Finite Group Databasebeta

← #11738577

Subgroups — #11738577

Order 896.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#90419C211yesmaximal, characteristic, complemented
2#90271C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89803C211yesmaximal, characteristic, complemented
6#89658C212yesmaximal
7#89649C212yesmaximal, complemented
8#89613C211yesmaximal, characteristic, complemented
9#89609C211yesmaximal, characteristic, complemented
10#89588C212yesmaximal, complemented
11#89570C212yesmaximal, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14#6510C2211yescharacteristic
15#6509—21no—
16#6509C2211yescharacteristic
17Q8×Dic7—21no—
18C2×#1144C2212yes—
19C2×#1132C2212yes—
20C2×#1131C2212yes—
21#6461—21no—
22#6460C2212yes—
23#6458C2211yescharacteristic
24#6457—21no—
25#6455—21no—
26#6453—21no—
27#6453C2211yescharacteristic
28#6450C2212yes—
29C4×D28—21no—
30C4×D28C2211yescharacteristic
31#6435—21no—
32C42×D7C2211yescharacteristic
33#6432C2211yescharacteristic
34C7×#107C2212yes—
35C7×C4⋊C8C2211yescharacteristic
36C7×#103C2212yes—
37#6390C2212yes—
38Dic7⋊C8C2212yes—
39#6384C2212yes—
40#6383C2212yes—
41#6382C2212yes—
42C4⋊(C7⋊C8)C2211yescharacteristic
43#3252—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C7×Q8C2311yescharacteristic
46C2×D28—21no—
47C2×D28C2311yescharacteristic
48C2×C4×D7—21no—
49C2×C4×D7D411yescharacteristic
50C2×C4×D7D411yescharacteristic
51C2×C4×D7C2311yescharacteristic
52C2×Dic14C2312yes—
53C2×C7×C8C2312yes—
54C7×C4⋊C4—21no—
55C7×C4⋊C4—21no—
56C7×C4⋊C4C2312yes—
57C7×C4⋊C4C2311yescharacteristic
58C42×C7—21no—
59C42×C7C2311yescharacteristic
60#1144—42no—
61(C2×D7)⋊C4—21no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—21no—
64(C2×D7)⋊C4—21no—
65(C2×D7)⋊C4—21no—
66(C2×D7)⋊C4—21no—
67C4⋊Dic7—21no—
68C4⋊Dic7—21no—
69C4⋊Dic7—21no—
70C4⋊Dic7—22no—
71C4⋊Dic7C2311yescharacteristic
72Dic7⋊C4—21no—
73Dic7⋊C4—21no—
74Dic7⋊C4—22no—
75C4×Dic7—22no—
76C4×Dic7D411yescharacteristic
77C4×Dic7D411yescharacteristic
78C4×Dic7C2311yescharacteristic
79C2×C7⋊C8C2312yes—
80#1132—42no—
81#1131—42no—
82#533—71no—
83#529—71no—
84#479—72no—
85#477D711yescharacteristic, complemented
86#476—71no—
87#470—72no—
88#468—72no—
89#464—72no—
90#462—71no—
91#461—71no—
92#423—71no—
93C22×D7—21no—
94C22×D7C2×D411yescharacteristic
95C7×Q8—21no—
96C7×Q8—21no—
97C7×Q8—42no—
98C2×C4×C7—22no—
99C2×C4×C7—21no—
100C2×C4×C7—21no—
101C2×C4×C7C2×D411yescharacteristic
102C2×C4×C7C2×D411yescharacteristic
103C2×C4×C7C2411yescharacteristic
104C2×Dic7—22no—
105C2×Dic7—21no—
106C2×Dic7C2×D411yescharacteristic
107C2×Dic7C2×D411yescharacteristic
108C2×Dic7C2×D411yescharacteristic
109D28—21no—
110D28—21no—
111D28—42no—
112C4×D7—22no—
113C4×D7—21no—
114C4×D7—21no—
115C4×D7—41no—
116C4×D7—42no—
117Dic14—22no—
118Dic14—22no—
119C7×C8—22no—
120C7⋊C8—22no—
121C2×Q16—72no—
122C2×SD16—72no—
123C2×M16—72no—
124C4⋊Q8—71no—
125C4⋊Q8C2×D711yescharacteristic
126#126—141no—
127#125—71no—
128#125—141no—
129#124—71no—
130#123—141no—
131C22⋊Q8—71no—
132C22⋊Q8—72no—
133C22⋊Q8—141no—
134#121—71no—
135C4×Q8—72no—
136C4×Q8—141no—
137C4×Q8C2×D711yescharacteristic
138C4×D4—71no—
139C4×D4—141no—
140#117—141no—
141#117—141no—
142C2×C42—71no—
143#107C2×D712yes—
144#106—72no—
145C4⋊C8—72no—
146C4⋊C8—71no—
147C4⋊C8C2×D711yescharacteristic
148#103—72no—
149#103—72no—
150#103C2×D712yes—
151#102—72no—
152C22×C7C22×D411yescharacteristic
153C2×D7—21no—
154C2×D7—21no—
155C2×D7—21no—
156C2×D7—41no—
157C2×D7—41no—
158C2×D7—41no—
159C4×C7—21no—
160C4×C7—41no—
161C4×C7—41no—
162C4×C7—42no—
163C4×C7#13712yes—
164C4×C7H(5,𝔽2)11yescharacteristic
165C4×C72−1+411yescharacteristic
166Dic7—21no—
167Dic7—21no—
168Dic7—21no—
169Dic7—42no—
170Dic7—41no—
171C2×Q8—21no—
172C2×Q8—72no—
173C2×Q8C22×D711yescharacteristic
174C2×D4—71no—
175C2×D4—141no—
176C22×C4—71no—
177C22×C4—71no—
178C22×C4—71no—
179C22×C4—141no—
180Q16—282no—
181SD16—282no—
182M16—282no—
183C2×C8—72no—
184C2×C8C22×D712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—71no—
188C4⋊C4—141no—
189C4⋊C4—141no—
190C4⋊C4—142no—
191C4⋊C4—141no—
192C4⋊C4—142no—
193C4⋊C4—141no—
194C4⋊C4—141no—
195C4⋊C4C22×D712yes—
196C4⋊C4C22×D711yescharacteristic
197C22⋊C4—141no—
198C22⋊C4—141no—
199C22⋊C4—141no—
200C22⋊C4—142no—
201C22⋊C4—141no—
202C22⋊C4—141no—
203C42—21no—
204C42—71no—
205C42—71no—
206C42—71no—
207C42—142no—
208C42C22×D711yescharacteristic
209C2×C7#53711yescharacteristic
210C2×C7C2×#13711yescharacteristic
211C2×C7#57511yescharacteristic
212D7—41no—
213D7—81no—
214C23—71no—
215C23—141no—
216Q8—21no—
217Q8—21no—
218Q8—42no—
219Q8—142no—
220Q8—142no—
221D4—141no—
222D4—141no—
223D4—282no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—71no—
228C2×C4—71no—
229C2×C4—71no—
230C2×C4—142no—
231C2×C4—141no—
232C2×C4—141no—
233C2×C4—142no—
234C2×C4—141no—
235C2×C4—281no—
236C2×C4—282no—
237C2×C4D4×D711yescharacteristic
238C2×C4D4×D711yescharacteristic
239C2×C4C23×D711yescharacteristic
240C8—22no—
241C8—142no—
242C7#325211yescharacteristic, complemented, Sylow
243C22—141no—
244C22—141no—
245C22—141no—
246C22—281no—
247C22—281no—
248C22—281no—
249C22C2×D4×D711yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—141no—
255C4—141no—
256C4—141no—
257C4—281no—
258C4—282no—
259C4#648112yes—
260C4#655111yescharacteristic
261C4#655311yescharacteristic
262C2—281no—
263C2—561no—
264C2#9032011yescharacteristic
265C2C2×#648111yescharacteristic
266C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#477D7
hypocenterC7#3252
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90419C2semidirect
3. C7×#477C2semidirect
4. #89856C2semidirect
5. #89803C2semidirect
7. #89649C2semidirect
8. #89613C2semidirect
9. #89609C2semidirect
10. #89588C2semidirect
11. #89570C2semidirect
85. #477D7semidirect
242. C7#3252semidirect

Maximal subgroups

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

subgroupindex
#904192
#902712
C7×#4772
#898562
#898032
#896582
#896492
#896132
#896092
#895882
#895702
#32527