Finite Group Databasebeta

← #11740417

Subgroups — #11740417

Order 896.

576 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×Q8×Dic7C211yesmaximal, characteristic, complemented
3C2×#6507C211yesmaximal, characteristic, complemented
4C2×#6499C211yesmaximal, characteristic
5#89993C211yesmaximal, characteristic
6#89720C214yesmaximal, complemented
7#89717C214yesmaximal, complemented
8C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
9#89701C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×D4C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C7×#124—24no—
14C2×C7×C22⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2214yes—
18C2×C22⋊Dic7C2212yes—
19Q8×Dic7—24no—
20#6507—24no—
21#6499—24no—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2214yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2214yes—
30C7⋊#95C2212yes—
31#2545—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8D414yes—
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C2×C7×D4—22no—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C2×Dic14—22no—
48C2×Dic14—22no—
49C2×Dic14—22no—
50C7×C22⋊C4—28no—
51C42×C7—22no—
52C22⋊Dic7—24no—
53C22⋊Dic7—24no—
54C22⋊Dic7—28no—
55C4⋊Dic7—22no—
56C4⋊Dic7—24no—
57Dic7⋊C4—28no—
58Dic7⋊C4—24no—
59C4×Dic7—22no—
60C4×Dic7—24no—
61C4×Dic7D414yes—
62C2×C4⋊Q8—71no—
63C2×#124—71no—
64C2×#124D711yescharacteristic
65C2×C4×Q8—71no—
66#396—74no—
67#391—71no—
68#390—71no—
69#388—74no—
70C4×C4⋊C4—71no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—24no—
74C23×C7—22no—
75C23×C7—24no—
76C23×C7C2411yescharacteristic
77C7×Q8—28no—
78C7×D4—44no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7C2×D412yes—
85C2×C4×C7C2×D414yes—
86C2×C4×C7C2×D412yes—
87C2×C4×C7#4112yes—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—24no—
96C2×Dic7—22no—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D412yes—
99C2×Dic7#4114yes—
100Dic14—44no—
101C22×Q8—71no—
102C22×Q8C2×D711yescharacteristic
103C22×D4C2×D711yescharacteristic
104C4⋊Q8—144no—
105#124—24no—
106#124—144no—
107C4×Q8—144no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—74no—
111C2×C4⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—74no—
115C2×C22⋊C4C2×D714yes—
116C2×C42—71no—
117C2×C42—72no—
118C2×C42—71no—
119C2×C42C2×D711yescharacteristic
120#95—72no—
121#95—74no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—44no—
126C22×C7—44no—
127C22×C7C22×D411yescharacteristic
128C22×C7C22×D411yescharacteristic
129C22×C7C2×#4111yescharacteristic
130C22×C7C2×#4112yes—
131C22×C72−1+412yes—
132C4×C7—24no—
133C4×C7—42no—
134C4×C7#12114yes—
135Dic7—22no—
136Dic7—22no—
137Dic7—24no—
138Dic7—42no—
139Dic7—42no—
140C24C22×D712yes—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—142no—
144C2×Q8—142no—
145C2×Q8—142no—
146C2×Q8C22⋊D714yes—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C22×C4—71no—
151C22×C4—72no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—144no—
159C4⋊C4—144no—
160C4⋊C4—142no—
161C4⋊C4—148no—
162C22⋊C4—28no—
163C22⋊C4—148no—
164C22⋊C4—144no—
165C22⋊C4—144no—
166C42—22no—
167C42—74no—
168C42—144no—
169C42—142no—
170C2×C7—82no—
171C2×C7C2×#12111yescharacteristic
172C2×C7#53211yescharacteristic
173C2×C7#53611yescharacteristic
174C2×C7#54812yes—
175C2×C7#55612yes—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—24no—
180C23—22no—
181C23C23×D711yescharacteristic
182Q8—28no—
183Q8—284no—
184D4—44no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—72no—
191C2×C4—72no—
192C2×C4—74no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—144no—
201C2×C4—142no—
202C2×C4D4×D712yes—
203C2×C4#115912yes—
204C2×C4C2×C22⋊D714yes—
205C2×C4C2×C22⋊D712yes—
206C7#254511yescharacteristic, complemented, Sylow
207C22—42no—
208C22—44no—
209C22—42no—
210C22—44no—
211C22—42no—
212C22C2×D4×D711yescharacteristic
213C22C2×#115911yescharacteristic
214C22D7×#4112yes—
215C22#655412yes—
216C22C22×C22⋊D711yescharacteristic
217C4—24no—
218C4—42no—
219C4—144no—
220C4—142no—
221C4—142no—
222C4—282no—
223C4—282no—
224C4#650214yes—
225C2—82no—
226C2#9035911yescharacteristic
227C2#9036212yes—
228C2#9036611yescharacteristic
229C2C2×#650211yescharacteristic
230C2#9052212yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#124C2
prime radicalC22×C7×D4C22
hypercenterC2×#124D7
hypocenterC7#2545
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic7C2semidirect
3. C2×#6507C2semidirect
6. #89720C2semidirect
7. #89717C2semidirect
8. C4×Dic7⋊C4C2semidirect
9. #89701C2semidirect
206. C7#2545semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1242
C2×Q8×Dic72
C2×#65072
C2×#64992
#899932
#897202
#897172
C4×Dic7⋊C42
#897012
#25457