Finite Group Databasebeta

← #22598985

Subgroups — #22598985

Order 9088.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#17986602C212yesmaximal, complemented
2#17986399C211yesmaximal, characteristic, complemented
3#17986374C211yesmaximal, characteristic, complemented
4#17986362C212yesmaximal, complemented
5#17986252C211yesmaximal, characteristic, complemented
6#17986233C212yesmaximal, complemented
7#17986111C211yesmaximal, characteristic, complemented
8#17986110C211yesmaximal, characteristic, complemented
9#17986079C212yesmaximal, complemented
10#17985957C211yesmaximal, characteristic, complemented
11#469×C71C211yesmaximal, characteristic, complemented
12C2×D568C2211yescharacteristic
13C2×#11765850C2212yes—
14C22×C71⋊C8C2211yescharacteristic
15#14556124C2212yes—
16#14556119C2212yes—
17#14556112—22no—
18#14556110—22no—
19#14556102C2212yes—
20#14556100—22no—
21#14556099—22no—
22D4×Dic71C2212yes—
23#14556090C2212yes—
24#14556087C2211yescharacteristic
25#14556061C2212yes—
26#14556060C2211yescharacteristic
27#14556050C2212yes—
28#14556049C2212yes—
29#14556048C2212yes—
30#14556038C2211yescharacteristic
31#14556038C2211yescharacteristic
32C8×Dic71C2211yescharacteristic
33C8×Dic71C2211yescharacteristic
34C2×D8×C71C2211yescharacteristic
35C2×M16×C71C2211yescharacteristic
36#121×C71C2212yes—
37#106×C71C2211yescharacteristic
38#102×C71C2212yes—
39C2×D284C2312yes—
40C2×C22⋊D71—22no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43C22×Dic71—22no—
44C2×C4×D71—22no—
45D568—22no—
46#11765850—24no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8C2311yescharacteristic
49C2×C71⋊C8C2311yescharacteristic
50C4⋊Dic71—22no—
51C4⋊Dic71D412yes—
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54C22⋊Dic71—22no—
55C22⋊Dic71D412yes—
56(C2×D71)⋊C4—22no—
57(C2×D71)⋊C4—22no—
58C4×Dic71C2311yescharacteristic
59C4×Dic71C2311yescharacteristic
60C2×D4×C71—22no—
61C2×D4×C71C2312yes—
62C22×C4×C71C2311yescharacteristic
63D8×C71—22no—
64M16×C71—22no—
65C2×C8×C71C2311yescharacteristic
66C2×C8×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C22⋊C4×C71—22no—
69C23×C71—22no—
70C23×C71C2×D411yescharacteristic
71C22×D71—22no—
72D4×C71—22no—
73D4×C71—22no—
74D4×C71—82no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—22no—
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2411yescharacteristic
80C22⋊D71—82no—
81C22⋊D71—82no—
82C22⋊D71—82no—
83C2×Dic71—22no—
84C2×Dic71—82no—
85C2×Dic71C2×D412yes—
86C2×Dic71C2×D412yes—
87D284—22no—
88D284—22no—
89C4×D71—82no—
90C8×C71—21no—
91C8×C71#4112yes—
92C71⋊C8—21no—
93C71⋊C8#4112yes—
94C22×C71—21no—
95C22×C71—22no—
96C22×C71—42no—
97C22×C71—42no—
98C22×C71—42no—
99C22×C71C22×D411yescharacteristic
100C2×D71—42no—
101C2×D71—42no—
102C2×D71—42no—
103C4×C71—21no—
104C4×C71—42no—
105C4×C71C2×#4111yescharacteristic
106C4×C71C2×#4111yescharacteristic
107Dic71—22no—
108Dic71—22no—
109Dic71—42no—
110C2×C71—41no—
111C2×C71—82no—
112C2×C71#53211yescharacteristic
113C2×C71#57612yes—
114D71—82no—
115#3176—711nomaximal, Sylow
116C71#317611yescharacteristic, complemented, Sylow
117#540—712no—
118#499—711no—
119#493—711no—
120#490—712no—
121#469—711no—
122#469D7111yescharacteristic, complemented
123#466—712no—
124#442—711no—
125#442—711no—
126C4×D8—712no—
127#405—711no—
128C2×D8—712no—
129C2×D8—711no—
130C2×D8C2×D7111yescharacteristic
131C2×M16C2×D7111yescharacteristic
132C22×C8—711no—
133#127—712no—
134#125—712no—
135#123—1422no—
136#123—1422no—
137#121—712no—
138#121—1422no—
139#121—1422no—
140#121C2×D7112yes—
141C4×D4—712no—
142C4×D4—712no—
143#117—711no—
144#107—712no—
145#106—711no—
146#106C2×D7111yescharacteristic
147#102—712no—
148#102—712no—
149#102—712no—
150#102C2×D7112yes—
151#97—711no—
152#97—711no—
153C4×C8—711no—
154C4×C8—711no—
155C2×D4—22no—
156C2×D4—712no—
157C2×D4—1422no—
158C2×D4—1422no—
159C2×D4—1422no—
160C2×D4C22×D7112yes—
161C22×C4—1422no—
162C22×C4—1422no—
163C22×C4C22×D7111yescharacteristic
164D8—22no—
165D8—1424no—
166D8—1422no—
167M16—22no—
168C2×C8—711no—
169C2×C8—711no—
170C2×C8—1422no—
171C2×C8C22×D7111yescharacteristic
172C2×C8C22×D7111yescharacteristic
173C4⋊C4—712no—
174C4⋊C4—712no—
175C4⋊C4—1422no—
176C4⋊C4—1422no—
177C4⋊C4C22×D7112yes—
178C22⋊C4—22no—
179C22⋊C4—712no—
180C22⋊C4—1422no—
181C22⋊C4—1422no—
182C22⋊C4—1422no—
183C42—711no—
184C42—711no—
185C23—22no—
186C23—1422no—
187C23D4×D7111yescharacteristic
188D4—22no—
189D4—22no—
190D4—82no—
191D4—5682no—
192D4—5682no—
193D4—5682no—
194D4—1422no—
195D4—1422no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—5682no—
200C2×C4—5682no—
201C2×C4—712no—
202C2×C4—712no—
203C2×C4—1422no—
204C2×C4D4×D7111yescharacteristic
205C2×C4C23×D7111yescharacteristic
206C8—21no—
207C8—712no—
208C8—1421no—
209C8#1176586412yes—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—2842no—
216C22—2842no—
217C22—2842no—
218C22C2×D4×D7111yescharacteristic
219C4—21no—
220C4—42no—
221C4—2842no—
222C4—1422no—
223C4—1422no—
224C4#41×D7111yescharacteristic
225C4C2×#1176586411yescharacteristic
226C2—41no—
227C2—82no—
228C2—5682no—
229C2#1798654411yescharacteristic
230C2#1798680312yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#469×C71C2
socleC22×C71C22×D4
hypercenter#469D71
hypocenterC71#3176
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #17986602C2semidirect
2. #17986399C2semidirect
3. #17986374C2semidirect
4. #17986362C2semidirect
5. #17986252C2semidirect
6. #17986233C2semidirect
7. #17986111C2semidirect
8. #17986110C2semidirect
9. #17986079C2semidirect
10. #17985957C2semidirect
11. #469×C71C2semidirect
116. C71#3176semidirect
122. #469D71semidirect