Finite Group Databasebeta

← #31885532

Subgroups — #31885532

Order 18176.

566 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588408C211yesmaximal, characteristic
2#22588408C211yesmaximal, characteristic
3#22586163C214yesmaximal, complemented
4C2×#389×C71C211yesmaximal, characteristic
5C2×C22⋊(C71⋊C8)—21no—
6C2×C22⋊(C71⋊C8)—21no—
7#17985754—21no—
8#17985754—21no—
9#17985754C2212yes—
10#17985754C2212yes—
11C22×C4⋊C4×C71—21no—
12C22×C4⋊C4×C71C411yescharacteristic
13C22×C4⋊C4×C71C411yescharacteristic
14C22×C42×C71C2211yescharacteristic
15#389×C71—21no—
16#389×C71—21no—
17#389×C71C412yes—
18#389×C71C2212yes—
19C2×#95×C71—21no—
20C22×C71⋊C8—22no—
21C22×C71⋊C8—22no—
22C22⋊(C71⋊C8)—42no—
23C22⋊(C71⋊C8)—42no—
24C23×C4×C71—21no—
25C23×C4×C71—21no—
26C23×C4×C71C2×C411yescharacteristic
27C23×C4×C71D411yescharacteristic
28C23×C4×C71D411yescharacteristic
29C2×C4⋊C4×C71—21no—
30C2×C4⋊C4×C71—22no—
31C2×C4⋊C4×C71—22no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—22no—
34C2×C4⋊C4×C71—44no—
35C2×C4⋊C4×C71—42no—
36C2×C4⋊C4×C71C814yescomplemented
37C2×C4⋊C4×C71C2×C412yes—
38C2×C4⋊C4×C71C2×C412yes—
39C2×C42×C71—42no—
40C2×C42×C71C2×C411yescharacteristic
41C2×C42×C71D411yescharacteristic
42C2×C42×C71D411yescharacteristic
43C2×C42×C71C2311yescharacteristic
44#95×C71—22no—
45#95×C71—22no—
46C2×C71⋊C8—42no—
47C2×C71⋊C8—42no—
48C2×C71⋊C8—42no—
49C2×C71⋊C8—42no—
50C2×C71⋊C8—42no—
51C2×C71⋊C8—42no—
52C24×C71C22⋊C411yescharacteristic
53C22×C4×C71—22no—
54C22×C4×C71—22no—
55C22×C4×C71—24no—
56C22×C4×C71—22no—
57C22×C4×C71—22no—
58C22×C4×C71—22no—
59C22×C4×C71—22no—
60C22×C4×C71—22no—
61C22×C4×C71—22no—
62C22×C4×C71—41no—
63C22×C4×C71—41no—
64C22×C4×C71—42no—
65C22×C4×C71—42no—
66C22×C4×C71—42no—
67C22×C4×C71—42no—
68C22×C4×C71—42no—
69C22×C4×C71C22⋊C411yescharacteristic
70C22×C4×C71C22⋊C411yescharacteristic
71C22×C4×C71C22⋊C411yescharacteristic
72C22×C4×C71C2×C812yes—
73C22×C4×C71M1612yes—
74C22×C4×C71C22×C411yescharacteristic
75C22×C4×C71C2×D411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C4⋊C4×C71—24no—
78C4⋊C4×C71—22no—
79C4⋊C4×C71—42no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—44no—
82C4⋊C4×C71—44no—
83C4⋊C4×C71C2×C814yes—
84C42×C71—41no—
85C42×C71—41no—
86C42×C71—41no—
87C42×C71—41no—
88C23×C71—21no—
89C23×C71—21no—
90C23×C71—21no—
91C23×C71—21no—
92C23×C71C22⋊C812yes—
93C23×C71#9912yes—
94C23×C71#10112yes—
95C23×C71C2×C22⋊C411yescharacteristic
96C2×C4×C71—24no—
97C2×C4×C71—24no—
98C2×C4×C71—21no—
99C2×C4×C71—21no—
100C2×C4×C71—21no—
101C2×C4×C71—21no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—21no—
106C2×C4×C71—24no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—44no—
114C2×C4×C71—44no—
115C2×C4×C71—48no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—84no—
122C2×C4×C71C22⋊C812yes—
123C2×C4×C71C22×C811yescharacteristic
124C2×C4×C71C2×M1611yescharacteristic
125C2×C4×C71AGL(1,ℤ/8)12yes—
126C2×C4×C71#13712yes—
127C71⋊C8—82no—
128C71⋊C8—82no—
129C22×C71—22no—
130C22×C71—22no—
131C22×C71—22no—
132C22×C71—22no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71#32414yes—
136C22×C71C2×C22⋊C811yescharacteristic
137C22×C71C2×#9911yescharacteristic
138C22×C71C2×#10111yescharacteristic
139C22×C71#41811yescharacteristic
140C22×C71#41911yescharacteristic
141C22×C71#42111yescharacteristic
142C22×C71#42111yescharacteristic
143C4×C71—22no—
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—22no—
147C4×C71—22no—
148C4×C71—44no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—82no—
152#7593—711nomaximal, Sylow
153C2×C71—24no—
154C2×C71C2×#32411yescharacteristic
155C2×C71#147811yescharacteristic
156C2×C71#147911yescharacteristic
157C2×C71#151011yescharacteristic
158C2×C71#151111yescharacteristic
159C2×C71#153411yescharacteristic
160C2×C71#155611yescharacteristic
161C2×#389D7111yescharacteristic
162#1817—711no—
163#1817—711no—
164#1271—714no—
165C71#759311yescharacteristic, complemented, Sylow
166C22×C4⋊C4—21no—
167C22×C4⋊C4Dic7111yescharacteristic
168C22×C4⋊C4Dic7111yescharacteristic
169C22×C42C2×D7111yescharacteristic
170C2×C22⋊C8—1421no—
171C2×C22⋊C8—1421no—
172#389—21no—
173#389—21no—
174#389Dic7112yes—
175#389C2×D7112yes—
176C2×#95—21no—
177#336—712no—
178#336—712no—
179#336—1421no—
180#336—1421no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C2×Dic7111yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C23×C4C22⋊D7111yescharacteristic
186C22×C8—1422no—
187C22×C8—1422no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4C71⋊C814yescomplemented
196C2×C4⋊C4C2×Dic7112yes—
197C2×C4⋊C4C2×Dic7112yes—
198C2×C42—42no—
199C2×C42C2×Dic7111yescharacteristic
200C2×C42C22⋊D7111yescharacteristic
201C2×C42C22⋊D7111yescharacteristic
202C2×C42C22×D7111yescharacteristic
203C22⋊C8—2842no—
204C22⋊C8—2842no—
205#95—22no—
206#95—22no—
207C24C22⋊Dic7111yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C22⋊Dic7111yescharacteristic
225C22×C4C22⋊Dic7111yescharacteristic
226C22×C4C22⋊Dic7111yescharacteristic
227C22×C4C2×C71⋊C812yes—
228C22×C4#1176584912yes—
229C22×C4C22×Dic7111yescharacteristic
230C22×C4C2×C22⋊D7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232C2×C8—2842no—
233C2×C8—2842no—
234C2×C8—2842no—
235C2×C8—2842no—
236C2×C8—2842no—
237C2×C8—2842no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4C2×C71⋊C814yes—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23C22⋊(C71⋊C8)12yes—
254C23#1455604312yes—
255C23#1455604712yes—
256C23C2×C22⋊Dic7111yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4C22⋊(C71⋊C8)12yes—
284C2×C4C22×C71⋊C811yescharacteristic
285C2×C4C2×#1176584911yescharacteristic
286C2×C4#1455615412yes—
287C2×C4#1455615912yes—
288C8—5682no—
289C8—5682no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1798572514yes—
297C22C2×C22⋊(C71⋊C8)11yescharacteristic
298C22C2×#1455604311yescharacteristic
299C22C2×#1455604711yescharacteristic
300C22#1798600411yescharacteristic
301C22#1798600811yescharacteristic
302C22#1798601611yescharacteristic
303C22#1798601611yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—41no—
311C4—41no—
312C4—82no—
313C2—24no—
314C2C2×#1798572511yescharacteristic
315C2#2258673111yescharacteristic
316C2#2258673511yescharacteristic
317C2#2258686911yescharacteristic
318C2#2258687311yescharacteristic
319C2#2258696711yescharacteristic
320C2#2258705711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC2×C4×C71C22×C8
FrattiniC2×C42C22×D71
FittingC2×#389×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenterC2×#389D71
hypocenterC71#7593
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #22586163C2semidirect
36. C2×C4⋊C4×C71C8semidirect
165. C71#7593semidirect
195. C2×C4⋊C4C71⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225861632
C2×#389×C712
#759371