Finite Group Databasebeta

← #11736715

Subgroups — #11736715

Order 896.

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

normal subgroups only

This group has 370 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#89619C214yesmaximal, complemented
5#89509C214yesmaximal, complemented
6#89466C211yesmaximal, characteristic, complemented
7#89466C211yesmaximal, characteristic, complemented
8#89454C211yesmaximal, characteristic, complemented
9#89452C211yesmaximal, characteristic, complemented
10C2×#1157—21no—
11C2×C42×C7C2211yescharacteristic
12#6493—21no—
13#6493C2212yes—
14#6489C2212yes—
15#6487—21no—
16#6487—21no—
17#6438—21no—
18#6437C2211yescharacteristic
19C4×D28—21no—
20C4×D28C2212yes—
21#6433C2212yes—
22#6432C2211yescharacteristic
23#6432C2211yescharacteristic
24#6432C2211yescharacteristic
25C4×Dic14—21no—
26C4×Dic14—21no—
27C7×C4⋊C8C2212yes—
28C7×C22⋊C8C2212yes—
29C4×C7×C8C2211yescharacteristic
30C4×C7×C8C2211yescharacteristic
31#6395C2214yes—
32#6395C2214yes—
33#6392C2214yes—
34#6391C2214yes—
35#3228—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×Dic14—21no—
47C2×Dic14—21no—
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⋊Dic7C2312yes—
61C4⋊Dic7C2312yes—
62C4⋊Dic7C2312yes—
63Dic7⋊C4—22no—
64Dic7⋊C4—22no—
65C4×Dic7—22no—
66#533—71no—
67#532—71no—
68#500—71no—
69#498—71no—
70#486—71no—
71#486—71no—
72#482—74no—
73#478—74no—
74#431D711yescharacteristic, complemented
75C23×C7C2×D411yescharacteristic
76C22×D7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D411yescharacteristic
81C2×C4×C7C2×D411yescharacteristic
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2×D411yescharacteristic
84C2×C4×C7C2×D411yescharacteristic
85C2×C4×C7C2411yescharacteristic
86C22⋊D7—44no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90D28—24no—
91D28—42no—
92D28—42no—
93C4×D7—44no—
94Dic14—42no—
95Dic14—42no—
96Dic14—42no—
97C7×C8—22no—
98C7×C8—22no—
99C2×#41—141no—
100C4⋊Q8—71no—
101C4⋊Q8—71no—
102C4⋊Q8—71no—
103#127—71no—
104#125—72no—
105#124—141no—
106C22⋊Q8—141no—
107C22⋊Q8—141no—
108#121—72no—
109#121—141no—
110C4×Q8—141no—
111C4×Q8—141no—
112C4×D4—72no—
113C4×D4—141no—
114#117—72no—
115C2×C42C2×D711yescharacteristic
116#107—74no—
117#106—74no—
118C4⋊C8C2×D712yes—
119#102—74no—
120#102—74no—
121C22⋊C8C2×D712yes—
122C4×C8C2×D711yescharacteristic
123C4×C8C2×D711yescharacteristic
124C22×C7—21no—
125C22×C7—21no—
126C22×C7—21no—
127C22×C7C22×D411yescharacteristic
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—42no—
131C4×C7—21no—
132C4×C7—21no—
133C4×C7—21no—
134C4×C7#13512yes—
135C4×C7AGL(1,ℤ/8)12yes—
136C4×C72−1+412yes—
137Dic7—42no—
138Dic7—42no—
139Dic7—42no—
140#41—284no—
141C2×Q8—141no—
142C2×Q8—141no—
143C2×Q8—141no—
144C2×D4—72no—
145C2×D4—142no—
146C2×D4—141no—
147C2×D4—141no—
148C22×C4—142no—
149C22×C4D2811yescharacteristic
150C22×C4D2811yescharacteristic
151C22×C4C22×D711yescharacteristic
152C2×C8C22×D712yes—
153C2×C8C22×D712yes—
154C4⋊C4—72no—
155C4⋊C4—72no—
156C4⋊C4—72no—
157C4⋊C4—142no—
158C4⋊C4—142no—
159C4⋊C4—141no—
160C4⋊C4—141no—
161C4⋊C4—141no—
162C22⋊C4—142no—
163C22⋊C4—142no—
164C22⋊C4—142no—
165C42—142no—
166C42D2811yescharacteristic
167C42D2811yescharacteristic
168C42C22×D711yescharacteristic
169C42C22×D711yescharacteristic
170C2×C7—41no—
171C2×C7#53611yescharacteristic
172C2×C7C2×#13511yescharacteristic
173C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
174D7—82no—
175C23—142no—
176C23C2×D2811yescharacteristic
177Q8—282no—
178Q8—282no—
179Q8—282no—
180D4—144no—
181D4—284no—
182D4—282no—
183D4—282no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—284no—
191C2×C4C2×D2811yescharacteristic
192C2×C4C2×D2811yescharacteristic
193C2×C4C2×D2811yescharacteristic
194C2×C4C2×D2811yescharacteristic
195C2×C4C2×D2811yescharacteristic
196C2×C4C23×D711yescharacteristic
197C8—22no—
198C8—22no—
199C7#322811yescharacteristic, complemented, Sylow
200C22—21no—
201C22—21no—
202C22—21no—
203C22—282no—
204C22—282no—
205C22—282no—
206C22C22×D2811yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—282no—
211C4—282no—
212C4—282no—
213C4#646712yes—
214C4#647112yes—
215C4#655412yes—
216C2—41no—
217C2—562no—
218C2#9021711yescharacteristic
219C2C2×#646711yescharacteristic
220C2C2×#647111yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#90168C2
hypercenter#431D7
hypocenterC7#3228
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. #89619C2semidirect
5. #89509C2semidirect
6. #89466C2semidirect
7. #89466C2semidirect
8. #89454C2semidirect
9. #89452C2semidirect
74. #431D7semidirect
199. C7#3228semidirect

Maximal subgroups

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

subgroupindex
#901682
#901612
C7×#4312
#896192
#895092
#894662
#894542
#894522
#32287