Finite Group Databasebeta

← #22599510

Subgroups — #22599510

Order 9088.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986533C211yesmaximal, characteristic
2C2×#14556120C211yesmaximal, characteristic, complemented
3#17986401C212yesmaximal, complemented
4#17986354C212yesmaximal, complemented
5#17986335C211yesmaximal, characteristic, complemented
6#17986330C211yesmaximal, characteristic, complemented
7#17986318C211yesmaximal, characteristic
8#17986313C211yesmaximal, characteristic
9#17986307C212yesmaximal, complemented
10#17986300C212yesmaximal, complemented
11#431×C71C211yesmaximal, characteristic, complemented
12#14556122—21no—
13#14556120—21no—
14#14556120—21no—
15#14556120C2212yes—
16#14556120C2212yes—
17#14556120C2211yescharacteristic
18#14556116C2211yescharacteristic
19#14556111—21no—
20#14556107C2211yescharacteristic
21#14556102C2211yescharacteristic
22C4×Dic142C2211yescharacteristic
23C4×D284C2211yescharacteristic
24C4×C22⋊D71—21no—
25#14556085—21no—
26C2×C4⋊Dic71C2211yescharacteristic
27C2×C4⋊Dic71C2211yescharacteristic
28C2×Dic71⋊C4—21no—
29C2×Dic71⋊C4—21no—
30#14556062C2212yes—
31#14556062C2212yes—
32#14556060C2212yes—
33#14556060C2212yes—
34#14556053C2212yes—
35#14556053C2212yes—
36#14556050C2212yes—
37#14556050C2212yes—
38C2×C42×C71C2211yescharacteristic
39C4⋊C8×C71C2212yes—
40C22⋊C8×C71C2211yescharacteristic
41C22⋊C8×C71C2211yescharacteristic
42C4×C8×C71C2212yes—
43C2×Dic142C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C22×Dic71—21no—
47C22×Dic71—21no—
48C2×C4×D71—21no—
49C4⋊Dic71—21no—
50C4⋊Dic71—21no—
51C4⋊Dic71C2312yes—
52C4⋊Dic71C2312yes—
53C4⋊Dic71C2311yescharacteristic
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4—21no—
60Dic71⋊C4—21no—
61Dic71⋊C4—21no—
62Dic71⋊C4—21no—
63C22⋊Dic71—21no—
64(C2×D71)⋊C4—21no—
65(C2×D71)⋊C4—21no—
66(C2×D71)⋊C4—21no—
67(C2×D71)⋊C4—21no—
68C4×Dic71—21no—
69C22×C4×C71D411yescharacteristic
70C22×C4×C71D411yescharacteristic
71C22×C4×C71C2311yescharacteristic
72C2×C8×C71C2312yes—
73C2×C8×C71C2312yes—
74C42×C71D411yescharacteristic
75C42×C71D411yescharacteristic
76C42×C71C2311yescharacteristic
77C42×C71C2311yescharacteristic
78C23×C71C2×D411yescharacteristic
79C22×D71—21no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—41no—
83C2×C4×C71C2×D411yescharacteristic
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2×D411yescharacteristic
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2×D411yescharacteristic
88C2×C4×C71C2411yescharacteristic
89C22⋊D71—42no—
90C2×Dic71—21no—
91C2×Dic71—21no—
92C2×Dic71—21no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—42no—
96C2×Dic71—42no—
97D284—22no—
98C4×D71—42no—
99Dic142—22no—
100C8×C71—22no—
101C8×C71—22no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71#13512yes—
105C22×C71C22×D411yescharacteristic
106C2×D71—41no—
107C2×D71—41no—
108C2×D71—41no—
109C4×C71—21no—
110C4×C71—21no—
111C4×C71—21no—
112C4×C71—21no—
113C4×C71—21no—
114C4×C712−1+412yes—
115Dic71—41no—
116Dic71—42no—
117Dic71—41no—
118Dic71—41no—
119Dic71—42no—
120C2×C71—22no—
121C2×C71#53611yescharacteristic
122C2×C71C2×#13511yescharacteristic
123C2×C71#57511yescharacteristic
124D71—81no—
125#3232—711nomaximal, Sylow
126C71#323211yescharacteristic, complemented, Sylow
127#529—711no—
128C2×#125—711no—
129#499—712no—
130#488—712no—
131#484—711no—
132#483—711no—
133#481—711no—
134#480—711no—
135#479—712no—
136#478—712no—
137#431D7111yescharacteristic, complemented
138#126—1421no—
139#125—712no—
140#125—712no—
141#125—711no—
142#125—1421no—
143#125—1421no—
144#124—711no—
145#123—1421no—
146C22⋊Q8—711no—
147#121—711no—
148C4×Q8—711no—
149C4×D4—711no—
150C4×D4—1421no—
151#117—1421no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—711no—
154C2×C4⋊C4—1421no—
155C2×C4⋊C4—1421no—
156C2×C42C2×D7111yescharacteristic
157#107—712no—
158#107—712no—
159#106—712no—
160#106—712no—
161C4⋊C8C2×D7112yes—
162#103—712no—
163#103—712no—
164#102—712no—
165#102—712no—
166C22⋊C8C2×D7111yescharacteristic
167C22⋊C8C2×D7111yescharacteristic
168C4×C8C2×D7112yes—
169C2×Q8—711no—
170C2×D4—711no—
171C2×D4—1421no—
172C22×C4—1421no—
173C22×C4—1421no—
174C22×C4—1421no—
175C22×C4D28411yescharacteristic
176C22×C4D28411yescharacteristic
177C22×C4C22×D7111yescharacteristic
178C2×C8C22×D7112yes—
179C2×C8C22×D7112yes—
180C4⋊C4—712no—
181C4⋊C4—712no—
182C4⋊C4—711no—
183C4⋊C4—711no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1422no—
188C4⋊C4—1421no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4—1421no—
193C4⋊C4—1421no—
194C22⋊C4—1421no—
195C22⋊C4—1421no—
196C22⋊C4—1421no—
197C22⋊C4—1421no—
198C22⋊C4—1421no—
199C42—1421no—
200C42D28411yescharacteristic
201C42D28411yescharacteristic
202C42C22×D7111yescharacteristic
203C42C22×D7111yescharacteristic
204C23—1421no—
205C23C2×D28411yescharacteristic
206Q8—1422no—
207D4—2842no—
208D4—1422no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—2842no—
213C2×C4—2842no—
214C2×C4—2842no—
215C2×C4—1421no—
216C2×C4—1421no—
217C2×C4—1421no—
218C2×C4—1422no—
219C2×C4—1422no—
220C2×C4C2×D28411yescharacteristic
221C2×C4C2×D28411yescharacteristic
222C2×C4C2×D28411yescharacteristic
223C2×C4C2×D28411yescharacteristic
224C2×C4C2×D28411yescharacteristic
225C2×C4C23×D7111yescharacteristic
226C8—22no—
227C8—22no—
228C22—21no—
229C22—21no—
230C22—2841no—
231C22—2841no—
232C22—2841no—
233C22#1455615012yes—
234C22C22×D28411yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—2841no—
241C4—2842no—
242C4—2841no—
243C4—2841no—
244C4—2842no—
245C4#1455617512yes—
246C2—22no—
247C2—5681no—
248C2#1798657511yescharacteristic
249C2C2×#1455615011yescharacteristic
250C2#1798679911yescharacteristic
subgroupG/H
centerC22C22×D284
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#431×C71C2
socleC22×C71C22×D4
hypercenter#431D71
hypocenterC71#3232
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. C2×#14556120C2semidirect
3. #17986401C2semidirect
4. #17986354C2semidirect
5. #17986335C2semidirect
6. #17986330C2semidirect
9. #17986307C2semidirect
10. #17986300C2semidirect
11. #431×C71C2semidirect
126. C71#3232semidirect
137. #431D71semidirect