Finite Group Databasebeta

← #13246433

Subgroups — #13246433

Order 1792.

1154 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×C7×#124C211yesmaximal, characteristic
2#11729887C212yesmaximal, complemented
3#11729293C214yesmaximal, complemented
4C23×C7×Q8C2211yescharacteristic
5C23×C7×D4C411yescharacteristic
6C2×C7×#124—24no—
7C2×C7×#124—28no—
8C22×C7×C22⋊C4C412yes—
9C22×C7×C22⋊C4C2212yes—
10C22×C42×C7C411yescharacteristic
11C2×#6410—24no—
12#89979C2214yes—
13#89320—28no—
14#8552—71nomaximal, Sylow
15C25×C7C2×C412yes—
16C22×C7×Q8—24no—
17C22×C7×Q8—44no—
18C22×C7×Q8D414yes—
19C22×C7×D4—24no—
20C22×C7×D4—22no—
21C22×C7×D4—44no—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7C2×C412yes—
25C23×C4×C7C2311yescharacteristic
26C7×#124—232no—
27C2×C7×C22⋊C4—24no—
28C2×C7×C22⋊C4—24no—
29C2×C7×C22⋊C4—24no—
30C2×C7×C22⋊C4—24no—
31C2×C7×C22⋊C4—24no—
32C2×C7×C22⋊C4D418yes—
33C2×C42×C7—24no—
34C2×C42×C7—22no—
35C2×#1136—24no—
36#6410—48no—
37C22⋊(C7⋊C8)—28no—
38C22×#124D711yescharacteristic
39#1784—72no—
40#1462—74no—
41C24×C7—22no—
42C24×C7—22no—
43C24×C7—44no—
44C24×C7—44no—
45C24×C7—44no—
46C24×C7C22⋊C414yes—
47C24×C7C22×C411yescharacteristic
48C2×C7×Q8—28no—
49C2×C7×Q8—28no—
50C2×C7×Q8—44no—
51C2×C7×Q8—416no—
52C2×C7×D4—28no—
53C2×C7×D4—416no—
54C2×C7×D4—44no—
55C2×C7×D4—44no—
56C22×C4×C7—22no—
57C22×C4×C7—24no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—42no—
62C22×C4×C7—42no—
63C22×C4×C7—44no—
64C22×C4×C7C22⋊C412yes—
65C22×C4×C7C22⋊C412yes—
66C22×C4×C7C22⋊C414yes—
67C22×C4×C7C2×D412yes—
68C22×C4×C7C2×D414yes—
69C7×C22⋊C4—216no—
70C7×C22⋊C4—216no—
71C42×C7—28no—
72#1136—48no—
73C2×C7⋊C8—44no—
74C23×Q8C2×D711yescharacteristic
75C23×D4Dic711yescharacteristic
76C2×#124—24no—
77C2×#124—28no—
78C22×C22⋊C4Dic712yes—
79C22×C22⋊C4C2×D712yes—
80C22×C42Dic711yescharacteristic
81C2×#101—144no—
82#407—74no—
83#323—148no—
84C23×C7—22no—
85C23×C7—24no—
86C23×C7—24no—
87C23×C7—28no—
88C23×C7—44no—
89C23×C7—44no—
90C23×C7—44no—
91C23×C7—44no—
92C23×C7—44no—
93C23×C7—44no—
94C23×C7—44no—
95C23×C7—44no—
96C23×C7—816no—
97C23×C7C2×C22⋊C411yescharacteristic
98C23×C7C2×C22⋊C412yes—
99C7×Q8—416no—
100C7×D4—416no—
101C2×C4×C7—28no—
102C2×C4×C7—22no—
103C2×C4×C7—24no—
104C2×C4×C7—22no—
105C2×C4×C7—24no—
106C2×C4×C7—22no—
107C2×C4×C7—24no—
108C2×C4×C7—44no—
109C2×C4×C7—44no—
110C2×C4×C7—44no—
111C2×C4×C7—44no—
112C2×C4×C7—44no—
113C2×C4×C7#9914yes—
114C2×C4×C7#12014yes—
115C7⋊C8—84no—
116C25C2×Dic712yes—
117C22×Q8—24no—
118C22×Q8—44no—
119C22×Q8C22⋊D714yes—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—44no—
123C23×C4C2×Dic711yescharacteristic
124C23×C4C2×Dic711yescharacteristic
125C23×C4C2×Dic712yes—
126C23×C4C22×D711yescharacteristic
127C2×M16—144no—
128#124—232no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4C22⋊D718yes—
135C2×C42—24no—
136C2×C42—22no—
137#101—288no—
138C22⋊C8—148no—
139C22×C7—28no—
140C22×C7—22no—
141C22×C7—22no—
142C22×C7—22no—
143C22×C7—44no—
144C22×C7—44no—
145C22×C7—44no—
146C22×C7—88no—
147C22×C7—88no—
148C22×C7—88no—
149C22×C7#37911yescharacteristic
150C22×C7C2×#9912yes—
151C22×C7#41314yes—
152C4×C7—24no—
153C4×C7—24no—
154C4×C7—42no—
155C4×C7—44no—
156C24—22no—
157C24—22no—
158C24—44no—
159C24—44no—
160C24—44no—
161C24C22⋊Dic714yes—
162C24C22×Dic711yescharacteristic
163C2×Q8—28no—
164C2×Q8—28no—
165C2×Q8—44no—
166C2×Q8—416no—
167C2×D4—28no—
168C2×D4—416no—
169C2×D4—44no—
170C2×D4—44no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4C22⋊Dic712yes—
180C22×C4C22⋊Dic712yes—
181C22×C4C22⋊Dic714yes—
182C22×C4C2×C22⋊D712yes—
183C22×C4C2×C22⋊D714yes—
184M16—288no—
185C2×C8—284no—
186C22⋊C4—216no—
187C22⋊C4—216no—
188C42—28no—
189C2×C7—22no—
190C2×C7—24no—
191C2×C7—84no—
192C2×C7#178111yescharacteristic
193C2×C7#178512yes—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—28no—
198C23—44no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23—816no—
207C23C2×C22⋊Dic711yescharacteristic
208C23C2×C22⋊Dic712yes—
209Q8—416no—
210D4—416no—
211C2×C4—28no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4#640814yes—
224C2×C4#651614yes—
225C8—564no—
226C7#855211yescharacteristic, complemented, Sylow
227C22—28no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—88no—
235C22—88no—
236C22—88no—
237C22C2×#640812yes—
238C22#9000914yes—
239C22#9001911yescharacteristic
240C4—24no—
241C4—24no—
242C4—42no—
243C4—44no—
244C2—22no—
245C2—24no—
246C2—84no—
247C2#1172988411yescharacteristic
248C2#1172989112yes—
subgroupG/H
centerC22#90019
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC22×C7×#124C2
socleC22×C7#379
prime radicalC23×C7×D4C4
cosocleC23×C4×C7C23
hypercenterC22×#124D7
hypocenterC7#8552
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#124

Complements

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

NH ≅ G/Ntype
2. #11729887C2semidirect
3. #11729293C2semidirect
226. C7#8552semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1242
#117298872
#117292932
#85527