Finite Group Databasebeta

← #13251631

Subgroups — #13251631

Order 1792.

518 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C7×#2484C211yesmaximal, characteristic
2#11739600C211yesmaximal, characteristic
3#11729344C212yesmaximal
4#11729306C211yesmaximal, characteristic
5#11725434C212yesmaximal
6C2×C7×#117C2211yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#400C412yescomplemented
9C7×#398C412yescomplemented
10C7×#387—22no—
11C7×#385—21no—
12C7×#381—21no—
13C7×#381C2211yescharacteristic
14#89984—21no—
15#89983—21no—
16C4×C22⋊Dic7C2212yes—
17#89979C2212yes—
18#89753—24no—
19#89327—24no—
20#89323—22no—
21#9482—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C7×#117—22no—
26C7×#117—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4D412yes—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C42×C7—21no—
37C2×C22⋊Dic7—22no—
38C2×C4⋊Dic7—22no—
39C2×C4×Dic7D414yes—
40C2×#1136—22no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95—21no—
45C7×#95C2×C412yes—
46C7⋊#95—24no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51#2484D711yescharacteristic
52#2313—71no—
53#1826—72no—
54#1463—71no—
55#1284—72no—
56C24×C7C2×D411yescharacteristic
57C22×C4×C7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—22no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7C22⋊C412yes—
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D412yes—
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7#4112yes—
73C22×Dic7—24no—
74C7×C4⋊C4—24no—
75C7×C4⋊C4—24no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—44no—
79C7×C22⋊C4—42no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C42×C7—42no—
83C22⋊Dic7—44no—
84C4⋊Dic7—82no—
85C4×Dic7—44no—
86#1136—44no—
87C2×C7⋊C8—42no—
88C2×#117C2×D711yescharacteristic
89C22×C4⋊C4C2×D711yescharacteristic
90#407—72no—
91#400Dic712yescomplemented
92#398Dic712yescomplemented
93#387—22no—
94#387—144no—
95#385—21no—
96#381—21no—
97#381—141no—
98#381C2×D711yescharacteristic
99#380—141no—
100C4×C22⋊C4—72no—
101#339—144no—
102#324—142no—
103C23×C7—22no—
104C23×C7—22no—
105C23×C7—41no—
106C23×C7C2×C22⋊C411yescharacteristic
107C23×C7#12011yescharacteristic
108C23×C7#12311yescharacteristic
109C2×C4×C7—24no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—81no—
129C2×C4×C7C4×D412yes—
130C2×C4×C7#12112yes—
131C2×Dic7—44no—
132C2×Dic7—44no—
133C2×Dic7—44no—
134C7⋊C8—82no—
135C23×C4C22⋊D711yescharacteristic
136C23×C4C22⋊D711yescharacteristic
137C23×C4C22×D711yescharacteristic
138C2×M16—142no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—142no—
145C2×C4⋊C4C2×Dic711yescharacteristic
146C2×C4⋊C4C2×Dic711yescharacteristic
147C2×C4⋊C4C2×Dic712yes—
148C2×C4⋊C4C22⋊D712yes—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—142no—
152C2×C42—21no—
153C2×C42—74no—
154C22⋊C8—142no—
155C22⋊C8—142no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—21no—
160#95—142no—
161#95—144no—
162#95—142no—
163#95C2×Dic712yes—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—44no—
172C22×C7#38611yescharacteristic
173C22×C7#42112yes—
174C22×C7#45812yes—
175C4×C7—22no—
176C4×C7—22no—
177C4×C7—42no—
178C4×C7—42no—
179C4×C7—82no—
180C4×C7—81no—
181C4×C7—81no—
182Dic7—84no—
183C24C2×C22⋊D711yescharacteristic
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—144no—
195C22×C4C22⋊Dic712yes—
196C22×C4C22⋊Dic712yes—
197C22×C4D4×D712yes—
198C22×C4#115912yes—
199C22×C4C22×Dic711yescharacteristic
200C22×C4C2×C22⋊D711yescharacteristic
201M16—284no—
202C2×C8—282no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—44no—
208C4⋊C4—562no—
209C22⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—284no—
213C42—42no—
214C42—284no—
215C2×C7—22no—
216C2×C7—22no—
217C2×C7—41no—
218C2×C7#189011yescharacteristic
219C2×C7#189711yescharacteristic
220C2×C7#191211yescharacteristic
221C23—22no—
222C23—22no—
223C23—41no—
224C23#649811yescharacteristic
225C23#650011yescharacteristic
226C23C2×C22⋊Dic711yescharacteristic
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—284no—
248C2×C4—284no—
249C2×C4—284no—
250C2×C4D4×Dic712yes—
251C2×C4#650212yes—
252C8—562no—
253C7#948211yescharacteristic, complemented, Sylow
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22#8982512yes—
263C22#8999211yescharacteristic
264C22#9000812yes—
265C4—22no—
266C4—22no—
267C4—42no—
268C4—42no—
269C4—82no—
270C4—81no—
271C4—81no—
272C4—564no—
273C2—22no—
274C2—22no—
275C2—41no—
276C2#1172945811yescharacteristic
277C2#1172946611yescharacteristic
278C2#1172961811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2484C2
socleC22×C7#386
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2484D7
hypocenterC7#9482
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
8. C7×#400C4semidirect
9. C7×#398C4semidirect
91. #400Dic7semidirect
92. #398Dic7semidirect
253. C7#9482semidirect

Maximal subgroups

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

subgroupindex
C7×#24842
#117396002
#117293442
#117293062
#117254342
#94827