Finite Group Databasebeta

← #11740623

Subgroups — #11740623

Order 896.

704 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 165 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×#127C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic, complemented
3C2×#6499C211yesmaximal, characteristic, complemented
4C2×D4×Dic7C211yesmaximal, characteristic, complemented
5#89994C214yesmaximal, complemented
6#89993C211yesmaximal, characteristic, complemented
7#89992C214yesmaximal, complemented
8C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
9#89701C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C7×D4C2214yes—
12C22×C7×D4C2211yescharacteristic
13C23×Dic7C2212yes—
14C22×Dic14C2211yescharacteristic
15C7×#127—24no—
16C2×C42×C7C2211yescharacteristic
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2214yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2214yes—
22#6499—24no—
23#6499—24no—
24D4×Dic7—24no—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2214yes—
30C7⋊#95C2212yes—
31#2544—71nomaximal, Sylow
32C24×C7C2312yes—
33C24×C7C2312yes—
34C2×C7×D4—28no—
35C2×C7×D4—22no—
36C2×C7×D4—22no—
37C2×C7×D4—22no—
38C2×C7×D4—22no—
39C2×C7×D4—28no—
40C2×C7×D4—22no—
41C2×C7×D4—44no—
42C2×C7×D4D414yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×Dic7—24no—
46C22×Dic7—44no—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C2×Dic14—22no—
53C2×Dic14—22no—
54C2×Dic14—22no—
55C42×C7—22no—
56C22⋊Dic7—28no—
57C22⋊Dic7—24no—
58C22⋊Dic7—24no—
59C22⋊Dic7—24no—
60C22⋊Dic7—44no—
61C4⋊Dic7—22no—
62Dic7⋊C4—24no—
63C4×Dic7—22no—
64C4×Dic7D414yescomplemented
65C2×#127D711yescharacteristic
66C2×#124—71no—
67C2×#124—71no—
68C2×C4×D4—71no—
69#394—74no—
70#391—71no—
71#390—71no—
72#386—74no—
73C4×C4⋊C4—71no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—24no—
77C23×C7—24no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—22no—
81C23×C7—24no—
82C23×C7—24no—
83C23×C7C2×D414yes—
84C23×C7C2411yescharacteristic
85C7×D4—28no—
86C7×D4—416no—
87C7×D4—44no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2×D412yes—
92C2×C4×C7#4114yes—
93C2×C4×C7#4112yes—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—24no—
104C2×Dic7—48no—
105C2×Dic7C2×D412yes—
106C2×Dic7C2×D412yes—
107Dic14—44no—
108C22×Q8—71no—
109C22×D4C2×D711yescharacteristic
110C22×D4C2×D714yes—
111C22×D4C2×D711yescharacteristic
112C23×C4—72no—
113#127—24no—
114#124—144no—
115#124—144no—
116C4×D4—144no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C22⋊C4—72no—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—74no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4—74no—
124C2×C42—71no—
125C2×C42—71no—
126C2×C42C2×D711yescharacteristic
127#95—72no—
128#95—74no—
129C22×C7—24no—
130C22×C7—28no—
131C22×C7—42no—
132C22×C7—42no—
133C22×C7—44no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—44no—
138C22×C7—44no—
139C22×C7C22×D411yescharacteristic
140C22×C7C22×D411yescharacteristic
141C22×C7C2×#4111yescharacteristic
142C22×C7C2×#4112yes—
143C22×C7H(5,𝔽2)12yes—
144C4×C7—24no—
145C4×C7#12114yes—
146Dic7—22no—
147Dic7—22no—
148Dic7—42no—
149Dic7—42no—
150Dic7—42no—
151C24C22×D712yes—
152C24C22×D712yes—
153C2×Q8—142no—
154C2×Q8—142no—
155C2×Q8—142no—
156C2×D4—28no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—28no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—44no—
164C2×D4C22⋊D714yes—
165C22×C4—71no—
166C22×C4—71no—
167C22×C4—72no—
168C22×C4—72no—
169C22×C4—72no—
170C22×C4—144no—
171C22×C4—284no—
172C22×C4C22×D712yes—
173C22×C4C22×D711yescharacteristic
174C4⋊C4—142no—
175C4⋊C4—144no—
176C22⋊C4—144no—
177C22⋊C4—144no—
178C22⋊C4—144no—
179C22⋊C4—148no—
180C22⋊C4—284no—
181C42—22no—
182C42—74no—
183C42—142no—
184C2×C7—44no—
185C2×C7—82no—
186C2×C7C2×#12111yescharacteristic
187C2×C7#53211yescharacteristic
188C2×C7#53511yescharacteristic
189C2×C7#54612yes—
190C2×C7#55912yes—
191C23—22no—
192C23—22no—
193C23—24no—
194C23—24no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—22no—
200C23C2×C22⋊D714yes—
201C23C23×D711yescharacteristic
202Q8—284no—
203D4—28no—
204D4—416no—
205D4—44no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—72no—
209C2×C4—72no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—144no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—288no—
221C2×C4D4×D712yes—
222C2×C4#115914yes—
223C2×C4#115912yes—
224C2×C4C2×C22⋊D712yes—
225C7#254411yescharacteristic, complemented, Sylow
226C22—24no—
227C22—28no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22C2×D4×D711yescharacteristic
237C22C2×#115911yescharacteristic
238C22C2×#115912yes—
239C22#654812yes—
240C22C22×C22⋊D711yescharacteristic
241C4—24no—
242C4—142no—
243C4—142no—
244C4—282no—
245C4—282no—
246C4—282no—
247C4#650214yes—
248C2—44no—
249C2—82no—
250C2#9040412yes—
251C2#9040611yescharacteristic
252C2#9040711yescharacteristic
253C2C2×#650211yescharacteristic
254C2#9049712yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#127C2
hypercenterC2×#127D7
hypocenterC7#2544
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
2. C2×#6499C2semidirect
3. C2×#6499C2semidirect
4. C2×D4×Dic7C2semidirect
5. #89994C2semidirect
6. #89993C2semidirect
7. #89992C2semidirect
8. C4×Dic7⋊C4C2semidirect
9. #89701C2semidirect
64. C4×Dic7D4semidirect
225. C7#2544semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1272
C2×#64992
C2×D4×Dic72
#899942
#899932
#899922
C4×Dic7⋊C42
#897012
#25447