Finite Group Databasebeta

← #22599407

Subgroups — #22599407

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#17986610C211yesmaximal, characteristic, complemented
2#17986567C211yesmaximal, characteristic
3#17986373C211yesmaximal, characteristic, complemented
4#17986364C211yesmaximal, characteristic, complemented
5#17986353C211yesmaximal, characteristic, complemented
6#17986223C212yesmaximal, complemented
7#17986197C212yesmaximal, complemented
8#17986160C212yesmaximal, complemented
9#17986148C212yesmaximal, complemented
10#17986061C211yesmaximal, characteristic, complemented
11#497×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765852C2212yes—
15C2×#11765852C2212yescomplemented
16#14556125C2211yescharacteristic
17#14556124C2211yescharacteristic
18#14556123—22no—
19#14556120C2211yescharacteristic, complemented
20#14556118—22no—
21#14556117C2211yescharacteristic
22#14556104C2212yes—
23#14556101—22no—
24#14556098—22no—
25Q8×Dic71C2212yescomplemented
26#14556085C2211yescharacteristic
27Dic71⋊C8C2212yescomplemented
28#14556052C2212yescomplemented
29#14556050C2212yes—
30#14556050C2212yescomplemented
31#14556049C2212yes—
32(C2×D71)⋊C8C2212yes—
33C4×C71⋊C8C2211yescharacteristic, complemented
34C2×Q16×C71C2212yescomplemented
35C2×SD16×C71C2212yes—
36C4⋊Q8×C71C2211yescharacteristic, complemented
37#124×C71C2211yescharacteristic
38#97×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40#11765866—42no—
41C2×D284—21no—
42C2×D284—22no—
43C2×D284C2311yescharacteristic
44C2×D284C2311yescharacteristic
45C2×C22⋊D71—21no—
46C2×C22⋊D71—22no—
47D4×D71—42no—
48C2×C4×D71—21no—
49C2×C4×D71C2311yescharacteristic
50#11765852—42no—
51#11765852—42no—
52C2×C71⋊C8C2312yes—
53C4⋊Dic71—22no—
54C4⋊Dic71C2312yes—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58(C2×D71)⋊C4—22no—
59(C2×D71)⋊C4—22no—
60(C2×D71)⋊C4D412yes—
61C4×Dic71—22no—
62C4×Dic71C2311yescharacteristic
63C2×Q8×C71C2311yescharacteristic
64C2×Q8×C71C2312yes—
65C2×D4×C71C2311yescharacteristic
66Q16×C71—42no—
67SD16×C71—42no—
68C2×C8×C71C2312yes—
69C4⋊C4×C71—22no—
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—
79Q8×C71—22no—
80Q8×C71—22no—
81D4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—22no—
84C2×C4×C71C2×D412yes—
85C2×C4×C71C2411yescharacteristic
86C22⋊D71—42no—
87C22⋊D71—82no—
88C2×Dic71—22no—
89C2×Dic71C2×D412yes—
90C2×Dic71C2×D411yescharacteristic
91D284—22no—
92D284—22no—
93D284—42no—
94D284—82no—
95C4×D71—22no—
96C4×D71—42no—
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#497D7111yescharacteristic, complemented
127#492—711no—
128#490—711no—
129#488—711no—
130#464—712no—
131#460—712no—
132#452—712no—
133#450—712no—
134#433—711no—
135C2×#41—711no—
136C22×D4—711no—
137C2×Q16C2×D7112yescomplemented
138C2×SD16—712no—
139C2×SD16—712no—
140C2×SD16C2×D7112yes—
141C4⋊Q8C2×D7111yescharacteristic, complemented
142#127—711no—
143#127—711no—
144#126—1422no—
145#125—711no—
146#125—1422no—
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#103—712no—
156#102—712no—
157#102—712no—
158#102—712no—
159C22⋊C8—712no—
160#97C2×D7111yescharacteristic, complemented
161C4×C8—711no—
162C24—1421no—
163#41—2842no—
164C2×Q8C22×D7111yescharacteristic
165C2×Q8C22×D7112yes—
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—42no—
177SD16—42no—
178SD16—2842no—
179SD16—2842no—
180C2×C8—712no—
181C2×C8C22×D7112yes—
182C4⋊C4—22no—
183C4⋊C4—712no—
184C4⋊C4—712no—
185C4⋊C4—1422no—
186C4⋊C4—1422no—
187C4⋊C4—1422no—
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—22no—
203Q8—22no—
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#1455617312yes—
239C2—81no—
240C2—2841no—
241C2—5681no—
242C2—5681no—
243C2#1798656211yescharacteristic
244C2#1798680812yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#497×C71C2
socleC22×C71C22×D4
hypercenter#497D71
hypocenterC71#3222
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #17986610C2semidirect
3. #17986373C2semidirect
4. #17986364C2semidirect
5. #17986353C2semidirect
6. #17986223C2semidirect
7. #17986197C2semidirect
8. #17986160C2semidirect
9. #17986148C2semidirect
10. #17986061C2semidirect
11. #497×C71C2semidirect
15. C2×#11765852C22semidirect
19. #14556120C22semidirect
25. Q8×Dic71C22semidirect
27. Dic71⋊C8C22semidirect
28. #14556052C22semidirect
30. #14556050C22semidirect
33. C4×C71⋊C8C22semidirect
34. C2×Q16×C71C22semidirect
36. C4⋊Q8×C71C22semidirect
38. #97×C71C22semidirect
123. C71#3222semidirect
126. #497D71semidirect
137. C2×Q16C2×D71semidirect
141. C4⋊Q8C2×D71semidirect
160. #97C2×D71semidirect