Finite Group Databasebeta

← #13265257

Subgroups — #13265257

Order 1792.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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#11739352C211yesmaximal, characteristic
2D7×C4⋊C16C211yesmaximal, characteristic, complemented
3#11726284C211yesmaximal, characteristic, complemented
4C7×#1363C211yesmaximal, characteristic, complemented
5#11725605C211yesmaximal, characteristic, complemented
6#11725368C211yesmaximal, characteristic, complemented
7#11725273C211yesmaximal, characteristic, complemented
8#90382—21no—
9C7×#499C2211yescharacteristic
10#89840—21no—
11C2×D7×C16—21no—
12#89659C411yescharacteristic
13#89655C411yescharacteristic
14#89645C411yescharacteristic
15#89644C411yescharacteristic
16#89636—21no—
17#89624—21no—
18#89607—21no—
19#89468C2211yescharacteristic
20C4×C8×D7C2211yescharacteristic
21C7×C4⋊C16C2211yescharacteristic
22C7×#334C2211yescharacteristic
23Dic7⋊C16—21no—
24#89274—21no—
25C4⋊(C7⋊C16)C2211yescharacteristic
26#89252C2211yescharacteristic
27#11978—71nomaximal, Sylow
28C7×#125—21no—
29C2×Dic28C2×C411yescharacteristic
30C2×D56C2×C411yescharacteristic
31C2×#1132—21no—
32C2×C8×D7C2×C411yescharacteristic
33C2×C8×D7D411yescharacteristic
34#6461—41no—
35#6460—21no—
36#6458—21no—
37#6457—41no—
38#6456—21no—
39#6456—41no—
40#6455—41no—
41#6450—21no—
42#6438—21no—
43C42×D7D411yescharacteristic
44C2×C7×C16—21no—
45C7×#107C2×C411yescharacteristic
46C7×#107C2×C411yescharacteristic
47C7×#106—21no—
48C7×C4⋊C8—21no—
49C4×C7×C8C2311yescharacteristic
50(C2×D7)⋊C8—41no—
51#6389—41no—
52C8×Dic7C2×C411yescharacteristic
53C8×Dic7D411yescharacteristic
54C2×C7⋊C16—21no—
55#6384—21no—
56#6384—21no—
57#6383—21no—
58#6383—21no—
59C4⋊(C7⋊C8)—21no—
60C4×C7⋊C8D411yescharacteristic
61D7×C16—42no—
62#3168—71no—
63C2×C4⋊C16—71no—
64#1568—71no—
65#1363—71no—
66#1363D711yescharacteristic, complemented
67#1360—71no—
68#1360—71no—
69C2×D28—21no—
70C2×D28—41no—
71C2×C4×D7—41no—
72C2×C4×D7C22⋊C411yescharacteristic
73C2×C4×D7D811yescharacteristic
74C2×C4×D7SD1611yescharacteristic
75C2×Dic14—21no—
76C2×C7×C8—41no—
77C2×C7×C8C22×C411yescharacteristic
78C2×C7×C8C2×D411yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C42×C7C2×D411yescharacteristic
84(C2×D7)⋊C4—41no—
85(C2×D7)⋊C4—41no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89C4⋊Dic7—41no—
90Dic7⋊C4—41no—
91C4×Dic7—41no—
92C4×Dic7C22⋊C411yescharacteristic
93C4×Dic7D811yescharacteristic
94C4×Dic7SD1611yescharacteristic
95C2×C7⋊C8—41no—
96C2×C7⋊C8C22⋊C411yescharacteristic
97C2×C7⋊C8C22⋊C411yescharacteristic
98Dic28—41no—
99D56—41no—
100#1132—41no—
101#1132—41no—
102C8×D7—21no—
103C8×D7—21no—
104C8×D7—41no—
105C7×C16—41no—
106C7⋊C16—41no—
107#529—141no—
108C22×C16—141no—
109#499C2×D711yescharacteristic
110#495—71no—
111#488—141no—
112#467—71no—
113#466—71no—
114#465—141no—
115C4×Q16—71no—
116C4×SD16—141no—
117C4×D8—71no—
118#432—141no—
119C2×C4×C8—71no—
120C4⋊C16—71no—
121C4⋊C16—141no—
122C4⋊C16C2×D711yescharacteristic
123#335—141no—
124#334—71no—
125#334C2×D711yescharacteristic
126C22×D7—41no—
127C22×D7#10211yescharacteristic
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7C2×C22⋊C411yescharacteristic
131C2×C4×C7C2×D811yescharacteristic
132C2×C4×C7C2×SD1611yescharacteristic
133C2×Dic7—41no—
134C2×Dic7#10211yescharacteristic
135C2×Dic7#10211yescharacteristic
136C2×Dic7#10211yescharacteristic
137D28—41no—
138D28—41no—
139D28—82no—
140C4×D7—22no—
141C4×D7—21no—
142C4×D7—21no—
143C4×D7—41no—
144C4×D7—82no—
145Dic14—41no—
146Dic14—41no—
147C7×C8—21no—
148C7×C8—81no—
149C7×C8AGL(1,ℤ/8)11yescharacteristic
150C7×C8#13711yescharacteristic
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8—81no—
154C2×Q16—71no—
155C2×SD16—141no—
156C2×D8—71no—
157C22×C8—71no—
158C22×C8—71no—
159#126—281no—
160#125—21no—
161#124—141no—
162#123—281no—
163C22⋊Q8—141no—
164#121—141no—
165C4×Q8—141no—
166C4×D4—141no—
167C4×D4—281no—
168#117—281no—
169C2×C42—71no—
170C2×C16—21no—
171C2×C16—141no—
172C2×C16—282no—
173#107C4×D711yescharacteristic
174#107C4×D711yescharacteristic
175#106—21no—
176C4⋊C8—21no—
177C4⋊C8—141no—
178#103—141no—
179#103—141no—
180#102—141no—
181#102—141no—
182C22⋊C8—281no—
183#97—281no—
184C4×C8—71no—
185C4×C8—71no—
186C4×C8—71no—
187C4×C8C22×D711yescharacteristic
188C22×C7C2×#10211yescharacteristic
189C2×D7—21no—
190C2×D7—21no—
191C2×D7—81no—
192C2×D7—81no—
193C2×D7—81no—
194C2×D7#35912yes—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—81no—
198C4×C7—81no—
199C4×C7#41311yescharacteristic
200C4×C7#41711yescharacteristic
201Dic7—21no—
202Dic7—21no—
203Dic7—21no—
204Dic7—81no—
205C2×Q8—141no—
206C2×D4—141no—
207C2×D4—281no—
208C22×C4—71no—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—281no—
212Q16—281no—
213SD16—281no—
214SD16—281no—
215D8—281no—
216C2×C8—41no—
217C2×C8—71no—
218C2×C8—71no—
219C2×C8—141no—
220C2×C8—141no—
221C2×C8—281no—
222C2×C8—281no—
223C2×C8C2×C4×D711yescharacteristic
224C2×C8D4×D711yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—281no—
230C4⋊C4—281no—
231C22⋊C4—281no—
232C22⋊C4—281no—
233C22⋊C4—281no—
234C22⋊C4—281no—
235C22⋊C4—281no—
236C42—71no—
237C42—71no—
238C42—71no—
239C42—281no—
240C42D4×D711yescharacteristic
241C16—41no—
242C16—281no—
243C2×C7#154711yescharacteristic
244C2×C7C2×#35911yescharacteristic
245C2×C7#214511yescharacteristic
246D7—22no—
247D7—161no—
248C23—71no—
249C23—281no—
250Q8—281no—
251Q8—281no—
252D4—281no—
253D4—281no—
254D4—562no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—71no—
258C2×C4—71no—
259C2×C4—71no—
260C2×C4—142no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—281no—
264C2×C4—281no—
265C2×C4—562no—
266C2×C4D7×C22⋊C411yescharacteristic
267C2×C4D7×D811yescharacteristic
268C2×C4D7×SD1611yescharacteristic
269C8—21no—
270C8—81no—
271C8—141no—
272C8—141no—
273C8—561no—
274C8#647711yescharacteristic
275C8#648111yescharacteristic
276C7#1197811yescharacteristic, complemented, Sylow
277C22—72no—
278C22—141no—
279C22—141no—
280C22—561no—
281C22—561no—
282C22—561no—
283C22D7×#10211yescharacteristic
284C4—21no—
285C4—21no—
286C4—81no—
287C4—81no—
288C4—141no—
289C4—141no—
290C4—141no—
291C4—561no—
292C4#8952311yescharacteristic
293C4#8957511yescharacteristic
294C2—142no—
295C2—1121no—
296C2#1172617511yescharacteristic
297C2D7×#35911yescharacteristic
298C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC4×C8C22×D7
FittingC7×#1363C2
socleC22×C7C2×#102
prime radical#89655C4
cosocleC4×C7×C8C23
hypercenter#1363D7
hypocenterC7#11978
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1363

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊C16C2semidirect
3. #11726284C2semidirect
4. C7×#1363C2semidirect
5. #11725605C2semidirect
6. #11725368C2semidirect
7. #11725273C2semidirect
66. #1363D7semidirect
276. C7#11978semidirect

Maximal subgroups

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

subgroupindex
#117393522
D7×C4⋊C162
#117262842
C7×#13632
#117256052
#117253682
#117252732
#119787