Finite Group Databasebeta

← #13264667

Subgroups — #13264667

Order 1792.

390 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11739703C211yesmaximal, characteristic
2C7×#1820C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11725297C212yesmaximal
5#11725296C212yesmaximal
6C2×C7×#117C2211yescharacteristic
7C2×C4×Dic14C411yescharacteristic, complemented
8C7×#381—21no—
9#89982—21no—
10#89755C411yescharacteristic, complemented
11#89751C411yescharacteristic, complemented
12#89750—21no—
13#89725C2211yescharacteristic
14#89719C2211yescharacteristic
15#89716—21no—
16#89709C411yescharacteristic, complemented
17#89705—21no—
18#89697—21no—
19C7×#342—21no—
20C7×#342C2212yes—
21#89323—21no—
22#89320C2212yes—
23#11842—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×Dic14C2×C411yescharacteristic
26C7×#117—41no—
27C7×#117—41no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4C2311yescharacteristic
33C2×C42×C7C2×C411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C4⋊Dic7C2×C411yescharacteristic
38C2×Dic7⋊C4—21no—
39C2×Dic7⋊C4—21no—
40C2×Dic7⋊C4—21no—
41C2×C4×Dic7—21no—
42C4×Dic14—41no—
43C4×Dic14—41no—
44C7×#95—41no—
45C7×#95—41no—
46C7⋊#95—21no—
47C7⋊#95—21no—
48C7⋊#95—21no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95C2×C411yescharacteristic
53C7⋊#95C2×C411yescharacteristic
54C22⋊(C7⋊C8)—22no—
55C4⋊(C7⋊C8)—41no—
56C4×C7⋊C8—41no—
57#2599—71no—
58#1820D711yescharacteristic
59#1458—71no—
60#1351—72no—
61#1344—72no—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—22no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—41no—
67C22×C4×C7—42no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—21no—
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7—21no—
78C2×Dic14—41no—
79C2×Dic14—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—21no—
90C22⋊Dic7—42no—
91C22⋊Dic7—42no—
92C4⋊Dic7—41no—
93Dic7⋊C4—42no—
94Dic7⋊C4—42no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97C4×Dic7—42no—
98C2×C7⋊C8—42no—
99C2×C4×Q8—71no—
100C2×#117C2×D711yescharacteristic
101#431—141no—
102#398—71no—
103#397—71no—
104#395—71no—
105#393—71no—
106#388—141no—
107#387—141no—
108#387—141no—
109#385—141no—
110#383—71no—
111#381—21no—
112#342—21no—
113#342C2×D712yes—
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—21no—
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—42no—
132C2×C4×C7—42no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—82no—
138C2×C4×C7—82no—
139C2×C4×C7—82no—
140C2×C4×C7#9911yescharacteristic
141C2×C4×C7#9911yescharacteristic
142C2×Dic7—22no—
143C2×Dic7—41no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—41no—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154Dic14—42no—
155Dic14—42no—
156C7⋊C8—82no—
157C22×Q8—71no—
158C23×C4C22⋊D711yescharacteristic
159C4×Q8—281no—
160C4×Q8—281no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—141no—
165C2×C4⋊C4—141no—
166C2×C4⋊C4—141no—
167C2×C4⋊C4C4×D711yescharacteristic
168C2×C4⋊C4D2811yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—141no—
171C2×C22⋊C4—141no—
172C2×C22⋊C4C22⋊D711yescharacteristic
173C2×C22⋊C4C22×D711yescharacteristic
174C2×C42—141no—
175C2×C42C4×D711yescharacteristic
176C2×C42D2811yescharacteristic
177C4⋊C8—281no—
178C22⋊C8—142no—
179C4×C8—281no—
180#95—41no—
181#95—41no—
182#95—71no—
183#95—71no—
184#95—141no—
185#95—141no—
186#95—141no—
187#95—141no—
188#95—141no—
189#95—141no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—22no—
193C22×C7—42no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—81no—
197C22×C7C2×#9911yescharacteristic
198C22×C7#41611yescharacteristic
199C22×C7#42111yescharacteristic
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—162no—
208Dic7—42no—
209Dic7—81no—
210Dic7—81no—
211Dic7—81no—
212C24C2×C22⋊D711yescharacteristic
213C2×Q8—281no—
214C2×Q8—282no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—141no—
221C22×C4—141no—
222C22×C4—141no—
223C22×C4—141no—
224C22×C4(C2×D7)⋊C411yescharacteristic
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4(C2×D7)⋊C411yescharacteristic
227C22×C4(C2×D7)⋊C411yescharacteristic
228C22×C4C2×C4×D711yescharacteristic
229C22×C4C2×D2811yescharacteristic
230C2×C8—282no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—281no—
234C4⋊C4—282no—
235C4⋊C4—282no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—282no—
243C22⋊C4—282no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—282no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—42no—
252C2×C7#151611yescharacteristic
253C2×C7#212011yescharacteristic
254C2×C7#212911yescharacteristic
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23C2×(C2×D7)⋊C411yescharacteristic
260C23#651312yes—
261Q8—282no—
262Q8—282no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—142no—
282C2×C4—281no—
283C2×C4—281no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—281no—
287C2×C4—281no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4#638011yescharacteristic
294C2×C4#638011yescharacteristic
295C8—562no—
296C7#1184211yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#638011yescharacteristic
305C22#8977411yescharacteristic
306C22#8977711yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—162no—
315C4—282no—
316C4—561no—
317C4—561no—
318C4—561no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#1172728611yescharacteristic
323C2#1172781011yescharacteristic
324C2#1172781311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1820C2
socleC22×C7C2×#99
prime radicalC24×C7C2×D4
cosocleC2×C7×C22⋊C4C23
hypercenter#1820D7
hypocenterC7#11842
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
7. C2×C4×Dic14C4semidirect
10. #89755C4semidirect
11. #89751C4semidirect
16. #89709C4semidirect
296. C7#11842semidirect

Maximal subgroups

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

subgroupindex
#117397032
C7×#18202
#117293252
#117252972
#117252962
#118427