Finite Group Databasebeta

← #31895142

Subgroups — #31895142

Order 18176.

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

normal subgroups only

This group has 498 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#22587057C212yesmaximal, complemented
3#22586681C211yesmaximal, characteristic
4#22586196C212yesmaximal, complemented
5C71×#2563C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8#17986056C2212yes—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)—21no—
11#17985745—24no—
12#17985723—21no—
13#17985723—21no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×C4×D4×C71C2211yescharacteristic
16#393×C71—21no—
17#393×C71—21no—
18#390×C71C412yes—
19#382×C71C412yes—
20#380×C71—21no—
21#380×C71—21no—
22C4×C4⋊C4×C71C2211yescharacteristic
23C2×#11765849—22no—
24C22×C71⋊C8—22no—
25C4⋊(C71⋊C8)—22no—
26C4⋊(C71⋊C8)D414yes—
27C22⋊(C71⋊C8)—22no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31#14556039—22no—
32#14556039—22no—
33C22×Q8×C71—21no—
34C22×D4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C4⋊Q8×C71—22no—
38C4⋊Q8×C71—22no—
39C4×D4×C71—22no—
40C4×D4×C71—22no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C412yes—
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C42×C71C2×C412yes—
50C2×C42×C71C2311yescharacteristic
51#95×C71—22no—
52#95×C71—22no—
53#95×C71—21no—
54#11765849—82no—
55C2×C71⋊C8—24no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—82no—
58C24×C71C22⋊C411yescharacteristic
59C24×C71C22⋊C411yescharacteristic
60C2×Q8×C71—42no—
61C2×Q8×C71—82no—
62C2×D4×C71—22no—
63C2×D4×C71—22no—
64C2×D4×C71—41no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—42no—
72C22×C4×C71—42no—
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22×C411yescharacteristic
76C22×C4×C71C2×D411yescharacteristic
77C22×C4×C71C2×D411yescharacteristic
78C4⋊C4×C71—22no—
79C4⋊C4×C71—24no—
80C4⋊C4×C71—42no—
81C4⋊C4×C71—41no—
82C4⋊C4×C71—41no—
83C22⋊C4×C71—22no—
84C22⋊C4×C71—22no—
85C22⋊C4×C71—82no—
86C22⋊C4×C71—82no—
87C42×C71—42no—
88C42×C71C2×D412yes—
89C42×C71#4112yes—
90C23×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
101Q8×C71—84no—
102D4×C71—44no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71—22no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—41no—
109C2×C4×C71—42no—
110C2×C4×C71—42no—
111C2×C4×C71—41no—
112C2×C4×C71—41no—
113C2×C4×C71—42no—
114C2×C4×C71—42no—
115C2×C4×C71—42no—
116C2×C4×C71—42no—
117C2×C4×C71—42no—
118C2×C4×C71—42no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71C4×D412yes—
122C2×C4×C71#12011yescharacteristic
123C2×C4×C71#12112yes—
124C2×C4×C71#12311yescharacteristic
125C71⋊C8—44no—
126C71⋊C8—82no—
127C22×C71—21no—
128C22×C71—21no—
129C22×C71—22no—
130C22×C71—42no—
131C22×C71—42no—
132C22×C71—41no—
133C22×C71—41no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—81no—
138C22×C71—81no—
139C22×C71#38611yescharacteristic
140C22×C71C2×#10111yescharacteristic
141C22×C71#41311yescharacteristic
142C4×C71—21no—
143C4×C71—22no—
144C4×C71—22no—
145C4×C71—41no—
146C4×C71—41no—
147C4×C71—82no—
148C4×C71—82no—
149C4×C71#45512yes—
150#9453—711nomaximal, Sylow
151C2×C71—21no—
152C2×C71—21no—
153C2×C71—42no—
154C2×C71—81no—
155C2×C71#188511yescharacteristic
156C2×C71#190111yescharacteristic
157C2×C71#191011yescharacteristic
158#2973—711no—
159#2563D7111yescharacteristic
160#1556—712no—
161#1466—711no—
162#1285—712no—
163C71#945311yescharacteristic, complemented, Sylow
164C2×C4⋊Q8C2×D7111yescharacteristic
165C2×C4×D4C2×D7111yescharacteristic
166#435—1424no—
167#432—712no—
168#432—712no—
169#407—1421no—
170C2×C22⋊C8—1421no—
171#393—21no—
172#393—21no—
173#390Dic7112yes—
174#382Dic7112yes—
175#380—21no—
176#380—21no—
177C4×C4⋊C4C2×D7111yescharacteristic
178#332—1424no—
179#323—1421no—
180#323—1421no—
181C22×Q8—21no—
182C22×D4C2×Dic7111yescharacteristic
183C23×C4C22⋊D7111yescharacteristic
184C23×C4C22⋊D7111yescharacteristic
185C2×M16—1422no—
186C22×C8—1422no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic7112yes—
194C2×C4⋊C4C2×Dic7111yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊D7111yescharacteristic
198C2×C22⋊C4C22⋊D7111yescharacteristic
199C2×C42C2×Dic7112yes—
200C2×C42C22×D7111yescharacteristic
201C4⋊C8—714no—
202C4⋊C8—1422no—
203C22⋊C8—1422no—
204C22⋊C8—1422no—
205C22⋊C8—1422no—
206C22⋊C8—1422no—
207#97—1422no—
208#97—1422no—
209#95—22no—
210#95—22no—
211#95—21no—
212C24C22⋊Dic7111yescharacteristic
213C24C22⋊Dic7111yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊Dic7111yescharacteristic
228C22×C4C22⋊Dic7111yescharacteristic
229C22×C4C22×Dic7111yescharacteristic
230C22×C4C2×C22⋊D7111yescharacteristic
231C22×C4C2×C22⋊D7111yescharacteristic
232M16—5682no—
233C2×C8—2842no—
234C2×C8—5682no—
235C2×C8—1424no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42D4×D7112yes—
247C42#1176586412yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1455604712yes—
258C23C2×C22⋊Dic7111yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4D4×Dic7112yes—
280C2×C4#1455609611yescharacteristic
281C2×C4#1455610012yes—
282C2×C4#1455610911yescharacteristic
283C8—2844no—
284C8—5682no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1798587911yescharacteristic
298C22C2×#1455604711yescharacteristic
299C22#1798598311yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4#1798617612yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2258872911yescharacteristic
313C2#2258880911yescharacteristic
314C2#2258886811yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2563C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2563D71
hypocenterC71#9453
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
2. #22587057C2semidirect
4. #22586196C2semidirect
163. C71#9453semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225870572
#225866812
#225861962
C71×#25632
#945371