Finite Group Databasebeta

← #11738474

Subgroups — #11738474

Order 896.

340 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 115 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#90423C211yesmaximal, characteristic
2#90275C211yesmaximal, characteristic
3C7×#475C211yesmaximal, characteristic, complemented
4#89850C211yesmaximal, characteristic, complemented
5C4×#1144C211yesmaximal, characteristic, complemented
6#89639C212yesmaximal, complemented
7#89632C212yesmaximal, complemented
8#89623C211yesmaximal, characteristic, complemented
9#89606C211yesmaximal, characteristic
10#89587C212yesmaximal, complemented
11#89585C212yesmaximal, complemented
12C7×C4⋊Q8C2211yescharacteristic
13C4×C7×Q8C2211yescharacteristic
14#6510—21no—
15#6509C2212yes—
16Q8×Dic7—21no—
17C2×#1144C2211yescharacteristic
18C2×C8×D7C2212yes—
19#6461—22no—
20#6461—22no—
21#6460C2212yes—
22#6455C2212yes—
23#6455C2212yes—
24#6438—21no—
25C4×D28C2211yescharacteristic
26#6432C2211yescharacteristic
27C4×Dic14—21no—
28C4×Dic14C2211yescharacteristic
29C7×#106C2212yes—
30C7×C4⋊C8C2211yescharacteristic
31C7×#103C2212yes—
32#6409C2211yescharacteristic
33(C2×D7)⋊C8C2212yes—
34#6391C2212yes—
35#6390C2212yes—
36#6384C2211yescharacteristic
37#6381C2211yescharacteristic
38#6381C2212yes—
39#6381C2212yes—
40C4×C7⋊C8C2211yescharacteristic
41#3342—71nomaximal, Sylow
42C2×C7×Q8—21no—
43C2×C7×Q8C2311yescharacteristic
44C2×D28D411yescharacteristic
45C2×C4×D7C2312yes—
46C2×Dic14—21no—
47C2×Dic14C2311yescharacteristic
48C2×C7×C8C2312yes—
49C7×C4⋊C4—21no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4C2312yes—
52C7×C4⋊C4C2311yescharacteristic
53C42×C7—21no—
54C42×C7C2311yescharacteristic
55#1144—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4D412yes—
61C4⋊Dic7—21no—
62C4⋊Dic7—22no—
63C4⋊Dic7D411yescharacteristic
64C4⋊Dic7C2311yescharacteristic
65C4⋊Dic7C2312yes—
66Dic7⋊C4—21no—
67Dic7⋊C4—22no—
68C4×Dic7—21no—
69C4×Dic7—22no—
70C4×Dic7—22no—
71C2×C7⋊C8C2311yescharacteristic
72C2×C7⋊C8C2311yescharacteristic
73C8×D7—24no—
74#556—71no—
75#556—71no—
76#500—71no—
77#485—72no—
78#485—72no—
79#475—71no—
80#475D711yescharacteristic, complemented
81#467—72no—
82C4×Q16—71no—
83C8×D4—71no—
84#427—72no—
85C22×D7C2×D412yes—
86C7×Q8—21no—
87C7×Q8—21no—
88C7×Q8—42no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7C2×D411yescharacteristic
93C2×C4×C7C2×D411yescharacteristic
94C2×C4×C7C2411yescharacteristic
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7C2×D412yes—
99D28—22no—
100C4×D7—22no—
101C4×D7—22no—
102Dic14—21no—
103Dic14—21no—
104Dic14—42no—
105C7×C8—22no—
106C7⋊C8—21no—
107C7⋊C8#4112yes—
108C2×Q16—71no—
109C22×C8—72no—
110C4⋊Q8—71no—
111C4⋊Q8C2×D711yescharacteristic
112#126—142no—
113#126—142no—
114#124—141no—
115#124—141no—
116C22⋊Q8—72no—
117C22⋊Q8—72no—
118C4×Q8—71no—
119C4×Q8—141no—
120C4×Q8—141no—
121C4×Q8C2×D711yescharacteristic
122C4×D4—71no—
123#117—72no—
124#117—72no—
125#107—71no—
126#107—72no—
127#107—72no—
128#106—72no—
129#106C2×D712yes—
130C4⋊C8C2×D711yescharacteristic
131#103—71no—
132#103—72no—
133#103—71no—
134#103C2×D712yes—
135C22⋊C8—72no—
136C4×C8—71no—
137C22×C7C22×D411yescharacteristic
138C2×D7—22no—
139C2×D7—22no—
140C2×D7—22no—
141C4×C7—21no—
142C4×C7—41no—
143C4×C7—42no—
144C4×C7—41no—
145C4×C7#13512yes—
146C4×C7C2×#4111yescharacteristic
147C4×C72−1+411yescharacteristic
148Dic7—22no—
149Dic7—41no—
150Dic7—42no—
151Dic7—41no—
152C2×Q8—21no—
153C2×Q8—71no—
154C2×Q8—141no—
155C2×Q8C22×D711yescharacteristic
156C2×D4—71no—
157C22×C4—72no—
158Q16—142no—
159C2×C8—71no—
160C2×C8—71no—
161C2×C8—144no—
162C2×C8C22×D712yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—71no—
166C4⋊C4—71no—
167C4⋊C4—72no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C4⋊C4—141no—
171C4⋊C4—141no—
172C4⋊C4C22×D711yescharacteristic
173C4⋊C4C22×D712yes—
174C22⋊C4—72no—
175C22⋊C4—142no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C22⋊C4—142no—
179C42—21no—
180C42—142no—
181C42—142no—
182C42—141no—
183C42C22×D711yescharacteristic
184C2×C7#54711yescharacteristic
185C2×C7C2×#13511yescharacteristic
186C2×C7#57811yescharacteristic
187D7—42no—
188C23—72no—
189Q8—21no—
190Q8—21no—
191Q8—42no—
192Q8—141no—
193Q8—141no—
194Q8—282no—
195D4—142no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—72no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4—142no—
204C2×C4—141no—
205C2×C4D4×D711yescharacteristic
206C2×C4D4×D711yescharacteristic
207C2×C4C23×D711yescharacteristic
208C8—22no—
209C8—72no—
210C8—141no—
211C7#334211yescharacteristic, complemented, Sylow
212C22—142no—
213C22—142no—
214C22—142no—
215C22C2×D4×D711yescharacteristic
216C4—21no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—142no—
221C4—282no—
222C4—281no—
223C4—281no—
224C4#647912yes—
225C4#655111yescharacteristic
226C4D7×#4111yescharacteristic
227C2—282no—
228C2#9032211yescharacteristic
229C2C2×#647911yescharacteristic
230C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#475C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#475D7
hypocenterC7#3342
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. C7×#475C2semidirect
4. #89850C2semidirect
5. C4×#1144C2semidirect
6. #89639C2semidirect
7. #89632C2semidirect
8. #89623C2semidirect
10. #89587C2semidirect
11. #89585C2semidirect
80. #475D7semidirect
211. C7#3342semidirect

Maximal subgroups

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

subgroupindex
#904232
#902752
C7×#4752
#898502
C4×#11442
#896392
#896322
#896232
#896062
#895872
#895852
#33427