Finite Group Databasebeta

← #11741538

Subgroups — #11741538

Order 896.

836 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 185 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
1C22×C7×#41C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic, complemented
3C2×#6499C211yesmaximal, characteristic, complemented
4C2×#6487C211yesmaximal, characteristic, complemented
5#90002C211yesmaximal, characteristic, complemented
6#89994C214yesmaximal, complemented
7#89993C211yesmaximal, characteristic, complemented
8#89982C214yesmaximal, complemented
9C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
10C2×C7×#41—24no—
11C2×C7×#41—24no—
12C2×C7×#41—24no—
13C22×C7×Q8C2211yescharacteristic
14C22×C7×D4C2212yes—
15C22×C7×D4C2211yescharacteristic
16C23×C4×C7C2212yes—
17C23×C4×C7C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2214yes—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2214yes—
23#6507—24no—
24#6499—24no—
25C2×C4⋊Dic7C2211yescharacteristic
26#6487—24no—
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2214yes—
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C7⋊#95C2212yes—
32C7⋊#95C2214yes—
33#2432—71nomaximal, Sylow
34C24×C7C2312yes—
35C24×C7C2311yescharacteristic
36C7×#41—232no—
37C2×C7×Q8—22no—
38C2×C7×Q8—22no—
39C2×C7×Q8D414yes—
40C2×C7×D4—24no—
41C2×C7×D4—24no—
42C2×C7×D4—24no—
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C2×C7×D4D414yes—
46C22×C4×C7—24no—
47C22×C4×C7—24no—
48C22×C4×C7—24no—
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7D414yes—
52C22×C4×C7C2311yescharacteristic
53C22×C4×C7C2312yes—
54C22×C4×C7C2311yescharacteristic
55C22×Dic7C2312yes—
56C22×Dic7C2312yes—
57C22×Dic7C2312yes—
58C22×Dic7C2312yes—
59C2×Dic14—22no—
60C2×Dic14—22no—
61C2×Dic14—22no—
62C22⋊Dic7—24no—
63C22⋊Dic7—28no—
64C22⋊Dic7—24no—
65C22⋊Dic7—28no—
66C4⋊Dic7—22no—
67Dic7⋊C4—24no—
68Dic7⋊C4—28no—
69C4×Dic7—22no—
70C4×Dic7—22no—
71C22×#41D711yescharacteristic
72C2×C4⋊Q8—71no—
73C2×#124—71no—
74C2×C22⋊Q8—71no—
75#394—74no—
76#391—71no—
77#390—71no—
78#385—74no—
79C4×C22⋊C4—71no—
80C23×C7—22no—
81C23×C7—21no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—24no—
86C23×C7—22no—
87C23×C7—24no—
88C23×C7C2×D414yes—
89C23×C7C2×D412yes—
90C23×C7C2411yescharacteristic
91C7×Q8—28no—
92C7×D4—216no—
93C7×D4—28no—
94C2×C4×C7—216no—
95C2×C4×C7—21no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—28no—
100C2×C4×C7C2×D412yes—
101C2×C4×C7C2×D414yes—
102C2×C4×C7C2×D412yes—
103C2×C4×C7C2×D412yes—
104C2×C4×C7C2×D412yes—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—24no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7#4114yescomplemented
116Dic14—44no—
117C2×#41—24no—
118C2×#41—24no—
119C2×#41—24no—
120C22×Q8—71no—
121C22×Q8C2×D711yescharacteristic
122C22×D4C2×D712yes—
123C22×D4C2×D711yescharacteristic
124C23×C4C2×D712yes—
125C23×C4C2×D711yescharacteristic
126C4⋊Q8—144no—
127#124—144no—
128C22⋊Q8—144no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—71no—
131C2×C4⋊C4—74no—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—74no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—74no—
136C2×C42—71no—
137C2×C42—71no—
138#95—72no—
139#95—74no—
140C22×C7—24no—
141C22×C7—22no—
142C22×C7—22no—
143C22×C7—22no—
144C22×C7—28no—
145C22×C7—42no—
146C22×C7—42no—
147C22×C7—42no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7C22×D411yescharacteristic
151C22×C7C22×D411yescharacteristic
152C22×C7C22×D411yescharacteristic
153C22×C7C2×#4112yes—
154C22×C72−1+412yes—
155C4×C7—22no—
156C4×C7—24no—
157C4×C7#12014yes—
158Dic7—24no—
159Dic7—42no—
160Dic7—42no—
161Dic7—42no—
162C24C22×D712yes—
163C24C22×D711yescharacteristic
164#41—232no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—142no—
170C2×Q8C22⋊D714yes—
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4C22⋊D714yes—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—72no—
183C22×C4—72no—
184C22×C4—72no—
185C22×C4—72no—
186C22×C4C22⋊D714yes—
187C22×C4C22×D711yescharacteristic
188C22×C4C22×D712yes—
189C22×C4C22×D711yescharacteristic
190C4⋊C4—144no—
191C4⋊C4—148no—
192C4⋊C4—142no—
193C22⋊C4—144no—
194C22⋊C4—148no—
195C22⋊C4—144no—
196C22⋊C4—148no—
197C42—142no—
198C42—142no—
199C2×C7—42no—
200C2×C7—44no—
201C2×C7C2×#12011yescharacteristic
202C2×C7#53211yescharacteristic
203C2×C7#53611yescharacteristic
204C2×C7#54712yes—
205C2×C7#54812yes—
206C23—24no—
207C23—22no—
208C23—24no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—21no—
214C23C2×C22⋊D714yes—
215C23C2×C22⋊D712yes—
216C23C23×D711yescharacteristic
217Q8—28no—
218Q8—284no—
219D4—216no—
220D4—28no—
221C2×C4—216no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—28no—
227C2×C4—74no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—142no—
231C2×C4—144no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4C2×C22⋊D712yes—
239C2×C4C2×C22⋊D714yes—
240C2×C4C2×C22⋊D712yes—
241C2×C4C2×C22⋊D712yes—
242C2×C4C2×C22⋊D712yes—
243C7#243211yescharacteristic, complemented, Sylow
244C22—28no—
245C22—22no—
246C22—22no—
247C22—24no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22D7×#4112yes—
255C22#655412yes—
256C22C22×C22⋊D711yescharacteristic
257C22C22×C22⋊D711yescharacteristic
258C22C22×C22⋊D711yescharacteristic
259C4—22no—
260C4—24no—
261C4—144no—
262C4—282no—
263C4—282no—
264C4—282no—
265C4#651614yes—
266C2—44no—
267C2—42no—
268C2#9051512yes—
269C2#9051611yescharacteristic
270C2#9052212yes—
271C2#9052311yescharacteristic
272C2C2×#651611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC22×C7×#41C2
hypercenterC22×#41D7
hypocenterC7#2432
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
2. C2×#6507C2semidirect
3. C2×#6499C2semidirect
4. C2×#6487C2semidirect
5. #90002C2semidirect
6. #89994C2semidirect
7. #89993C2semidirect
8. #89982C2semidirect
9. C4×C22⋊Dic7C2semidirect
115. C2×Dic7#41semidirect
243. C7#2432semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#412
C2×#65072
C2×#64992
C2×#64872
#900022
#899942
#899932
#899822
C4×C22⋊Dic72
#24327