Finite Group Databasebeta

← #11740143

Subgroups — #11740143

Order 896.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×C7×#121C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic, complemented
3C2×#6487C211yesmaximal, characteristic, complemented
4C22×C4×Dic7C211yesmaximal, characteristic, complemented
5#89993C211yesmaximal, characteristic, complemented
6#89992C214yesmaximal, complemented
7#89749C211yesmaximal, characteristic, complemented
8#89746C211yesmaximal, characteristic, complemented
9#89717C214yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C7×D4C2212yes—
12C23×C4×C7C2211yescharacteristic
13C23×Dic7C2212yes—
14C22×Dic14C2211yescharacteristic
15C7×#121—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2214yes—
20C2×C22⋊Dic7C2214yes—
21#6499—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23#6487—24no—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7—24no—
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2214yes—
30C7⋊#95C2214yes—
31#2850—71nomaximal, Sylow
32C24×C7C2312yes—
33C24×C7C2311yescharacteristic
34C2×C7×D4—24no—
35C2×C7×D4—28no—
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C2×C7×D4—22no—
39C22×C4×C7—22no—
40C22×C4×C7—22no—
41C22×C4×C7D414yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7—24no—
46C22×Dic7—24no—
47C22×Dic7—24no—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C2×Dic14—22no—
53C2×Dic14—22no—
54C2×Dic14—22no—
55C7×C4⋊C4—22no—
56C7×C22⋊C4—24no—
57C22⋊Dic7—24no—
58C22⋊Dic7—28no—
59C22⋊Dic7—28no—
60C4⋊Dic7—22no—
61Dic7⋊C4—24no—
62C4×Dic7—22no—
63C4×Dic7—24no—
64C4×Dic7D414yescomplemented
65C2×#124—71no—
66C2×C22⋊Q8—71no—
67C2×#121D711yescharacteristic
68C22×C42—71no—
69#391—71no—
70#390—71no—
71#388—74no—
72#386—74no—
73#382—71no—
74C23×C7—21no—
75C23×C7—22no—
76C23×C7—21no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—24no—
80C23×C7—22no—
81C23×C7—24no—
82C23×C7C2×D412yes—
83C23×C7#4114yes—
84C23×C7C2411yescharacteristic
85C7×D4—48no—
86C7×D4—44no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—28no—
90C2×C4×C7—22no—
91C2×C4×C7—24no—
92C2×C4×C7C2×D412yes—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×D412yes—
95C2×C4×C7#4112yes—
96C2×Dic7—216no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—24no—
105C2×Dic7—22no—
106C2×Dic7C2×D414yes—
107C2×Dic7#4114yes—
108Dic14—44no—
109C22×Q8—71no—
110C22×D4C2×D711yescharacteristic
111C22×D4C2×D712yes—
112C23×C4—72no—
113C23×C4C2×D711yescharacteristic
114#124—144no—
115C22⋊Q8—144no—
116#121—24no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C22⋊C4—72no—
121C2×C22⋊C4—74no—
122C2×C22⋊C4—74no—
123C2×C22⋊C4C2×D712yes—
124C2×C42—71no—
125C2×C42—72no—
126C2×C42—71no—
127C2×C42—144no—
128#95—74no—
129#95—74no—
130C22×C7—22no—
131C22×C7—24no—
132C22×C7—24no—
133C22×C7—22no—
134C22×C7—42no—
135C22×C7—44no—
136C22×C7—42no—
137C22×C7—44no—
138C22×C7—42no—
139C22×C7#12414yescomplemented
140C22×C7C22×D411yescharacteristic
141C22×C7C22×D411yescharacteristic
142C22×C7C2×#4111yescharacteristic
143C22×C7C2×#4112yes—
144C22×C7C2×#4112yes—
145C4×C7—22no—
146C4×C7—42no—
147C4×C7#12114yes—
148Dic7—24no—
149Dic7—24no—
150Dic7—42no—
151Dic7—42no—
152C24C22×D712yes—
153C24C22×D711yescharacteristic
154C2×Q8—142no—
155C2×Q8—142no—
156C2×Q8—142no—
157C2×D4—28no—
158C2×D4—24no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—72no—
168C22×C4—144no—
169C22×C4—144no—
170C22×C4—144no—
171C22×C4C22⋊D714yes—
172C22×C4C22×D712yes—
173C22×C4C22×D711yescharacteristic
174C22×C4C22×D711yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—144no—
177C4⋊C4—142no—
178C22⋊C4—24no—
179C22⋊C4—148no—
180C22⋊C4—148no—
181C22⋊C4—144no—
182C42—74no—
183C42—144no—
184C42—142no—
185C2×C7—24no—
186C2×C7—82no—
187C2×C7C2×#12111yescharacteristic
188C2×C7#52512yes—
189C2×C7C2×#12411yescharacteristic
190C2×C7#52912yes—
191C2×C7#53211yescharacteristic
192C23—21no—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23#115914yes—
201C23C2×C22⋊D712yes—
202C23C23×D711yescharacteristic
203Q8—284no—
204D4—48no—
205D4—44no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—28no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—74no—
212C2×C4—74no—
213C2×C4—1416no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—144no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4D4×D712yes—
224C2×C4#115912yes—
225C2×C4C2×C22⋊D712yes—
226C2×C4C2×C22⋊D712yes—
227C7#285011yescharacteristic, complemented, Sylow
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—44no—
237C22#649914yescomplemented
238C22C2×D4×D711yescharacteristic
239C22C2×#115912yes—
240C22C2×#115911yescharacteristic
241C22D7×#4112yes—
242C22C22×C22⋊D711yescharacteristic
243C4—22no—
244C4—42no—
245C4—144no—
246C4—144no—
247C4—282no—
248C4—282no—
249C4#650214yes—
250C2—24no—
251C2—82no—
252C2#9029112yes—
253C2#9029712yes—
254C2C2×#649911yescharacteristic
255C2C2×#650211yescharacteristic
256C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2850
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#6499C2semidirect
3. C2×#6487C2semidirect
4. C22×C4×Dic7C2semidirect
5. #89993C2semidirect
6. #89992C2semidirect
7. #89749C2semidirect
8. #89746C2semidirect
9. #89717C2semidirect
64. C4×Dic7D4semidirect
139. C22×C7#124semidirect
227. C7#2850semidirect
237. C22#6499semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#64992
C2×#64872
C22×C4×Dic72
#899932
#899922
#897492
#897462
#897172
#28507