Finite Group Databasebeta

← #31896466

Subgroups — #31896466

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588193C211yesmaximal, characteristic, complemented
2C4×#14556039C211yesmaximal, characteristic, complemented
3C2×#17985741C212yesmaximal, complemented
4C2×#17985739C212yesmaximal
5C71×#2861C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×#14556039—21no—
8C2×#14556039C2212yes—
9C2×#14556039C2212yes—
10C2×C4×C71⋊C8—21no—
11#17985754—21no—
12#17985741—24no—
13#17985739—24no—
14C2×#125×C71C2211yescharacteristic
15C2×#124×C71C2211yescharacteristic
16#391×C71C411yescharacteristic
17#391×C71C411yescharacteristic
18#390×C71C411yescharacteristic
19#390×C71C411yescharacteristic
20#388×C71—21no—
21#388×C71—21no—
22#388×C71—21no—
23#388×C71—21no—
24C43×C71C2211yescharacteristic
25C22×C71⋊C8—22no—
26C22×C71⋊C8—22no—
27C4⋊(C71⋊C8)—24no—
28#14556039—24no—
29#14556039—22no—
30#14556039—22no—
31#14556039—22no—
32#14556039D414yes—
33C4×C71⋊C8—24no—
34C22×Q8×C71C2×C411yescharacteristic
35C22×D4×C71C2×C411yescharacteristic
36#125×C71—22no—
37#125×C71—22no—
38#124×C71—22no—
39#124×C71—22no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71C2×C411yescharacteristic
52C2×C42×C71D411yescharacteristic
53C2×C42×C71D411yescharacteristic
54C2×C42×C71D411yescharacteristic
55C2×C42×C71D411yescharacteristic
56C2×C42×C71C2311yescharacteristic
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—24no—
64C2×C71⋊C8—24no—
65C2×C71⋊C8—42no—
66C2×C71⋊C8—42no—
67C24×C71—21no—
68C2×Q8×C71—22no—
69C2×Q8×C71—41no—
70C2×Q8×C71—41no—
71C2×D4×C71—22no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C4⋊C4×C71—22no—
85C4⋊C4×C71—22no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C22⋊C4×C71—42no—
94C22⋊C4×C71—41no—
95C22⋊C4×C71—41no—
96C42×C71—22no—
97C42×C71—22no—
98C42×C71—22no—
99C42×C71—41no—
100C42×C71—41no—
101C42×C71—41no—
102C42×C71C2×D412yes—
103C42×C71#4112yes—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71—41no—
110C23×C71—41no—
111C23×C71C2×C22⋊C411yescharacteristic
112Q8×C71—82no—
113D4×C71—82no—
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—21no—
128C2×C4×C71—21no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
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#10412yes—
139C2×C4×C71#10412yes—
140C2×C4×C71#10412yes—
141C2×C4×C71C4×D412yes—
142C2×C4×C71#12112yes—
143C2×C4×C71#12411yescharacteristic
144C2×C4×C71#12711yescharacteristic
145C71⋊C8—28no—
146C71⋊C8—44no—
147C22×C71—81no—
148C22×C71—81no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71—81no—
153C22×C71—81no—
154C22×C71#39011yescharacteristic
155C22×C71#41311yescharacteristic
156C22×C71#41311yescharacteristic
157C22×C71C2×#10411yescharacteristic
158C22×C71C2×#10411yescharacteristic
159C22×C71C2×#10411yescharacteristic
160C22×C71#42111yescharacteristic
161C4×C71—22no—
162C4×C71—22no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—22no—
166C4×C71—22no—
167C4×C71—22no—
168C4×C71—41no—
169C4×C71—81no—
170C4×C71—81no—
171C4×C71—81no—
172#9687—711nomaximal, Sylow
173C2×C71—161no—
174C2×C71#152411yescharacteristic
175C2×C71#152411yescharacteristic
176C2×C71#152911yescharacteristic
177C2×C71#153211yescharacteristic
178C2×C71#196211yescharacteristic
179C2×C71#196711yescharacteristic
180C2×C71#198011yescharacteristic
181#2861D7111yescharacteristic
182#1767—711no—
183C4×#97—711no—
184C2×#330—712no—
185C2×#329—712no—
186C71#968711yescharacteristic, complemented, Sylow
187C2×#125C2×D7111yescharacteristic
188C2×#124C2×D7111yescharacteristic
189C2×C4⋊C8—1421no—
190C2×#97—712no—
191C2×#97—712no—
192C2×#97—1421no—
193C2×C4×C8—1421no—
194#391Dic7111yescharacteristic
195#391Dic7111yescharacteristic
196#390Dic7111yescharacteristic
197#390Dic7111yescharacteristic
198#388—21no—
199#388—21no—
200#388—21no—
201#388—21no—
202C43C2×D7111yescharacteristic
203#336—1421no—
204#330—1424no—
205#329—1424no—
206C22×Q8C2×Dic7111yescharacteristic
207C22×D4C2×Dic7111yescharacteristic
208C22×C8—1422no—
209C22×C8—1422no—
210#125—22no—
211#125—22no—
212#124—22no—
213#124—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic7111yescharacteristic
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42C2×Dic7111yescharacteristic
225C2×C42C2×Dic7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C2×C42C22⋊D7111yescharacteristic
228C2×C42C22⋊D7111yescharacteristic
229C2×C42C22⋊D7111yescharacteristic
230C2×C42C22×D7111yescharacteristic
231C4⋊C8—1424no—
232#97—714no—
233#97—1424no—
234#97—1422no—
235#97—1422no—
236#97—1422no—
237C4×C8—1424no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4C22⋊Dic7111yescharacteristic
253C22×C4C22⋊Dic7111yescharacteristic
254C22×C4C22⋊Dic7111yescharacteristic
255C22×C4C22⋊Dic7111yescharacteristic
256C22×C4C22×Dic7111yescharacteristic
257C22×C4C2×C22⋊D7111yescharacteristic
258C22×C4C2×C22⋊D7111yescharacteristic
259C2×C8—2842no—
260C2×C8—2842no—
261C2×C8—1424no—
262C2×C8—1424no—
263C2×C8—1424no—
264C2×C8—1424no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42D4×D7112yes—
284C42#1176586412yes—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic7111yescharacteristic
293Q8—82no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
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—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#1455605512yes—
320C2×C4#1455605512yes—
321C2×C4#1455605512yes—
322C2×C4D4×Dic7112yes—
323C2×C4#1455609912yes—
324C2×C4#1455611511yescharacteristic
325C2×C4#1455612411yescharacteristic
326C8—2844no—
327C8—1428no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#1798590211yescharacteristic
336C22#1798598311yescharacteristic
337C22#1798598311yescharacteristic
338C22C2×#1455605511yescharacteristic
339C22C2×#1455605511yescharacteristic
340C22C2×#1455605511yescharacteristic
341C22#1798601611yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C2—161no—
354C2#2258692811yescharacteristic
355C2#2258692811yescharacteristic
356C2#2258694711yescharacteristic
357C2#2258696011yescharacteristic
358C2#2258916911yescharacteristic
359C2#2258919011yescharacteristic
360C2#2258926011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2861C2
socleC23×C71C2×C22⋊C4
prime radicalC22×D4×C71C2×C4
cosocleC2×C42×C71C23
hypercenter#2861D71
hypocenterC71#9687
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

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

NH ≅ G/Ntype
1. #22588193C2semidirect
2. C4×#14556039C2semidirect
3. C2×#17985741C2semidirect
186. C71#9687semidirect

Maximal subgroups

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

subgroupindex
#225881932
C4×#145560392
C2×#179857412
C2×#179857392
C71×#28612
#968771