Finite Group Databasebeta

← #11733820

Subgroups — #11733820

Order 896.

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

normal subgroups only

This group has 420 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
1C7×#390C211yesmaximal, characteristic
2#89993C211yesmaximal, characteristic
3#89993C211yesmaximal, characteristic
4#89757C211yesmaximal, characteristic, complemented
5#89745C211yesmaximal, characteristic, complemented
6#89725C212yesmaximal, complemented
7#89724C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89720C212yesmaximal, complemented
10#89704C211yesmaximal, characteristic, complemented
11#89700C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2211yescharacteristic
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35#2644—71nomaximal, Sylow
36C24×C7C2312yes—
37C2×C7×D4—22no—
38C2×C7×D4—22no—
39C2×C7×D4—22no—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C7×C4⋊C4—22no—
51C7×C22⋊C4—42no—
52C7×C22⋊C4—42no—
53C42×C7—22no—
54C22⋊Dic7—42no—
55C22⋊Dic7—42no—
56C22⋊Dic7—42no—
57C22⋊Dic7—42no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7—42no—
61Dic7⋊C4—42no—
62Dic7⋊C4—42no—
63C4×Dic7—22no—
64C4×Dic7—22no—
65#397—72no—
66#397—72no—
67#397—72no—
68#396—72no—
69#390—71no—
70#390—71no—
71#390D711yescharacteristic
72#389—71no—
73#389—71no—
74#389—71no—
75#376—71no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—22no—
81C23×C7—22no—
82C23×C7—22no—
83C23×C7C2411yescharacteristic
84C7×D4—44no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7#4112yes—
95C2×C4×C7#4112yes—
96C2×C4×C7#4112yes—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—21no—
108C2×Dic7—22no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C22×D4C2×D711yescharacteristic
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C22⋊C4—72no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C22⋊C4C2×D712yes—
127C2×C22⋊C4C2×D712yes—
128C2×C42—71no—
129C2×C42—71no—
130C2×C42C2×D711yescharacteristic
131#95—71no—
132#95—71no—
133#95—71no—
134#95—71no—
135#95—72no—
136#95—72no—
137#95—72no—
138C22×C7—42no—
139C22×C7—42no—
140C22×C7—42no—
141C22×C7—42no—
142C22×C7—42no—
143C22×C7—42no—
144C22×C7—42no—
145C22×C7C2×#4111yescharacteristic
146C22×C7C2×#4111yescharacteristic
147C22×C7C2×#4111yescharacteristic
148C22×C7H(5,𝔽2)11yescharacteristic
149C22×C7H(5,𝔽2)11yescharacteristic
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+411yescharacteristic
152C4×C7—42no—
153C4×C7—41no—
154C4×C7—41no—
155C4×C7#12614yes—
156Dic7—41no—
157Dic7—41no—
158Dic7—42no—
159Dic7—42no—
160Dic7—41no—
161Dic7—41no—
162C24C22×D712yes—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C22×C4—71no—
167C22×C4—71no—
168C22×C4—71no—
169C22×C4—71no—
170C22×C4—72no—
171C22×C4—72no—
172C22×C4C22×D711yescharacteristic
173C22×C4C22×D711yescharacteristic
174C22×C4C22×D712yes—
175C22×C4C22×D711yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—142no—
178C4⋊C4—142no—
179C4⋊C4—282no—
180C4⋊C4—282no—
181C4⋊C4—282no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—282no—
185C22⋊C4—282no—
186C22⋊C4—282no—
187C22⋊C4—282no—
188C42—22no—
189C42—142no—
190C42—142no—
191C2×C7—82no—
192C2×C7C2×#12611yescharacteristic
193C2×C7#54011yescharacteristic
194C2×C7#54011yescharacteristic
195C2×C7#54011yescharacteristic
196C2×C7#54211yescharacteristic
197C2×C7#54211yescharacteristic
198C2×C7#54211yescharacteristic
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23C23×D711yescharacteristic
207D4—44no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—142no—
220C2×C4—141no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—141no—
228C2×C4—141no—
229C2×C4—141no—
230C2×C4—141no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—141no—
234C2×C4—141no—
235C2×C4#115712yes—
236C2×C4#115912yes—
237C2×C4#115912yes—
238C7#264411yescharacteristic, complemented, Sylow
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22C2×#115711yescharacteristic
247C22C2×#115911yescharacteristic
248C22C2×#115911yescharacteristic
249C22#654811yescharacteristic
250C22#654811yescharacteristic
251C22#655311yescharacteristic
252C22#655411yescharacteristic
253C4—42no—
254C4—41no—
255C4—41no—
256C4—281no—
257C4—281no—
258C4—281no—
259C4—281no—
260C4—282no—
261C4—282no—
262C4#644214yes—
263C2—82no—
264C2C2×#644211yescharacteristic
265C2#9025211yescharacteristic
266C2#9025311yescharacteristic
267C2#9029411yescharacteristic
268C2#9031011yescharacteristic
269C2#9037311yescharacteristic
270C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2644
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
4. #89757C2semidirect
5. #89745C2semidirect
6. #89725C2semidirect
7. #89724C2semidirect
8. #89724C2semidirect
9. #89720C2semidirect
10. #89704C2semidirect
11. #89700C2semidirect
238. C7#2644semidirect

Maximal subgroups

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

subgroupindex
C7×#3902
#899932
#897572
#897452
#897252
#897242
#897202
#897042
#897002
#26447