Finite Group Databasebeta

← #31895422

Subgroups — #31895422

Order 18176.

524 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 111 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#22597369C211yesmaximal, characteristic
2#22588420C212yesmaximal, complemented
3#22586678C211yesmaximal, characteristic
4#22586246C212yesmaximal, complemented
5C71×#2487C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8#17985963C2212yes—
9#17985963C2212yes—
10C2×C22⋊(C71⋊C8)—21no—
11#17985768—24no—
12#17985725—21no—
13#17985723—21no—
14C2×C4×D4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#400×C71—21no—
17#399×C71—21no—
18#389×C71—21no—
19#386×C71C412yes—
20#385×C71C412yes—
21#380×C71—21no—
22#380×C71C2211yescharacteristic
23C2×#11765849—22no—
24C2×#11765849D414yes—
25C22×C71⋊C8—22no—
26C4⋊(C71⋊C8)—22no—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32#14556039—22no—
33C22×D4×C71C2×C411yescharacteristic
34C23×C4×C71C2×C412yes—
35C23×C4×C71D411yescharacteristic
36C23×C4×C71C2311yescharacteristic
37C4×D4×C71—22no—
38C4×D4×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—22no—
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71C2×C411yescharacteristic
48C2×C22⋊C4×C71C2×C412yes—
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C42×C71D411yescharacteristic
51#95×C71—22no—
52#95×C71—22no—
53#95×C71—21no—
54#11765849—44no—
55#11765849—44no—
56C2×C71⋊C8—24no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—82no—
59C24×C71C22⋊C411yescharacteristic
60C24×C71C22×C411yescharacteristic
61C2×D4×C71—22no—
62C2×D4×C71—22no—
63C2×D4×C71—41no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—22no—
67C22×C4×C71—22no—
68C22×C4×C71—24no—
69C22×C4×C71—42no—
70C22×C4×C71—42no—
71C22×C4×C71—82no—
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C2×D412yes—
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71#4112yes—
79C4⋊C4×C71—22no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—42no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—82no—
85C22⋊C4×C71—22no—
86C22⋊C4×C71—22no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—82no—
89C42×C71—21no—
90C42×C71—21no—
91C23×C71—21no—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—41no—
96C23×C71—41no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71#10112yes—
100C23×C71C2×C22⋊C411yescharacteristic
101C23×C71C4×D412yes—
102D4×C71—44no—
103C2×C4×C71—22no—
104C2×C4×C71—24no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—41no—
111C2×C4×C71—42no—
112C2×C4×C71—42no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—42no—
120C2×C4×C71—42no—
121C2×C4×C71—81no—
122C2×C4×C71—84no—
123C2×C4×C71#12011yescharacteristic
124C2×C4×C71#12112yes—
125C2×C4×C71#12311yescharacteristic
126C71⋊C8—44no—
127C71⋊C8—82no—
128C22×C71—21no—
129C22×C71—21no—
130C22×C71—22no—
131C22×C71—22no—
132C22×C71—22no—
133C22×C71—22no—
134C22×C71—42no—
135C22×C71—42no—
136C22×C71—42no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—81no—
140C22×C71#38611yescharacteristic
141C22×C71C2×#10111yescharacteristic
142C22×C71#41311yescharacteristic
143C4×C71—22no—
144C4×C71—21no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71—41no—
148C4×C71—82no—
149C4×C71—82no—
150C4×C71#45812yes—
151#9509—711nomaximal, Sylow
152C2×C71—21no—
153C2×C71—21no—
154C2×C71—42no—
155C2×C71—42no—
156C2×C71#189011yescharacteristic
157C2×C71#190111yescharacteristic
158C2×C71#191411yescharacteristic
159#2973—711no—
160#2487D7111yescharacteristic
161#1821—712no—
162#1465—711no—
163#1305—712no—
164C71#950911yescharacteristic, complemented, Sylow
165C2×C4×D4C2×D7111yescharacteristic
166C22×C4⋊C4C2×D7111yescharacteristic
167#435—1424no—
168#432—1421no—
169#407—712no—
170#407—712no—
171C2×C22⋊C8—1421no—
172#400—21no—
173#399—21no—
174#389—21no—
175#386Dic7112yes—
176#385Dic7112yes—
177#380—21no—
178#380C2×D7111yescharacteristic
179#343—1424no—
180#324—1421no—
181#323—1421no—
182C22×D4C2×Dic7111yescharacteristic
183C23×C4C2×Dic7112yes—
184C23×C4C22⋊D7111yescharacteristic
185C23×C4C22×D7111yescharacteristic
186C2×M16—714no—
187C2×M16—1422no—
188C22×C8—1422no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D7111yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic7111yescharacteristic
200C2×C22⋊C4C2×Dic7112yes—
201C2×C22⋊C4C22⋊D7111yescharacteristic
202C2×C42C22⋊D7111yescharacteristic
203C4⋊C8—1422no—
204C22⋊C8—1422no—
205C22⋊C8—1422no—
206C22⋊C8—1422no—
207C22⋊C8—1422no—
208C22⋊C8—1422no—
209#97—1422no—
210#95—22no—
211#95—22no—
212#95—21no—
213C24C22⋊Dic7111yescharacteristic
214C24C22×Dic7111yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic7111yescharacteristic
227C22×C4C22⋊Dic7111yescharacteristic
228C22×C4C22⋊Dic7111yescharacteristic
229C22×C4D4×D7112yes—
230C22×C4C2×C22⋊D7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232C22×C4#1176586412yes—
233M16—2844no—
234M16—2844no—
235C2×C8—2842no—
236C2×C8—5682no—
237C2×C8—1424no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1455604712yes—
259C23C2×C22⋊Dic7111yescharacteristic
260C23D4×Dic7112yes—
261D4—44no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4#1455609611yescharacteristic
283C2×C4#1455610012yes—
284C2×C4#1455610911yescharacteristic
285C8—2844no—
286C8—5682no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1798587911yescharacteristic
300C22C2×#1455604711yescharacteristic
301C22#1798598311yescharacteristic
302C4—22no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1798618812yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#2258875411yescharacteristic
315C2#2258880911yescharacteristic
316C2#2258888811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2487C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#2487D71
hypocenterC71#9509
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #22588420C2semidirect
4. #22586246C2semidirect
164. C71#9509semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225884202
#225866782
#225862462
C71×#24872
#950971