteristic
182C22×C4C22×D712yes—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—144no—
186C4⋊C4—142no—
187C4⋊C4—142no—
188C4⋊C4—282no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—144no—
192C22⋊C4—142no—
193C22⋊C4—142no—
194C22⋊C4—144no—
195C22⋊C4—282no—
196C42—282no—
197C2×C7—81no—
198C2×C7#53211yescharacteristic
199C2×C7#53411yescharacteristic
200C2×C7#53611yescharacteristic
201C2×C7#53912yes—
202C2×C7#54212yes—
203D7—42no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—72no—
210C23—142no—
211C23—141no—
212C23—142no—
213C23—141no—
214C23C23×D711yescharacteristic
215Q8—284no—
216D4—284no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—72no—
226C2×C4—142no—
227C2×C4—141no—
228C2×C4—142no—
229C2×C4—141no—
230C2×C4—142no—
231C2×C4—141no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—141no—
238C2×C4—284no—
239C2×C4D4×D712yes—
240C2×C4#115914yes—
241C7#266311yescharacteristic, complemented, Sylow
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—142no—
248C22—142no—
249C22—142no—
250C22—282no—
251C22—282no—
252C22C2×D4×D711yescharacteristic
253C22C2×#115912yes—
254C22#655111yescharacteristic
255C22#655312yes—
256C22#655411yescharacteristic
257C4—22no—
258C4—24no—
259C4—41no—
260C4—42no—
261C4—142no—
262C4—281no—
263C4—282no—
264C4—281no—
265C4—282no—
266C2—81no—
267C2—282no—
268C2#9031911yescharacteristic
269C2#9033412yes—
270C2#9034411yescharacteristic
271C2#9035612yes—
272C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2663
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
3. C7×#399C2semidirect
4. #89765C2semidirect
5. #89757C2semidirect
6. #89754C2semidirect
8. #89723C2semidirect
9. #89719C2semidirect
10. #89438C2semidirect
11. #89428C2semidirect
72. #399D7semidirect
241. C7#2663semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64462
C7×#3992
#897652
#897572
#897542
#897332
#897232
#897192
#894382
#894282
#26637