Finite Group Databasebeta

← #31895132

Subgroups — #31895132

Order 18176.

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

normal subgroups only

This group has 518 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#22587066C212yesmaximal, complemented
3#22586681C211yesmaximal, characteristic
4#22586196C212yesmaximal, complemented
5C71×#2547C211yesmaximal, characteristic
6#17986071—24no—
7#17986056C2212yes—
8#17986056C2212yes—
9#17985963—21no—
10C2×C22⋊(C71⋊C8)—21no—
11#17985748—24no—
12#17985723—21no—
13#17985723—21no—
14C2×C4⋊Q8×C71C2211yescharacteristic
15C2×#124×C71C412yes—
16C2×C4×D4×C71C2211yescharacteristic
17#397×C71—21no—
18#397×C71—21no—
19#391×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×C71C2×C412yes—
34C22×D4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37C4⋊Q8×C71—22no—
38C4⋊Q8×C71—22no—
39#124×C71—44no—
40C4×D4×C71—22no—
41C4×D4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—21no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71—22no—
48C2×C22⋊C4×C71D411yescharacteristic
49C2×C22⋊C4×C71D411yescharacteristic
50C2×C42×C71C2×C412yes—
51C2×C42×C71C2311yescharacteristic
52#95×C71—22no—
53#95×C71—22no—
54#95×C71—21no—
55#11765849—82no—
56C2×C71⋊C8—24no—
57C2×C71⋊C8—42no—
58C2×C71⋊C8—82no—
59C24×C71C22⋊C411yescharacteristic
60C24×C71C22⋊C411yescharacteristic
61C2×Q8×C71—24no—
62C2×Q8×C71—42no—
63C2×Q8×C71—42no—
64C2×D4×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—21no—
71C22×C4×C71—22no—
72C22×C4×C71—22no—
73C22×C4×C71—42no—
74C22×C4×C71—42no—
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22×C411yescharacteristic
78C22×C4×C71C2×D411yescharacteristic
79C22×C4×C71C2×D411yescharacteristic
80C4⋊C4×C71—22no—
81C4⋊C4×C71—42no—
82C4⋊C4×C71—41no—
83C4⋊C4×C71—42no—
84C4⋊C4×C71—41no—
85C22⋊C4×C71—22no—
86C22⋊C4×C71—22no—
87C22⋊C4×C71—44no—
88C22⋊C4×C71—44no—
89C42×C71—42no—
90C42×C71C2×D412yes—
91C42×C71#4112yes—
92C23×C71—21no—
93C23×C71—21no—
94C23×C71—21no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—41no—
98C23×C71—41no—
99C23×C71—41no—
100C23×C71—41no—
101C23×C71#10012yes—
102C23×C71C2×C22⋊C411yescharacteristic
103Q8×C71—84no—
104D4×C71—44no—
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—22no—
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—81no—
123C2×C4×C71C4×D412yes—
124C2×C4×C71#12011yescharacteristic
125C2×C4×C71#12112yes—
126C2×C4×C71#12311yescharacteristic
127C71⋊C8—44no—
128C71⋊C8—82no—
129C22×C71—21no—
130C22×C71—21no—
131C22×C71—22no—
132C22×C71—42no—
133C22×C71—42no—
134C22×C71—41no—
135C22×C71—41no—
136C22×C71—41no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—81no—
140C22×C71—81no—
141C22×C71#38611yescharacteristic
142C22×C71C2×#10011yescharacteristic
143C22×C71#41311yescharacteristic
144C4×C71—21no—
145C4×C71—22no—
146C4×C71—22no—
147C4×C71—41no—
148C4×C71—41no—
149C4×C71—82no—
150C4×C71—82no—
151C4×C71#45512yes—
152#9451—711nomaximal, Sylow
153C2×C71—21no—
154C2×C71—21no—
155C2×C71—42no—
156C2×C71—81no—
157C2×C71#188511yescharacteristic
158C2×C71#190011yescharacteristic
159C2×C71#190911yescharacteristic
160#2973—711no—
161#2547D7111yescharacteristic
162#1558—712no—
163#1466—711no—
164#1285—712no—
165C71#945111yescharacteristic, complemented, Sylow
166C2×C4⋊Q8C2×D7111yescharacteristic
167C2×#124Dic7112yes—
168C2×C4×D4C2×D7111yescharacteristic
169#435—1424no—
170#432—712no—
171#432—712no—
172#407—1421no—
173C2×C22⋊C8—1421no—
174#397—21no—
175#397—21no—
176#391Dic7112yes—
177#380—21no—
178#380—21no—
179C4×C4⋊C4C2×D7111yescharacteristic
180#333—1424no—
181#323—1421no—
182#323—1421no—
183C22×Q8C2×Dic7112yes—
184C22×D4C2×Dic7111yescharacteristic
185C23×C4C22⋊D7111yescharacteristic
186C23×C4C22⋊D7111yescharacteristic
187C2×M16—1422no—
188C22×C8—1422no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#124—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic7111yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊D7111yescharacteristic
201C2×C22⋊C4C22⋊D7111yescharacteristic
202C2×C42C2×Dic7112yes—
203C2×C42C22×D7111yescharacteristic
204C4⋊C8—714no—
205C4⋊C8—1422no—
206C22⋊C8—1422no—
207C22⋊C8—1422no—
208C22⋊C8—1422no—
209C22⋊C8—1422no—
210#97—1422no—
211#97—1422no—
212#95—22no—
213#95—22no—
214#95—21no—
215C24C22⋊Dic7111yescharacteristic
216C24C22⋊Dic7111yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic7111yescharacteristic
232C22×C4C22⋊Dic7111yescharacteristic
233C22×C4C22×Dic7111yescharacteristic
234C22×C4C2×C22⋊D7111yescharacteristic
235C22×C4C2×C22⋊D7111yescharacteristic
236M16—5682no—
237C2×C8—2842no—
238C2×C8—5682no—
239C2×C8—1424no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D4×D7112yes—
251C42#1176586412yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1455604512yes—
262C23C2×C22⋊Dic7111yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4D4×Dic7112yes—
284C2×C4#1455609611yescharacteristic
285C2×C4#1455610012yes—
286C2×C4#1455610911yescharacteristic
287C8—2844no—
288C8—5682no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1798587911yescharacteristic
302C22C2×#1455604511yescharacteristic
303C22#1798598311yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#1798617612yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2258872911yescharacteristic
317C2#2258880411yescharacteristic
318C2#2258886311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2547C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2547D71
hypocenterC71#9451
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #22587066C2semidirect
4. #22586196C2semidirect
165. C71#9451semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225870662
#225866812
#225861962
C71×#25472
#945171