Finite Group Databasebeta

← #31905318

Subgroups — #31905318

Order 18176.

1042 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 227 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×#17985963C212yesmaximal
2C2×#17985768C214yesmaximal, complemented
3C2×#380×C71C211yesmaximal, characteristic
4C22×#11765849C2214yes—
5#17985963—28no—
6C2×C22⋊(C71⋊C8)—22no—
7C2×C22⋊(C71⋊C8)—22no—
8#17985768—28no—
9C24×C4×C71C2211yescharacteristic
10C22×C4⋊C4×C71C2212yes—
11C22×C22⋊C4×C71C412yes—
12#380×C71—22no—
13#380×C71—22no—
14C2×#95×C71C412yes—
15C2×#11765849—24no—
16C2×#11765849—24no—
17C2×#11765849—24no—
18C2×#11765849—24no—
19C2×#11765849—24no—
20C2×#11765849D418yes—
21C22×C71⋊C8—24no—
22C22⋊(C71⋊C8)—28no—
23C22⋊(C71⋊C8)—28no—
24C25×C71C2×C411yescharacteristic
25C23×C4×C71—24no—
26C23×C4×C71—24no—
27C23×C4×C71—24no—
28C23×C4×C71C2×C411yescharacteristic
29C23×C4×C71C2×C414yes—
30C23×C4×C71D412yes—
31C23×C4×C71D412yes—
32C23×C4×C71C2311yescharacteristic
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—22no—
35C2×C4⋊C4×C71—24no—
36C2×C4⋊C4×C71—24no—
37C2×C22⋊C4×C71—22no—
38C2×C22⋊C4×C71—22no—
39C2×C22⋊C4×C71—44no—
40#95×C71—24no—
41#11765849—28no—
42#11765849—28no—
43#11765849—44no—
44#11765849—44no—
45C2×C71⋊C8—28no—
46C2×C71⋊C8—44no—
47C2×C71⋊C8—44no—
48C24×C71—22no—
49C24×C71—22no—
50C24×C71—22no—
51C24×C71—24no—
52C24×C71—42no—
53C24×C71C22⋊C411yescharacteristic
54C24×C71C22⋊C411yescharacteristic
55C24×C71C22×C411yescharacteristic
56C22×C4×C71—22no—
57C22×C4×C71—24no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—24no—
61C22×C4×C71—24no—
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71—24no—
65C22×C4×C71—24no—
66C22×C4×C71—24no—
67C22×C4×C71—28no—
68C22×C4×C71—44no—
69C22×C4×C71—44no—
70C22×C4×C71—44no—
71C22×C4×C71—44no—
72C22×C4×C71—44no—
73C22×C4×C71—44no—
74C22×C4×C71C22⋊C412yes—
75C22×C4×C71C2×D414yes—
76C22×C4×C71C2×D412yes—
77C22×C4×C71#4114yes—
78C22×C4×C71#4114yes—
79C4⋊C4×C71—42no—
80C4⋊C4×C71—44no—
81C4⋊C4×C71—42no—
82C22⋊C4×C71—44no—
83C22⋊C4×C71—44no—
84C23×C71—21no—
85C23×C71—21no—
86C23×C71—22no—
87C23×C71—28no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—22no—
91C23×C71—22no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—22no—
95C23×C71—22no—
96C23×C71—48no—
97C23×C71—48no—
98C23×C71—44no—
99C23×C71—44no—
100C23×C71#10012yes—
101C23×C71#10112yes—
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71#11712yes—
104C23×C71C4×D414yes—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—22no—
108C2×C4×C71—44no—
109C2×C4×C71—44no—
110C2×C4×C71—44no—
111C2×C4×C71—44no—
112C2×C4×C71—44no—
113C2×C4×C71—44no—
114C2×C4×C71—44no—
115C2×C4×C71—44no—
116C2×C4×C71—88no—
117C2×C4×C71#12012yes—
118C2×C4×C71#12114yes—
119C2×C4×C71#12114yes—
120C2×C4×C71#12312yes—
121C2×C4×C71#12314yes—
122C2×C4×C71#12314yes—
123C2×C4×C71#12414yes—
124C2×C4×C71#12614yes—
125C71⋊C8—48no—
126C22×C71—22no—
127C22×C71—22no—
128C22×C71—24no—
129C22×C71—24no—
130C22×C71—22no—
131C22×C71—22no—
132C22×C71—24no—
133C22×C71—24no—
134C22×C71—416no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—42no—
138C22×C71—42no—
139C22×C71—41no—
140C22×C71—41no—
141C22×C71#38111yescharacteristic
142C22×C71#38612yes—
143C22×C71#38814yes—
144C22×C71C2×#10011yescharacteristic
145C22×C71C2×#10111yescharacteristic
146C22×C71#41312yes—
147C4×C71—42no—
148C4×C71—84no—
149C4×C71#39414yes—
150C4×C71#39714yes—
151#11198—711nomaximal, Sylow
152C2×C71—24no—
153C2×C71—42no—
154C2×C71—42no—
155C2×C71#144011yescharacteristic
156C2×C71#182111yescharacteristic
157C2×C71#182211yescharacteristic
158C2×C71#190012yes—
159C2×C71#190112yes—
160C2×#407—712no—
161C2×#380D7111yescharacteristic
162C2×#343—714no—
163C71#1119811yescharacteristic, complemented, Sylow
164C24×C4C2×D7111yescharacteristic
165C22×M16—714no—
166C22×C4⋊C4C2×D7112yes—
167C22×C22⋊C4Dic7112yes—
168#407—1428no—
169C2×C22⋊C8—1422no—
170C2×C22⋊C8—1422no—
171#380—22no—
172#380—22no—
173C2×#95Dic7112yes—
174#343—1428no—
175C25C2×Dic7111yescharacteristic
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4C2×Dic7111yescharacteristic
180C23×C4C2×Dic7114yes—
181C23×C4C22⋊D7112yes—
182C23×C4C22⋊D7112yes—
183C23×C4C22×D7111yescharacteristic
184C2×M16—718no—
185C2×M16—1424no—
186C2×M16—1424no—
187C2×M16—1424no—
188C2×M16—1424no—
189C2×M16—1424no—
190C22×C8—1424no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C22⋊C8—1428no—
199C22⋊C8—1428no—
200#95—24no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—24no—
205C24—42no—
206C24C22⋊Dic7111yescharacteristic
207C24C22⋊Dic7111yescharacteristic
208C24C22×Dic7111yescharacteristic
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4C22⋊Dic7112yes—
228C22×C4D4×D7114yes—
229C22×C4C2×C22⋊D7112yes—
230C22×C4#1176586414yes—
231C22×C4#1176586514yes—
232M16—2844no—
233M16—2844no—
234M16—1428no—
235M16—1428no—
236C2×C8—2844no—
237C2×C8—2844no—
238C2×C8—1428no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C23—21no—
245C23—21no—
246C23—22no—
247C23—28no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23#1455604512yes—
261C23#1455604712yes—
262C23C2×C22⋊Dic7111yescharacteristic
263C23#1455608712yes—
264C23D4×Dic7114yes—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4#1455609612yes—
278C2×C4#1455610014yes—
279C2×C4#1455610114yes—
280C2×C4#1455610912yes—
281C2×C4#1455611014yes—
282C2×C4#1455611114yes—
283C2×C4#1455611414yes—
284C2×C4#1455612114yes—
285C8—2848no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—416no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1798586211yescharacteristic
302C22#1798587912yes—
303C22#1798589214yes—
304C22C2×#1455604511yescharacteristic
305C22C2×#1455604711yescharacteristic
306C22#1798598312yes—
307C4—42no—
308C4—84no—
309C4#1798591414yes—
310C4#1798592714yes—
311C2—24no—
312C2—42no—
313C2—42no—
314C2#2258658911yescharacteristic
315C2#2258842011yescharacteristic
316C2#2258842311yescharacteristic
317C2#2258880412yes—
318C2#2258880912yes—
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC2×#380×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC25×C71C2×C4
cosocleC23×C4×C71C23
hypercenterC2×#380D71
hypocenterC71#11198
LCSG ▷ C24×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#17985768C2semidirect
163. C71#11198semidirect

Maximal subgroups

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

subgroupindex
C2×#179859632
C2×#179857682
C2×#380×C712
#1119871