Finite Group Databasebeta

← #31893078

Subgroups — #31893078

Order 18176.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986056C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic
3C2×#17985741C212yesmaximal, complemented
4#22586221C212yesmaximal, complemented
5C71×#2851C211yesmaximal, characteristic
6#17986056—24no—
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9C2×C22⋊(C71⋊C8)—21no—
10C2×#14556039C2212yes—
11C2×#14556039C2212yes—
12#17985754—21no—
13#17985741—24no—
14C2×#125×C71C2211yescharacteristic
15C22×C42×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C411yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#385×C71—21no—
22#382×C71C2211yescharacteristic
23#380×C71C411yescharacteristic
24#380×C71C411yescharacteristic
25C22×C71⋊C8—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—24no—
28C22⋊(C71⋊C8)—24no—
29C22⋊(C71⋊C8)—24no—
30#14556039—22no—
31#14556039—22no—
32#14556039—22no—
33#14556039D414yes—
34C23×C4×C71C2×C411yescharacteristic
35C23×C4×C71D411yescharacteristic
36C23×C4×C71D411yescharacteristic
37#125×C71—22no—
38#125×C71—22no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71—21no—
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C42×C71—22no—
50C2×C42×C71—22no—
51C2×C42×C71C2×C411yescharacteristic
52C2×C42×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54C2×C42×C71C2311yescharacteristic
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61C2×C71⋊C8—24no—
62C2×C71⋊C8—24no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C24×C71C22⋊C411yescharacteristic
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—41no—
79C22×C4×C71—41no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C4⋊C4×C71—22no—
89C4⋊C4×C71—22no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—42no—
96C4⋊C4×C71—41no—
97C22⋊C4×C71—42no—
98C22⋊C4×C71—42no—
99C42×C71—22no—
100C42×C71—21no—
101C42×C71—22no—
102C42×C71—21no—
103C42×C71C2×D412yes—
104C42×C71#4112yes—
105C23×C71—21no—
106C23×C71—21no—
107C23×C71—21no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71#10412yes—
112C23×C71C2×C22⋊C411yescharacteristic
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—44no—
128C2×C4×C71—44no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—41no—
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#10112yes—
144C2×C4×C71C4×D412yes—
145C2×C4×C71#12011yescharacteristic
146C2×C4×C71#12112yes—
147C2×C4×C71#12311yescharacteristic
148C71⋊C8—44no—
149C71⋊C8—44no—
150C22×C71—22no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—41no—
154C22×C71—41no—
155C22×C71—41no—
156C22×C71—41no—
157C22×C71#38611yescharacteristic
158C22×C71C2×#10111yescharacteristic
159C22×C71#41311yescharacteristic
160C22×C71C2×#10411yescharacteristic
161C22×C71#42111yescharacteristic
162C22×C71#42111yescharacteristic
163C22×C71#42111yescharacteristic
164C4×C71—22no—
165C4×C71—21no—
166C4×C71—21no—
167C4×C71—22no—
168C4×C71—22no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—81no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71—81no—
175#9125—711nomaximal, Sylow
176C2×C71—42no—
177C2×C71#152811yescharacteristic
178C2×C71#153211yescharacteristic
179C2×C71#153511yescharacteristic
180C2×C71#153711yescharacteristic
181C2×C71#187111yescharacteristic
182C2×C71#189711yescharacteristic
183C2×C71#190111yescharacteristic
184C2×#432—711no—
185#2851D7111yescharacteristic
186#1749—711no—
187C2×#330—712no—
188#1295—712no—
189C71#912511yescharacteristic, complemented, Sylow
190C2×#125C2×D7111yescharacteristic
191C22×C42C2×D7111yescharacteristic
192#432—1424no—
193C2×C4⋊C8—1421no—
194C2×C22⋊C8—1421no—
195C2×C22⋊C8—1421no—
196C2×#97—712no—
197C2×#97—712no—
198#389Dic7111yescharacteristic
199#389Dic7111yescharacteristic
200#387—21no—
201#387—21no—
202#385—21no—
203#385—21no—
204#382C2×D7111yescharacteristic
205#380Dic7111yescharacteristic
206#380Dic7111yescharacteristic
207#336—1421no—
208#330—1424no—
209C23×C4C2×Dic7111yescharacteristic
210C23×C4C22⋊D7111yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C22×C8—1422no—
213C22×C8—1422no—
214#125—22no—
215#125—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic7111yescharacteristic
221C2×C4⋊C4C2×Dic7111yescharacteristic
222C2×C4⋊C4C2×Dic7111yescharacteristic
223C2×C4⋊C4C2×Dic7111yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C2×Dic7111yescharacteristic
229C2×C42C22⋊D7111yescharacteristic
230C2×C42C22⋊D7111yescharacteristic
231C2×C42C22×D7111yescharacteristic
232C4⋊C8—1424no—
233C22⋊C8—1424no—
234C22⋊C8—1424no—
235#97—714no—
236#97—1422no—
237#97—1422no—
238#97—1422no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C24C22⋊Dic7111yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4C22⋊Dic7111yescharacteristic
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C22×Dic7111yescharacteristic
264C22×C4C2×C22⋊D7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266C2×C8—2842no—
267C2×C8—2842no—
268C2×C8—2842no—
269C2×C8—2842no—
270C2×C8—1424no—
271C2×C8—1424no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—22no—
284C42—21no—
285C42—22no—
286C42—21no—
287C42D4×D7112yes—
288C42#1176586412yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1455605512yes—
296C23C2×C22⋊Dic7111yescharacteristic
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
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#1455604712yes—
328C2×C4D4×Dic7112yes—
329C2×C4#1455609611yescharacteristic
330C2×C4#1455610012yes—
331C2×C4#1455610911yescharacteristic
332C8—2844no—
333C8—2844no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#1798587911yescharacteristic
342C22C2×#1455604711yescharacteristic
343C22#1798598311yescharacteristic
344C22C2×#1455605511yescharacteristic
345C22#1798601611yescharacteristic
346C22#1798601611yescharacteristic
347C22#1798601611yescharacteristic
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C2—42no—
360C2#2258694211yescharacteristic
361C2#2258696011yescharacteristic
362C2#2258697111yescharacteristic
363C2#2258697811yescharacteristic
364C2#2258864311yescharacteristic
365C2#2258878911yescharacteristic
366C2#2258880911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2851C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2851D71
hypocenterC71#9125
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#17985741C2semidirect
4. #22586221C2semidirect
189. C71#9125semidirect

Maximal subgroups

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

subgroupindex
C2×#179860562
#225881022
C2×#179857412
#225862212
C71×#28512
#912571