Finite Group Databasebeta

← #22599415

Subgroups — #22599415

Order 9088.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#17986612C211yesmaximal, characteristic, complemented
2#17986567C211yesmaximal, characteristic
3#17986394C211yesmaximal, characteristic, complemented
4#17986372C211yesmaximal, characteristic, complemented
5#17986365C211yesmaximal, characteristic, complemented
6#17986224C212yesmaximal, complemented
7#17986198C212yesmaximal, complemented
8#17986164C212yesmaximal, complemented
9#17986151C212yesmaximal, complemented
10#17986058C211yesmaximal, characteristic, complemented
11#488×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765855C2212yescomplemented
15C2×#11765854C2212yes—
16C2×#11765854C2212yescomplemented
17C2×#11765853C2212yes—
18#14556128C2211yescharacteristic, complemented
19#14556125C2211yescharacteristic
20#14556124C2211yescharacteristic
21#14556123—22no—
22#14556119—22no—
23#14556117C2211yescharacteristic
24#14556108C2212yes—
25#14556101—22no—
26#14556098—22no—
27#14556093C2212yescomplemented
28#14556085C2211yescharacteristic
29Dic71⋊C8C2212yescomplemented
30#14556048C2212yes—
31#14556048C2212yescomplemented
32(C2×D71)⋊C8C2212yes—
33#14556039C2211yescharacteristic, complemented
34#125×C71C2211yescharacteristic, complemented
35#124×C71C2211yescharacteristic
36#103×C71C2212yescomplemented
37#102×C71C2212yes—
38C4×C8×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40#11765866—42no—
41C2×Dic142C2312yes—
42C2×D284—21no—
43C2×D284—22no—
44C2×D284C2311yescharacteristic
45C2×D284C2311yescharacteristic
46C2×C22⋊D71—21no—
47C2×C22⋊D71—22no—
48D4×D71—42no—
49C2×C4×D71—21no—
50C2×C4×D71C2311yescharacteristic
51#11765855—42no—
52#11765854—42no—
53#11765854—42no—
54#11765853—42no—
55C2×C71⋊C8C2312yes—
56C4⋊Dic71—22no—
57C4⋊Dic71—22no—
58Dic71⋊C4—22no—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4D412yes—
63C4×Dic71—22no—
64C4×Dic71C2311yescharacteristic
65C2×Q8×C71C2311yescharacteristic
66C2×D4×C71C2311yescharacteristic
67C2×C8×C71C2312yes—
68C4⋊C4×C71—22no—
69C4⋊C4×C71C2312yes—
70C22⋊C4×C71—22no—
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C22×D71—21no—
74C22×D71—21no—
75C22×D71—42no—
76C22×D71—82no—
77C22×D71C2×D411yescharacteristic
78Q8×C71—22no—
79D4×C71—22no—
80C2×C4×C71—21no—
81C2×C4×C71—22no—
82C2×C4×C71C2×D412yes—
83C2×C4×C71C2411yescharacteristic
84C22⋊D71—42no—
85C22⋊D71—82no—
86C2×Dic71—22no—
87C2×Dic71C2×D412yes—
88C2×Dic71C2×D411yescharacteristic
89D284—22no—
90D284—22no—
91D284—42no—
92D284—82no—
93C4×D71—22no—
94C4×D71—42no—
95Dic142—22no—
96Dic142—22no—
97C8×C71—22no—
98C71⋊C8—22no—
99C22×C71—41no—
100C22×C71—42no—
101C22×C71C22×D411yescharacteristic
102C2×D71—21no—
103C2×D71—22no—
104C2×D71—41no—
105C2×D71—41no—
106C2×D71—42no—
107C2×D71—42no—
108C2×D71—84no—
109C4×C71—22no—
110C4×C71—41no—
111C4×C71—42no—
112C4×C71H(5,𝔽2)12yes—
113Dic71—21no—
114Dic71—22no—
115Dic71—42no—
116C2×C71—81no—
117C2×C71#53511yescharacteristic
118C2×C71#57712yes—
119D71—41no—
120D71—81no—
121D71—81no—
122#3222—711nomaximal, Sylow
123C71#322211yescharacteristic, complemented, Sylow
124#541—711no—
125#535—711no—
126#497—711no—
127#492—711no—
128#490—711no—
129#488D7111yescharacteristic, complemented
130#464—712no—
131#460—712no—
132#452—712no—
133#450—712no—
134#433—711no—
135C2×#41—711no—
136C22×D4—711no—
137C2×Q16—712no—
138C2×SD16—712no—
139C2×SD16—712no—
140C2×SD16—712no—
141C4⋊Q8—711no—
142#127—711no—
143#127—711no—
144#126—1422no—
145#125—1422no—
146#125C2×D7111yescharacteristic, complemented
147#124—711no—
148#124C2×D7111yescharacteristic
149C22⋊Q8—712no—
150#121—1422no—
151#120—1422no—
152C4×Q8—712no—
153#117—711no—
154C4⋊C8—712no—
155#103C2×D7112yescomplemented
156#102—712no—
157#102—712no—
158#102C2×D7112yes—
159C22⋊C8—712no—
160#97—711no—
161C4×C8C2×D7111yescharacteristic, complemented
162C24—1421no—
163#41—2842no—
164C2×Q8—712no—
165C2×Q8C22×D7111yescharacteristic
166C2×D4—2842no—
167C2×D4—711no—
168C2×D4—711no—
169C2×D4—1421no—
170C2×D4—1421no—
171C2×D4—1422no—
172C2×D4—1422no—
173C2×D4C22×D7111yescharacteristic
174C22×C4—711no—
175C22×C4—1421no—
176Q16—2842no—
177SD16—2842no—
178SD16—2842no—
179SD16—2842no—
180C2×C8—712no—
181C2×C8C22×D7112yes—
182C4⋊C4—22no—
183C4⋊C4—712no—
184C4⋊C4—1422no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4C22×D7112yes—
188C22⋊C4—22no—
189C22⋊C4—712no—
190C22⋊C4—1422no—
191C22⋊C4—1422no—
192C42—711no—
193C42—1422no—
194C42C22×D7111yescharacteristic
195C23—21no—
196C23—2842no—
197C23—5682no—
198C23—711no—
199C23—1421no—
200C23—1421no—
201Q8—22no—
202Q8—1422no—
203Q8—1422no—
204D4—22no—
205D4—2842no—
206D4—2842no—
207D4—5682no—
208D4—5682no—
209D4—1422no—
210D4—1422no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—2842no—
214C2×C4—712no—
215C2×C4—711no—
216C2×C4—1422no—
217C2×C4—1422no—
218C2×C4D4×D7112yes—
219C2×C4C23×D7111yescharacteristic
220C8—22no—
221C8—1422no—
222C22—41no—
223C22—42no—
224C22—2841no—
225C22—2841no—
226C22—2842no—
227C22—2842no—
228C22—5684no—
229C22—1421no—
230C22—1422no—
231C22C2×D4×D7111yescharacteristic
232C4—22no—
233C4—41no—
234C4—42no—
235C4—2842no—
236C4—1421no—
237C4—1422no—
238C4#1455617412yes—
239C2—81no—
240C2—2841no—
241C2—5681no—
242C2—5681no—
243C2#1798656711yescharacteristic
244C2#1798680612yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#488×C71C2
socleC22×C71C22×D4
hypercenter#488D71
hypocenterC71#3222
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #17986612C2semidirect
3. #17986394C2semidirect
4. #17986372C2semidirect
5. #17986365C2semidirect
6. #17986224C2semidirect
7. #17986198C2semidirect
8. #17986164C2semidirect
9. #17986151C2semidirect
10. #17986058C2semidirect
11. #488×C71C2semidirect
14. C2×#11765855C22semidirect
16. C2×#11765854C22semidirect
18. #14556128C22semidirect
27. #14556093C22semidirect
29. Dic71⋊C8C22semidirect
31. #14556048C22semidirect
33. #14556039C22semidirect
34. #125×C71C22semidirect
36. #103×C71C22semidirect
38. C4×C8×C71C22semidirect
123. C71#3222semidirect
129. #488D71semidirect
146. #125C2×D71semidirect
155. #103C2×D71semidirect
161. C4×C8C2×D71semidirect