Finite Group Databasebeta

← #13265113

Subgroups — #13265113

Order 1792.

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

normal subgroups only

This group has 302 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#11739254C211yesmaximal, characteristic
2#11727029C211yesmaximal, characteristic, complemented
3#11726281C211yesmaximal, characteristic, complemented
4C7×#1360C211yesmaximal, characteristic, complemented
5#11725602C211yesmaximal, characteristic, complemented
6#11725369C211yesmaximal, characteristic, complemented
7#11725274C211yesmaximal, characteristic, complemented
8#90371—21no—
9C7×#495C2211yescharacteristic
10#89960C411yescharacteristic
11Q16×Dic7C411yescharacteristic
12#89943—21no—
13SD16×Dic7—21no—
14#89929C411yescharacteristic
15D8×Dic7C411yescharacteristic
16#89828—21no—
17#89605—21no—
18C4×C8×D7C2211yescharacteristic
19#89455C2211yescharacteristic
20C7×C4⋊C16C2211yescharacteristic
21C7×#334C2211yescharacteristic
22(C2×D7)⋊C16—21no—
23#89297—21no—
24#89274—21no—
25C4⋊(C7⋊C16)C2211yescharacteristic
26#89252C2211yescharacteristic
27#11946—71nomaximal, Sylow
28C2×C7×Q16C2×C411yescharacteristic
29C2×C7×SD16—21no—
30C2×C7×D8C2×C411yescharacteristic
31C7×#124—21no—
32#6509—21no—
33Q8×Dic7—21no—
34#6501—21no—
35D4×Dic7—21no—
36C2×C8×D7C2×C411yescharacteristic
37C2×C8×D7D411yescharacteristic
38#6446—41no—
39(C22⋊D7)⋊C4—41no—
40#6442—41no—
41#6440—41no—
42C42×D7D411yescharacteristic
43#6433—21no—
44C2×C7×C16—21no—
45C7×C4⋊C8—21no—
46C4×C7×C8C2311yescharacteristic
47#6409—21no—
48#6409—21no—
49#6406—21no—
50#6406—21no—
51(C2×D7)⋊C8—41no—
52#6392C2×C411yescharacteristic
53#6392C2×C411yescharacteristic
54#6391—21no—
55C8×Dic7—41no—
56C8×Dic7C2×C411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59C4⋊(C7⋊C8)—21no—
60C4×C7⋊C8D411yescharacteristic
61#3168—71no—
62#2163—71no—
63#1564—71no—
64#1363—71no—
65#1363—71no—
66#1360—71no—
67#1360D711yescharacteristic, complemented
68C2×C7×Q8—21no—
69C2×C7×D4—21no—
70C2×C22⋊D7—41no—
71C22×Dic7—41no—
72C2×C4×D7C22⋊C411yescharacteristic
73C2×C4×D7D811yescharacteristic
74C2×C4×D7SD1611yescharacteristic
75C7×Q16—41no—
76C7×SD16—41no—
77C7×SD16—41no—
78C7×D8—41no—
79C2×C7×C8—41no—
80C2×C7×C8C22×C411yescharacteristic
81C2×C7×C8C2×D411yescharacteristic
82C7×C22⋊C4—41no—
83C7×C22⋊C4—41no—
84C42×C7C2×D411yescharacteristic
85C22⋊Dic7—41no—
86C22⋊Dic7—41no—
87(C2×D7)⋊C4—41no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—41no—
91Dic7⋊C4—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94C4×Dic7—41no—
95C4×Dic7C22⋊C411yescharacteristic
96C4×Dic7D811yescharacteristic
97C4×Dic7SD1611yescharacteristic
98C2×C7⋊C8—41no—
99C2×C7⋊C8C22⋊C411yescharacteristic
100C2×C7⋊C8C22⋊C411yescharacteristic
101C8×D7—21no—
102C8×D7—21no—
103C8×D7—41no—
104C7×C16—41no—
105C7⋊C16—41no—
106#529—141no—
107#499—71no—
108#495C2×D711yescharacteristic
109#488—141no—
110#467—71no—
111#466—71no—
112#465—141no—
113C4×Q16—71no—
114C4×SD16—141no—
115C4×D8—71no—
116#431—141no—
117C2×C4×C8—71no—
118C4⋊C16—71no—
119C4⋊C16C2×D711yescharacteristic
120C22⋊C16—141no—
121#346—141no—
122#335—141no—
123#334—71no—
124#334C2×D711yescharacteristic
125C23×C7—41no—
126C22×D7#10211yescharacteristic
127C7×Q8—41no—
128C7×Q8—41no—
129C7×D4—41no—
130C7×D4—41no—
131C2×C4×C7—41no—
132C2×C4×C7C2×C22⋊C411yescharacteristic
133C2×C4×C7C2×D811yescharacteristic
134C2×C4×C7C2×SD1611yescharacteristic
135C22⋊D7—82no—
136C2×Dic7—41no—
137C2×Dic7—41no—
138C2×Dic7—82no—
139C2×Dic7#10211yescharacteristic
140C2×Dic7#10211yescharacteristic
141C2×Dic7#10211yescharacteristic
142C4×D7—21no—
143C4×D7—21no—
144C4×D7—41no—
145C4×D7—41no—
146C7×C8—21no—
147C7×C8—81no—
148C7×C8#13511yescharacteristic
149C7×C8#13511yescharacteristic
150C7⋊C8—21no—
151C7⋊C8—21no—
152C7⋊C8—81no—
153C2×Q16C4×D711yescharacteristic
154C2×SD16—21no—
155C2×D8C4×D711yescharacteristic
156C22×C8—71no—
157C22×C8—71no—
158#126—281no—
159#125—141no—
160#124—21no—
161#123—281no—
162C22⋊Q8—141no—
163#121—141no—
164C4×Q8—141no—
165C4×D4—141no—
166C4×D4—281no—
167#117—281no—
168C2×C42—71no—
169C2×C16—21no—
170C2×C16—141no—
171#107—71no—
172#107—71no—
173#106—141no—
174C4⋊C8—21no—
175C4⋊C8—141no—
176#103—141no—
177#103—141no—
178#102—141no—
179#102—141no—
180C22⋊C8—281no—
181C4×C8—71no—
182C4×C8—71no—
183C4×C8—71no—
184C4×C8—281no—
185C4×C8C22×D711yescharacteristic
186C22×C7—81no—
187C22×C7—81no—
188C22×C7—81no—
189C22×C7C2×#10211yescharacteristic
190C2×D7—21no—
191C2×D7—21no—
192C2×D7—21no—
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—81no—
196C4×C7#41311yescharacteristic
197C4×C7#41611yescharacteristic
198Dic7—21no—
199Dic7—21no—
200Dic7—21no—
201Dic7—81no—
202Dic7—81no—
203C2×Q8—21no—
204C2×D4—21no—
205C2×D4—281no—
206C22×C4—71no—
207C22×C4—71no—
208C22×C4—71no—
209C22×C4—281no—
210Q16—41no—
211SD16—41no—
212SD16—41no—
213D8—41no—
214C2×C8—41no—
215C2×C8—71no—
216C2×C8—71no—
217C2×C8—141no—
218C2×C8—141no—
219C2×C8—281no—
220C2×C8—281no—
221C2×C8C2×C4×D711yescharacteristic
222C2×C8D4×D711yescharacteristic
223C4⋊C4—141no—
224C4⋊C4—141no—
225C4⋊C4—281no—
226C4⋊C4—281no—
227C4⋊C4—281no—
228C4⋊C4—281no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—281no—
232C22⋊C4—281no—
233C22⋊C4—281no—
234C42—71no—
235C42—71no—
236C42—71no—
237C42—281no—
238C42D4×D711yescharacteristic
239C16—41no—
240C16—281no—
241C2×C7—161no—
242C2×C7#154011yescharacteristic
243C2×C7#214311yescharacteristic
244C2×C7#214511yescharacteristic
245D7—41no—
246C23—41no—
247C23—71no—
248Q8—41no—
249Q8—41no—
250D4—41no—
251D4—41no—
252D4—562no—
253C2×C4—41no—
254C2×C4—71no—
255C2×C4—71no—
256C2×C4—71no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—281no—
260C2×C4—281no—
261C2×C4—281no—
262C2×C4—281no—
263C2×C4—562no—
264C2×C4D7×C22⋊C411yescharacteristic
265C2×C4D7×D811yescharacteristic
266C2×C4D7×SD1611yescharacteristic
267C8—21no—
268C8—81no—
269C8—141no—
270C8—141no—
271C8—561no—
272C8#647511yescharacteristic
273C8#647911yescharacteristic
274C7#1194611yescharacteristic, complemented, Sylow
275C22—81no—
276C22—81no—
277C22—81no—
278C22—141no—
279C22—141no—
280C22—141no—
281C22D7×#10211yescharacteristic
282C4—21no—
283C4—21no—
284C4—81no—
285C4—141no—
286C4—141no—
287C4—141no—
288C4—561no—
289C4—561no—
290C4#8951711yescharacteristic
291C4#8954411yescharacteristic
292C2—161no—
293C2—281no—
294C2#1172613511yescharacteristic
295C2#1172699511yescharacteristic
296C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC4×C8C22×D7
FittingC7×#1360C2
socleC22×C7C2×#102
prime radical#89929C4
cosocleC4×C7×C8C23
hypercenter#1360D7
hypocenterC7#11946
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1360

Complements

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

NH ≅ G/Ntype
2. #11727029C2semidirect
3. #11726281C2semidirect
4. C7×#1360C2semidirect
5. #11725602C2semidirect
6. #11725369C2semidirect
7. #11725274C2semidirect
67. #1360D7semidirect
274. C7#11946semidirect

Maximal subgroups

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

subgroupindex
#117392542
#117270292
#117262812
C7×#13602
#117256022
#117253692
#117252742
#119467