Finite Group Databasebeta

← #31894803

Subgroups — #31894803

Order 18176.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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#22596978C211yesmaximal, characteristic
2#22586942C212yesmaximal, complemented
3#22586674C211yesmaximal, characteristic
4#22586233C212yesmaximal, complemented
5C71×#2611C211yesmaximal, characteristic
6#17986060—24no—
7#17986056C2212yes—
8#17986023—21no—
9#17985963—21no—
10C4×#11765849C2212yes—
11#17985741—24no—
12#17985725—21no—
13#17985723—21no—
14C2×#125×C71C2211yescharacteristic
15C2×#117×C71C2211yescharacteristic
16#400×C71C411yescharacteristic
17#399×C71C411yescharacteristic
18#398×C71C411yescharacteristic
19#397×C71C411yescharacteristic
20#389×C71—21no—
21#388×C71—21no—
22#387×C71—21no—
23#385×C71—21no—
24#376×C71C2211yescharacteristic
25C2×#11765849—22no—
26C4⋊(C71⋊C8)—24no—
27C4⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—24no—
29C22⋊(C71⋊C8)—22no—
30#14556039—22no—
31#14556039D414yes—
32C4×C71⋊C8—22no—
33C23×C4×C71D411yescharacteristic
34#125×C71—22no—
35#125×C71—22no—
36#117×C71—22no—
37#117×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71D411yescharacteristic
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C42×C71—21no—
51C2×C42×C71D411yescharacteristic
52C2×C42×C71C2311yescharacteristic
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71C2×C411yescharacteristic
59#95×C71C2×C411yescharacteristic
60#11765849—42no—
61#11765849—42no—
62C2×C71⋊C8—24no—
63C2×C71⋊C8—42no—
64C24×C71C22⋊C411yescharacteristic
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—41no—
72C22×C4×C71—41no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22×C411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C4⋊C4×C71—22no—
80C4⋊C4×C71—22no—
81C4⋊C4×C71—22no—
82C4⋊C4×C71—22no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—41no—
88C22⋊C4×C71—22no—
89C22⋊C4×C71—22no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C42×C71—21no—
93C42×C71—21no—
94C42×C71—42no—
95C42×C71C2×D412yes—
96C42×C71#4112yes—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—41no—
101C23×C71—41no—
102C23×C71C2×C22⋊C411yescharacteristic
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—41no—
119C2×C4×C71—41no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71C4×D412yes—
129C2×C4×C71#12011yescharacteristic
130C2×C4×C71#12112yes—
131C2×C4×C71#12311yescharacteristic
132C71⋊C8—44no—
133C71⋊C8—82no—
134C22×C71—21no—
135C22×C71—21no—
136C22×C71—42no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—81no—
141C22×C71#38611yescharacteristic
142C22×C71#42111yescharacteristic
143C22×C71#42111yescharacteristic
144C4×C71—22no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—22no—
148C4×C71—42no—
149C4×C71—41no—
150C4×C71—41no—
151C4×C71—41no—
152C4×C71—81no—
153C4×C71—81no—
154C4×C71—81no—
155#9390—711nomaximal, Sylow
156C2×C71—21no—
157C2×C71—21no—
158C2×C71—81no—
159C2×C71#188411yescharacteristic
160C2×C71#189711yescharacteristic
161C2×C71#191511yescharacteristic
162#2921—711no—
163#2611D7111yescharacteristic
164#1528—712no—
165#1464—711no—
166#1300—712no—
167C71#939011yescharacteristic, complemented, Sylow
168C2×#125C2×D7111yescharacteristic
169C2×#117C2×D7111yescharacteristic
170#433—1424no—
171#432—712no—
172#423—1421no—
173#407—1421no—
174C4×M16—712no—
175#400Dic7111yescharacteristic
176#399Dic7111yescharacteristic
177#398Dic7111yescharacteristic
178#397Dic7111yescharacteristic
179#389—21no—
180#388—21no—
181#387—21no—
182#385—21no—
183#376C2×D7111yescharacteristic
184#330—1424no—
185#324—1421no—
186#323—1421no—
187C23×C4C22⋊D7111yescharacteristic
188C2×M16—1422no—
189#125—22no—
190#125—22no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic7111yescharacteristic
198C2×C4⋊C4C2×Dic7111yescharacteristic
199C2×C4⋊C4C2×Dic7111yescharacteristic
200C2×C4⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C2×Dic7111yescharacteristic
204C2×C22⋊C4C22⋊D7111yescharacteristic
205C2×C42—21no—
206C2×C42C22⋊D7111yescharacteristic
207C2×C42C22×D7111yescharacteristic
208C4⋊C8—1424no—
209C4⋊C8—1422no—
210C22⋊C8—1422no—
211C22⋊C8—1424no—
212#97—714no—
213#97—1422no—
214C4×C8—1422no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×Dic7111yescharacteristic
221#95C2×Dic7111yescharacteristic
222C24C22⋊Dic7111yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22⋊Dic7111yescharacteristic
234C22×C4C22×Dic7111yescharacteristic
235C22×C4C2×C22⋊D7111yescharacteristic
236C22×C4C2×C22⋊D7111yescharacteristic
237M16—2842no—
238M16—2842no—
239C2×C8—2842no—
240C2×C8—1424no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42D4×D7112yes—
258C42#1176586412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic7111yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4D4×Dic7112yes—
291C2×C4#1455609611yescharacteristic
292C2×C4#1455610012yes—
293C2×C4#1455610911yescharacteristic
294C8—2844no—
295C8—5682no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1798587911yescharacteristic
304C22#1798601611yescharacteristic
305C22#1798601611yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2258872411yescharacteristic
321C2#2258878911yescharacteristic
322C2#2258889311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2611C2
socleC22×C71#386
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2611D71
hypocenterC71#9390
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #22586942C2semidirect
4. #22586233C2semidirect
167. C71#9390semidirect

Maximal subgroups

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

subgroupindex
#225969782
#225869422
#225866742
#225862332
C71×#26112
#939071