Finite Group Databasebeta

← #13263160

Subgroups — #13263160

Order 1792.

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

normal subgroups only

This group has 662 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
1C7×#2280C211yesmaximal, characteristic
2#11729326C212yesmaximal, complemented
3#11725433C212yesmaximal, complemented
4#11725428C212yesmaximal
5C24×C4×C7C411yescharacteristic
6C2×C7×#117C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#381C411yescharacteristic
9C7×#381C2211yescharacteristic
10C7×#380—22no—
11C7×#379C411yescharacteristic
12C7×#379C2211yescharacteristic
13C4×C7×C22⋊C4—22no—
14#89698—22no—
15#89323—22no—
16#89320C2214yes—
17#11501—71nomaximal, Sylow
18C25×C7C2×C411yescharacteristic
19C23×C4×C7—22no—
20C23×C4×C7—21no—
21C23×C4×C7—21no—
22C23×C4×C7—22no—
23C23×C4×C7—42no—
24C23×C4×C7—42no—
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C7×#117—42no—
28C7×#117—42no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4C2×C411yescharacteristic
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7D412yes—
39C7×#95—22no—
40C7×#95—22no—
41C22⋊(C7⋊C8)—24no—
42C4⋊(C7⋊C8)—42no—
43#6377—42no—
44#2280D711yescharacteristic
45#1464—72no—
46#1317—72no—
47#1316—72no—
48C24×C7—21no—
49C24×C7—21no—
50C24×C7—41no—
51C24×C7—41no—
52C24×C7—44no—
53C24×C7M1612yes—
54C24×C7C22×C411yescharacteristic
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—24no—
60C22×C4×C7—24no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—48no—
66C22×C4×C7—48no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C412yes—
76C22×C4×C7M1612yes—
77C22×C4×C7C2×D412yes—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—44no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C42×C7—22no—
87C42×C7—22no—
88C42×C7—42no—
89C2×C7⋊C8—44no—
90C24×C4Dic711yescharacteristic
91C2×#117Dic711yescharacteristic
92C2×#117C2×D711yescharacteristic
93#432—142no—
94#381Dic711yescharacteristic
95#381C2×D711yescharacteristic
96#380—22no—
97#379Dic711yescharacteristic
98#379C2×D711yescharacteristic
99C4×C22⋊C4—22no—
100#324—142no—
101#323—74no—
102C23×C7—24no—
103C23×C7—22no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—42no—
107C23×C7—44no—
108C23×C7—42no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7—44no—
112C23×C7—44no—
113C23×C7—48no—
114C23×C7—44no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7C2×C22⋊C411yescharacteristic
118C23×C7C2×M1611yescharacteristic
119C23×C7C2×M1611yescharacteristic
120C2×C4×C7—24no—
121C2×C4×C7—24no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—48no—
130C2×C4×C7—48no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7#9912yes—
142C2×C4×C7#10012yes—
143C7⋊C8—84no—
144C25C2×Dic711yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×Dic711yescharacteristic
152C23×C4C2×Dic711yescharacteristic
153#117—42no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C22⋊D712yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C2×Dic711yescharacteristic
160C2×C22⋊C4C2×Dic711yescharacteristic
161C2×C22⋊C4C2×Dic711yescharacteristic
162C2×C22⋊C4C22×D711yescharacteristic
163C2×C42—21no—
164C2×C42C22⋊D712yes—
165C4⋊C8—282no—
166C22⋊C8—144no—
167#97—282no—
168#95—22no—
169#95—22no—
170C22×C7—22no—
171C22×C7—24no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—44no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7—44no—
180C22×C7—44no—
181C22×C7—48no—
182C22×C7—44no—
183C22×C7—42no—
184C22×C7—42no—
185C22×C7#40711yescharacteristic
186C22×C7C2×#9911yescharacteristic
187C22×C7C2×#10011yescharacteristic
188C4×C7—22no—
189C4×C7—22no—
190C4×C7—42no—
191C4×C7—42no—
192C4×C7—42no—
193C4×C7—81no—
194C4×C7—81no—
195C24—21no—
196C24—21no—
197C24—41no—
198C24—41no—
199C24—44no—
200C24#113612yes—
201C24C22×Dic711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4#113612yes—
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22⋊Dic712yes—
223C22×C4C22⋊Dic711yescharacteristic
224C22×C4C2×C22⋊D712yes—
225C2×C8—284no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C42—22no—
235C42—22no—
236C42—42no—
237C2×C7—21no—
238C2×C7—21no—
239C2×C7—44no—
240C2×C7—41no—
241C2×C7—41no—
242C2×C7#146211yescharacteristic
243C2×C7#212012yes—
244C23—24no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—42no—
249C23—44no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—44no—
257C23—41no—
258C23—41no—
259C23C2×#113611yescharacteristic
260C23C2×#113611yescharacteristic
261C23C2×C22⋊Dic711yescharacteristic
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4#640712yes—
284C2×C4#640812yes—
285C8—564no—
286C7#1150111yescharacteristic, complemented, Sylow
287C22—22no—
288C22—24no—
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—44no—
297C22—44no—
298C22—48no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22#8997911yescharacteristic
303C22C2×#640711yescharacteristic
304C22C2×#640811yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—41no—
316C2—41no—
317C2#1172929311yescharacteristic
318C2#1172947012yes—
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2280C2
socleC22×C7C2×#99
prime radicalC25×C7C2×C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2280D7
hypocenterC7#11501
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2280

Complements

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

NH ≅ G/Ntype
2. #11729326C2semidirect
3. #11725433C2semidirect
286. C7#11501semidirect

Maximal subgroups

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

subgroupindex
C7×#22802
#117293262
#117254332
#117254282
#115017