Finite Group Databasebeta

← #31896471

Subgroups — #31896471

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588193C211yesmaximal, characteristic, complemented
2#22588089C211yesmaximal, characteristic, complemented
3C2×#17985741C212yesmaximal, complemented
4C2×#17985739C212yesmaximal
5C71×#2861C211yesmaximal, characteristic
6C2×C4⋊(C71⋊C8)—21no—
7C2×#14556039C2212yes—
8C2×#14556039C2212yes—
9#17985754—21no—
10#17985754—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—22no—
29#14556039—22no—
30#14556039—22no—
31#14556039D414yes—
32C22×Q8×C71C2×C411yescharacteristic
33C22×D4×C71C2×C411yescharacteristic
34#125×C71—22no—
35#125×C71—22no—
36#124×C71—22no—
37#124×C71—22no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C22⋊C4×C71—21no—
48C2×C42×C71C2×C411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
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—
59C2×C71⋊C8—24no—
60C2×C71⋊C8—24no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C2×C71⋊C8—42no—
65C24×C71—21no—
66C2×Q8×C71—22no—
67C2×Q8×C71—41no—
68C2×Q8×C71—41no—
69C2×D4×C71—22no—
70C2×D4×C71—41no—
71C2×D4×C71—41no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—22no—
83C4⋊C4×C71—22no—
84C4⋊C4×C71—42no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C22⋊C4×C71—42no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—42no—
90C22⋊C4×C71—42no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—41no—
93C22⋊C4×C71—41no—
94C42×C71—22no—
95C42×C71—41no—
96C42×C71—41no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71—41no—
100C42×C71C2×D412yes—
101C42×C71#4112yes—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71—41no—
105C23×C71—41no—
106C23×C71—41no—
107C23×C71—41no—
108C23×C71—41no—
109C23×C71C2×C22⋊C411yescharacteristic
110Q8×C71—82no—
111D4×C71—82no—
112C2×C4×C71—21no—
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—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×C71C4×D412yes—
140C2×C4×C71#12112yes—
141C2×C4×C71#12411yescharacteristic
142C2×C4×C71#12711yescharacteristic
143C71⋊C8—44no—
144C71⋊C8—44no—
145C22×C71—81no—
146C22×C71—81no—
147C22×C71—81no—
148C22×C71—81no—
149C22×C71—81no—
150C22×C71—81no—
151C22×C71—81no—
152C22×C71#39011yescharacteristic
153C22×C71#41311yescharacteristic
154C22×C71#41311yescharacteristic
155C22×C71C2×#10411yescharacteristic
156C22×C71#42111yescharacteristic
157C22×C71#42111yescharacteristic
158C22×C71#42111yescharacteristic
159C4×C71—22no—
160C4×C71—22no—
161C4×C71—21no—
162C4×C71—21no—
163C4×C71—22no—
164C4×C71—41no—
165C4×C71—41no—
166C4×C71—41no—
167C4×C71—81no—
168C4×C71—81no—
169C4×C71—81no—
170#9688—711nomaximal, Sylow
171C2×C71—161no—
172C2×C71#152911yescharacteristic
173C2×C71#153211yescharacteristic
174C2×C71#153511yescharacteristic
175C2×C71#153511yescharacteristic
176C2×C71#196311yescharacteristic
177C2×C71#196711yescharacteristic
178C2×C71#198011yescharacteristic
179#2861D7111yescharacteristic
180#1767—711no—
181#1745—711no—
182C2×#330—712no—
183C2×#329—712no—
184C71#968811yescharacteristic, complemented, Sylow
185C2×#125C2×D7111yescharacteristic
186C2×#124C2×D7111yescharacteristic
187C2×C4⋊C8—1421no—
188C2×#97—712no—
189C2×#97—712no—
190#391Dic7111yescharacteristic
191#391Dic7111yescharacteristic
192#390Dic7111yescharacteristic
193#390Dic7111yescharacteristic
194#388—21no—
195#388—21no—
196#388—21no—
197#388—21no—
198C43C2×D7111yescharacteristic
199#336—1421no—
200#336—1421no—
201#336—1421no—
202#330—1424no—
203#329—1424no—
204C22×Q8C2×Dic7111yescharacteristic
205C22×D4C2×Dic7111yescharacteristic
206C22×C8—1422no—
207C22×C8—1422no—
208#125—22no—
209#125—22no—
210#124—22no—
211#124—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C4⋊C4C2×Dic7111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42C2×Dic7111yescharacteristic
223C2×C42C2×Dic7111yescharacteristic
224C2×C42C22⋊D7111yescharacteristic
225C2×C42C22⋊D7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C2×C42C22⋊D7111yescharacteristic
228C2×C42C22×D7111yescharacteristic
229C4⋊C8—1424no—
230#97—714no—
231#97—1422no—
232#97—1422no—
233#97—1422no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238C24—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C22⋊Dic7111yescharacteristic
249C22×C4C22⋊Dic7111yescharacteristic
250C22×C4C22⋊Dic7111yescharacteristic
251C22×C4C22⋊Dic7111yescharacteristic
252C22×C4C22×Dic7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4C2×C22⋊D7111yescharacteristic
255C2×C8—2842no—
256C2×C8—2842no—
257C2×C8—2842no—
258C2×C8—2842no—
259C2×C8—1424no—
260C2×C8—1424no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42—22no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42D4×D7112yes—
280C42#1176586412yes—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊Dic7111yescharacteristic
289Q8—82no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
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—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#1455605512yes—
318C2×C4D4×Dic7112yes—
319C2×C4#1455609912yes—
320C2×C4#1455611511yescharacteristic
321C2×C4#1455612411yescharacteristic
322C8—2844no—
323C8—2844no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#1798590211yescharacteristic
332C22#1798598311yescharacteristic
333C22#1798598311yescharacteristic
334C22C2×#1455605511yescharacteristic
335C22#1798601611yescharacteristic
336C22#1798601611yescharacteristic
337C22#1798601611yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—161no—
350C2#2258694711yescharacteristic
351C2#2258696011yescharacteristic
352C2#2258697111yescharacteristic
353C2#2258697111yescharacteristic
354C2#2258917311yescharacteristic
355C2#2258919011yescharacteristic
356C2#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#9688
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. #22588089C2semidirect
3. C2×#17985741C2semidirect
184. C71#9688semidirect

Maximal subgroups

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

subgroupindex
#225881932
#225880892
C2×#179857412
C2×#179857392
C71×#28612
#968871