Finite Group Databasebeta

← #32024855

Subgroups — #32024855

Order 18176.

1378 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 275 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#22590719C213yesmaximal
2#22589506C214yesmaximal, complemented
3#22589503C214yesmaximal
4#22588073C213yesmaximal, complemented
5C71×#3458C211yesmaximal, characteristic, complemented
6C23×C4×D71C2211yescharacteristic
7C2×#14556087C2213yes—
8#17985860C2216yes—
9#17985856—26no—
10#17985854C2216yes—
11C4×(C2×D71)⋊C4—26no—
12#17985766C22112yes—
13#544×C71—26no—
14#543×C71C2214yes—
15#533×C71—26no—
16C2×#117×C71C2213yes—
17C24×D71C2311yescharacteristic
18C23×Dic71C2311yescharacteristic
19C22×C4×D71—23no—
20C22×C4×D71—23no—
21C22×C4×D71—26no—
22C22×C4×D71Q814yescomplemented
23#14556087—212no—
24C2×C4⋊Dic71—23no—
25C2×C22⋊Dic71C2316yes—
26C2×(C2×D71)⋊C4—212no—
27C2×C4×Dic71—23no—
28C71⋊#95—212no—
29C23×C4×C71C2311yescharacteristic
30C4⋊Q8×C71—212no—
31#125×C71—212no—
32C22⋊Q8×C71—224no—
33C4×Q8×C71—28no—
34#117×C71—26no—
35#117×C71—26no—
36C2×C4⋊C4×C71D416yes—
37C2×C22⋊C4×C71C2316yes—
38C2×C42×C71D416yes—
39C23×D71—23no—
40C23×D71—23no—
41C23×D71—44no—
42C23×D71C2×Q812yes—
43C22×Dic71—23no—
44C22×Dic71—23no—
45C22×Dic71—26no—
46C22×Dic71C2×Q812yes—
47C2×C4×D71—212no—
48C2×C4×D71—212no—
49C2×C4×D71—43no—
50C2×C4×D71—41no—
51C2×C4×D71—46no—
52C2×C4×D71—46no—
53C4⋊Dic71—46no—
54C22⋊Dic71—212no—
55(C2×D71)⋊C4—424no—
56C4×Dic71—46no—
57C24×C71C2411yescharacteristic
58C2×Q8×C71—28no—
59C22×C4×C71C2×D4112yes—
60C22×C4×C71C2×D413yes—
61C22×C4×C71C2×D413yes—
62C22×C4×C71C2×Q812yes—
63C22×C4×C71#4116yes—
64C4⋊C4×C71—224no—
65C4⋊C4×C71—212no—
66C22⋊C4×C71—212no—
67C42×C71—26no—
68C42×C71—26no—
69C23×C71—42no—
70C23×C71C22×D413yes—
71C23×C71C22×Q811yescharacteristic
72C23×C71C2×#4113yes—
73C22×D71—212no—
74C22×D71—26no—
75C22×D71—44no—
76C22×D71—412no—
77C22×D71—46no—
78C22×D71—41no—
79C22×D71—43no—
80Q8×C71—88no—
81C2×C4×C71—212no—
82C2×C4×C71—412no—
83C2×C4×C71#12013yes—
84C2×C4×C71#12011yescharacteristic
85C2×C4×C71C22⋊Q816yes—
86C2×C4×C71C22⋊Q816yes—
87C2×C4×C71#12416yes—
88C2×C4×C71#12416yes—
89C2×Dic71—26no—
90C2×Dic71—41no—
91C2×Dic71—43no—
92C2×Dic71—412no—
93C2×Dic71—46no—
94C4×D71—412no—
95C4×D71—44no—
96C22×C71—26no—
97C22×C71—41no—
98C22×C71—43no—
99C22×C71C2×#12011yescharacteristic
100C22×C71C2×C22⋊Q813yes—
101C22×C71C2×#12413yes—
102C2×D71—212no—
103C2×D71—412no—
104C2×D71—46no—
105C2×D71—42no—
106C2×D71—44no—
107C4×C71—412no—
108C4×C71#39316yes—
109C4×C71#39312yes—
110Dic71—42no—
111Dic71—86no—
112#23467—711nomaximal, Sylow
113C2×C71—23no—
114C2×C71—23no—
115C2×C71C2×#39311yescharacteristic
116C2×C71#302512yes—
117D71—44no—
118#3458D7111yescharacteristic, complemented
119#2280—713no—
120#2030—714no—
121#2029—714no—
122#1740—713no—
123C71#2346711yescharacteristic, complemented, Sylow
124C24×C4—711no—
125#544—26no—
126#543C2×D7114yes—
127#533—26no—
128C2×#117—713no—
129C2×#117C2×D7113yes—
130#381—716no—
131#380—1426no—
132#379—716no—
133C4×C22⋊C4—1426no—
134#342—7112no—
135C25—711no—
136C23×C4—711no—
137C23×C4—714no—
138C23×C4—1423no—
139C23×C4—1423no—
140C23×C4—1426no—
141C23×C4C22×D7111yescharacteristic
142C4⋊Q8—212no—
143#125—212no—
144C22⋊Q8—224no—
145C4×Q8—28no—
146#117—26no—
147#117—26no—
148#117—14212no—
149C2×C4⋊C4—1423no—
150C2×C4⋊C4C22⋊D7116yes—
151C2×C22⋊C4—716no—
152C2×C22⋊C4—14212no—
153C2×C22⋊C4C22×D7116yes—
154C2×C42—1423no—
155C2×C42C22⋊D7116yes—
156#95—14212no—
157C24—2844no—
158C24—712no—
159C24—1423no—
160C24—1423no—
161C24C23×D7111yescharacteristic
162C2×Q8—28no—
163C22×C4—2843no—
164C22×C4—2841no—
165C22×C4—2846no—
166C22×C4—2846no—
167C22×C4—712no—
168C22×C4—1426no—
169C22×C4—1423no—
170C22×C4—1423no—
171C22×C4—14212no—
172C22×C4—14212no—
173C22×C4C2×C22⋊D71112yes—
174C22×C4C2×C22⋊D7113yes—
175C22×C4C2×C22⋊D7113yes—
176C22×C4Q8×D7112yes—
177C22×C4#1176586616yes—
178C4⋊C4—224no—
179C4⋊C4—212no—
180C4⋊C4—2846no—
181C22⋊C4—212no—
182C22⋊C4—28424no—
183C22⋊C4—14212no—
184C42—26no—
185C42—26no—
186C42—2846no—
187C23—42no—
188C23—2846no—
189C23—28412no—
190C23—2844no—
191C23—2841no—
192C23—2843no—
193C23—14212no—
194C23—1426no—
195C23C22×C22⋊D7113yes—
196C23C2×Q8×D7111yescharacteristic
197C23C2×#1176586613yes—
198Q8—88no—
199C2×C4—212no—
200C2×C4—412no—
201C2×C4—2841no—
202C2×C4—2843no—
203C2×C4—28412no—
204C2×C4—2844no—
205C2×C4—28412no—
206C2×C4—2846no—
207C2×C4—1426no—
208C2×C4#1455609713yes—
209C2×C4#1455609711yescharacteristic
210C2×C4#1455610416yes—
211C2×C4#1455610416yes—
212C2×C4#1455611716yes—
213C2×C4#1455611716yes—
214C22—26no—
215C22—41no—
216C22—43no—
217C22—28412no—
218C22—2846no—
219C22—2842no—
220C22—2844no—
221C22—14212no—
222C22C2×#1455609711yescharacteristic
223C22C2×#1455610413yes—
224C22C2×#1455611713yes—
225C4—412no—
226C4—2842no—
227C4—5686no—
228C4#1798591016yes—
229C4#1798591012yes—
230C2—23no—
231C2—23no—
232C2—2844no—
233C2C2×#1798591011yescharacteristic
234C2#2259773212yes—
subgroupG/H
centerC2×C4#14556097
commutator, cosocleC24×C71C24
FrattiniC24C23×D71
FittingC71×#3458C2
socleC22×C71C2×#120
prime radicalC24×D71C23
hypercenter#3458D71
hypocenterC71#23467
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #22589506C2semidirect
4. #22588073C2semidirect
5. C71×#3458C2semidirect
22. C22×C4×D71Q8semidirect
118. #3458D71semidirect
123. C71#23467semidirect

Maximal subgroups

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

subgroupindex
#225907192
#225895062
#225895032
#225880732
C71×#34582
#2346771