Finite Group Databasebeta

← #2842×D71 #32203861

Subgroups — #2842×D71 #32203861

Order 18176.

3294 conjugacy classes of subgroups; only the 579 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 579 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#22596605C214yesmaximal, complemented
2#22596604C213yesmaximal, complemented
3#22596603C218yesmaximal, complemented
4#399×D71C214yesmaximal, complemented
5#396×D71C218yesmaximal, complemented
6#379×D71C213yesmaximal
7C71×#2842C211yesmaximal, characteristic, complemented
8C25×D71C2211yescharacteristic
9C2×C4⋊C4×D71C22112yescomplemented
10C2×C22⋊C4×D71C22112yes—
11C2×C22⋊C4×D71C2216yes—
12#17985936C2214yescomplemented
13#17985935C22112yescomplemented
14#17985934C22112yescomplemented
15#17985923C2218yescomplemented
16#17985922C22124yescomplemented
17#17985921C22124yescomplemented
18#17985855C2213yes—
19#17985854C22112yes—
20#17985854C2216yes—
21#95×D71C2214yescomplemented
22#399×C71C2214yescomplemented
23#396×C71C2218yescomplemented
24#379×C71C2213yes—
25C24×D71—412no—
26C24×D71Q818yes—
27C24×D71C2312yes—
28C24×D71C2311yescharacteristic
29C24×D71C2312yes—
30C24×D71C2311yescharacteristic
31C22×C4×D71C23112yes—
32C2×C4⋊Dic71C23112yescomplemented
33C2×Dic71⋊C4C23124yescomplemented
34C4⋊C4×D71—424no—
35C2×C22⋊Dic71C23112yes—
36C2×C22⋊Dic71C2316yes—
37C2×(C2×D71)⋊C4C23124yes—
38C2×(C2×D71)⋊C4C23112yes—
39C22⋊C4×D71—424no—
40C22⋊C4×D71—412no—
41C71⋊#95C23112yescomplemented
42C25×C71C2311yescharacteristic
43C2×C4⋊C4×C71C23112yescomplemented
44C2×C22⋊C4×C71C23112yes—
45C2×C22⋊C4×C71C2316yes—
46#95×C71C2314yescomplemented
47C23×D71—224no—
48C23×D71—212no—
49C23×D71—448no—
50C23×D71—412no—
51C23×D71—46no—
52C23×D71—464no—
53C23×D71C2×Q818yes—
54C23×D71C2×Q814yes—
55C23×D71C2×Q818yes—
56C23×D71C2×Q814yes—
57C23×D71C2412yes—
58C23×D71C2411yescharacteristic
59C23×D71C2411yescharacteristic
60C22×Dic71C24112yes—
61C2×C4×D71—224no—
62C2×C4×D71—248no—
63C4⋊Dic71—412no—
64Dic71⋊C4—424no—
65C22⋊Dic71—46no—
66C22⋊Dic71—412no—
67(C2×D71)⋊C4—412no—
68(C2×D71)⋊C4—424no—
69C24×C71—46no—
70C24×C71C2×Q814yes—
71C24×C71C2412yes—
72C24×C71C2411yescharacteristic
73C22×C4×C71C24112yes—
74C4⋊C4×C71—412no—
75C22⋊C4×C71—412no—
76C22⋊C4×C71—46no—
77C23×C71—212no—
78C23×C71—26no—
79C23×C71—412no—
80C23×C71—416no—
81C23×C71C22×Q814yes—
82C23×C71C22×Q812yes—
83C23×C71C2511yescharacteristic
84C22×D71—26no—
85C22×D71—212no—
86C22×D71—248no—
87C22×D71—224no—
88C22×D71—424no—
89C22×D71—412no—
90C22×D71—432no—
91C22×D71—416no—
92C22×D71—4128no—
93C22×D71—432no—
94C22×D71—416no—
95C22×D71C22×Q814yes—
96C22×D71C22×Q812yes—
97C22×D71C22×Q812yes—
98C22×D71H(5,𝔽2)112yes—
99C2×C4×C71—224no—
100C2×C4×C71—212no—
101C2×Dic71—212no—
102C2×Dic71—224no—
103C4×D71—448no—
104C22×C71—212no—
105C22×C71—26no—
106C22×C71—416no—
107C22×C71—48no—
108C22×C71—44no—
109C22×C71C23×Q811yescharacteristic
110C22×C71C2×H(5,𝔽2)16yes—
111C2×D71—26no—
112C2×D71—212no—
113C2×D71—432no—
114C2×D71—416no—
115C2×D71—44no—
116C2×D71—48no—
117C2×D71—432no—
118C2×D71—416no—
119C2×D71#543112yes—
120C2×D71#560116yes—
121C4×C71—412no—
122Dic71—412no—
123C2×#2842—711nomaximal, Sylow
124C2×C71—44no—
125C2×C71—42no—
126C2×C71C2×#54313yes—
127C2×C71C2×#56014yes—
128D71—42no—
129D71—44no—
130D71#284218yesdirect product factor
131#2842—714no—
132#2842—718no—
133#2842—713no—
134#2842D7111yescharacteristic, direct product factor
135C2×#399—714no—
136C2×#396—718no—
137C2×#379—713no—
138C71C2×#284211yescharacteristic, complemented, Sylow
139C26—711no—
140C22×C4⋊C4—7112no—
141C22×C22⋊C4—7112no—
142C22×C22⋊C4—716no—
143#399—7112no—
144#399—7112no—
145#399—714no—
146#399C2×D7114yescomplemented
147#396—7124no—
148#396—7124no—
149#396—718no—
150#396C2×D7118yescomplemented
151#379—713no—
152#379—7112no—
153#379—716no—
154#379C2×D7113yes—
155C2×#95—714no—
156C25—28412no—
157C25—718no—
158C25—711no—
159C25—712no—
160C25—712no—
161C25—711no—
162C25C22×D7111yescharacteristic
163C23×C4—7112no—
164C2×C4⋊C4—28424no—
165C2×C4⋊C4—7124no—
166C2×C4⋊C4—7112no—
167C2×C4⋊C4C22×D71112yescomplemented
168C2×C22⋊C4—28424no—
169C2×C22⋊C4—28412no—
170C2×C22⋊C4—7112no—
171C2×C22⋊C4—716no—
172C2×C22⋊C4—7124no—
173C2×C22⋊C4—7112no—
174C2×C22⋊C4C22×D71112yes—
175C2×C22⋊C4C22×D7116yes—
176#95—7112no—
177#95C22×D7114yescomplemented
178C24—46no—
179C24—28412no—
180C24—2846no—
181C24—28464no—
182C24—28448no—
183C24—714no—
184C24—718no—
185C24—711no—
186C24—718no—
187C24—714no—
188C24—712no—
189C24—711no—
190C24—14224no—
191C24—14212no—
192C24Q8×D7114yes—
193C24C23×D7112yes—
194C24C23×D7111yescharacteristic
195C22×C4—7112no—
196C22×C4—14248no—
197C22×C4—14224no—
198C22×C4C23×D71112yes—
199C4⋊C4—412no—
200C4⋊C4—28424no—
201C4⋊C4—28412no—
202C22⋊C4—412no—
203C22⋊C4—46no—
204C22⋊C4—28424no—
205C22⋊C4—28412no—
206C22⋊C4—28412no—
207C22⋊C4—2846no—
208C23—212no—
209C23—26no—
210C23—412no—
211C23—416no—
212C23—28424no—
213C23—28412no—
214C23—284128no—
215C23—28432no—
216C23—28416no—
217C23—28432no—
218C23—28416no—
219C23—712no—
220C23—7112no—
221C23—714no—
222C23—712no—
223C23—14248no—
224C23—14224no—
225C23—14212no—
226C23—1426no—
227C23C2×Q8×D7114yes—
228C23C2×Q8×D7112yes—
229C23C24×D7111yescharacteristic
230C2×C4—224no—
231C2×C4—212no—
232C2×C4—28448no—
233C2×C4—14212no—
234C2×C4—14224no—
235C22—212no—
236C22—26no—
237C22—416no—
238C22—48no—
239C22—44no—
240C22—28432no—
241C22—28416no—
242C22—28432no—
243C22—28416no—
244C22—2848no—
245C22—2844no—
246C22—7112no—
247C22—7116no—
248C22—14212no—
249C22—1426no—
250C22C22×Q8×D7111yescharacteristic
251C22H(5,𝔽2)×D7116yes—
252C4—412no—
253C4—28412no—
254C2—44no—
255C2—42no—
256C2—2844no—
257C2—2842no—
258C2—718no—
259C2#543×D7113yes—
260C2#560×D7114yes—
subgroupG/H
center, FrattiniC23C24×D71
commutator, socle, cosocleC23×C71C25
FittingC71×#2842C2
prime radicalC25×D71C22
hypercenter#2842D71
hypocenterC71C2×#2842
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2842

Complements

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

NH ≅ G/Ntype
1. #22596605C2semidirect
2. #22596604C2semidirect
3. #22596603C2semidirect
4. #399×D71C2semidirect
5. #396×D71C2semidirect
7. C71×#2842C2semidirect
9. C2×C4⋊C4×D71C22semidirect
12. #17985936C22semidirect
13. #17985935C22semidirect
14. #17985934C22semidirect
15. #17985923C22semidirect
16. #17985922C22semidirect
17. #17985921C22semidirect
21. #95×D71C22semidirect
22. #399×C71C22semidirect
23. #396×C71C22semidirect
32. C2×C4⋊Dic71C23semidirect
33. C2×Dic71⋊C4C23semidirect
41. C71⋊#95C23semidirect
43. C2×C4⋊C4×C71C23semidirect
46. #95×C71C23semidirect
130. D71#2842direct (N × H)
134. #2842D71direct (N × H)
138. C71C2×#2842semidirect
146. #399C2×D71semidirect
150. #396C2×D71semidirect
167. C2×C4⋊C4C22×D71semidirect
177. #95C22×D71semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225966052
#225966042
#225966032
#399×D712
#396×D712
#379×D712
C71×#28422
C2×#284271