Finite Group Databasebeta

← #31885699

Subgroups — #31885699

Order 18176.

590 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 131 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#22588034C212yesmaximal
2#22586163C214yesmaximal, complemented
3C71×#2296C211yesmaximal, characteristic
4C22×#11765849—22no—
5#17985754—22no—
6#17985754C2214yes—
7C2×#117×C71C411yescharacteristic
8C2×#117×C71C411yescharacteristic
9C22×C42×C71C2211yescharacteristic
10#389×C71C412yes—
11#389×C71C2212yes—
12#385×C71—22no—
13C4×C4⋊C4×C71—22no—
14C2×#11765849—28no—
15C2×#11765849—44no—
16C2×#11765849—44no—
17C22×C71⋊C8—24no—
18C23×C4×C71C2×C411yescharacteristic
19C23×C4×C71D412yes—
20#117×C71—24no—
21#117×C71—42no—
22C2×C4⋊C4×C71—22no—
23C2×C4⋊C4×C71—22no—
24C2×C4⋊C4×C71C2×C412yes—
25C2×C4⋊C4×C71C2×C412yes—
26C2×C22⋊C4×C71—21no—
27C2×C22⋊C4×C71—21no—
28C2×C42×C71—21no—
29C2×C42×C71—21no—
30C2×C42×C71—42no—
31C2×C42×C71C2×C411yescharacteristic
32C2×C42×C71D412yes—
33C2×C42×C71C2311yescharacteristic
34#95×C71—22no—
35#95×C71—22no—
36#11765849—416no—
37C2×C71⋊C8—28no—
38C2×C71⋊C8—44no—
39C2×C71⋊C8—44no—
40C24×C71C22⋊C411yescharacteristic
41C22×C4×C71—22no—
42C22×C4×C71—24no—
43C22×C4×C71—22no—
44C22×C4×C71—24no—
45C22×C4×C71—22no—
46C22×C4×C71—22no—
47C22×C4×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71—22no—
50C22×C4×C71C22⋊C411yescharacteristic
51C22×C4×C71C22⋊C412yes—
52C22×C4×C71C22×C411yescharacteristic
53C22×C4×C71C2×D412yes—
54C4⋊C4×C71—22no—
55C4⋊C4×C71—42no—
56C4⋊C4×C71—44no—
57C4⋊C4×C71—42no—
58C4⋊C4×C71—42no—
59C4⋊C4×C71M1614yescomplemented
60C22⋊C4×C71—24no—
61C22⋊C4×C71—42no—
62C42×C71—24no—
63C42×C71—41no—
64C42×C71—42no—
65C42×C71—41no—
66C42×C71—42no—
67C23×C71—21no—
68C23×C71—21no—
69C23×C71—22no—
70C23×C71—22no—
71C23×C71#9912yes—
72C23×C71C2×C22⋊C411yescharacteristic
73C2×C4×C71—28no—
74C2×C4×C71—28no—
75C2×C4×C71—28no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71—22no—
84C2×C4×C71—24no—
85C2×C4×C71—42no—
86C2×C4×C71—42no—
87C2×C4×C71—42no—
88C2×C4×C71—42no—
89C2×C4×C71—42no—
90C2×C4×C71#10112yes—
91C2×C4×C71#10414yes—
92C2×C4×C71C2×M1612yes—
93C2×C4×C71AGL(1,ℤ/8)12yes—
94C2×C4×C71#13712yes—
95C71⋊C8—48no—
96C22×C71—22no—
97C22×C71—24no—
98C22×C71—41no—
99C22×C71—42no—
100C22×C71—41no—
101C22×C71#40711yescharacteristic
102C22×C71C2×#9911yescharacteristic
103C22×C71C2×#10111yescharacteristic
104C22×C71#41712yes—
105C22×C71C2×#10412yes—
106C4×C71—22no—
107C4×C71—22no—
108C4×C71—22no—
109C4×C71—24no—
110C4×C71—22no—
111C4×C71—44no—
112C4×C71—82no—
113#7632—711nomaximal, Sylow
114C2×C71—42no—
115C2×C71#146311yescharacteristic
116C2×C71#148912yes—
117C2×C71#150412yes—
118C2×C71#152611yescharacteristic
119C2×C71#156111yescharacteristic
120#2296D7111yescharacteristic
121#1728—712no—
122#1271—714no—
123C71#763211yescharacteristic, complemented, Sylow
124C22×M16—1422no—
125C2×#117Dic7111yescharacteristic
126C2×#117Dic7111yescharacteristic
127C22×C42C2×D7111yescharacteristic
128#389Dic7112yes—
129#389C2×D7112yes—
130#385—22no—
131C4×C4⋊C4—22no—
132#336—714no—
133#336—1422no—
134C23×C4C2×Dic7111yescharacteristic
135C23×C4C22⋊D7112yes—
136C2×M16—2844no—
137C2×M16—2844no—
138C2×M16—1428no—
139C22×C8—1424no—
140#117—24no—
141#117—42no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4C2×Dic7112yes—
145C2×C4⋊C4C2×Dic7112yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—42no—
151C2×C42C2×Dic7111yescharacteristic
152C2×C42C22⋊D7112yes—
153C2×C42C22×D7111yescharacteristic
154#95—22no—
155#95—22no—
156C24C22⋊Dic7111yescharacteristic
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C22⋊Dic7111yescharacteristic
167C22×C4C22⋊Dic7112yes—
168C22×C4C22×Dic7111yescharacteristic
169C22×C4C2×C22⋊D7112yes—
170M16—28416no—
171C2×C8—2844no—
172C2×C8—2844no—
173C2×C8—1428no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4#1176584914yescomplemented
180C22⋊C4—24no—
181C22⋊C4—42no—
182C42—24no—
183C42—41no—
184C42—42no—
185C42—41no—
186C42—42no—
187C23—21no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23#1455604312yes—
192C23C2×C22⋊Dic7111yescharacteristic
193C2×C4—28no—
194C2×C4—28no—
195C2×C4—28no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4#1455604712yes—
211C2×C4#1455605514yes—
212C2×C4C2×#1176584912yes—
213C2×C4#1455615312yes—
214C2×C4#1455615812yes—
215C8—2848no—
216C22—22no—
217C22—24no—
218C22—41no—
219C22—42no—
220C22—41no—
221C22#1798596311yescharacteristic
222C22C2×#1455604311yescharacteristic
223C22C2×#1455604711yescharacteristic
224C22#1798600012yes—
225C22C2×#1455605512yes—
226C4—22no—
227C4—22no—
228C4—22no—
229C4—24no—
230C4—22no—
231C4—44no—
232C4—82no—
233C2—42no—
234C2#2258667011yescharacteristic
235C2#2258677912yes—
236C2#2258684412yes—
237C2#2258693511yescharacteristic
238C2#2258707811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2296C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2296D71
hypocenterC71#7632
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
2. #22586163C2semidirect
59. C4⋊C4×C71M16semidirect
123. C71#7632semidirect
179. C4⋊C4#11765849semidirect

Maximal subgroups

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

subgroupindex
#225880342
#225861632
C71×#22962
#763271