Finite Group Databasebeta

← #31928506

Subgroups — #31928506

Order 18176.

1530 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 371 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
1C22×#14556122C211yesmaximal, characteristic
2C2×#17985867C211yesmaximal, characteristic, complemented
3#22586590C216yesmaximal, complemented
4#22586582C216yesmaximal
5C2×#384×C71C211yesmaximal, characteristic, complemented
6C2×#14556122—24no—
7C2×#14556122—24no—
8C2×#14556122C418yescomplemented
9C22×Dic71⋊C4C2213yes—
10C22×(C2×D71)⋊C4C2213yes—
11C22×(C2×D71)⋊C4C2214yes—
12#17985867—24no—
13C2×C71⋊#95C2216yes—
14C2×C71⋊#95C22112yes—
15C22×C4⋊C4×C71C2216yes—
16C22×C42×C71C2211yescharacteristic
17#384×C71—24no—
18C24×D71C2311yescharacteristic
19C23×Dic71C2314yes—
20C23×Dic71C2313yes—
21#14556122—232no—
22C2×Dic71⋊C4—26no—
23C2×Dic71⋊C4—26no—
24C2×Dic71⋊C4C2×C4112yes—
25C2×(C2×D71)⋊C4—26no—
26C2×(C2×D71)⋊C4—26no—
27C2×(C2×D71)⋊C4—412no—
28C2×(C2×D71)⋊C4C2×C4112yes—
29C71⋊#95—212no—
30C71⋊#95—412no—
31C23×C4×C71C2314yes—
32C23×C4×C71C2313yes—
33C2×C4⋊C4×C71—212no—
34C2×C4⋊C4×C71—412no—
35C2×C42×C71C2×C414yes—
36C2×C42×C71D414yes—
37C2×C42×C71D414yes—
38C23×D71—21no—
39C23×D71—21no—
40C23×D71—26no—
41C23×D71—26no—
42C23×D71C22×C412yes—
43C22×Dic71—23no—
44C22×Dic71—23no—
45C22×Dic71—26no—
46C22×Dic71—26no—
47C22×Dic71—24no—
48C22×Dic71—212no—
49C22×Dic71—212no—
50C22×Dic71C22×C416yes—
51Dic71⋊C4—224no—
52(C2×D71)⋊C4—224no—
53(C2×D71)⋊C4—416no—
54C24×C71C2411yescharacteristic
55C22×C4×C71—26no—
56C22×C4×C71—26no—
57C22×C4×C71—24no—
58C22×C4×C71—212no—
59C22×C4×C71—212no—
60C22×C4×C71C22×C416yes—
61C22×C4×C71C2×D416yes—
62C22×C4×C71C2×D416yes—
63C4⋊C4×C71—424no—
64C42×C71C22⋊C4116yescomplemented
65C23×C71C23×C411yescharacteristic
66C23×C71C22×D411yescharacteristic
67C23×C71C22×D411yescharacteristic
68C23×C71H(5,𝔽2)16yes—
69C23×C712−1+416yes—
70C22×D71—22no—
71C22×D71—212no—
72C22×D71—43no—
73C22×D71—43no—
74C22×D71—46no—
75C22×D71—44no—
76C22×D71—412no—
77C2×C4×C71—212no—
78C2×C4×C71—26no—
79C2×C4×C71—26no—
80C2×C4×C71—212no—
81C2×C4×C71—412no—
82C2×C4×C71—412no—
83C2×C4×C71—44no—
84C2×C4×C71C2×C22⋊C4112yes—
85C2×Dic71—26no—
86C2×Dic71—212no—
87C2×Dic71—43no—
88C2×Dic71—43no—
89C2×Dic71—46no—
90C2×Dic71—412no—
91C2×Dic71—44no—
92C2×Dic71—412no—
93C22×C71C22×C22⋊C411yescharacteristic
94C22×C71#52016yes—
95C22×C71#53513yes—
96C22×C71#53613yes—
97C22×C71#53716yes—
98C22×C71#56014yes—
99C22×C71#563112yes—
100C2×D71—46no—
101C2×D71—48no—
102C2×D71—81no—
103C2×D71—83no—
104C2×D71—84no—
105C4×C71—212no—
106C4×C71—84no—
107Dic71—46no—
108Dic71—84no—
109#14665—711nomaximal, Sylow
110C2×C71#231113yes—
111C2×C71#238314yes—
112C2×C71#281811yescharacteristic
113C2×C71#282113yes—
114C2×C71#282314yes—
115D71—82no—
116C22×#126—711no—
117C2×#384D7111yescharacteristic, complemented
118C2×#383—711no—
119#1441—716no—
120#1439—716no—
121C71#1466511yescharacteristic, complemented, Sylow
122C2×#126—718no—
123C2×#126—1424no—
124C2×#126—1424no—
125C22×C4⋊C4—713no—
126C22×C4⋊C4C2×D7116yes—
127C22×C22⋊C4—714no—
128C22×C22⋊C4—713no—
129C22×C42C2×D7111yescharacteristic
130#384—24no—
131#383—1424no—
132C2×#95—7112no—
133C2×#95—716no—
134C25—711no—
135C23×C4—713no—
136C23×C4—714no—
137C23×C4C22×D7114yes—
138C23×C4C22×D7113yes—
139#126—14232no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—412no—
142C2×C4⋊C4—7112no—
143C2×C4⋊C4—1426no—
144C2×C4⋊C4—1426no—
145C2×C22⋊C4—28412no—
146C2×C22⋊C4—7112no—
147C2×C22⋊C4—1426no—
148C2×C22⋊C4—1426no—
149C2×C42C4×D7114yes—
150C2×C42D28414yes—
151C2×C42C22⋊D7114yes—
152#95—28412no—
153#95—14212no—
154C24—712no—
155C24—1421no—
156C24—1421no—
157C24—1426no—
158C24—1426no—
159C24C23×D7111yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—24no—
163C22×C4—212no—
164C22×C4—212no—
165C22×C4—716no—
166C22×C4—1424no—
167C22×C4—14212no—
168C22×C4—14212no—
169C22×C4—1423no—
170C22×C4—1423no—
171C22×C4—1426no—
172C22×C4—1426no—
173C22×C4C2×C4×D7116yes—
174C22×C4C2×C22⋊D7116yes—
175C22×C4C2×D28416yes—
176C4⋊C4—424no—
177C4⋊C4—14224no—
178C22⋊C4—28416no—
179C22⋊C4—14224no—
180C42(C2×D71)⋊C4116yescomplemented
181C23—2843no—
182C23—2843no—
183C23—2846no—
184C23—2844no—
185C23—28412no—
186C23—1422no—
187C23—14212no—
188C23C22×C4×D7111yescharacteristic
189C23C22×C22⋊D7111yescharacteristic
190C23C22×D28411yescharacteristic
191C23#1455617316yes—
192C23#1455617616yes—
193C2×C4—26no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—212no—
197C2×C4—412no—
198C2×C4—44no—
199C2×C4—412no—
200C2×C4—2843no—
201C2×C4—2843no—
202C2×C4—2846no—
203C2×C4—28412no—
204C2×C4—2844no—
205C2×C4—28412no—
206C2×C4—1426no—
207C2×C4—14212no—
208C2×C4C2×(C2×D71)⋊C4112yes—
209C22—2848no—
210C22—2846no—
211C22—5681no—
212C22—5683no—
213C22—5684no—
214C22C22×(C2×D71)⋊C411yescharacteristic
215C22#1798648016yes—
216C22#1798656613yes—
217C22#1798657313yes—
218C22#1798658016yes—
219C22#1798673614yes—
220C22#17986748112yes—
221C4—212no—
222C4—84no—
223C4—2846no—
224C4—5684no—
225C2—5682no—
226C2#2259093313yes—
227C2#2259153314yes—
228C2#2259638111yescharacteristic
229C2#2259639913yes—
230C2#2259641414yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#384×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#384D71
hypocenterC71#14665
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#17985867C2semidirect
3. #22586590C2semidirect
5. C2×#384×C71C2semidirect
8. C2×#14556122C4semidirect
64. C42×C71C22⋊C4semidirect
117. C2×#384D71semidirect
121. C71#14665semidirect
180. C42(C2×D71)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145561222
C2×#179858672
#225865902
#225865822
C2×#384×C712
#1466571