Finite Group Databasebeta

← #22599505

Subgroups — #22599505

Order 9088.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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#17986530C211yesmaximal, characteristic
2C2×#14556119C211yesmaximal, characteristic, complemented
3#17986400C212yesmaximal, complemented
4#17986352C212yesmaximal, complemented
5#17986335C211yesmaximal, characteristic, complemented
6#17986329C211yesmaximal, characteristic, complemented
7#17986315C211yesmaximal, characteristic
8#17986305C212yesmaximal, complemented
9#17986297C212yesmaximal, complemented
10#17986049C211yesmaximal, characteristic, complemented
11#480×C71C211yesmaximal, characteristic
12#14556121—21no—
13#14556119—21no—
14#14556119—21no—
15#14556119C2212yes—
16#14556119C2212yes—
17#14556119C2211yescharacteristic
18#14556115C2211yescharacteristic
19#14556109—21no—
20#14556107C2211yescharacteristic
21#14556093C2211yescharacteristic
22D4×Dic71—21no—
23D4×Dic71C2211yescharacteristic
24#14556084—21no—
25C2×C4⋊Dic71—21no—
26C2×C4⋊Dic71C2211yescharacteristic
27C2×Dic71⋊C4—21no—
28C2×C4×Dic71C2211yescharacteristic
29#14556062C2212yes—
30#14556059C2212yes—
31#14556059C2212yes—
32Dic71⋊C8C2212yes—
33#14556053C2212yes—
34#14556051C2212yes—
35#14556049C2212yes—
36C22⋊(C71⋊C8)C2211yescharacteristic
37C8×Dic71C2212yes—
38#121×C71C2211yescharacteristic
39C2×C4⋊C4×C71C2211yescharacteristic
40#107×C71C2212yes—
41#102×C71C2212yes—
42C22⋊C8×C71C2211yescharacteristic
43C2×Dic142C2311yescharacteristic
44C22×Dic71—21no—
45C22×Dic71—21no—
46C22×Dic71D411yescharacteristic
47C22×Dic71D411yescharacteristic
48C2×C71⋊C8C2312yes—
49C4⋊Dic71—21no—
50C4⋊Dic71—22no—
51C4⋊Dic71—22no—
52C4⋊Dic71—21no—
53C4⋊Dic71C2312yes—
54C4⋊Dic71C2311yescharacteristic
55Dic71⋊C4—21no—
56Dic71⋊C4—22no—
57Dic71⋊C4—22no—
58Dic71⋊C4—21no—
59Dic71⋊C4—21no—
60C22⋊Dic71—21no—
61C22⋊Dic71—21no—
62C22⋊Dic71—21no—
63C22⋊Dic71—21no—
64C4×Dic71—21no—
65C4×Dic71D411yescharacteristic
66C4×Dic71D411yescharacteristic
67C4×Dic71C2311yescharacteristic
68C4×Dic71C2311yescharacteristic
69C2×D4×C71—21no—
70C2×D4×C71C2311yescharacteristic
71C22×C4×C71—21no—
72C22×C4×C71C2311yescharacteristic
73C2×C8×C71C2312yes—
74C4⋊C4×C71—21no—
75C4⋊C4×C71C2312yes—
76C4⋊C4×C71C2311yescharacteristic
77C22⋊C4×C71—21no—
78C23×C71—21no—
79C23×C71C2×D411yescharacteristic
80D4×C71—22no—
81D4×C71—42no—
82C2×C4×C71—22no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—42no—
86C2×C4×C71C2×D411yescharacteristic
87C2×C4×C71C2411yescharacteristic
88C2×Dic71—22no—
89C2×Dic71—21no—
90C2×Dic71—21no—
91C2×Dic71—22no—
92C2×Dic71—41no—
93C2×Dic71—42no—
94C2×Dic71—42no—
95C2×Dic71C2×D411yescharacteristic
96C2×Dic71C2×D411yescharacteristic
97C2×Dic71C2×D411yescharacteristic
98C2×Dic71C2×D411yescharacteristic
99Dic142—22no—
100C8×C71—22no—
101C71⋊C8—22no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—41no—
105C22×C71—41no—
106C22×C71—41no—
107C22×C71#13512yes—
108C22×C71C22×D411yescharacteristic
109C4×C71—21no—
110C4×C71—42no—
111C4×C71—41no—
112C4×C712−1+412yes—
113Dic71—21no—
114Dic71—21no—
115Dic71—21no—
116Dic71—21no—
117Dic71—41no—
118Dic71—41no—
119Dic71—42no—
120C2×C71—22no—
121C2×C71—81no—
122C2×C71#53611yescharacteristic
123C2×C71C2×#13511yescharacteristic
124C2×C71#57511yescharacteristic
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#480D7111yescharacteristic
135#479—712no—
136#478—712no—
137#431—711no—
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#121C2×D7111yescharacteristic
148C4×Q8—711no—
149C4×D4—711no—
150C4×D4—1421no—
151#117—1421no—
152C2×C4⋊C4—711no—
153C2×C4⋊C4—1421no—
154C2×C4⋊C4—1421no—
155C2×C4⋊C4C2×D7111yescharacteristic
156C2×C42—711no—
157#107—712no—
158#107C2×D7112yes—
159#106—712no—
160#106—712no—
161C4⋊C8—712no—
162#103—712no—
163#103—712no—
164#102—712no—
165#102C2×D7112yes—
166C22⋊C8—711no—
167C22⋊C8C2×D7111yescharacteristic
168C4×C8—712no—
169C2×Q8—711no—
170C2×D4—21no—
171C2×D4C22×D7111yescharacteristic
172C22×C4—21no—
173C22×C4—711no—
174C22×C4—711no—
175C22×C4—1421no—
176C22×C4—1421no—
177C22×C4C22×D7111yescharacteristic
178C2×C8—712no—
179C2×C8C22×D7112yes—
180C4⋊C4—21no—
181C4⋊C4—712no—
182C4⋊C4—711no—
183C4⋊C4—1422no—
184C4⋊C4—1422no—
185C4⋊C4—1421no—
186C4⋊C4—1421no—
187C4⋊C4—1422no—
188C4⋊C4—1422no—
189C4⋊C4—1421no—
190C4⋊C4—1421no—
191C4⋊C4—1421no—
192C4⋊C4C22×D7112yes—
193C4⋊C4C22×D7111yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—1421no—
196C22⋊C4—1421no—
197C22⋊C4—1421no—
198C22⋊C4—1421no—
199C42—711no—
200C42—711no—
201C42—711no—
202C42—711no—
203C42—1421no—
204C23—21no—
205C23D4×D7111yescharacteristic
206Q8—1422no—
207D4—22no—
208D4—42no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—2842no—
214C2×C4—2842no—
215C2×C4—2841no—
216C2×C4—711no—
217C2×C4—711no—
218C2×C4—711no—
219C2×C4—711no—
220C2×C4—1421no—
221C2×C4—1421no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4D4×D7111yescharacteristic
225C2×C4C23×D7111yescharacteristic
226C8—22no—
227C8—1422no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22#1455614712yes—
234C22C2×D4×D7111yescharacteristic
235C4—21no—
236C4—42no—
237C4—41no—
238C4—2841no—
239C4—2842no—
240C4—2841no—
241C4—1421no—
242C4—1421no—
243C4—1421no—
244C4—1421no—
245C4#1455617512yes—
246C2—22no—
247C2—81no—
248C2#1798657111yescharacteristic
249C2C2×#1455614711yescharacteristic
250C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#480×C71C2
socleC22×C71C22×D4
hypercenter#480D71
hypocenterC71#3232
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#14556119C2semidirect
3. #17986400C2semidirect
4. #17986352C2semidirect
5. #17986335C2semidirect
6. #17986329C2semidirect
8. #17986305C2semidirect
9. #17986297C2semidirect
10. #17986049C2semidirect
126. C71#3232semidirect