Finite Group Databasebeta

← #11733898

Subgroups — #11733898

Order 896.

568 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6501C211yesmaximal, characteristic
2C2×#6455C211yesmaximal, characteristic, complemented
3C7×#390C211yesmaximal, characteristic, complemented
4#89993C211yesmaximal, characteristic, complemented
5#89749C211yesmaximal, characteristic, complemented
6#89733C214yesmaximal, complemented
7#89723C214yesmaximal, complemented
8C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
9#89707C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×C4×D7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2214yes—
18#6501—24no—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2214yes—
24C2×C4×Dic7C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26#6455—24no—
27C7⋊#95C2212yes—
28C7⋊#95C2214yes—
29#2500—71nomaximal, Sylow
30C24×C7C2312yes—
31C23×D7C2311yescharacteristic
32C2×C7×D4—22no—
33C2×C7×D4—22no—
34C2×C7×D4—22no—
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—24no—
39C2×C22⋊D7—44no—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2311yescharacteristic
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7D414yes—
47C7×C4⋊C4—22no—
48C7×C22⋊C4—28no—
49C42×C7—22no—
50C22⋊Dic7—24no—
51C22⋊Dic7—44no—
52(C2×D7)⋊C4—24no—
53(C2×D7)⋊C4—24no—
54C4⋊Dic7—22no—
55C4⋊Dic7—22no—
56C4⋊Dic7—44no—
57C4×Dic7—22no—
58C4×Dic7—24no—
59C2×#121—71no—
60C2×#117—71no—
61#394—74no—
62#390—71no—
63#390D711yescharacteristic, complemented
64#388—74no—
65#384—71no—
66#382—71no—
67C4×C22⋊C4—71no—
68C23×C7—22no—
69C23×C7—24no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—24no—
73C23×C7C2411yescharacteristic
74C22×D7—22no—
75C22×D7—21no—
76C22×D7—21no—
77C22×D7—22no—
78C22×D7C2×D412yes—
79C7×D4—44no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—24no—
84C2×C4×C7C2×D412yes—
85C2×C4×C7#4112yes—
86C2×C4×C7#4114yes—
87C2×C4×C7#4112yes—
88C22⋊D7—48no—
89C2×Dic7—21no—
90C2×Dic7—21no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—24no—
96C2×Dic7—24no—
97C2×Dic7—22no—
98C2×Dic7C2×D412yes—
99C4×D7—28no—
100C22×D4—72no—
101C22×D4C2×D711yescharacteristic
102C23×C4—71no—
103#121—144no—
104#117—144no—
105C2×C4⋊C4—71no—
106C2×C4⋊C4—71no—
107C2×C4⋊C4—74no—
108C2×C4⋊C4C2×D711yescharacteristic
109C2×C22⋊C4—72no—
110C2×C22⋊C4—72no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—74no—
113C2×C22⋊C4C2×D714yes—
114C2×C42—72no—
115C2×C42—71no—
116C2×C42C2×D711yescharacteristic
117#95—72no—
118#95—74no—
119C22×C7—42no—
120C22×C7—42no—
121C22×C7—42no—
122C22×C7—44no—
123C22×C7—44no—
124C22×C7C22×D411yescharacteristic
125C22×C7C2×#4111yescharacteristic
126C22×C7C2×#4111yescharacteristic
127C22×C7C2×#4112yes—
128C22×C7H(5,𝔽2)12yes—
129C2×D7—22no—
130C2×D7—22no—
131C2×D7—22no—
132C2×D7—42no—
133C2×D7—42no—
134C4×C7—24no—
135C4×C7—42no—
136C4×C7#12314yes—
137Dic7—22no—
138Dic7—42no—
139Dic7—42no—
140Dic7—42no—
141C24—71no—
142C24C22×D712yes—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—144no—
147C2×D4—284no—
148C22×C4—71no—
149C22×C4—74no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4—142no—
154C22×C4—142no—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—142no—
160C4⋊C4—142no—
161C4⋊C4—284no—
162C22⋊C4—28no—
163C22⋊C4—144no—
164C22⋊C4—144no—
165C22⋊C4—144no—
166C22⋊C4—284no—
167C42—22no—
168C42—144no—
169C42—142no—
170C2×C7—82no—
171C2×C7C2×#12311yescharacteristic
172C2×C7#53211yescharacteristic
173C2×C7#53511yescharacteristic
174C2×C7#55312yes—
175C2×C7#55512yes—
176D7—42no—
177C23—22no—
178C23—22no—
179C23—24no—
180C23—22no—
181C23—24no—
182C23—72no—
183C23—141no—
184C23—142no—
185C23—141no—
186C23—142no—
187C23C23×D711yescharacteristic
188D4—44no—
189D4—288no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—72no—
195C2×C4—141no—
196C2×C4—141no—
197C2×C4—148no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—144no—
204C2×C4—144no—
205C2×C4#115712yes—
206C2×C4D4×D712yes—
207C2×C4#115914yes—
208C2×C4#115912yes—
209C7#250011yescharacteristic, complemented, Sylow
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—42no—
215C22—142no—
216C22—142no—
217C22—142no—
218C22—282no—
219C22—282no—
220C22C2×#115711yescharacteristic
221C22C2×D4×D711yescharacteristic
222C22C2×#115912yes—
223C22C2×#115911yescharacteristic
224C22#655312yes—
225C4—24no—
226C4—42no—
227C4—142no—
228C4—282no—
229C4—282no—
230C4—282no—
231C4#644614yes—
232C2—82no—
233C2—282no—
234C2C2×#644611yescharacteristic
235C2#9025412yes—
236C2#9030912yes—
237C2#9036511yescharacteristic
238C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC2×#6501C2
hypercenter#390D7
hypocenterC7#2500
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6455C2semidirect
3. C7×#390C2semidirect
4. #89993C2semidirect
5. #89749C2semidirect
6. #89733C2semidirect
7. #89723C2semidirect
8. C4×(C2×D7)⋊C4C2semidirect
9. #89707C2semidirect
63. #390D7semidirect
209. C7#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#64552
C7×#3902
#899932
#897492
#897332
#897232
C4×(C2×D7)⋊C42
#897072
#25007