Finite Group Databasebeta

← #11740108

Subgroups — #11740108

Order 896.

568 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 153 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×#121C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic, complemented
3#89994C214yesmaximal, complemented
4#89993C211yesmaximal, characteristic, complemented
5#89993C211yesmaximal, characteristic, complemented
6C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
7#89749C211yesmaximal, characteristic, complemented
8#89748C211yesmaximal, characteristic, complemented
9#89716C214yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C7×D4C2212yes—
12C23×C4×C7C2211yescharacteristic
13C7×#121—24no—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2214yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2214yes—
20#6489—24no—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2214yes—
24C2×C4×Dic7C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2214yes—
29#2500—71nomaximal, Sylow
30C24×C7C2312yes—
31C24×C7C2311yescharacteristic
32C2×C7×D4—24no—
33C2×C7×D4—22no—
34C2×C7×D4—22no—
35C2×C7×D4—22no—
36C2×C7×D4—44no—
37C22×C4×C7—22no—
38C22×C4×C7—22no—
39C22×C4×C7D414yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C7×C4⋊C4—22no—
48C7×C22⋊C4—24no—
49C22⋊Dic7—24no—
50C22⋊Dic7—24no—
51C22⋊Dic7—28no—
52C22⋊Dic7—44no—
53C4⋊Dic7—22no—
54C4⋊Dic7—22no—
55Dic7⋊C4—44no—
56C4×Dic7—24no—
57C4×Dic7—22no—
58C4×Dic7—22no—
59C2×#121D711yescharacteristic
60C2×#117—71no—
61#394—74no—
62#390—71no—
63#390—71no—
64#388—74no—
65#384—71no—
66#382—71no—
67C4×C22⋊C4—71no—
68C23×C7—22no—
69C23×C7—21no—
70C23×C7—21no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—24no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—24no—
77C23×C7C2×D412yes—
78C23×C7C2411yescharacteristic
79C7×D4—48no—
80C7×D4—44no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—28no—
84C2×C4×C7—22no—
85C2×C4×C7—24no—
86C2×C4×C7C2×D412yes—
87C2×C4×C7C2×D412yes—
88C2×C4×C7#4112yes—
89C2×C4×C7#4112yes—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—24no—
96C2×Dic7—22no—
97C2×Dic7—24no—
98C2×Dic7—22no—
99C2×Dic7#4114yes—
100C22×D4C2×D711yescharacteristic
101C22×D4C2×D712yes—
102C23×C4C2×D711yescharacteristic
103#121—24no—
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—74no—
112C2×C22⋊C4—74no—
113C2×C22⋊C4C2×D712yes—
114C2×C42—72no—
115C2×C42—71no—
116C2×C42—71no—
117#95—72no—
118#95—74no—
119C22×C7—22no—
120C22×C7—22no—
121C22×C7—22no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—42no—
126C22×C7—42no—
127C22×C7—44no—
128C22×C7—44no—
129C22×C7C22×D411yescharacteristic
130C22×C7C2×#4111yescharacteristic
131C22×C7C2×#4111yescharacteristic
132C22×C7C2×#4112yes—
133C22×C7H(5,𝔽2)12yes—
134C4×C7—22no—
135C4×C7—42no—
136C4×C7#12314yes—
137Dic7—24no—
138Dic7—42no—
139Dic7—42no—
140Dic7—42no—
141C24C22×D712yes—
142C24C22×D711yescharacteristic
143C2×D4—24no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—44no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4C22⋊D714yes—
155C22×C4C22×D712yes—
156C22×C4C22×D711yescharacteristic
157C22×C4C22×D711yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—142no—
160C4⋊C4—142no—
161C4⋊C4—284no—
162C22⋊C4—24no—
163C22⋊C4—144no—
164C22⋊C4—148no—
165C22⋊C4—144no—
166C22⋊C4—284no—
167C42—142no—
168C42—142no—
169C42—144no—
170C2×C7—42no—
171C2×C7—82no—
172C2×C7C2×#12311yescharacteristic
173C2×C7#53211yescharacteristic
174C2×C7#53511yescharacteristic
175C2×C7#55312yes—
176C2×C7#55512yes—
177C23—21no—
178C23—21no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—24no—
184C23—24no—
185C23—22no—
186C23C2×C22⋊D712yes—
187C23C23×D711yescharacteristic
188D4—48no—
189D4—44no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—28no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—74no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—144no—
201C2×C4—144no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4#115912yes—
206C2×C4#115912yes—
207C2×C4C2×C22⋊D712yes—
208C2×C4C2×C22⋊D712yes—
209C7#250011yescharacteristic, complemented, Sylow
210C22—22no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—44no—
219C22—44no—
220C22C2×#115911yescharacteristic
221C22C2×#115911yescharacteristic
222C22#654812yes—
223C22D7×#4112yes—
224C22C22×C22⋊D711yescharacteristic
225C4—22no—
226C4—42no—
227C4—144no—
228C4—282no—
229C4—282no—
230C4—282no—
231C4#649814yes—
232C2—42no—
233C2—82no—
234C2#9030512yes—
235C2#9030712yes—
236C2C2×#649811yescharacteristic
237C2#9049611yescharacteristic
238C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2500
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#6489C2semidirect
3. #89994C2semidirect
4. #89993C2semidirect
5. #89993C2semidirect
6. C4×C22⋊Dic7C2semidirect
7. #89749C2semidirect
8. #89748C2semidirect
9. #89716C2semidirect
209. C7#2500semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#64892
#899942
#899932
C4×C22⋊Dic72
#897492
#897482
#897162
#25007