Finite Group Databasebeta

← #31895462

Subgroups — #31895462

Order 18176.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#22586678C211yesmaximal, characteristic
3#22586670C212yesmaximal, complemented
4#22586246C212yesmaximal, complemented
5C71×#2579C211yesmaximal, characteristic
6#17986071—24no—
7#17986056—21no—
8#17985963C2212yes—
9#17985963C2212yes—
10C2×C22⋊(C71⋊C8)—21no—
11#17985725—24no—
12#17985725—21no—
13#17985723—21no—
14C2×C4×D4×C71C2211yescharacteristic
15C22×C4⋊C4×C71C2211yescharacteristic
16#398×C71—21no—
17#396×C71—21no—
18#387×C71—21no—
19#386×C71C411yescharacteristic
20#386×C71C411yescharacteristic
21#385×C71—21no—
22#385×C71C411yescharacteristic
23#385×C71C411yescharacteristic
24#380×C71C2211yescharacteristic
25C2×#11765849—22no—
26C2×#11765849—22no—
27C22×C71⋊C8—22no—
28C4⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)—22no—
30C22⋊(C71⋊C8)—22no—
31C22⋊(C71⋊C8)—22no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)D414yes—
34#14556039—22no—
35C22×D4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39C23×C4×C71C2311yescharacteristic
40C4×D4×C71—22no—
41C4×D4×C71—22no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71—22no—
44C2×C4⋊C4×C71—21no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—22no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71C2×C411yescharacteristic
53C2×C22⋊C4×C71C2×C411yescharacteristic
54C2×C22⋊C4×C71C2×C411yescharacteristic
55C2×C22⋊C4×C71D411yescharacteristic
56C2×C42×C71D411yescharacteristic
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61#95×C71—21no—
62#11765849—82no—
63#11765849—82no—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—82no—
67C24×C71C22⋊C411yescharacteristic
68C24×C71C22×C411yescharacteristic
69C2×D4×C71—22no—
70C2×D4×C71—22no—
71C2×D4×C71—41no—
72C22×C4×C71—21no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—42no—
81C22×C4×C71—81no—
82C22×C4×C71—81no—
83C22×C4×C71—81no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C2×D412yes—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71#4112yes—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—42no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—81no—
97C4⋊C4×C71—81no—
98C22⋊C4×C71—22no—
99C22⋊C4×C71—22no—
100C22⋊C4×C71—42no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—81no—
104C42×C71—21no—
105C42×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—41no—
111C23×C71—41no—
112C23×C71—41no—
113C23×C71—41no—
114C23×C71#10112yes—
115C23×C71C2×C22⋊C411yescharacteristic
116C23×C71C4×D412yes—
117D4×C71—44no—
118C2×C4×C71—22no—
119C2×C4×C71—24no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—41no—
131C2×C4×C71—42no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—81no—
145C2×C4×C71—82no—
146C2×C4×C71#12011yescharacteristic
147C2×C4×C71#12112yes—
148C2×C4×C71#12311yescharacteristic
149C71⋊C8—44no—
150C71⋊C8—82no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71—22no—
157C22×C71—42no—
158C22×C71—42no—
159C22×C71—42no—
160C22×C71—41no—
161C22×C71—41no—
162C22×C71—81no—
163C22×C71#38611yescharacteristic
164C22×C71C2×#10111yescharacteristic
165C22×C71#41311yescharacteristic
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—21no—
169C4×C71—42no—
170C4×C71—41no—
171C4×C71—41no—
172C4×C71—41no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176#9517—711nomaximal, Sylow
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—42no—
180C2×C71—42no—
181C2×C71#189111yescharacteristic
182C2×C71#190111yescharacteristic
183C2×C71#191311yescharacteristic
184#2973—711no—
185#2579D7111yescharacteristic
186#1465—711no—
187#1463—712no—
188#1305—712no—
189C71#951711yescharacteristic, complemented, Sylow
190C2×C4×D4C2×D7111yescharacteristic
191C22×C4⋊C4C2×D7111yescharacteristic
192#435—1424no—
193#432—1421no—
194#407—712no—
195#407—712no—
196C2×C22⋊C8—1421no—
197#398—21no—
198#396—21no—
199#387—21no—
200#386Dic7111yescharacteristic
201#386Dic7111yescharacteristic
202#385—21no—
203#385Dic7111yescharacteristic
204#385Dic7111yescharacteristic
205#380C2×D7111yescharacteristic
206#324—1424no—
207#324—1421no—
208#323—1421no—
209C22×D4C2×Dic7111yescharacteristic
210C23×C4C2×Dic7111yescharacteristic
211C23×C4C2×Dic7111yescharacteristic
212C23×C4C22⋊D7111yescharacteristic
213C23×C4C22×D7111yescharacteristic
214C2×M16—1422no—
215C2×M16—1422no—
216C22×C8—1422no—
217C4×D4—22no—
218C4×D4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C22⋊D7111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4C2×Dic7111yescharacteristic
230C2×C22⋊C4C2×Dic7111yescharacteristic
231C2×C22⋊C4C2×Dic7111yescharacteristic
232C2×C22⋊C4C22⋊D7111yescharacteristic
233C2×C42C22⋊D7111yescharacteristic
234C4⋊C8—1422no—
235C22⋊C8—714no—
236C22⋊C8—1422no—
237C22⋊C8—1422no—
238C22⋊C8—1422no—
239C22⋊C8—1422no—
240#97—1422no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24C22⋊Dic7111yescharacteristic
247C24C22×Dic7111yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22⋊Dic7111yescharacteristic
265C22×C4C22⋊Dic7111yescharacteristic
266C22×C4D4×D7112yes—
267C22×C4C2×C22⋊D7111yescharacteristic
268C22×C4C2×C22⋊D7111yescharacteristic
269C22×C4#1176586412yes—
270M16—5682no—
271M16—5682no—
272C2×C8—2842no—
273C2×C8—5682no—
274C2×C8—1424no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1455604712yes—
299C23C2×C22⋊Dic7111yescharacteristic
300C23D4×Dic7112yes—
301D4—44no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4#1455609611yescharacteristic
331C2×C4#1455610012yes—
332C2×C4#1455610911yescharacteristic
333C8—2844no—
334C8—5682no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22#1798587911yescharacteristic
348C22C2×#1455604711yescharacteristic
349C22#1798598311yescharacteristic
350C4—21no—
351C4—22no—
352C4—21no—
353C4—42no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#2258875911yescharacteristic
365C2#2258880911yescharacteristic
366C2#2258888311yescharacteristic
subgroupG/H
centerC22#17985879
commutatorC24×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2579C2
socleC22×C71#386
prime radicalC22×D4×C71C2×C4
cosocleC23×C4×C71C23
hypercenter#2579D71
hypocenterC71#9517
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2579

Complements

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

NH ≅ G/Ntype
3. #22586670C2semidirect
4. #22586246C2semidirect
189. C71#9517semidirect

Maximal subgroups

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

subgroupindex
#225973692
#225866782
#225866702
#225862462
C71×#25792
#951771