Finite Group Databasebeta

← #31965198

Subgroups — #31965198

Order 18176.

1202 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 355 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#22602111C211yesmaximal, characteristic
2#22597421C211yesmaximal, characteristic, complemented
3#22586899C216yesmaximal, complemented
4#22586834C216yesmaximal, complemented
5C71×#2351C211yesmaximal, characteristic, complemented
6C2×#14556176—21no—
7#17986671C414yescomplemented
8Q8×D284C414yescomplemented
9#17986661—24no—
10#17986580—23no—
11#17986549C2216yes—
12Q8×C71⋊C8—24no—
13#17986052C2216yes—
14#17985732C22112yes—
15#17985732C2214yes—
16C2×C4×Q8×C71—21no—
17C2×C4×Q8×C71C2211yescharacteristic
18#389×C71—23no—
19#382×C71C2213yes—
20C4×C4⋊C4×C71C2213yes—
21#14556176—44no—
22#14556176—44no—
23C2×#11765866—24no—
24C2×#11765865—26no—
25C2×Q8×D71—24no—
26#14556128C2×C416yes—
27#14556125C2×C416yes—
28#14556116—26no—
29#14556108—212no—
30#14556103—212no—
31#14556102—26no—
32#14556102C2314yes—
33C4×D284—26no—
34C4×D284C2×C4112yes—
35C4⋊(C71⋊C8)—26no—
36C22⋊(C71⋊C8)C2314yes—
37C4×C71⋊C8—26no—
38C22×Q8×C71D411yescharacteristic
39C4×Q8×C71—44no—
40C4×Q8×C71C2×C414yes—
41C4×Q8×C71D414yes—
42C2×C4⋊C4×C71—23no—
43C2×C4⋊C4×C71—23no—
44C2×C4⋊C4×C71D413yes—
45C2×C42×C71—23no—
46C2×C42×C71C2313yes—
47#95×C71—23no—
48#95×C71C2311yescharacteristic
49#95×C71C2313yes—
50#11765866—416no—
51#11765865—412no—
52#11765865—412no—
53C2×Dic142—26no—
54Q8×D71—416no—
55C2×D284—26no—
56C2×D284—26no—
57C2×D284C22×C414yes—
58C2×C22⋊D71—24no—
59C2×C4×D71—212no—
60C2×C71⋊C8—24no—
61C4⋊Dic71—26no—
62C4⋊Dic71C22×C414yes—
63(C2×D71)⋊C4—212no—
64(C2×D71)⋊C4—24no—
65C2×Q8×C71C22⋊C414yes—
66C2×Q8×C71D814yes—
67C2×Q8×C71SD1614yes—
68C22×C4×C71—21no—
69C22×C4×C71—23no—
70C22×C4×C71C2×D413yes—
71C22×C4×C71C2×D413yes—
72C22×C4×C71C2411yescharacteristic
73C4⋊C4×C71—46no—
74C4⋊C4×C71—46no—
75C4⋊C4×C71C22⋊C4112yes—
76C42×C71—46no—
77C42×C71C22×C416yes—
78C42×C71C2×D416yes—
79C23×C71C22×D411yescharacteristic
80C22×D71—24no—
81Q8×C71#102116yes—
82C2×C4×C71—22no—
83C2×C4×C71—43no—
84C2×C4×C71—41no—
85C2×C4×C71—41no—
86C2×C4×C71—41no—
87C2×C4×C71—43no—
88C2×C4×C71—43no—
89C2×C4×C71—43no—
90C2×C4×C71C2×C22⋊C416yes—
91C2×C4×C71C2×C22⋊C416yes—
92C2×C4×C71C2×D816yes—
93C2×C4×C71C2×SD1616yes—
94C2×C4×C71C23×C411yescharacteristic
95C2×C4×C71C22×D411yescharacteristic
96C22⋊D71—44no—
97C22⋊D71—44no—
98C2×Dic71—24no—
99D284—44no—
100D284—412no—
101D284—86no—
102C4×D71—424no—
103Dic142—412no—
104C71⋊C8—44no—
105C22×C71—21no—
106C22×C71—21no—
107C22×C71C22×C22⋊C411yescharacteristic
108C22×C71C22×D811yescharacteristic
109C22×C71C22×SD1611yescharacteristic
110C2×D71—44no—
111C2×D71—44no—
112C2×D71—44no—
113C4×C71—21no—
114C4×C71—21no—
115C4×C71—26no—
116C4×C71—42no—
117C4×C71—83no—
118C4×C71C2×#102112yes—
119Dic71—44no—
120#17618—711nomaximal, Sylow
121C2×C71—21no—
122C2×C71—21no—
123C2×C71#288411yescharacteristic
124C2×C71C22×#10211yescharacteristic
125C2×C71#290111yescharacteristic
126D71—84no—
127#3497—711no—
128#2982—711no—
129#2351D7111yescharacteristic, complemented
130#1517—716no—
131#1502—716no—
132C71#1761811yescharacteristic, complemented, Sylow
133C2×2−1+4—1421no—
134#550—714no—
135D4×Q8—714no—
136#548—1424no—
137#537—1423no—
138#532—716no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2×D7111yescharacteristic
141C8×Q8—1424no—
142#431—716no—
143#389—23no—
144#382C2×D7113yes—
145C4×C4⋊C4C2×D7113yes—
146#327—714no—
147#327—7112no—
1482−1+4—2844no—
1492−1+4—2844no—
150C2×#41—1424no—
151C2×#41—1426no—
152C22×Q8—1424no—
153C22×Q8C22⋊D7111yescharacteristic
154C4⋊Q8—716no—
155#127—716no—
156#124—1426no—
157C22⋊Q8—14212no—
158#121—714no—
159#121—1426no—
160#121—14212no—
161C4×Q8—44no—
162C4×Q8C4×D7114yes—
163C4×Q8D28414yes—
164C4×D4—7112no—
165C4×D4—1426no—
166C2×C4⋊C4—23no—
167C2×C4⋊C4—23no—
168C2×C4⋊C4C22⋊D7113yes—
169C2×C42—23no—
170C2×C42C22×D7113yes—
171C4⋊C8—1426no—
172C22⋊C8—714no—
173C4×C8—1426no—
174#95—23no—
175#95C22×D7111yescharacteristic
176#95C22×D7113yes—
177#41—28416no—
178#41—28412no—
179#41—28412no—
180C2×Q8—28416no—
181C2×Q8—1426no—
182C2×Q8(C2×D71)⋊C414yes—
183C2×Q8#1176585014yes—
184C2×Q8#1176585214yes—
185C2×D4—714no—
186C2×D4—1424no—
187C2×D4—1426no—
188C2×D4—1426no—
189C22×C4—21no—
190C22×C4—23no—
191C22×C4—14212no—
192C22×C4C2×C22⋊D7113yes—
193C22×C4C2×C22⋊D7113yes—
194C22×C4C23×D7111yescharacteristic
195C2×C8—1424no—
196C4⋊C4—46no—
197C4⋊C4—46no—
198C4⋊C4—714no—
199C4⋊C4—1426no—
200C4⋊C4(C2×D71)⋊C4112yes—
201C22⋊C4—1424no—
202C22⋊C4—14212no—
203C42—46no—
204C42C2×C4×D7116yes—
205C42C2×D28416yes—
206C23—1424no—
207C23C22×C22⋊D7111yescharacteristic
208Q8—28412no—
209Q8#14556048116yes—
210D4—2844no—
211D4—2844no—
212D4—2844no—
213D4—28412no—
214D4—5686no—
215C2×C4—22no—
216C2×C4—43no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—43no—
221C2×C4—43no—
222C2×C4—43no—
223C2×C4—28424no—
224C2×C4—1424no—
225C2×C4C2×(C2×D71)⋊C416yes—
226C2×C4C2×(C2×D71)⋊C416yes—
227C2×C4C2×#1176585016yes—
228C2×C4C2×#1176585216yes—
229C2×C4C22×C4×D7111yescharacteristic
230C2×C4C22×D28411yescharacteristic
231C8—2844no—
232C22—21no—
233C22—21no—
234C22—2844no—
235C22—2844no—
236C22—2844no—
237C22C22×(C2×D71)⋊C411yescharacteristic
238C22C22×#1176585011yescharacteristic
239C22C22×#1176585211yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—83no—
245C4—2844no—
246C4C2×#14556048112yes—
247C2—21no—
248C2—21no—
249C2—5684no—
250C2#2259680311yescharacteristic
251C2C22×#1455604811yescharacteristic
252C2#2259688811yescharacteristic
subgroupG/H
centerC22C22×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2351C2
socleC22×C71C22×C22⋊C4
prime radical#22602111C2
cosocleC22×C4×C71C24
hypercenter#2351D71
hypocenterC71#17618
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #22597421C2semidirect
3. #22586899C2semidirect
4. #22586834C2semidirect
5. C71×#2351C2semidirect
7. #17986671C4semidirect
8. Q8×D284C4semidirect
129. #2351D71semidirect
132. C71#17618semidirect

Maximal subgroups

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

subgroupindex
#226021112
#225974212
#225868992
#225868342
C71×#23512
#1761871