Finite Group Databasebeta

← #11733796

Subgroups — #11733796

Order 896.

576 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6499C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic
3C2×#6450C211yesmaximal, characteristic, complemented
4C2×#6432C211yesmaximal, characteristic, complemented
5C7×#390C211yesmaximal, characteristic
6#89746C211yesmaximal, characteristic, complemented
7#89723C214yesmaximal, complemented
8#89720C214yesmaximal, complemented
9C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2214yes—
18C2×C22⋊Dic7C2212yes—
19#6499—24no—
20#6499—24no—
21C2×C4⋊Dic7C2214yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2211yescharacteristic
27#6450—24no—
28#6432—24no—
29C7⋊#95C2212yes—
30C7⋊#95C2214yes—
31#2545—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—22no—
34C2×C7×D4—22no—
35C2×C7×D4—22no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2311yescharacteristic
42C22×Dic7C2311yescharacteristic
43C22×Dic7C2312yes—
44C2×Dic14—22no—
45C2×Dic14—22no—
46C2×Dic14—22no—
47C2×Dic14—22no—
48C2×Dic14—22no—
49C2×Dic14D414yes—
50C7×C4⋊C4—22no—
51C7×C22⋊C4—28no—
52C42×C7—22no—
53C22⋊Dic7—28no—
54C22⋊Dic7—24no—
55C22⋊Dic7—24no—
56C4⋊Dic7—28no—
57Dic7⋊C4—24no—
58Dic7⋊C4—24no—
59C4×Dic7—22no—
60C4×Dic7—24no—
61C4×Dic7D414yes—
62C2×C4⋊Q8—71no—
63C2×#124—71no—
64C2×#124—71no—
65C2×C4×Q8—71no—
66#396—74no—
67#391—71no—
68#390D711yescharacteristic
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×D4—44no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—24no—
82C2×C4×C7C2×D412yes—
83C2×C4×C7C2×D412yes—
84C2×C4×C7#4112yes—
85C2×C4×C7#4114yes—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7C2×D412yes—
97C2×Dic7C2×D412yes—
98C2×Dic7C2×D414yes—
99Dic14—28no—
100Dic14—44no—
101C22×Q8—71no—
102C22×Q8—71no—
103C22×D4C2×D711yescharacteristic
104C4⋊Q8—144no—
105#124—144no—
106#124—144no—
107C4×Q8—144no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—74no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D711yescharacteristic
112C2×C22⋊C4—72no—
113C2×C22⋊C4—74no—
114C2×C22⋊C4—72no—
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—24no—
136Dic7—22no—
137Dic7—22no—
138Dic7—42no—
139Dic7—42no—
140C24C22×D712yes—
141C2×Q8—74no—
142C2×Q8—142no—
143C2×Q8—142no—
144C2×Q8—142no—
145C2×Q8—142no—
146C2×Q8—142no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—144no—
160C4⋊C4—148no—
161C4⋊C4—144no—
162C22⋊C4—28no—
163C22⋊C4—144no—
164C22⋊C4—144no—
165C22⋊C4—148no—
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—22no—
179C23—24no—
180C23—24no—
181C23C23×D711yescharacteristic
182Q8—148no—
183Q8—284no—
184D4—44no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—72no—
190C2×C4—72no—
191C2×C4—74no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4#115712yes—
203C2×C4D4×D712yes—
204C2×C4D4×D712yes—
205C2×C4#115914yes—
206C7#254511yescharacteristic, complemented, Sylow
207C22—42no—
208C22—42no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22C2×#115711yescharacteristic
213C22C2×D4×D711yescharacteristic
214C22C2×D4×D711yescharacteristic
215C22C2×#115912yes—
216C22#655412yes—
217C4—24no—
218C4—42no—
219C4—142no—
220C4—144no—
221C4—142no—
222C4—282no—
223C4—282no—
224C4#644714yes—
225C2—82no—
226C2C2×#644711yescharacteristic
227C2#9023312yes—
228C2#9028912yes—
229C2#9036611yescharacteristic
230C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2545
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
3. C2×#6450C2semidirect
4. C2×#6432C2semidirect
6. #89746C2semidirect
7. #89723C2semidirect
8. #89720C2semidirect
9. C4×Dic7⋊C4C2semidirect
206. C7#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64502
C2×#64322
C7×#3902
#897462
#897232
#897202
C4×Dic7⋊C42
#25457