Finite Group Databasebeta

← #11740567

Subgroups — #11740567

Order 896.

332 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 121 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×#529C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3#89836C212yesmaximal, complemented
4#89828C212yesmaximal, complemented
5#89817C212yesmaximal, complemented
6#89807C212yesmaximal
7#89793C211yesmaximal, characteristic, complemented
8#89792C211yesmaximal, characteristic, complemented
9#89780C211yesmaximal, characteristic, complemented
10#89779C211yesmaximal, characteristic, complemented
11#89697C211yesmaximal, characteristic, complemented
12C7×#126—21no—
13C7×#125C2211yescharacteristic
14C7×#124C2211yescharacteristic
15C7×#123—21no—
16C7×C22⋊Q8C2211yescharacteristic
17C7×#121C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4—21no—
20C4×C7×D4C2211yescharacteristic
21C7×#117—21no—
22C2×C42×C7C2211yescharacteristic
23#6489C2212yes—
24#6487—21no—
25#6487—21no—
26#6433C2212yes—
27#6432C2211yescharacteristic
28#6432C2211yescharacteristic
29C4×Dic14—21no—
30C4×Dic14—21no—
31#6409C2212yes—
32#6409C2212yes—
33#6406C2212yes—
34#6406C2212yes—
35C22⋊(C7⋊C8)C2212yes—
36#6382C2212yes—
37#6382C2212yes—
38#6381C2212yes—
39#6381C2212yes—
40C4⋊(C7⋊C8)C2211yescharacteristic
41C4⋊(C7⋊C8)C2211yescharacteristic
42C4×C7⋊C8C2212yes—
43#3233—71nomaximal, Sylow
44C2×C7×Q8C2311yescharacteristic
45C2×C7×D4—21no—
46C2×C7×D4C2311yescharacteristic
47C22×C4×C7—21no—
48C22×C4×C7D411yescharacteristic
49C22×C4×C7D411yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C2×Dic14—21no—
52C2×Dic14—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4—21no—
57C7×C4⋊C4C2311yescharacteristic
58C7×C4⋊C4C2311yescharacteristic
59C7×C22⋊C4—21no—
60C7×C22⋊C4—21no—
61C7×C22⋊C4—21no—
62C7×C22⋊C4—21no—
63C7×C22⋊C4—21no—
64C42×C7—21no—
65C42×C7D411yescharacteristic
66C42×C7D411yescharacteristic
67C42×C7C2311yescharacteristic
68C42×C7C2311yescharacteristic
69C22⋊Dic7—22no—
70C4⋊Dic7—21no—
71C4⋊Dic7—21no—
72C4⋊Dic7C2312yes—
73C4⋊Dic7C2312yes—
74Dic7⋊C4—22no—
75Dic7⋊C4—22no—
76C4×Dic7—22no—
77C2×C7⋊C8C2312yes—
78C2×C7⋊C8C2312yes—
79#533—71no—
80#529D711yescharacteristic
81#499—72no—
82#488—72no—
83#485—72no—
84#482—72no—
85#477—71no—
86#476—71no—
87#475—71no—
88#474—71no—
89#431—71no—
90C23×C7—21no—
91C23×C7C2×D411yescharacteristic
92C7×Q8—21no—
93C7×Q8—21no—
94C7×D4—21no—
95C7×D4—21no—
96C7×D4—42no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7C2×D411yescharacteristic
106C2×C4×C7C2×D411yescharacteristic
107C2×C4×C7C2×D411yescharacteristic
108C2×C4×C7C2×D411yescharacteristic
109C2×C4×C7C2×D411yescharacteristic
110C2×C4×C7C2411yescharacteristic
111C2×Dic7—22no—
112C2×Dic7—22no—
113Dic14—42no—
114Dic14—42no—
115C7⋊C8—22no—
116C7⋊C8—22no—
117C4⋊Q8—71no—
118C4⋊Q8—71no—
119#126—21no—
120#125—72no—
121#125C2×D711yescharacteristic
122#124C2×D711yescharacteristic
123#123—21no—
124C22⋊Q8—141no—
125C22⋊Q8—141no—
126C22⋊Q8C2×D711yescharacteristic
127#121C2×D711yescharacteristic
128C4×Q8—141no—
129C4×Q8—141no—
130C4×Q8C2×D711yescharacteristic
131C4×D4—21no—
132C4×D4C2×D711yescharacteristic
133#117—21no—
134#117—72no—
135C2×C42C2×D711yescharacteristic
136#107—72no—
137#107—72no—
138#106—72no—
139#106—72no—
140C4⋊C8—71no—
141C4⋊C8—71no—
142#103—72no—
143#103—72no—
144#102—72no—
145#102—72no—
146C22⋊C8—72no—
147C4×C8—72no—
148C22×C7—21no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7C22×D411yescharacteristic
155C4×C7—21no—
156C4×C7—21no—
157C4×C7—21no—
158C4×C7—21no—
159C4×C7—41no—
160C4×C7—41no—
161C4×C7—41no—
162C4×C7#13512yes—
163C4×C72−1+411yescharacteristic
164C4×C72−1+411yescharacteristic
165Dic7—42no—
166Dic7—42no—
167C2×Q8—141no—
168C2×Q8—141no—
169C2×Q8C22×D711yescharacteristic
170C2×D4—21no—
171C2×D4C22×D711yescharacteristic
172C22×C4—21no—
173C22×C4C22⋊D711yescharacteristic
174C22×C4C22⋊D711yescharacteristic
175C22×C4C22×D711yescharacteristic
176C2×C8—72no—
177C2×C8—72no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—72no—
183C4⋊C4—72no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—141no—
187C4⋊C4—141no—
188C4⋊C4C22×D711yescharacteristic
189C4⋊C4C22×D711yescharacteristic
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—142no—
196C42—21no—
197C42—142no—
198C42C22⋊D711yescharacteristic
199C42C22⋊D711yescharacteristic
200C42C22×D711yescharacteristic
201C42C22×D711yescharacteristic
202C2×C7—41no—
203C2×C7—81no—
204C2×C7#53611yescharacteristic
205C2×C7C2×#13511yescharacteristic
206C2×C7#57511yescharacteristic
207C23—21no—
208C23C2×C22⋊D711yescharacteristic
209Q8—21no—
210Q8—21no—
211Q8—282no—
212Q8—282no—
213D4—21no—
214D4—21no—
215D4—42no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4C2×C22⋊D711yescharacteristic
227C2×C4C2×C22⋊D711yescharacteristic
228C2×C4C2×C22⋊D711yescharacteristic
229C2×C4C2×C22⋊D711yescharacteristic
230C2×C4C2×C22⋊D711yescharacteristic
231C2×C4C23×D711yescharacteristic
232C8—142no—
233C8—142no—
234C7#323311yescharacteristic, complemented, Sylow
235C22—21no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22C22×C22⋊D711yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—282no—
250C4—282no—
251C4#651312yes—
252C4#655411yescharacteristic
253C4#655411yescharacteristic
254C2—41no—
255C2—81no—
256C2C2×#651311yescharacteristic
257C2#9051311yescharacteristic
258C2#9051611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#529C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#529D7
hypocenterC7#3233
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. #89836C2semidirect
4. #89828C2semidirect
5. #89817C2semidirect
7. #89793C2semidirect
8. #89792C2semidirect
9. #89780C2semidirect
10. #89779C2semidirect
11. #89697C2semidirect
234. C7#3233semidirect

Maximal subgroups

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

subgroupindex
C7×#5292
#901612
#898362
#898282
#898172
#898072
#897932
#897922
#897802
#897792
#896972
#32337