Finite Group Databasebeta

← #11736751

Subgroups — #11736751

Order 896.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#90168C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89617C214yesmaximal, complemented
5#89505C214yesmaximal, complemented
6#89471C211yesmaximal, characteristic, complemented
7#89466C211yesmaximal, characteristic, complemented
8#89465C211yesmaximal, characteristic, complemented
9#89453C211yesmaximal, characteristic, complemented
10C2×#1157C2212yes—
11C2×C42×C7C2211yescharacteristic
12#6493—21no—
13#6493—21no—
14#6489—21no—
15#6487—21no—
16#6487C2212yes—
17C2×Dic28C2214yes—
18C2×#1132C2214yes—
19#6438C2212yes—
20#6437C2211yescharacteristic
21C4×D28—21no—
22C4×D28—21no—
23#6433—21no—
24#6432C2211yescharacteristic
25#6432C2211yescharacteristic
26#6432C2211yescharacteristic
27C4×Dic14—21no—
28C4×Dic14C2212yes—
29C7×C4⋊C8C2212yes—
30C7×C22⋊C8C2212yes—
31C4×C7×C8C2211yescharacteristic
32C4×C7×C8C2211yescharacteristic
33#6395C2214yes—
34#6390C2214yes—
35#3210—71nomaximal, Sylow
36C22×C4×C7D411yescharacteristic
37C22×C4×C7D411yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C2×C22⋊D7—22no—
40#1157—44no—
41C2×D28—21no—
42C2×D28—21no—
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×Dic14—21no—
46C2×Dic14C2312yes—
47C2×Dic14C2312yes—
48C2×C7×C8C2312yes—
49C2×C7×C8C2312yes—
50C42×C7D411yescharacteristic
51C42×C7D411yescharacteristic
52C42×C7C2311yescharacteristic
53C42×C7C2311yescharacteristic
54C22⋊Dic7—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57C4⋊Dic7—21no—
58C4⋊Dic7—21no—
59C4⋊Dic7—21no—
60C4⋊Dic7—21no—
61C4⋊Dic7—21no—
62C4⋊Dic7C2312yes—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65C4×Dic7—22no—
66Dic28—44no—
67#1132—44no—
68#533—71no—
69#532—71no—
70#494—71no—
71#492—71no—
72#487—71no—
73#486—71no—
74#464—74no—
75#452—74no—
76#431D711yescharacteristic, complemented
77C23×C7C2×D411yescharacteristic
78C22×D7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2×D411yescharacteristic
86C2×C4×C7C2×D411yescharacteristic
87C2×C4×C7C2411yescharacteristic
88C22⋊D7—44no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92D28—24no—
93D28—42no—
94D28—42no—
95C4×D7—44no—
96Dic14—24no—
97Dic14—24no—
98Dic14—42no—
99C7×C8—22no—
100C7×C8—22no—
101C2×#41—72no—
102C2×Q16—74no—
103C2×SD16—74no—
104C4⋊Q8—71no—
105C4⋊Q8—71no—
106C4⋊Q8—71no—
107#127—71no—
108#125—141no—
109#124—72no—
110C22⋊Q8—72no—
111C22⋊Q8—141no—
112#121—141no—
113#121—141no—
114C4×Q8—72no—
115C4×Q8—141no—
116C4×D4—141no—
117C4×D4—141no—
118#117—141no—
119C2×C42C2×D711yescharacteristic
120C4⋊C8C2×D712yes—
121#103—74no—
122#102—74no—
123C22⋊C8C2×D712yes—
124C4×C8C2×D711yescharacteristic
125C4×C8C2×D711yescharacteristic
126C22×C7—21no—
127C22×C7—21no—
128C22×C7—21no—
129C22×C7C22×D411yescharacteristic
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—42no—
133C4×C7—21no—
134C4×C7—21no—
135C4×C7—21no—
136C4×C7#13512yes—
137C4×C7#13712yes—
138C4×C7H(5,𝔽2)12yes—
139Dic7—42no—
140Dic7—42no—
141Dic7—42no—
142#41—284no—
143C2×Q8—72no—
144C2×Q8—72no—
145C2×Q8—141no—
146C2×D4—72no—
147C2×D4—142no—
148C2×D4—141no—
149C2×D4—141no—
150C22×C4—142no—
151C22×C4D2811yescharacteristic
152C22×C4D2811yescharacteristic
153C22×C4C22×D711yescharacteristic
154Q16—284no—
155SD16—284no—
156C2×C8C22×D712yes—
157C2×C8C22×D712yes—
158C4⋊C4—72no—
159C4⋊C4—142no—
160C4⋊C4—142no—
161C4⋊C4—141no—
162C4⋊C4—141no—
163C4⋊C4—141no—
164C4⋊C4—141no—
165C4⋊C4—141no—
166C22⋊C4—142no—
167C22⋊C4—142no—
168C22⋊C4—142no—
169C42—142no—
170C42D2811yescharacteristic
171C42D2811yescharacteristic
172C42C22×D711yescharacteristic
173C42C22×D711yescharacteristic
174C2×C7—41no—
175C2×C7#53511yescharacteristic
176C2×C7C2×#13511yescharacteristic
177C2×C7C2×#13711yescharacteristic
178D7—82no—
179C23—142no—
180C23C2×D2811yescharacteristic
181Q8—144no—
182Q8—144no—
183Q8—282no—
184D4—144no—
185D4—284no—
186D4—282no—
187D4—282no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—284no—
195C2×C4C2×D2811yescharacteristic
196C2×C4C2×D2811yescharacteristic
197C2×C4C2×D2811yescharacteristic
198C2×C4C2×D2811yescharacteristic
199C2×C4C2×D2811yescharacteristic
200C2×C4C23×D711yescharacteristic
201C8—22no—
202C8—22no—
203C7#321011yescharacteristic, complemented, Sylow
204C22—21no—
205C22—21no—
206C22—21no—
207C22—282no—
208C22—282no—
209C22—282no—
210C22C22×D2811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—282no—
215C4—282no—
216C4—282no—
217C4#646712yes—
218C4#647212yes—
219C4#655312yes—
220C2—41no—
221C2—562no—
222C2#9021611yescharacteristic
223C2C2×#646711yescharacteristic
224C2C2×#647211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#90168C2
hypercenter#431D7
hypocenterC7#3210
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. C7×#431C2semidirect
4. #89617C2semidirect
5. #89505C2semidirect
6. #89471C2semidirect
7. #89466C2semidirect
8. #89465C2semidirect
9. #89453C2semidirect
76. #431D7semidirect
203. C7#3210semidirect

Maximal subgroups

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

subgroupindex
#901682
#901612
C7×#4312
#896172
#895052
#894712
#894662
#894652
#894532
#32107