Finite Group Databasebeta

← #11734249

Subgroups — #11734249

Order 896.

520 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 141 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×Q8×Dic7C211yesmaximal, characteristic, complemented
2C2×#6448C211yesmaximal, characteristic
3C7×#393C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic, complemented
5#89729C212yesmaximal
6#89725C211yesmaximal, characteristic, complemented
7#89723C212yesmaximal, complemented
8#89446C211yesmaximal, characteristic, complemented
9#89442C212yesmaximal, complemented
10#89421C211yesmaximal, characteristic, complemented
11#89418C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19Q8×Dic7—24no—
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×C4⋊Dic7C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30#6448—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2592—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C2×C7×Q8—22no—
41C2×C7×Q8—24no—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2312yes—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—24no—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C2×Dic14—22no—
54C2×Dic14—24no—
55C7×C22⋊C4—22no—
56C7×C22⋊C4—24no—
57C22⋊Dic7—22no—
58C22⋊Dic7—42no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—42no—
63C4⋊Dic7—22no—
64C4⋊Dic7—24no—
65Dic7⋊C4—24no—
66C4×Dic7—24no—
67C4×Dic7—42no—
68C4×Dic7D414yes—
69C2×#124—71no—
70C2×C4×Q8—71no—
71#397—71no—
72#397—71no—
73#393—71no—
74#393D711yescharacteristic, complemented
75#391—72no—
76#390—72no—
77#388—72no—
78#388—72no—
79C4×C4⋊C4—71no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7C2411yescharacteristic
86C22×D7—21no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—21no—
90C22×D7—21no—
91C7×Q8—28no—
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—44no—
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×Dic7C2×D412yes—
114C2×Dic7C2×D412yes—
115C2×Dic7#4114yes—
116Dic14—44no—
117C22×Q8—71no—
118C22×Q8C2×D711yescharacteristic
119C22×D4—71no—
120#124—144no—
121C4×Q8—144no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—72no—
125C2×C22⋊C4—71no—
126C2×C22⋊C4—71no—
127C2×C22⋊C4—71no—
128C2×C22⋊C4—72no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4C2×D711yescharacteristic
132C2×C22⋊C4C2×D712yes—
133C2×C42—71no—
134C2×C42—72no—
135C2×C42—72no—
136#95—71no—
137#95—72no—
138#95—72no—
139#95—72no—
140#95C2×D711yescharacteristic
141#95C2×D712yes—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7—41no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7C2×#4112yes—
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+411yescharacteristic
152C2×D7—41no—
153C2×D7—41no—
154C2×D7—41no—
155C2×D7—42no—
156C2×D7—42no—
157C4×C7—22no—
158C4×C7—24no—
159C4×C7—41no—
160C4×C7—42no—
161Dic7—22no—
162Dic7—41no—
163Dic7—42no—
164Dic7—42no—
165Dic7#12414yes—
166C24—71no—
167C24C22×D711yescharacteristic
168C2×Q8—22no—
169C2×Q8—24no—
170C2×Q8—142no—
171C2×Q8—144no—
172C2×D4—142no—
173C2×D4—144no—
174C22×C4—71no—
175C22×C4—71no—
176C22×C4—71no—
177C22×C4—72no—
178C22×C4—72no—
179C22×C4C22×D711yescharacteristic
180C22×C4C22×D712yes—
181C22×C4C22×D711yescharacteristic
182C22×C4C22×D712yes—
183C4⋊C4—144no—
184C4⋊C4—144no—
185C4⋊C4—142no—
186C22⋊C4—22no—
187C22⋊C4—24no—
188C22⋊C4—142no—
189C22⋊C4—142no—
190C22⋊C4—142no—
191C22⋊C4—144no—
192C22⋊C4—282no—
193C22⋊C4—282no—
194C42—74no—
195C42—144no—
196C42—282no—
197C2×C7—81no—
198C2×C7#52511yescharacteristic
199C2×C7C2×#12411yescharacteristic
200C2×C7#53711yescharacteristic
201C2×C7#55612yes—
202C2×C7#55912yes—
203D7—81no—
204C23—21no—
205C23—22no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—141no—
210C23—142no—
211C23—141no—
212C23—142no—
213C23—141no—
214C23C23×D711yescharacteristic
215Q8—28no—
216Q8—284no—
217D4—284no—
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—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×C4D4×D712yes—
240C2×C4#115914yes—
241C7#259211yescharacteristic, complemented, Sylow
242C22—42no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—41no—
247C22—282no—
248C22—281no—
249C22—281no—
250C22—282no—
251C22—281no—
252C22C2×D4×D711yescharacteristic
253C22C2×#115912yes—
254C22#655111yescharacteristic
255C22D7×#4112yes—
256C22#655311yescharacteristic
257C4—22no—
258C4—24no—
259C4—42no—
260C4—41no—
261C4—74no—
262C4—142no—
263C4—282no—
264C4—282no—
265C4—281no—
266C2—81no—
267C2—561no—
268C2#9027711yescharacteristic
269C2#9032011yescharacteristic
270C2#9033212yes—
271C2D7×#12411yescharacteristic
272C2#9037212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC22×C22⋊D7C22
hypercenter#393D7
hypocenterC7#2592
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic7C2semidirect
3. C7×#393C2semidirect
4. #90003C2semidirect
6. #89725C2semidirect
7. #89723C2semidirect
8. #89446C2semidirect
9. #89442C2semidirect
10. #89421C2semidirect
11. #89418C2semidirect
74. #393D7semidirect
241. C7#2592semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic72
C2×#64482
C7×#3932
#900032
#897292
#897252
#897232
#894462
#894422
#894212
#894182
#25927