Finite Group Databasebeta

← #11733890

Subgroups — #11733890

Order 896.

836 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 173 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×#1159C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic, complemented
3C2×#6460C211yesmaximal, characteristic, complemented
4C2×#6432C211yesmaximal, characteristic, complemented
5C7×#390C211yesmaximal, characteristic, complemented
6#89746C211yesmaximal, characteristic, complemented
7#89733C214yesmaximal, complemented
8#89722C214yesmaximal, complemented
9C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C23×Dic7C2212yes—
13C2×#1159—24no—
14C2×#1159—24no—
15C2×#1159—24no—
16C22×C4×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2214yes—
21C2×C42×C7C2211yescharacteristic
22C2×C22⋊Dic7C2214yes—
23#6499—24no—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×C4⋊Dic7C2214yes—
27C2×C4⋊Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6460—24no—
30#6432—24no—
31C7⋊#95C2212yes—
32C7⋊#95C2214yes—
33#2432—71nomaximal, Sylow
34C24×C7C2312yes—
35C23×D7C2311yescharacteristic
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C2×C7×D4D414yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C2×C22⋊D7—24no—
43C2×C22⋊D7—24no—
44C2×C22⋊D7—24no—
45C22×Dic7—24no—
46C22×Dic7—24no—
47C22×Dic7—24no—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52#1159—232no—
53C2×C4×D7—22no—
54C2×C4×D7—22no—
55C2×C4×D7D414yes—
56C2×Dic14—22no—
57C2×Dic14—22no—
58C2×Dic14—22no—
59C2×Dic14—22no—
60C2×Dic14—22no—
61C2×Dic14D414yes—
62C7×C4⋊C4—22no—
63C7×C22⋊C4—28no—
64C42×C7—22no—
65C22⋊Dic7—28no—
66(C2×D7)⋊C4—24no—
67(C2×D7)⋊C4—24no—
68C4⋊Dic7—28no—
69C4⋊Dic7—24no—
70C4×Dic7—22no—
71C22×#41—71no—
72C2×C4⋊Q8—71no—
73C2×#124—71no—
74C2×C22⋊Q8—71no—
75#394—74no—
76#391—71no—
77#390D711yescharacteristic, complemented
78#385—74no—
79C4×C22⋊C4—71no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7—24no—
83C23×C7—24no—
84C23×C7C2×D414yes—
85C23×C7C2411yescharacteristic
86C22×D7—21no—
87C22×D7—21no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7C2×D412yes—
91C7×D4—28no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—24no—
96C2×C4×C7C2×D412yes—
97C2×C4×C7C2×D412yes—
98C2×C4×C7C2×D412yes—
99C2×C4×C7#4114yescomplemented
100C22⋊D7—216no—
101C2×Dic7—216no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7C2×D414yes—
113C2×Dic7C2×D412yes—
114C4×D7—28no—
115Dic14—28no—
116Dic14—44no—
117C2×#41—144no—
118C2×#41—144no—
119C2×#41—144no—
120C22×Q8—71no—
121C22×Q8—71no—
122C22×D4—72no—
123C22×D4C2×D711yescharacteristic
124C23×C4—71no—
125C23×C4—72no—
126C4⋊Q8—144no—
127#124—144no—
128C22⋊Q8—144no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—74no—
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C22⋊C4—74no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—72no—
135C2×C22⋊C4C2×D714yes—
136C2×C42—71no—
137C2×C42C2×D711yescharacteristic
138#95—72no—
139#95—74no—
140C22×C7—24no—
141C22×C7—28no—
142C22×C7—42no—
143C22×C7—42no—
144C22×C7—44no—
145C22×C7C22×D411yescharacteristic
146C22×C7C22×D411yescharacteristic
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4112yes—
149C22×C72−1+412yes—
150C2×D7—22no—
151C2×D7—22no—
152C2×D7—22no—
153C2×D7—42no—
154C2×D7—42no—
155C4×C7—24no—
156C4×C7—42no—
157C4×C7#12014yes—
158Dic7—24no—
159Dic7—22no—
160Dic7—42no—
161Dic7—42no—
162C24—71no—
163C24C22×D712yes—
164#41—1432no—
165C2×Q8—74no—
166C2×Q8—142no—
167C2×Q8—142no—
168C2×Q8—142no—
169C2×Q8—142no—
170C2×Q8—142no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—144no—
174C2×D4—144no—
175C2×D4—144no—
176C2×D4D2814yes—
177C22×C4—71no—
178C22×C4—72no—
179C22×C4—74no—
180C22×C4—72no—
181C22×C4—72no—
182C22×C4—144no—
183C22×C4—144no—
184C22×C4—144no—
185C22×C4—142no—
186C22×C4—142no—
187C22×C4C22×D712yes—
188C22×C4C22×D712yes—
189C22×C4C22×D711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—144no—
192C4⋊C4—148no—
193C22⋊C4—28no—
194C22⋊C4—144no—
195C22⋊C4—148no—
196C22⋊C4—144no—
197C42—22no—
198C42—142no—
199C2×C7—44no—
200C2×C7C2×#12011yescharacteristic
201C2×C7#53211yescharacteristic
202C2×C7#53611yescharacteristic
203C2×C7#54712yes—
204C2×C7#54812yes—
205D7—42no—
206C23—22no—
207C23—22no—
208C23—24no—
209C23—24no—
210C23—72no—
211C23—141no—
212C23—141no—
213C23—142no—
214C23—142no—
215C23C2×D2814yes—
216C23C23×D711yescharacteristic
217Q8—148no—
218Q8—284no—
219D4—28no—
220D4—1416no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—72no—
226C2×C4—74no—
227C2×C4—1416no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—148no—
233C2×C4—142no—
234C2×C4—142no—
235C2×C4—142no—
236C2×C4—142no—
237C2×C4—142no—
238C2×C4—142no—
239C2×C4C2×D2812yes—
240C2×C4D4×D712yes—
241C2×C4D4×D712yes—
242C2×C4#115914yescomplemented
243C7#243211yescharacteristic, complemented, Sylow
244C22—24no—
245C22—28no—
246C22—42no—
247C22—44no—
248C22—42no—
249C22—142no—
250C22—142no—
251C22—142no—
252C22—282no—
253C22—282no—
254C22C22×D2811yescharacteristic
255C22C2×D4×D711yescharacteristic
256C22C2×D4×D711yescharacteristic
257C22C2×#115912yes—
258C22#655412yes—
259C4—24no—
260C4—42no—
261C4—142no—
262C4—144no—
263C4—282no—
264C4—282no—
265C4#644514yes—
266C2—44no—
267C2—282no—
268C2C2×#644511yescharacteristic
269C2#9024612yes—
270C2#9028912yes—
271C2#9036611yescharacteristic
272C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×#1159C2
hypercenter#390D7
hypocenterC7#2432
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6499C2semidirect
3. C2×#6460C2semidirect
4. C2×#6432C2semidirect
5. C7×#390C2semidirect
6. #89746C2semidirect
7. #89733C2semidirect
8. #89722C2semidirect
9. C4×(C2×D7)⋊C4C2semidirect
77. #390D7semidirect
99. C2×C4×C7#41semidirect
242. C2×C4#1159semidirect
243. C7#2432semidirect

Maximal subgroups

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

subgroupindex
C22×#11592
C2×#64992
C2×#64602
C2×#64322
C7×#3902
#897462
#897332
#897222
C4×(C2×D7)⋊C42
#24327