Finite Group Databasebeta

← #13247299

Subgroups — #13247299

Order 1792.

786 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11741489C211yesmaximal, characteristic
2C7×#1781C211yesmaximal, characteristic
3#11729884C211yesmaximal, characteristic
4C7⋊#1304C214yesmaximal, complemented
5C23×C7×D4C2211yescharacteristic
6C2×#6489C412yescomplemented
7C2×C7×#99—22no—
8C7×#379C2212yes—
9#90019C412yescomplemented
10#90019C2212yes—
11#90019C2212yes—
12#89993—22no—
13#89993—22no—
14C2×#6408—22no—
15#8748—71nomaximal, Sylow
16C25×C7C2×C411yescharacteristic
17C25×C7C2311yescharacteristic
18C22×C7×D4—24no—
19C22×C7×D4—44no—
20C22×C7×D4D412yes—
21C22×C7×D4D412yes—
22C23×C4×C7C2×C411yescharacteristic
23C2×C7×C22⋊C4—22no—
24C2×C7×C22⋊C4—24no—
25C2×C22⋊Dic7—22no—
26C2×C22⋊Dic7—24no—
27C2×C22⋊Dic7—22no—
28C2×C22⋊Dic7—22no—
29C2×C22⋊Dic7—24no—
30C2×C22⋊Dic7C2×C414yes—
31#6489—44no—
32C2×C4⋊Dic7—22no—
33C2×C4×Dic7—22no—
34C7×#99—44no—
35#6408—44no—
36#2309—71no—
37#1781—71no—
38#1781D711yescharacteristic
39#1304—74no—
40C24×C7—22no—
41C24×C7—42no—
42C24×C7—42no—
43C24×C7—82no—
44C24×C7—82no—
45C24×C7C22⋊C412yes—
46C24×C7C22×C411yescharacteristic
47C24×C7C2×D411yescharacteristic
48C24×C7C2×D411yescharacteristic
49C24×C7#4114yes—
50C2×C7×D4—24no—
51C2×C7×D4—24no—
52C2×C7×D4—24no—
53C2×C7×D4—24no—
54C2×C7×D4—42no—
55C2×C7×D4—42no—
56C2×C7×D4—416no—
57C22×C4×C7—22no—
58C22×C4×C7—24no—
59C22×C4×C7—42no—
60C22×C4×C7C22⋊C412yes—
61C22×Dic7—24no—
62C22×Dic7—42no—
63C7×C22⋊C4—44no—
64C7×C22⋊C4—84no—
65C22⋊Dic7—44no—
66C22⋊Dic7—44no—
67C22⋊Dic7—82no—
68C22⋊Dic7—82no—
69C22⋊Dic7—84no—
70C22⋊Dic7—84no—
71C4⋊Dic7—44no—
72C4×Dic7—44no—
73C23×D4C2×D711yescharacteristic
74C2×#117—72no—
75C2×#99—22no—
76C2×#99—142no—
77#390—142no—
78#390—142no—
79#379—72no—
80#379—72no—
81#379—72no—
82#379C2×D712yes—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—42no—
90C23×C7—42no—
91C23×C7—42no—
92C23×C7—44no—
93C23×C7—44no—
94C23×C7—44no—
95C23×C7—44no—
96C23×C7—44no—
97C23×C7—82no—
98C23×C7—82no—
99C23×C7—88no—
100C23×C7—88no—
101C23×C7C2×C22⋊C411yescharacteristic
102C23×C7#11712yes—
103C23×C7#12312yes—
104C23×C7#12312yes—
105C7×D4—48no—
106C7×D4—48no—
107C2×C4×C7—28no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—82no—
111C2×C4×C7—84no—
112C2×C4×C7#9914yes—
113C2×Dic7—44no—
114C2×Dic7—44no—
115C2×Dic7—44no—
116C2×Dic7—82no—
117C2×Dic7—84no—
118C25C4×D711yescharacteristic
119C25C22×D711yescharacteristic
120C22×D4—24no—
121C22×D4—44no—
122C22×D4D2812yes—
123C22×D4C22⋊D712yes—
124C23×C4C4×D711yescharacteristic
125#117—284no—
126C2×C4⋊C4—142no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—74no—
130C2×C22⋊C4—142no—
131C2×C22⋊C4—144no—
132C2×C22⋊C4—142no—
133C2×C22⋊C4—142no—
134C2×C22⋊C4—144no—
135C2×C42—142no—
136#99—44no—
137#99—284no—
138C22×C7—22no—
139C22×C7—22no—
140C22×C7—22no—
141C22×C7—22no—
142C22×C7—24no—
143C22×C7—24no—
144C22×C7—22no—
145C22×C7—22no—
146C22×C7—44no—
147C22×C7—42no—
148C22×C7—44no—
149C22×C7—82no—
150C22×C7—82no—
151C22×C7—82no—
152C22×C7—82no—
153C22×C7—88no—
154C22×C7—88no—
155C22×C7#38111yescharacteristic
156C22×C7C2×#9912yes—
157C4×C7—24no—
158C4×C7—162no—
159Dic7—84no—
160Dic7—162no—
161C24—22no—
162C24—42no—
163C24—42no—
164C24—82no—
165C24—82no—
166C24(C2×D7)⋊C412yes—
167C24C2×C4×D711yescharacteristic
168C24C2×D2811yescharacteristic
169C24#115914yes—
170C24C2×C22⋊D711yescharacteristic
171C2×D4—24no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—416no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—42no—
181C22×C4—144no—
182C22×C4—282no—
183C22×C4(C2×D7)⋊C412yes—
184C4⋊C4—284no—
185C22⋊C4—44no—
186C22⋊C4—84no—
187C22⋊C4—284no—
188C22⋊C4—284no—
189C22⋊C4—562no—
190C22⋊C4—562no—
191C22⋊C4—564no—
192C22⋊C4—564no—
193C42—284no—
194C2×C7—22no—
195C2×C7—42no—
196C2×C7—42no—
197C2×C7—42no—
198C2×C7—82no—
199C2×C7#181911yescharacteristic
200C2×C7#182812yes—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—42no—
208C23—44no—
209C23—42no—
210C23—44no—
211C23—42no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—82no—
216C23—82no—
217C23—88no—
218C23—88no—
219C23#644012yes—
220C23#644912yes—
221C23C2×(C2×D7)⋊C411yescharacteristic
222C23#649812yes—
223D4—48no—
224D4—48no—
225C2×C4—28no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—82no—
229C2×C4—84no—
230C2×C4—284no—
231C2×C4—284no—
232C2×C4—284no—
233C2×C4—562no—
234C2×C4—564no—
235C2×C4#638014yes—
236C7#874811yescharacteristic, complemented, Sylow
237C22—22no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—24no—
242C22—24no—
243C22—22no—
244C22—22no—
245C22—44no—
246C22—42no—
247C22—44no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22—88no—
253C22—88no—
254C22#8971511yescharacteristic
255C22C2×#638012yes—
256C4—24no—
257C4—162no—
258C4—564no—
259C4—1122no—
260C2—22no—
261C2—42no—
262C2—42no—
263C2—42no—
264C2—82no—
265C2#1172724911yescharacteristic
266C2#1172777412yes—
subgroupG/H
centerC22#89715
commutatorC24×C7C22×C4
FrattiniC25C22×D7
FittingC7×#1781C2
socleC22×C7#381
prime radicalC23×C7×D4C22
cosocleC25×C7C23
hypercenter#1781D7
hypocenterC7#8748
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
4. C7⋊#1304C2semidirect
6. C2×#6489C4semidirect
9. #90019C4semidirect
236. C7#8748semidirect

Maximal subgroups

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

subgroupindex
#117414892
C7×#17812
#117298842
C7⋊#13042
#87487