Finite Group Databasebeta

← #13264199

Subgroups — #13264199

Order 1792.

356 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 83 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#11726538C211yesmaximal, 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—
15C4⋊C4×Dic7—21no—
16C2×#6382—21no—
17#89650C411yescharacteristic, complemented
18#89644C411yescharacteristic, complemented
19#89572—21no—
20#89560—21no—
21C7×C22⋊C16C2211yescharacteristic
22C7×#340C2211yescharacteristic
23#89346C2211yescharacteristic
24C22⋊(C7⋊C16)C2211yescharacteristic
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11741—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×Dic7—41no—
38C2×C4×Dic7D411yescharacteristic
39C22×C7⋊C8D411yescharacteristic
40C2×D56C2×C411yescharacteristic
41#6461—41no—
42#6456—21no—
43#6455—41no—
44#6453—21no—
45#6449—41no—
46(C22⋊D7)⋊C4—41no—
47C2×C7×C16—21no—
48C7×#107C2×C411yescharacteristic
49C7×#103—21no—
50#6409—21no—
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—24no—
61#6382—21no—
62#3168—71no—
63#2162—71no—
64C4×#106—71no—
65#1348—71no—
66#1348D711yescharacteristic, complemented
67#1347—71no—
68#1347—71no—
69C2×C7×Q8—21no—
70C22×C4×C7—41no—
71C22×C4×C7C2×D411yescharacteristic
72C2×C22⋊D7—41no—
73C22×Dic7—41no—
74C22×Dic7D811yescharacteristic
75C22×Dic7D811yescharacteristic
76C2×D28—21no—
77C2×C4×D7—41no—
78C7×Q16—41no—
79C2×C7×C8—41no—
80C2×C7×C8C22×C411yescharacteristic
81C2×C7×C8C2×D411yescharacteristic
82C7×C4⋊C4—22no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C22⋊C4—41no—
87#1143—42no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4—41no—
90(C2×D7)⋊C4—41no—
91(C2×D7)⋊C4—41no—
92C4⋊Dic7—22no—
93C4⋊Dic7—21no—
94C4⋊Dic7—41no—
95C4⋊Dic7—41no—
96C4⋊Dic7—41no—
97Dic7⋊C4—41no—
98C4×Dic7—41no—
99C4×Dic7—82no—
100C4×Dic7C22⋊C411yescharacteristic
101C4×Dic7C22⋊C411yescharacteristic
102C4×Dic7SD1611yescharacteristic
103C4×Dic7SD1611yescharacteristic
104C2×C7⋊C8—22no—
105C2×C7⋊C8C22⋊C411yescharacteristic
106C2×C7⋊C8C22⋊C411yescharacteristic
107D56—41no—
108C7×C16—41no—
109C7⋊C16—41no—
110#529—141no—
111#499—71no—
112#495—71no—
113#488—141no—
114#467C2×D711yescharacteristic
115#466—71no—
116#465—141no—
117C4×Q16—71no—
118C4×SD16—141no—
119C4×D8—71no—
120C2×#106—141no—
121C2×C4×C8—71no—
122C4×C4⋊C4—141no—
123C4⋊C16—141no—
124C22⋊C16—71no—
125C22⋊C16C2×D711yescharacteristic
126C4×C16—141no—
127#340—71no—
128#340C2×D711yescharacteristic
129C23×C7C2×D811yescharacteristic
130C22×D7—41no—
131C7×Q8—41no—
132C7×Q8—41no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7C2×C22⋊C411yescharacteristic
141C2×C4×C7C2×SD1611yescharacteristic
142C22⋊D7—81no—
143C22⋊D7—81no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—22no—
147C2×Dic7—42no—
148C2×Dic7—41no—
149C2×Dic7—81no—
150C2×Dic7—81no—
151C2×Dic7#10211yescharacteristic
152C2×Dic7#10211yescharacteristic
153C2×Dic7#10211yescharacteristic
154C2×Dic7#10211yescharacteristic
155D28—41no—
156D28—41no—
157C4×D7—82no—
158C7×C8—21no—
159C7×C8—21no—
160C7⋊C8—21no—
161C7⋊C8#10412yes—
162C2×Q16C4×D711yescharacteristic
163C2×SD16—141no—
164C2×D8—71no—
165C22×C8—71no—
166C22×C8C22×D711yescharacteristic
167#126—281no—
168#125—141no—
169#124—141no—
170#123—281no—
171C22⋊Q8—21no—
172#121—141no—
173C4×Q8—141no—
174C4×D4—141no—
175C4×D4—281no—
176#117—281no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—141no—
179C2×C42—71no—
180C2×C42—281no—
181C2×C16—21no—
182C2×C16—141no—
183#107—71no—
184#107C4×D711yescharacteristic
185#106—141no—
186#106—144no—
187#103—21no—
188#103—141no—
189#102—141no—
190#102—141no—
191C4×C8—71no—
192C4×C8—71no—
193C4×C8—71no—
194C4×C8—71no—
195#95—281no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7C2×#10211yescharacteristic
199C22×C7#50811yescharacteristic
200C22×C7#50811yescharacteristic
201C2×D7—81no—
202C2×D7—81no—
203C2×D7—81no—
204C4×C7—21no—
205C4×C7—82no—
206C4×C7—81no—
207C4×C7—81no—
208C4×C7#41011yescharacteristic
209C4×C7C2×#10411yescharacteristic
210Dic7—21no—
211Dic7—21no—
212Dic7—21no—
213Dic7—82no—
214Dic7—81no—
215Dic7#36012yes—
216C2×Q8—21no—
217C2×D4—141no—
218C2×D4—281no—
219C22×C4—41no—
220C22×C4—71no—
221C22×C4—71no—
222C22×C4—281no—
223C22×C4—281no—
224C22×C4D4×D711yescharacteristic
225Q16—41no—
226SD16—282no—
227D8—281no—
228C2×C8—41no—
229C2×C8—71no—
230C2×C8—71no—
231C2×C8—142no—
232C2×C8C2×C4×D711yescharacteristic
233C2×C8D4×D711yescharacteristic
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—142no—
239C4⋊C4—141no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C4⋊C4—281no—
243C4⋊C4—281no—
244C22⋊C4—41no—
245C22⋊C4—281no—
246C22⋊C4—281no—
247C22⋊C4—281no—
248C22⋊C4—281no—
249C42—71no—
250C42—71no—
251C42—71no—
252C42—71no—
253C42—281no—
254C42—562no—
255C16—41no—
256C16—281no—
257C2×C7—21no—
258C2×C7—21no—
259C2×C7#150211yescharacteristic
260C2×C7#213811yescharacteristic
261C2×C7C2×#36011yescharacteristic
262D7—161no—
263C23—281no—
264C23D7×D811yescharacteristic
265Q8—41no—
266Q8—41no—
267D4—281no—
268D4—281no—
269D4—561no—
270D4—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—142no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—282no—
286C2×C4—281no—
287C2×C4—562no—
288C2×C4—561no—
289C2×C4—561no—
290C2×C4D7×C22⋊C411yescharacteristic
291C2×C4D7×SD1611yescharacteristic
292C8—21no—
293C8—21no—
294C8—72no—
295C8—141no—
296C7#1174111yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—561no—
300C22—561no—
301C22—561no—
302C22D7×#10211yescharacteristic
303C22#8968811yescharacteristic
304C22#8969111yescharacteristic
305C4—21no—
306C4—72no—
307C4—82no—
308C4—81no—
309C4—81no—
310C4—141no—
311C4—141no—
312C4—141no—
313C4—562no—
314C4—561no—
315C4#8951211yescharacteristic
316C4D7×#10411yescharacteristic
317C2—21no—
318C2—21no—
319C2—1121no—
320C2#1172603111yescharacteristic
321C2#1172697411yescharacteristic
322C2D7×#36011yescharacteristic
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#11741
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. #11726538C2semidirect
4. C7×#1348C2semidirect
5. #11725582C2semidirect
10. #89963C4semidirect
11. Q16×Dic7C4semidirect
17. #89650C4semidirect
18. #89644C4semidirect
66. #1348D7semidirect
296. C7#11741semidirect

Maximal subgroups

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

subgroupindex
#117381852
#117268462
#117265382
C7×#13482
#117255822
#117253322
#117252912
#117417