Finite Group Databasebeta

← #11739426

Subgroups — #11739426

Order 896.

322 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 111 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
2#90379C211yesmaximal, characteristic
3C7×#501C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89841C211yesmaximal, characteristic, complemented
6#89660C212yesmaximal
7#89647C212yesmaximal, complemented
8#89639C212yesmaximal
9#89626C212yesmaximal, complemented
10#89477C211yesmaximal, characteristic
11#89475C211yesmaximal, characteristic, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C7×#125C2211yescharacteristic
14#6509—21no—
15Q8×Dic7—21no—
16#6460C2212yes—
17#6459—21no—
18#6459—21no—
19#6455—21no—
20#6455C2211yescharacteristic
21#6455C2211yescharacteristic
22#6453—21no—
23#6451C2211yescharacteristic
24#6451C2211yescharacteristic
25#6450—21no—
26#6450—21no—
27#6450C2212yes—
28C42×D7C2211yescharacteristic
29#6433C2211yescharacteristic
30#6432C2211yescharacteristic
31C7×#107C2212yes—
32C7×#106C2212yes—
33C7×#97C2211yescharacteristic
34(C2×D7)⋊C8C2212yes—
35#6390C2212yes—
36#6390C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2212yes—
39#6384C2212yes—
40#6382C2212yes—
41#6381C2212yes—
42#6377C2211yescharacteristic
43#3237—71nomaximal, Sylow
44C2×C7×Q8—21no—
45C2×C4×D7D411yescharacteristic
46C2×C4×D7D411yescharacteristic
47C2×C4×D7C2311yescharacteristic
48C2×Dic14—21no—
49C2×Dic14—21no—
50C2×Dic14C2312yes—
51C2×C7×C8C2312yes—
52C7×C4⋊C4—21no—
53C7×C4⋊C4—21no—
54C7×C4⋊C4—21no—
55C7×C4⋊C4C2311yescharacteristic
56C7×C4⋊C4C2312yes—
57C7×C4⋊C4C2311yescharacteristic
58C42×C7C2311yescharacteristic
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—21no—
61(C2×D7)⋊C4—21no—
62C4⋊Dic7—21no—
63C4⋊Dic7—21no—
64C4⋊Dic7—22no—
65C4⋊Dic7C2311yescharacteristic
66C4⋊Dic7C2311yescharacteristic
67Dic7⋊C4—21no—
68Dic7⋊C4—21no—
69Dic7⋊C4—21no—
70Dic7⋊C4—21no—
71Dic7⋊C4—22no—
72C4×Dic7—21no—
73C4×Dic7—21no—
74C4×Dic7—22no—
75C4×Dic7D411yescharacteristic
76C4×Dic7D411yescharacteristic
77C4×Dic7C2311yescharacteristic
78C2×C7⋊C8C2312yes—
79#533—71no—
80#533—71no—
81#501—71no—
82#501D711yescharacteristic, complemented
83#491—71no—
84#491—71no—
85#485—72no—
86#485—72no—
87#477—72no—
88#475—72no—
89#432—71no—
90C22×D7C2×D411yescharacteristic
91C7×Q8—42no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7C2×D411yescharacteristic
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2411yescharacteristic
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7C2×D411yescharacteristic
102C2×Dic7C2×D411yescharacteristic
103C2×Dic7C2×D411yescharacteristic
104C4×D7—22no—
105C4×D7—22no—
106C4×D7—41no—
107Dic14—24no—
108Dic14—42no—
109Dic14—42no—
110C7×C8—22no—
111C7⋊C8—22no—
112C4⋊Q8—71no—
113C4⋊Q8—71no—
114C4⋊Q8—71no—
115C4⋊Q8C2×D711yescharacteristic
116#125—71no—
117#125—141no—
118#125C2×D711yescharacteristic
119C22⋊Q8—72no—
120C22⋊Q8—141no—
121C22⋊Q8—141no—
122C22⋊Q8—141no—
123C4×Q8—72no—
124C4×Q8—141no—
125C4×Q8—141no—
126C4×Q8—141no—
127#117—71no—
128#117—71no—
129#117—141no—
130C2×C42—71no—
131#107—72no—
132#107C2×D712yes—
133#106—72no—
134#106C2×D712yes—
135C4⋊C8—72no—
136#103—72no—
137#103—72no—
138#103—72no—
139#103—72no—
140C22⋊C8—72no—
141#97—71no—
142#97C2×D711yescharacteristic
143C22×C7C22×D411yescharacteristic
144C2×D7—21no—
145C2×D7—21no—
146C2×D7—21no—
147C4×C7—21no—
148C4×C7—21no—
149C4×C7—41no—
150C4×C7—41no—
151C4×C7—42no—
152C4×C72−1+412yes—
153Dic7—21no—
154Dic7—21no—
155Dic7—42no—
156Dic7—41no—
157Dic7—41no—
158Dic7#13712yes—
159C2×Q8—21no—
160C2×Q8—72no—
161C2×Q8—141no—
162C2×Q8—141no—
163C22×C4—71no—
164C22×C4—71no—
165C22×C4—71no—
166C2×C8—72no—
167C2×C8C22×D712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—71no—
172C4⋊C4—71no—
173C4⋊C4—142no—
174C4⋊C4—141no—
175C4⋊C4—141no—
176C4⋊C4—142no—
177C4⋊C4—141no—
178C4⋊C4—141no—
179C4⋊C4—141no—
180C4⋊C4—141no—
181C4⋊C4C22×D711yescharacteristic
182C4⋊C4C22×D711yescharacteristic
183C4⋊C4C22×D712yes—
184C22⋊C4—141no—
185C22⋊C4—141no—
186C22⋊C4—142no—
187C42—71no—
188C42—71no—
189C42—71no—
190C42—141no—
191C42—141no—
192C42—142no—
193C42C22×D711yescharacteristic
194C2×C7#53611yescharacteristic
195C2×C7C2×#13711yescharacteristic
196C2×C7#57511yescharacteristic
197D7—41no—
198C23—71no—
199Q8—42no—
200Q8—144no—
201Q8—282no—
202Q8—282no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—71no—
207C2×C4—71no—
208C2×C4—71no—
209C2×C4—142no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—142no—
214C2×C4—281no—
215C2×C4D4×D711yescharacteristic
216C2×C4D4×D711yescharacteristic
217C2×C4C23×D711yescharacteristic
218C8—22no—
219C8—142no—
220C7#323711yescharacteristic, complemented, Sylow
221C22—141no—
222C22—141no—
223C22—141no—
224C22C2×D4×D711yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—72no—
231C4—141no—
232C4—141no—
233C4—282no—
234C4—281no—
235C4—281no—
236C4#655112yes—
237C2—281no—
238C2#9041511yescharacteristic
239C2#9046411yescharacteristic
240C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#501C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#501D7
hypocenterC7#3237
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. C7×#501C2semidirect
4. #89852C2semidirect
5. #89841C2semidirect
7. #89647C2semidirect
9. #89626C2semidirect
11. #89475C2semidirect
82. #501D7semidirect
220. C7#3237semidirect

Maximal subgroups

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

subgroupindex
#904192
#903792
C7×#5012
#898522
#898412
#896602
#896472
#896392
#896262
#894772
#894752
#32377