Finite Group Databasebeta

← #13251486

Subgroups — #13251486

Order 1792.

498 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2563C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3#11729530C212yesmaximal, complemented
4#11729357C211yesmaximal, characteristic
5#11725427C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#393—21no—
9C7×#393—21no—
10C7×#390C412yes—
11C7×#382C412yes—
12C7×#380—21no—
13C7×#380—21no—
14C4×C7×C4⋊C4C2211yescharacteristic
15#89979—21no—
16C2×C22⋊(C7⋊C8)—21no—
17#89783—24no—
18#89698C2212yes—
19#89698C2212yes—
20#89342—24no—
21#89320—21no—
22#89320—21no—
23#9453—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×C4⋊Q8—22no—
29C7×C4⋊Q8—22no—
30C4×C7×D4—22no—
31C4×C7×D4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7C2×C412yes—
41C2×C42×C7C2311yescharacteristic
42C2×#1136—22no—
43C22×C7⋊C8—22no—
44C7×#95—22no—
45C7×#95—22no—
46C7×#95—21no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)D414yes—
53#6377—22no—
54#6377—22no—
55#2973—71no—
56#2563D711yescharacteristic
57#1556—72no—
58#1466—71no—
59#1285—72no—
60C24×C7C22⋊C411yescharacteristic
61C24×C7C22⋊C411yescharacteristic
62C2×C7×Q8—42no—
63C2×C7×Q8—82no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×C4⋊C4—22no—
81C7×C4⋊C4—24no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—82no—
88C7×C22⋊C4—82no—
89C42×C7—42no—
90C42×C7C2×D412yes—
91C42×C7#4112yes—
92#1136—82no—
93C2×C7⋊C8—24no—
94C2×C7⋊C8—42no—
95C2×C7⋊C8—82no—
96C2×C4⋊Q8C2×D711yescharacteristic
97C2×C4×D4C2×D711yescharacteristic
98#435—144no—
99#432—72no—
100#432—72no—
101#407—141no—
102C2×C22⋊C8—141no—
103#393—21no—
104#393—21no—
105#390Dic712yes—
106#382Dic712yes—
107#380—21no—
108#380—21no—
109C4×C4⋊C4C2×D711yescharacteristic
110#332—144no—
111#323—141no—
112#323—141no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10112yes—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—84no—
125C7×D4—44no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7C4×D412yes—
145C2×C4×C7#12011yescharacteristic
146C2×C4×C7#12112yes—
147C2×C4×C7#12311yescharacteristic
148C7⋊C8—44no—
149C7⋊C8—82no—
150C22×Q8—21no—
151C22×D4C2×Dic711yescharacteristic
152C23×C4C22⋊D711yescharacteristic
153C23×C4C22⋊D711yescharacteristic
154C2×M16—142no—
155C22×C8—142no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic712yes—
163C2×C4⋊C4C2×Dic711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊D711yescharacteristic
167C2×C22⋊C4C22⋊D711yescharacteristic
168C2×C42C2×Dic712yes—
169C2×C42C22×D711yescharacteristic
170C4⋊C8—74no—
171C4⋊C8—142no—
172C22⋊C8—142no—
173C22⋊C8—142no—
174C22⋊C8—142no—
175C22⋊C8—142no—
176#97—142no—
177#97—142no—
178#95—22no—
179#95—22no—
180#95—21no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—81no—
192C22×C7—81no—
193C22×C7#38611yescharacteristic
194C22×C7C2×#10111yescharacteristic
195C22×C7#41311yescharacteristic
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7—82no—
202C4×C7—82no—
203C4×C7#45512yes—
204C24C22⋊Dic711yescharacteristic
205C24C22⋊Dic711yescharacteristic
206C2×Q8—42no—
207C2×Q8—82no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C22⋊Dic711yescharacteristic
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22×Dic711yescharacteristic
222C22×C4C2×C22⋊D711yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224M16—562no—
225C2×C8—144no—
226C2×C8—282no—
227C2×C8—562no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C42—42no—
238C42D4×D712yes—
239C42#115912yes—
240C2×C7—21no—
241C2×C7—21no—
242C2×C7—42no—
243C2×C7—81no—
244C2×C7#188511yescharacteristic
245C2×C7#190111yescharacteristic
246C2×C7#191011yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23#641012yes—
257C23C2×C22⋊Dic711yescharacteristic
258Q8—84no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4D4×Dic712yes—
279C2×C4#649811yescharacteristic
280C2×C4#650011yescharacteristic
281C2×C4#650212yes—
282C8—284no—
283C8—562no—
284C7#945311yescharacteristic, complemented, Sylow
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#8999211yescharacteristic
298C22C2×#641011yescharacteristic
299C22#9000911yescharacteristic
300C4—22no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4#8983312yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#1172945211yescharacteristic
313C2#1172947711yescharacteristic
314C2#1172960811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2563C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2563D7
hypocenterC7#9453
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #11729530C2semidirect
5. #11725427C2semidirect
284. C7#9453semidirect

Maximal subgroups

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

subgroupindex
C7×#25632
#117397472
#117295302
#117293572
#117254272
#94537