Finite Group Databasebeta

← #13264733

Subgroups — #13264733

Order 1792.

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

normal subgroups only

This group has 324 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#11738185C211yesmaximal, characteristic
2#11726846C211yesmaximal, characteristic, complemented
3#11726602C211yesmaximal, characteristic, complemented
4C7×#1348C211yesmaximal, characteristic, complemented
5#11725582C211yesmaximal, characteristic, complemented
6#11725332C211yesmaximal, characteristic
7#11725291C211yesmaximal, characteristic
8#90315—21no—
9C7×#467C2211yescharacteristic
10#89963C411yescharacteristic, complemented
11Q16×Dic7C411yescharacteristic, complemented
12#89887C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14#89820—21no—
15#89749—21no—
16#89650C411yescharacteristic, complemented
17#89644C411yescharacteristic, complemented
18#89572—21no—
19#89560—21no—
20C7×C22⋊C16C2211yescharacteristic
21C7×#340C2211yescharacteristic
22#89346C2211yescharacteristic
23C22⋊(C7⋊C16)C2211yescharacteristic
24#89324—21no—
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11855—71nomaximal, Sylow
28C2×C7×Q16C2×C411yescharacteristic
29C22×C7×C8C2311yescharacteristic
30C7×C22⋊Q8—21no—
31C2×C7×C4⋊C4—21no—
32#6510—21no—
33Q8×Dic7—21no—
34C2×#1143—21no—
35#6493—21no—
36C2×C4⋊Dic7—21no—
37C2×C4×Dic7D411yescharacteristic
38C22×C7⋊C8D411yescharacteristic
39C2×D56C2×C411yescharacteristic
40#6461—41no—
41#6456—21no—
42#6455—41no—
43#6453—21no—
44#6449—41no—
45(C22⋊D7)⋊C4—41no—
46C2×C7×C16—21no—
47C7×#107C2×C411yescharacteristic
48C7×#103—21no—
49#6409—21no—
50C7⋊#95—41no—
51C7⋊#95—41no—
52#6395—21no—
53#6392C2×C411yescharacteristic
54C8×Dic7C2×C411yescharacteristic
55C8×Dic7C2×C411yescharacteristic
56C8×Dic7D411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59#6383—21no—
60#6382—21no—
61#3168—71no—
62#2162—71no—
63#1835—71no—
64#1348—71no—
65#1348D711yescharacteristic, complemented
66#1347—71no—
67#1347—71no—
68C2×C7×Q8—21no—
69C22×C4×C7—41no—
70C22×C4×C7C2×D411yescharacteristic
71C2×C22⋊D7—41no—
72C22×Dic7—41no—
73C22×Dic7D811yescharacteristic
74C22×Dic7D811yescharacteristic
75C2×D28—21no—
76C2×C4×D7—41no—
77C7×Q16—41no—
78C2×C7×C8—41no—
79C2×C7×C8C22×C411yescharacteristic
80C2×C7×C8C2×D411yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C22⋊C4—41no—
86#1143—42no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91C4⋊Dic7—21no—
92C4⋊Dic7—41no—
93C4⋊Dic7—41no—
94C4⋊Dic7—41no—
95C4⋊Dic7—41no—
96Dic7⋊C4—41no—
97C4×Dic7—41no—
98C4×Dic7C22⋊C411yescharacteristic
99C4×Dic7C22⋊C411yescharacteristic
100C4×Dic7SD1611yescharacteristic
101C4×Dic7SD1611yescharacteristic
102C2×C7⋊C8—41no—
103C2×C7⋊C8C22⋊C411yescharacteristic
104C2×C7⋊C8C22⋊C411yescharacteristic
105D56—41no—
106C7×C16—41no—
107C7⋊C16—41no—
108#529—141no—
109#499—71no—
110#495—71no—
111#488—141no—
112#467C2×D711yescharacteristic
113#466—71no—
114#465—141no—
115C4×Q16—71no—
116C4×SD16—141no—
117C4×D8—71no—
118C2×C4×C8—71no—
119#382—141no—
120C4⋊C16—141no—
121C22⋊C16—71no—
122C22⋊C16C2×D711yescharacteristic
123C4×C16—141no—
124#340—71no—
125#340—141no—
126#340C2×D711yescharacteristic
127C23×C7C2×D811yescharacteristic
128C22×D7—41no—
129C7×Q8—41no—
130C7×Q8—41no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—81no—
136C2×C4×C7—81no—
137C2×C4×C7—81no—
138C2×C4×C7C2×C22⋊C411yescharacteristic
139C2×C4×C7C2×SD1611yescharacteristic
140C22⋊D7—81no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—41no—
145C2×Dic7—41no—
146C2×Dic7—81no—
147C2×Dic7—81no—
148C2×Dic7—81no—
149C2×Dic7#10211yescharacteristic
150C2×Dic7#10211yescharacteristic
151C2×Dic7#10211yescharacteristic
152C2×Dic7#10211yescharacteristic
153D28—41no—
154D28—41no—
155C4×D7—82no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C2×Q16C4×D711yescharacteristic
161C2×SD16—141no—
162C2×D8—71no—
163C22×C8—71no—
164C22×C8C22×D711yescharacteristic
165#126—281no—
166#125—141no—
167#124—141no—
168#123—281no—
169C22⋊Q8—21no—
170#121—141no—
171C4×Q8—141no—
172C4×D4—141no—
173C4×D4—281no—
174#117—281no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—141no—
177C2×C42—71no—
178C2×C16—21no—
179C2×C16—141no—
180#107—71no—
181#107C4×D711yescharacteristic
182#106—141no—
183#103—21no—
184#103—141no—
185#102—141no—
186#102—141no—
187C4×C8—71no—
188C4×C8—71no—
189C4×C8—71no—
190C4×C8—71no—
191#95—281no—
192#95—281no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7C2×#10211yescharacteristic
196C22×C7#50811yescharacteristic
197C22×C7#50811yescharacteristic
198C2×D7—81no—
199C2×D7—81no—
200C2×D7—81no—
201C4×C7—21no—
202C4×C7—81no—
203C4×C7—81no—
204C4×C7—161no—
205C4×C7#41011yescharacteristic
206C4×C7#42111yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—81no—
212Dic7—161no—
213C2×Q8—21no—
214C2×D4—141no—
215C2×D4—281no—
216C22×C4—41no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—281no—
220C22×C4—281no—
221C22×C4D4×D711yescharacteristic
222Q16—41no—
223SD16—282no—
224D8—281no—
225C2×C8—41no—
226C2×C8—71no—
227C2×C8—71no—
228C2×C8—281no—
229C2×C8C2×C4×D711yescharacteristic
230C2×C8D4×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—141no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C22⋊C4—41no—
242C22⋊C4—281no—
243C22⋊C4—281no—
244C22⋊C4—281no—
245C22⋊C4—281no—
246C42—71no—
247C42—71no—
248C42—71no—
249C42—71no—
250C42—281no—
251C16—41no—
252C16—281no—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7#151711yescharacteristic
256C2×C7#213811yescharacteristic
257C2×C7#214511yescharacteristic
258D7—161no—
259C23—281no—
260C23D7×D811yescharacteristic
261Q8—41no—
262Q8—41no—
263D4—281no—
264D4—281no—
265D4—561no—
266D4—561no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—71no—
272C2×C4—71no—
273C2×C4—71no—
274C2×C4—71no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—281no—
281C2×C4—281no—
282C2×C4—562no—
283C2×C4—561no—
284C2×C4—561no—
285C2×C4—561no—
286C2×C4D7×C22⋊C411yescharacteristic
287C2×C4D7×SD1611yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#1185511yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—561no—
296C22—561no—
297C22—561no—
298C22D7×#10211yescharacteristic
299C22#8968811yescharacteristic
300C22#8969111yescharacteristic
301C4—21no—
302C4—81no—
303C4—81no—
304C4—141no—
305C4—141no—
306C4—141no—
307C4—141no—
308C4—161no—
309C4—561no—
310C4—1121no—
311C4#8951211yescharacteristic
312C4#8959311yescharacteristic
313C2—21no—
314C2—21no—
315C2—1121no—
316C2#1172603611yescharacteristic
317C2#1172697411yescharacteristic
318C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1348C2
socleC22×C7C2×#102
prime radical#89887C22
cosocleC22×C7×C8C23
hypercenter#1348D7
hypocenterC7#11855
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #11726846C2semidirect
3. #11726602C2semidirect
4. C7×#1348C2semidirect
5. #11725582C2semidirect
10. #89963C4semidirect
11. Q16×Dic7C4semidirect
16. #89650C4semidirect
17. #89644C4semidirect
65. #1348D7semidirect
292. C7#11855semidirect

Maximal subgroups

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

subgroupindex
#117381852
#117268462
#117266022
C7×#13482
#117255822
#117253322
#117252912
#118557