Finite Group Databasebeta

← #13263387

Subgroups — #13263387

Order 1792.

366 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2499C211yesmaximal, characteristic
2#11729326C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725449C212yesmaximal, complemented
5#11725438C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#397—21no—
9C7×#397—21no—
10C7×#391C411yescharacteristic
11C7×#391C2211yescharacteristic
12C7×#387—21no—
13C7×#387—21no—
14C7×#384C411yescharacteristic
15C7×#382C2211yescharacteristic
16C4×C7×C22⋊C4C411yescharacteristic
17#89698—21no—
18#89697—21no—
19#89323C2212yes—
20#89323C2212yes—
21#89320—21no—
22#89320—21no—
23#11556—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C7×C22⋊Q8—41no—
27C7×C22⋊Q8—41no—
28C7×#117—41no—
29C7×#117—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C7×#95—21no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—41no—
54C4⋊(C7⋊C8)—41no—
55#6377—41no—
56C4×C7⋊C8—41no—
57#2499D711yescharacteristic
58#1464—71no—
59#1458—71no—
60#1324—72no—
61#1319—72no—
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—41no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7—22no—
96C42×C7—41no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8—42no—
99C2×C22⋊Q8Dic711yescharacteristic
100C2×#117C2×D711yescharacteristic
101#432—141no—
102#431—141no—
103#397—21no—
104#397—21no—
105#391Dic711yescharacteristic
106#391C2×D711yescharacteristic
107#387—21no—
108#387—21no—
109#384Dic711yescharacteristic
110#382C2×D711yescharacteristic
111C4×C22⋊C4Dic711yescharacteristic
112#324—72no—
113#324—72no—
114#323—141no—
115#323—141no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7C2×C22⋊C411yescharacteristic
122C7×Q8—82no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7#13112yes—
152C7⋊C8—82no—
153C7⋊C8—82no—
154C22×Q8C2×Dic711yescharacteristic
155C23×C4C2×Dic711yescharacteristic
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic711yescharacteristic
164C2×C4⋊C4C2×Dic711yescharacteristic
165C2×C4⋊C4C22×D711yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4C22⋊D711yescharacteristic
169C2×C22⋊C4C22⋊D711yescharacteristic
170C2×C42C2×Dic711yescharacteristic
171C2×C42C2×Dic711yescharacteristic
172C2×C42C22⋊D711yescharacteristic
173C2×C42C22⋊D711yescharacteristic
174C4⋊C8—281no—
175C4⋊C8—281no—
176C22⋊C8—142no—
177C22⋊C8—142no—
178#97—281no—
179C4×C8—281no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—81no—
193C22×C7—81no—
194C22×C7#40811yescharacteristic
195C22×C7C2×#9911yescharacteristic
196C22×C7#41311yescharacteristic
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—22no—
200C4×C7—42no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—81no—
206C4×C7—81no—
207C4×C7—81no—
208C24C22⋊Dic711yescharacteristic
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22⋊Dic711yescharacteristic
223C22×C4C22×Dic711yescharacteristic
224C22×C4C2×C22⋊D711yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C2×C8—282no—
227C2×C8—282no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—22no—
244C42—41no—
245C2×C7—21no—
246C2×C7—21no—
247C2×C7—81no—
248C2×C7#146811yescharacteristic
249C2×C7#212911yescharacteristic
250C2×C7#213211yescharacteristic
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic711yescharacteristic
257Q8—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#640811yescharacteristic
285C2×C4#640811yescharacteristic
286C2×C4#651112yes—
287C8—562no—
288C8—562no—
289C7#1155611yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22C2×#640811yescharacteristic
298C22#9000611yescharacteristic
299C22#9000911yescharacteristic
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—81no—
310C4—81no—
311C2—21no—
312C2—21no—
313C2—81no—
314C2#1172945611yescharacteristic
315C2#1172957411yescharacteristic
316C2#1172962511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2499C2
socleC22×C7C2×#99
prime radicalC24×C7C22⋊C4
cosocleC2×C7×C4⋊C4C23
hypercenter#2499D7
hypocenterC7#11556
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2499

Complements

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

NH ≅ G/Ntype
4. #11725449C2semidirect
5. #11725438C2semidirect
289. C7#11556semidirect

Maximal subgroups

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

subgroupindex
C7×#24992
#117293262
#117293252
#117254492
#117254382
#115567