Finite Group Databasebeta

← #31913727

Subgroups — #31913727

Order 18176.

614 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 107 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#22588268C212yesmaximal
2#22586467C214yesmaximal, complemented
3C71×#2311C211yesmaximal, characteristic
4C2×#14556047C2214yes—
5C2×#14556045—22no—
6#17985963—22no—
7C22×#41×C71C2211yescharacteristic
8C2×#117×C71C411yescharacteristic
9C22×C4⋊C4×C71C411yescharacteristic
10#391×C71C2212yes—
11#390×C71C412yes—
12#386×C71—22no—
13#380×C71—22no—
14C2×#11765849—24no—
15#14556047—28no—
16#14556045—44no—
17C22⋊(C71⋊C8)—44no—
18C2×#41×C71—24no—
19C2×#41×C71—42no—
20C2×#41×C71—42no—
21C22×Q8×C71C2311yescharacteristic
22C22×D4×C71C2×C411yescharacteristic
23C22×D4×C71D412yes—
24C23×C4×C71—21no—
25C23×C4×C71C2×C411yescharacteristic
26C23×C4×C71D412yes—
27#117×C71—42no—
28C2×C4⋊C4×C71—21no—
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—42no—
31C2×C4⋊C4×C71C2×C412yes—
32C2×C22⋊C4×C71—21no—
33C2×C22⋊C4×C71—24no—
34C2×C42×C71C2×C412yes—
35#95×C71—24no—
36#11765849—44no—
37#11765849—44no—
38C2×C71⋊C8—44no—
39C24×C71C22⋊C412yes—
40C24×C71C22⋊C411yescharacteristic
41#41×C71—416no—
42C2×Q8×C71—41no—
43C2×Q8×C71—41no—
44C2×Q8×C71#4114yes—
45C2×D4×C71—24no—
46C2×D4×C71—22no—
47C2×D4×C71—24no—
48C2×D4×C71—41no—
49C2×D4×C71—41no—
50C2×D4×C71—42no—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71—22no—
54C22×C4×C71—24no—
55C22×C4×C71—24no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—42no—
59C22×C4×C71—42no—
60C22×C4×C71—82no—
61C22×C4×C71C22⋊C411yescharacteristic
62C22×C4×C71C22×C411yescharacteristic
63C22×C4×C71C2×D412yes—
64C4⋊C4×C71—42no—
65C4⋊C4×C71—42no—
66C4⋊C4×C71—42no—
67C22⋊C4×C71—42no—
68C22⋊C4×C71—84no—
69C42×C71—42no—
70C23×C71—21no—
71C23×C71—21no—
72C23×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—41no—
76C23×C71—41no—
77C23×C71—42no—
78C23×C71—42no—
79C23×C71#9912yes—
80C23×C71C2×C22⋊C411yescharacteristic
81Q8×C71—44no—
82D4×C71—48no—
83D4×C71—44no—
84C2×C4×C71—28no—
85C2×C4×C71—21no—
86C2×C4×C71—21no—
87C2×C4×C71—21no—
88C2×C4×C71—21no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—48no—
92C2×C4×C71—42no—
93C2×C4×C71—44no—
94C2×C4×C71—42no—
95C2×C4×C71—42no—
96C2×C4×C71—42no—
97C2×C4×C71—84no—
98C2×C4×C71#9912yes—
99C2×C4×C71#11712yes—
100C2×C4×C71#12314yes—
101C71⋊C8—84no—
102C22×C71—21no—
103C22×C71—21no—
104C22×C71—22no—
105C22×C71—42no—
106C22×C71—42no—
107C22×C71—44no—
108C22×C71—41no—
109C22×C71—41no—
110C22×C71—42no—
111C22×C71—42no—
112C22×C71—42no—
113C22×C71—82no—
114C22×C71#38111yescharacteristic
115C22×C71C2×#9911yescharacteristic
116C22×C71C2×#9911yescharacteristic
117C4×C71—24no—
118C4×C71—22no—
119C4×C71—22no—
120C4×C71—82no—
121C4×C71—82no—
122#12863—711nomaximal, Sylow
123C2×C71—21no—
124C2×C71—21no—
125C2×C71—42no—
126C2×C71—82no—
127C2×C71#181911yescharacteristic
128C2×C71#213411yescharacteristic
129C2×C71#213511yescharacteristic
130#2311D7111yescharacteristic
131#1785—712no—
132#1394—714no—
133C71#1286311yescharacteristic, complemented, Sylow
134C22×#41C2×D7111yescharacteristic
135C2×#117Dic7111yescharacteristic
136C22×C4⋊C4Dic7111yescharacteristic
137C2×#101—714no—
138C2×#100—1422no—
139#407—1422no—
140#391C2×D7112yes—
141#390Dic7112yes—
142#386—22no—
143#380—22no—
144C2×#41—24no—
145C2×#41—42no—
146C2×#41—42no—
147C22×Q8C22×D7111yescharacteristic
148C22×D4C2×Dic7111yescharacteristic
149C22×D4C22⋊D7112yes—
150C23×C4—21no—
151C23×C4C2×Dic7111yescharacteristic
152C23×C4C22⋊D7112yes—
153C2×M16—1424no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C2×Dic7112yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C42C2×Dic7112yes—
162#101—1428no—
163#100—2844no—
164C22⋊C8—2844no—
165#95—24no—
166C24C22⋊Dic7112yes—
167C24C22⋊Dic7111yescharacteristic
168#41—416no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8#1176586514yes—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—24no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic7111yescharacteristic
189C22×C4C22×Dic7111yescharacteristic
190C22×C4C2×C22⋊D7112yes—
191M16—2844no—
192M16—2844no—
193C2×C8—2844no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—84no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23#1455604312yes—
210C23C2×C22⋊Dic7111yescharacteristic
211Q8—44no—
212D4—48no—
213D4—44no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4#1455604312yes—
229C2×C4#1455608712yes—
230C2×C4#1455611114yes—
231C8—5684no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—44no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22#1798586211yescharacteristic
245C22C2×#1455604311yescharacteristic
246C22C2×#1455604311yescharacteristic
247C4—24no—
248C4—22no—
249C4—22no—
250C4—82no—
251C4—82no—
252C2—21no—
253C2—21no—
254C2—42no—
255C2—82no—
256C2#2258841411yescharacteristic
257C2#2259003211yescharacteristic
258C2#2259003511yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC22×C4×C71C22×C4
FrattiniC22×Q8C22×D71
FittingC71×#2311C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC22×Q8×C71C23
hypercenter#2311D71
hypocenterC71#12863
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #22586467C2semidirect
133. C71#12863semidirect

Maximal subgroups

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

subgroupindex
#225882682
#225864672
C71×#23112
#1286371