Finite Group Databasebeta

← #13264689

Subgroups — #13264689

Order 1792.

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

normal subgroups only

This group has 382 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×#2611C211yesmaximal, characteristic
2#11729348C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725457C212yesmaximal
5#11725455C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C2×C7×#117C2211yescharacteristic
8C7×#400C411yescharacteristic, complemented
9C7×#399C2211yescharacteristic
10C7×#398C411yescharacteristic, complemented
11C7×#397C2211yescharacteristic
12C7×#389—21no—
13C7×#388—21no—
14C7×#387—21no—
15C7×#385—21no—
16C7×#376C411yescharacteristic, complemented
17#89983—21no—
18#89697—21no—
19#89323—21no—
20#89321—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11846—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C7×#125—41no—
26C7×#125—41no—
27C7×#117—41no—
28C7×#117—41no—
29C2×C7×C4⋊C4—21no—
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⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C7×C22⋊C4C2311yescharacteristic
41C2×C42×C7—21no—
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C22⋊Dic7—22no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95C2×C411yescharacteristic
51C7×#95C2×C411yescharacteristic
52C7⋊#95—41no—
53C7⋊#95—41no—
54C22⋊(C7⋊C8)—22no—
55C4⋊(C7⋊C8)—41no—
56C4×C7⋊C8—41no—
57#2611D711yescharacteristic
58#1820—71no—
59#1458—71no—
60#1352—72no—
61#1344—72no—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—41no—
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—42no—
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—42no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C2×C7⋊C8—42no—
99C2×#125Dic711yescharacteristic, complemented
100C2×#117C2×D711yescharacteristic
101#431—141no—
102#400Dic711yescharacteristic, complemented
103#399C2×D711yescharacteristic
104#398Dic711yescharacteristic, complemented
105#397C2×D711yescharacteristic
106#389—21no—
107#388—21no—
108#387—21no—
109#385—21no—
110#381—141no—
111#376Dic711yescharacteristic, complemented
112#342—72no—
113#342—141no—
114#324—141no—
115#323—72no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#13512yes—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
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×Dic7—82no—
152C2×Dic7—82no—
153C2×Dic7—82no—
154C7⋊C8—82no—
155C23×C4C22⋊D711yescharacteristic
156#125—41no—
157#125—41no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Dic711yescharacteristic
165C2×C4⋊C4C2×Dic711yescharacteristic
166C2×C4⋊C4C2×Dic711yescharacteristic
167C2×C4⋊C4C22⋊D711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—142no—
171C2×C22⋊C4C22⋊D711yescharacteristic
172C2×C22⋊C4C22×D711yescharacteristic
173C2×C42—21no—
174C2×C42C2×Dic711yescharacteristic
175C2×C42C22⋊D711yescharacteristic
176C4⋊C8—281no—
177C22⋊C8—142no—
178C4×C8—281no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—281no—
185#95—281no—
186#95C2×Dic711yescharacteristic
187#95C2×Dic711yescharacteristic
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—22no—
191C22×C7—42no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7C2×#9911yescharacteristic
196C22×C7#41611yescharacteristic
197C22×C7#42111yescharacteristic
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—42no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—81no—
207C4×C7—81no—
208C4×C7—81no—
209Dic7—162no—
210C24C2×C22⋊D711yescharacteristic
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—282no—
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22⋊Dic711yescharacteristic
223C22×C4C22⋊Dic711yescharacteristic
224C22×C4C22×Dic711yescharacteristic
225C22×C4C2×C22⋊D711yescharacteristic
226C2×C8—282no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—282no—
241C22⋊C4—282no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—42no—
247C2×C7—21no—
248C2×C7—21no—
249C2×C7—42no—
250C2×C7#151611yescharacteristic
251C2×C7#212311yescharacteristic
252C2×C7#212611yescharacteristic
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23#651312yes—
258C23C2×C22⋊Dic711yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
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—41no—
285C2×C4—41no—
286C2×C4—562no—
287C2×C4—562no—
288C2×C4—562no—
289C2×C4#640811yescharacteristic
290C2×C4#640811yescharacteristic
291C8—562no—
292C7#1184611yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#640811yescharacteristic
301C22#9000511yescharacteristic
302C22#9000811yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—1122no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#1172946411yescharacteristic
319C2#1172957311yescharacteristic
320C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2611C2
socleC22×C7C2×#99
prime radicalC24×C7C2×D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2611D7
hypocenterC7#11846
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
8. C7×#400C4semidirect
10. C7×#398C4semidirect
16. C7×#376C4semidirect
99. C2×#125Dic7semidirect
102. #400Dic7semidirect
104. #398Dic7semidirect
111. #376Dic7semidirect
292. C7#11846semidirect

Maximal subgroups

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

subgroupindex
C7×#26112
#117293482
#117293252
#117254572
#117254552
#118467