Finite Group Databasebeta

← #31912642

Subgroups — #31912642

Order 18176.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#17985796C212yesmaximal
2C4×#14556047C212yesmaximal, complemented
3#22586467C212yesmaximal, complemented
4C71×#2326C211yesmaximal, characteristic
5C2×#14556047C2214yes—
6C4×#11765849—22no—
7#17985796C418yescomplemented
8#17985768—22no—
9C2×#124×C71C2211yescharacteristic
10C2×C4×Q8×C71C2211yescharacteristic
11C2×C4×D4×C71C411yescharacteristic
12#391×C71C2211yescharacteristic
13#390×C71C411yescharacteristic
14#388×C71—22no—
15#385×C71—22no—
16#384×C71C411yescharacteristic
17#376×C71C411yescharacteristic
18C2×#11765849—24no—
19#14556047—24no—
20#14556047C2×C418yescomplemented
21#14556039—42no—
22C4×C71⋊C8—42no—
23C22×Q8×C71C2311yescharacteristic
24C22×D4×C71C2×C411yescharacteristic
25C23×C4×C71—21no—
26#124×C71—22no—
27#124×C71C2×C414yes—
28C4×Q8×C71—42no—
29C4×D4×C71—42no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71C2×C411yescharacteristic
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71D412yes—
34C2×C22⋊C4×C71—22no—
35C2×C22⋊C4×C71—21no—
36C2×C22⋊C4×C71—22no—
37C2×C42×C71C2×C411yescharacteristic
38C2×C42×C71C2×C411yescharacteristic
39C2×C42×C71C2×C411yescharacteristic
40C2×C42×C71D412yes—
41#95×C71—21no—
42#95×C71—21no—
43#95×C71—22no—
44#11765849—28no—
45#11765849—44no—
46C2×C71⋊C8—44no—
47C24×C71—21no—
48C2×Q8×C71—41no—
49C2×Q8×C71—41no—
50C2×Q8×C71C22×C412yes—
51C2×Q8×C71#4112yes—
52C2×D4×C71—21no—
53C2×D4×C71—41no—
54C2×D4×C71—41no—
55C2×D4×C71C4212yes—
56C22×C4×C71—21no—
57C22×C4×C71—21no—
58C22×C4×C71—21no—
59C22×C4×C71—42no—
60C22×C4×C71—82no—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22⋊C412yes—
63C22×C4×C71C22⋊C411yescharacteristic
64C22×C4×C71C22×C411yescharacteristic
65C22×C4×C71C2×D412yes—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—42no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—82no—
70C22⋊C4×C71—28no—
71C22⋊C4×C71—42no—
72C22⋊C4×C71—82no—
73C42×C71—22no—
74C42×C71—22no—
75C42×C71—22no—
76C42×C71—21no—
77C42×C71—41no—
78C42×C71C4212yes—
79C23×C71—22no—
80C23×C71—42no—
81C23×C71—42no—
82C23×C71—41no—
83C23×C71—41no—
84C23×C71C2×C22⋊C411yescharacteristic
85Q8×C71—42no—
86Q8×C71—42no—
87D4×C71—42no—
88D4×C71—42no—
89C2×C4×C71—21no—
90C2×C4×C71—22no—
91C2×C4×C71—21no—
92C2×C4×C71—22no—
93C2×C4×C71—22no—
94C2×C4×C71—21no—
95C2×C4×C71—22no—
96C2×C4×C71—41no—
97C2×C4×C71—41no—
98C2×C4×C71—41no—
99C2×C4×C71—42no—
100C2×C4×C71—41no—
101C2×C4×C71—41no—
102C2×C4×C71—42no—
103C2×C4×C71—41no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71—84no—
107C2×C4×C71#9912yes—
108C2×C4×C71C2×C4211yescharacteristic
109C2×C4×C71#11711yescharacteristic
110C2×C4×C71C4×D414yes—
111C71⋊C8—48no—
112C22×C71—21no—
113C22×C71—21no—
114C22×C71—44no—
115C22×C71—42no—
116C22×C71—82no—
117C22×C71—81no—
118C22×C71—81no—
119C22×C71C4×C22⋊C411yescharacteristic
120C22×C71C2×#9911yescharacteristic
121C22×C71#41011yescharacteristic
122C4×C71—22no—
123C4×C71—24no—
124C4×C71—21no—
125C4×C71—21no—
126C4×C71—42no—
127C4×C71—42no—
128C4×C71—41no—
129C4×C71—81no—
130C4×C71—81no—
131#12632—711nomaximal, Sylow
132C2×C71—21no—
133C2×C71—21no—
134C2×C71—82no—
135C2×C71C4×#9911yescharacteristic
136C2×C71#213311yescharacteristic
137C2×C71#213411yescharacteristic
138#2326D7111yescharacteristic
139C2×#355—712no—
140C4×#101—712no—
141#1394—712no—
142C71#1263211yescharacteristic, complemented, Sylow
143C2×#124C2×D7111yescharacteristic
144C2×C4×Q8C2×D7111yescharacteristic
145C2×C4×D4Dic7111yescharacteristic
146C2×#101—714no—
147C4×M16—1422no—
148#391C2×D7111yescharacteristic
149#390Dic7111yescharacteristic
150#388—22no—
151#385—22no—
152#384Dic7111yescharacteristic
153#376Dic7111yescharacteristic
154#355—718no—
155#343—1422no—
156C22×Q8C22×D7111yescharacteristic
157C22×D4C2×Dic7111yescharacteristic
158C23×C4—21no—
159C2×M16—1424no—
160#124—22no—
161#124C4×D7114yes—
162C4×Q8—42no—
163C4×D4—42no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×Dic7111yescharacteristic
166C2×C4⋊C4C2×Dic7111yescharacteristic
167C2×C4⋊C4C22⋊D7112yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C42C2×Dic7111yescharacteristic
172C2×C42C2×Dic7111yescharacteristic
173C2×C42C2×Dic7111yescharacteristic
174C2×C42C22⋊D7112yes—
175#101—718no—
176#101—1424no—
177#97—2842no—
178C4×C8—2842no—
179#95—21no—
180#95—21no—
181#95—22no—
182C24—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×C4×D7112yes—
186C2×Q8#1176586512yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×Dic7112yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C22⋊Dic7111yescharacteristic
197C22×C4C22⋊Dic7112yes—
198C22×C4C22⋊Dic7111yescharacteristic
199C22×C4C22×Dic7111yescharacteristic
200C22×C4C2×C22⋊D7112yes—
201M16—2844no—
202M16—1428no—
203C2×C8—2844no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42C4×Dic7112yes—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊Dic7111yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#1455604312yes—
246C2×C4C2×C4×Dic7111yescharacteristic
247C2×C4#1455608711yescharacteristic
248C2×C4C4×C22⋊D7114yes—
249C8—2848no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊Dic7111yescharacteristic
258C22C2×#1455604311yescharacteristic
259C22#1798597311yescharacteristic
260C4—22no—
261C4—24no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#1455604311yescharacteristic
273C2#2259002911yescharacteristic
274C2#2259003211yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC2×C4×C71C2×C42
FrattiniC22×Q8C22×D71
FittingC71×#2326C2
socleC22×C71C2×#99
prime radicalC22×D4×C71C2×C4
cosocleC22×Q8×C71C23
hypercenter#2326D71
hypocenterC71#12632
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C4×#14556047C2semidirect
3. #22586467C2semidirect
7. #17985796C4semidirect
20. #14556047C2×C4semidirect
142. C71#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#179857962
C4×#145560472
#225864672
C71×#23262
#1263271