Finite Group Databasebeta

← #31965201

Subgroups — #31965201

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#22591261C211yesmaximal, characteristic, complemented
3#22586901C216yesmaximal, complemented
4#22586836C216yesmaximal, complemented
5C71×#2982C211yesmaximal, characteristic, complemented
6C2×#14556176—21no—
7#17986671C414yescomplemented
8Q8×D284C414yescomplemented
9#17986661—24no—
10#17986580—23no—
11#17986549C2216yes—
12C2×Q8×Dic71—21no—
13#17985898—23no—
14#17985865C2213yes—
15C4×C4⋊Dic71C2213yes—
16#17985734C22112yes—
17#17985734C2214yes—
18C2×C4×Q8×C71C2211yescharacteristic
19C8×Q8×C71—24no—
20#431×C71C2216yes—
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—
33Q8×Dic71—44no—
34C4×D284—26no—
35C4×D284C2×C4112yes—
36C2×C4⋊Dic71—23no—
37C2×C4⋊Dic71—23no—
38C2×C4×Dic71—23no—
39C71⋊#95—23no—
40C71⋊#95C2311yescharacteristic
41C71⋊#95C2313yes—
42C22×Q8×C71D411yescharacteristic
43C4×Q8×C71C2×C414yes—
44C4×Q8×C71D414yes—
45C2×C4⋊C4×C71D413yes—
46C2×C42×C71C2313yes—
47C4⋊C8×C71—26no—
48C22⋊C8×C71C2314yes—
49C4×C8×C71—26no—
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—
59C22×Dic71—21no—
60C22×Dic71—23no—
61C2×C4×D71—212no—
62C4⋊Dic71—26no—
63C4⋊Dic71—46no—
64C4⋊Dic71—46no—
65C4⋊Dic71C22×C414yes—
66(C2×D71)⋊C4—212no—
67(C2×D71)⋊C4—24no—
68C4×Dic71—46no—
69C2×Q8×C71C22⋊C414yes—
70C2×Q8×C71D814yes—
71C2×Q8×C71SD1614yes—
72C22×C4×C71C2×D413yes—
73C22×C4×C71C2×D413yes—
74C22×C4×C71C2411yescharacteristic
75C2×C8×C71—24no—
76C4⋊C4×C71C22⋊C4112yes—
77C42×C71C22×C416yes—
78C42×C71C2×D416yes—
79C23×C71C22×D411yescharacteristic
80C22×D71—24no—
81Q8×C71#102116yes—
82C2×C4×C71—43no—
83C2×C4×C71—41no—
84C2×C4×C71C2×C22⋊C416yes—
85C2×C4×C71C2×C22⋊C416yes—
86C2×C4×C71C2×D816yes—
87C2×C4×C71C2×SD1616yes—
88C2×C4×C71C23×C411yescharacteristic
89C2×C4×C71C22×D411yescharacteristic
90C22⋊D71—44no—
91C22⋊D71—44no—
92C2×Dic71—22no—
93C2×Dic71—24no—
94C2×Dic71—41no—
95C2×Dic71—41no—
96C2×Dic71—43no—
97C2×Dic71—43no—
98C2×Dic71—43no—
99D284—44no—
100D284—412no—
101D284—86no—
102C4×D71—424no—
103Dic142—412no—
104C8×C71—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×C71C2×#102112yes—
117Dic71—42no—
118Dic71—44no—
119Dic71—83no—
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#2982D7111yescharacteristic, complemented
129#2351—711no—
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—1421no—
140C2×C4×Q8C2×D7111yescharacteristic
141C8×Q8—24no—
142#431C2×D7116yes—
143#389—1423no—
144#382—713no—
145C4×C4⋊C4—713no—
146#327—714no—
147#327—7112no—
1482−1+4—2844no—
1492−1+4—2844no—
150C2×#41—1424no—
151C2×#41—1426no—
152C22×Q8—1424no—
153C22×Q8D28411yescharacteristic
154C4⋊Q8—716no—
155#127—716no—
156#124—1426no—
157C22⋊Q8—14212no—
158#121—714no—
159#121—1426no—
160#121—14212no—
161C4×Q8—2844no—
162C4×Q8C4×D7114yes—
163C4×Q8C22⋊D7114yes—
164C4×D4—7112no—
165C4×D4—1426no—
166C2×C4⋊C4—1423no—
167C2×C4⋊C4—1423no—
168C2×C4⋊C4D28413yes—
169C2×C42—1423no—
170C2×C42C22×D7113yes—
171C4⋊C8—26no—
172C22⋊C8C22×D7114yes—
173C4×C8—26no—
174#95—711no—
175#95—713no—
176#95—1423no—
177#41—28416no—
178#41—28412no—
179#41—28412no—
180C2×Q8—28416no—
181C2×Q8—1426no—
182C2×Q8(C2×D71)⋊C414yes—
183C2×Q8D56814yes—
184C2×Q8#1176585414yes—
185C2×D4—714no—
186C2×D4—1424no—
187C2×D4—1426no—
188C2×D4—1426no—
189C22×C4—1423no—
190C22×C4—1421no—
191C22×C4—14212no—
192C22×C4C2×D28413yes—
193C22×C4C2×D28413yes—
194C22×C4C23×D7111yescharacteristic
195C2×C8—24no—
196C4⋊C4—2846no—
197C4⋊C4—2846no—
198C4⋊C4—714no—
199C4⋊C4—1426no—
200C4⋊C4(C2×D71)⋊C4112yes—
201C22⋊C4—1424no—
202C22⋊C4—14212no—
203C42—2846no—
204C42C2×C4×D7116yes—
205C42C2×C22⋊D7116yes—
206C23—1424no—
207C23C22×D28411yescharacteristic
208Q8—28412no—
209Q8#14556050116yes—
210D4—2844no—
211D4—2844no—
212D4—2844no—
213D4—28412no—
214D4—5686no—
215C2×C4—43no—
216C2×C4—41no—
217C2×C4—2841no—
218C2×C4—2841no—
219C2×C4—28424no—
220C2×C4—2843no—
221C2×C4—2843no—
222C2×C4—2843no—
223C2×C4—1422no—
224C2×C4—1424no—
225C2×C4C2×(C2×D71)⋊C416yes—
226C2×C4C2×(C2×D71)⋊C416yes—
227C2×C4C2×D56816yes—
228C2×C4C2×#1176585416yes—
229C2×C4C22×C4×D7111yescharacteristic
230C2×C4C22×C22⋊D7111yescharacteristic
231C8—44no—
232C22—21no—
233C22—21no—
234C22—2844no—
235C22—2844no—
236C22—2844no—
237C22C22×(C2×D71)⋊C411yescharacteristic
238C22C22×D56811yescharacteristic
239C22C22×#1176585411yescharacteristic
240C4—26no—
241C4—21no—
242C4—21no—
243C4—2842no—
244C4—2844no—
245C4—5683no—
246C4C2×#14556050112yes—
247C2—21no—
248C2—21no—
249C2—5684no—
250C2#2259680311yescharacteristic
251C2C22×#1455605011yescharacteristic
252C2#2259688511yescharacteristic
subgroupG/H
centerC22C22×(C2×D71)⋊C4
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2982C2
socleC22×C71C22×C22⋊C4
prime radical#22602111C2
cosocleC22×C4×C71C24
hypercenter#2982D71
hypocenterC71#17618
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #22591261C2semidirect
3. #22586901C2semidirect
4. #22586836C2semidirect
5. C71×#2982C2semidirect
7. #17986671C4semidirect
8. Q8×D284C4semidirect
128. #2982D71semidirect
132. C71#17618semidirect

Maximal subgroups

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

subgroupindex
#226021112
#225912612
#225869012
#225868362
C71×#29822
#1761871