Finite Group Databasebeta

← #11740438

Subgroups — #11740438

Order 896.

752 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×C7×#124C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3C2×#6507C211yesmaximal, characteristic, complemented
4C2×#6503C211yesmaximal, characteristic
5C2×#6499C211yesmaximal, characteristic, complemented
6C2×#6438C211yesmaximal, characteristic, complemented
7#89729C218yesmaximal, complemented
8C42×Dic7C211yesmaximal, characteristic, complemented
9C22×C7×Q8C2211yescharacteristic
10C22×C7×D4C2211yescharacteristic
11C22×C22⋊D7C2214yes—
12C22×D28C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C7×#124—24no—
15C2×C7×C22⋊C4C2214yes—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2214yes—
18#6510—24no—
19#6507—24no—
20#6503—24no—
21#6499—24no—
22C2×(C2×D7)⋊C4C2214yes—
23C2×(C2×D7)⋊C4C2214yes—
24C2×Dic7⋊C4C2214yes—
25C2×C4×Dic7C2214yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28#6438—24no—
29#2858—71nomaximal, Sylow
30C24×C7C2312yes—
31C23×D7C2312yes—
32C2×C7×Q8—22no—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×D4—22no—
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C2×C22⋊D7—28no—
42C2×C22⋊D7—216no—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C2×D28—22no—
47C2×D28—22no—
48C2×D28—22no—
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—22no—
52C7×C22⋊C4—28no—
53C42×C7D414yescomplemented
54C22⋊Dic7—28no—
55(C2×D7)⋊C4—28no—
56(C2×D7)⋊C4—28no—
57Dic7⋊C4—28no—
58C4×Dic7—28no—
59C4×Dic7D414yes—
60C4×Dic7D414yescomplemented
61C2×C4⋊Q8—71no—
62C2×#127—71no—
63C2×#124—71no—
64C2×#124—71no—
65C2×#124—71no—
66C2×#124D711yescharacteristic, complemented
67#390—78no—
68C43—71no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—28no—
73C23×C7C2411yescharacteristic
74C22×D7—22no—
75C22×D7—22no—
76C22×D7—22no—
77C22×D7—28no—
78C7×Q8—44no—
79C7×D4—44no—
80C2×C4×C7—24no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7C2×D412yes—
85C2×C4×C7C2×D412yes—
86C2×C4×C7C2×D414yes—
87C2×C4×C7C2×D412yes—
88C22⋊D7—416no—
89C2×Dic7—24no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7C2×D414yes—
94C2×Dic7C2×D414yes—
95C2×Dic7#4118yes—
96D28—44no—
97Dic14—44no—
98C22×Q8—71no—
99C22×Q8C2×D711yescharacteristic
100C22×D4—74no—
101C22×D4—71no—
102C22×D4C2×D711yescharacteristic
103C4⋊Q8—144no—
104#127—144no—
105#124—24no—
106#124—144no—
107#124—144no—
108#124—144no—
109C2×C4⋊C4—74no—
110C2×C22⋊C4—74no—
111C2×C22⋊C4—74no—
112C2×C22⋊C4—74no—
113C2×C22⋊C4C2×D714yes—
114C2×C42—71no—
115C2×C42—71no—
116C2×C42—74no—
117C2×C42C2×D711yescharacteristic
118C22×C7—44no—
119C22×C7—42no—
120C22×C7—44no—
121C22×C7—44no—
122C22×C7C22×D411yescharacteristic
123C22×C7C22×D411yescharacteristic
124C22×C7C22×D411yescharacteristic
125C22×C7C2×#4114yes—
126C2×D7—44no—
127C2×D7—42no—
128C2×D7—44no—
129C2×D7—44no—
130C4×C7—24no—
131C4×C7—42no—
132C4×C7#12714yes—
133Dic7—24no—
134Dic7—42no—
135Dic7#12418yescomplemented
136C24—72no—
137C24C22×D712yes—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—142no—
142C2×Q8—142no—
143C2×Q8—142no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—148no—
148C2×D4—1416no—
149C2×D4—142no—
150C2×D4—142no—
151C2×D4—142no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—148no—
159C22⋊C4—28no—
160C22⋊C4—148no—
161C22⋊C4—148no—
162C22⋊C4—148no—
163C42—74no—
164C42—74no—
165C42—148no—
166C42C22⋊D714yescomplemented
167C2×C7—82no—
168C2×C7C2×#12412yes—
169C2×C7C2×#12711yescharacteristic
170C2×C7#53212yes—
171C2×C7#53212yes—
172D7—82no—
173C23—22no—
174C23—22no—
175C23—28no—
176C23—22no—
177C23—142no—
178C23—142no—
179C23—148no—
180C23—142no—
181C23C23×D711yescharacteristic
182Q8—44no—
183Q8—284no—
184D4—44no—
185D4—2816no—
186D4—284no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—74no—
192C2×C4—78no—
193C2×C4—74no—
194C2×C4—144no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4D4×D712yes—
199C2×C4D4×D712yes—
200C2×C4C2×C22⋊D714yes—
201C2×C4C2×C22⋊D712yes—
202C7#285811yescharacteristic, complemented, Sylow
203C22—42no—
204C22—44no—
205C22—44no—
206C22—44no—
207C22—282no—
208C22—284no—
209C22—284no—
210C22—284no—
211C22C2×D4×D711yescharacteristic
212C22C2×D4×D711yescharacteristic
213C22D7×#4114yes—
214C22C22×C22⋊D711yescharacteristic
215C4—24no—
216C4—42no—
217C4—78no—
218C4—144no—
219C4—282no—
220C4#650314yes—
221C2—82no—
222C2—562no—
223C2#9035912yes—
224C2D7×#12412yes—
225C2C2×#650311yescharacteristic
226C2#9052312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC2×#6503C2
hypercenterC2×#124D7
hypocenterC7#2858
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C7×#124C2semidirect
2. C2×#6510C2semidirect
3. C2×#6507C2semidirect
5. C2×#6499C2semidirect
6. C2×#6438C2semidirect
7. #89729C2semidirect
8. C42×Dic7C2semidirect
53. C42×C7D4semidirect
60. C4×Dic7D4semidirect
66. C2×#124D7semidirect
135. Dic7#124semidirect
166. C42C22⋊D7semidirect
202. C7#2858semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×#65102
C2×#65072
C2×#65032
C2×#64992
C2×#64382
#897292
C42×Dic72
#28587