Finite Group Databasebeta

← #11739860

Subgroups — #11739860

Order 896.

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

normal subgroups only

This group has 732 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×C4×C7×D4C211yesmaximal, characteristic, complemented
2C2×#6493C212yesmaximal, complemented
3C2×C4×D28C211yesmaximal, characteristic, complemented
4#89995C212yesmaximal, complemented
5#89984C212yesmaximal, complemented
6#89765C212yesmaximal, complemented
7#89728C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C22×C7×D4C2211yescharacteristic
10C23×C4×C7C2212yes—
11C22×C22⋊D7C2214yes—
12C22×D28C2211yescharacteristic
13C22×C4×D7C2212yes—
14C4×C7×D4—24no—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C42×C7C2211yescharacteristic
18C2×C22⋊Dic7C2214yes—
19#6493—28no—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2214yes—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C4×D28—24no—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28#2535—71nomaximal, Sylow
29C24×C7C2312yes—
30C23×D7C2312yes—
31C2×C7×D4—24no—
32C2×C7×D4—22no—
33C22×C4×C7—24no—
34C22×C4×C7—24no—
35C22×C4×C7—24no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—28no—
41C2×C22⋊D7—28no—
42C2×C22⋊D7—44no—
43C22×Dic7C2314yes—
44C22×Dic7C2312yes—
45C2×D28—22no—
46C2×D28—22no—
47C2×D28D414yescomplemented
48C2×C4×D7—24no—
49C2×C4×D7—44no—
50C7×C4⋊C4—22no—
51C7×C22⋊C4—24no—
52C42×C7D414yescomplemented
53C22⋊Dic7—44no—
54(C2×D7)⋊C4—28no—
55(C2×D7)⋊C4—24no—
56C4⋊Dic7—22no—
57C4⋊Dic7—24no—
58C2×#121—72no—
59C2×C4×D4—71no—
60C2×C4×D4D711yescharacteristic, complemented
61#397—72no—
62#392—72no—
63#386—74no—
64#382—71no—
65#380—72no—
66C23×C7—22no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7#4114yes—
73C23×C7C2411yescharacteristic
74C22×D7—22no—
75C22×D7—22no—
76C22×D7—24no—
77C22×D7—24no—
78C22×D7C2×D414yes—
79C7×D4—28no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—48no—
86C2×C4×C7C2×D412yes—
87C2×C4×C7C2×D412yes—
88C2×C4×C7C2×D412yes—
89C2×C4×C7C2×D412yes—
90C2×C4×C7#4112yes—
91C22⋊D7—416no—
92C2×Dic7—22no—
93C2×Dic7—24no—
94C2×Dic7—24no—
95C2×Dic7—24no—
96C2×Dic7—24no—
97D28—28no—
98C4×D7—48no—
99C22×D4—74no—
100C22×D4—71no—
101C22×D4C2×D711yescharacteristic
102C23×C4—72no—
103C23×C4C2×D712yes—
104#121—148no—
105C4×D4—24no—
106C4×D4—144no—
107C2×C4⋊C4—71no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—74no—
112C2×C22⋊C4—74no—
113C2×C22⋊C4C2×D712yes—
114C2×C42C2×D711yescharacteristic
115#95—74no—
116#95—72no—
117#95—72no—
118C22×C7—24no—
119C22×C7—24no—
120C22×C7—24no—
121C22×C7—42no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7C22×D411yescharacteristic
126C22×C7C22×D411yescharacteristic
127C22×C7C2×#4112yes—
128C22×C7C2×#4111yescharacteristic
129C22×C7H(5,𝔽2)12yes—
130C2×D7—24no—
131C2×D7—28no—
132C2×D7—42no—
133C2×D7—42no—
134C2×D7—44no—
135C4×C7—22no—
136C4×C7—22no—
137C4×C7—42no—
138C4×C7#12114yes—
139Dic7—42no—
140Dic7—44no—
141C24—72no—
142C24C22×D712yes—
143C2×D4—24no—
144C2×D4—22no—
145C2×D4—74no—
146C2×D4—148no—
147C2×D4—148no—
148C2×D4—142no—
149C2×D4—142no—
150C2×D4—284no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—74no—
155C22×C4—72no—
156C22×C4—144no—
157C22×C4—284no—
158C22×C4C22×D712yes—
159C22×C4C22×D711yescharacteristic
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D711yescharacteristic
162C4⋊C4—22no—
163C4⋊C4—144no—
164C4⋊C4—142no—
165C22⋊C4—24no—
166C22⋊C4—148no—
167C22⋊C4—144no—
168C22⋊C4—284no—
169C42C22⋊D714yescomplemented
170C2×C7—44no—
171C2×C7C2×#12111yescharacteristic
172C2×C7#52511yescharacteristic
173C2×C7#53511yescharacteristic
174C2×C7#54612yes—
175C2×C7#55312yes—
176D7—44no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—74no—
184C23—144no—
185C23—142no—
186C23—144no—
187C23—142no—
188C23#115714yes—
189C23C23×D711yescharacteristic
190D4—28no—
191D4—148no—
192D4—2816no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—48no—
199C2×C4—144no—
200C2×C4—144no—
201C2×C4—144no—
202C2×C4—144no—
203C2×C4—142no—
204C2×C4—288no—
205C2×C4D4×D712yes—
206C2×C4#115912yes—
207C2×C4C2×C22⋊D712yes—
208C2×C4C2×C22⋊D712yes—
209C2×C4C2×C22⋊D712yes—
210C7#253511yescharacteristic, complemented, Sylow
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22—144no—
219C22—148no—
220C22—282no—
221C22—284no—
222C22—282no—
223C22C2×#115712yes—
224C22C2×D4×D711yescharacteristic
225C22C2×#115911yescharacteristic
226C22#655312yes—
227C22C22×C22⋊D711yescharacteristic
228C4—22no—
229C4—22no—
230C4—42no—
231C4—284no—
232C4—282no—
233C4#650114yes—
234C2—44no—
235C2—284no—
236C2#9024112yes—
237C2#9025612yes—
238C2#9048211yescharacteristic
239C2C2×#650111yescharacteristic
240C2#9052111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
hypercenterC2×C4×D4D7
hypocenterC7#2535
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×D4C2semidirect
2. C2×#6493C2semidirect
3. C2×C4×D28C2semidirect
4. #89995C2semidirect
5. #89984C2semidirect
6. #89765C2semidirect
7. #89728C2semidirect
8. #89708C2semidirect
47. C2×D28D4semidirect
52. C42×C7D4semidirect
60. C2×C4×D4D7semidirect
169. C42C22⋊D7semidirect
210. C7#2535semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×#64932
C2×C4×D282
#899952
#899842
#897652
#897282
#897082
#25357