Finite Group Databasebeta

← #31885909

Subgroups — #31885909

Order 18176.

568 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588034C211yesmaximal, characteristic
2C2×#17985754C211yesmaximal, characteristic
3#22586166C212yesmaximal, complemented
4#22586163C212yesmaximal, complemented
5C71×#2855C211yesmaximal, characteristic
6C22×#11765849—21no—
7C23×C71⋊C8—21no—
8#17985754—21no—
9#17985754—21no—
10#17985754C2212yes—
11#17985754C2212yes—
12C2×C22⋊Q8×C71C411yescharacteristic
13C22×C42×C71C2211yescharacteristic
14#391×C71C411yescharacteristic
15#391×C71C2211yescharacteristic
16#389×C71C411yescharacteristic
17#389×C71C2211yescharacteristic
18#387×C71—21no—
19#387×C71—21no—
20#385×C71—21no—
21#381×C71—21no—
22#380×C71C411yescharacteristic
23C2×#11765849—24no—
24C2×#11765849—42no—
25C2×#11765849—42no—
26C22×C71⋊C8—24no—
27C22×C71⋊C8—22no—
28C22×C71⋊C8—22no—
29C22×C71⋊C8—42no—
30C22×C71⋊C8—42no—
31C22×Q8×C71C2×C411yescharacteristic
32C23×C4×C71C2×C411yescharacteristic
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C22⋊Q8×C71—42no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C22⋊C4×C71—21no—
43C2×C22⋊C4×C71—21no—
44C2×C42×C71—42no—
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71D411yescharacteristic
47C2×C42×C71D411yescharacteristic
48C2×C42×C71C2311yescharacteristic
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#11765849—48no—
57C2×C71⋊C8—24no—
58C2×C71⋊C8—24no—
59C2×C71⋊C8—48no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C24×C71C22⋊C411yescharacteristic
65C2×Q8×C71—22no—
66C2×Q8×C71—41no—
67C2×Q8×C71—41no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—22no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—41no—
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22⋊C411yescharacteristic
84C22×C4×C71SD1612yes—
85C22×C4×C71Q1612yes—
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
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—41no—
96C22⋊C4×C71—42no—
97C22⋊C4×C71—42no—
98C42×C71—41no—
99C42×C71—41no—
100C42×C71—41no—
101C42×C71—41no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71—21no—
106C23×C71#9912yes—
107C23×C71#10312yes—
108C23×C71#10412yes—
109C23×C71C2×C22⋊C411yescharacteristic
110Q8×C71—82no—
111C2×C4×C71—24no—
112C2×C4×C71—24no—
113C2×C4×C71—24no—
114C2×C4×C71—24no—
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—44no—
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—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71#10312yes—
142C2×C4×C71#10412yes—
143C2×C4×C71#13112yes—
144C2×C4×C71C2×SD1611yescharacteristic
145C2×C4×C71C2×Q1611yescharacteristic
146C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C71#13711yescharacteristic
148C71⋊C8—44no—
149C71⋊C8—44no—
150C22×C71—22no—
151C22×C71—22no—
152C22×C71—22no—
153C22×C71—22no—
154C22×C71—22no—
155C22×C71—22no—
156C22×C71#32814yes—
157C22×C71#40811yescharacteristic
158C22×C71C2×#9911yescharacteristic
159C22×C71#41311yescharacteristic
160C22×C71C2×#10311yescharacteristic
161C22×C71#41711yescharacteristic
162C22×C71C2×#10411yescharacteristic
163C22×C71C2×#10411yescharacteristic
164C4×C71—22no—
165C4×C71—21no—
166C4×C71—22no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—22no—
170C4×C71—21no—
171C4×C71—41no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71—81no—
175C4×C71—81no—
176#7681—711nomaximal, Sylow
177C2×C71—24no—
178C2×C71#146811yescharacteristic
179C2×C71#148311yescharacteristic
180C2×C71#149311yescharacteristic
181C2×C71C2×#32811yescharacteristic
182C2×C71#150411yescharacteristic
183C2×C71#152411yescharacteristic
184C2×C71#155011yescharacteristic
185#2855D7111yescharacteristic
186#1728—711no—
187C2×#336—711no—
188#1272—712no—
189#1271—712no—
190C71#768111yescharacteristic, complemented, Sylow
191C22×M16—1421no—
192C23×C8—1421no—
193C2×C22⋊Q8Dic7111yescharacteristic
194C22×C42C2×D7111yescharacteristic
195#391Dic7111yescharacteristic
196#391C2×D7111yescharacteristic
197#389Dic7111yescharacteristic
198#389C2×D7111yescharacteristic
199#387—21no—
200#387—21no—
201#385—21no—
202#381—21no—
203#380Dic7111yescharacteristic
204#336—712no—
205#336—712no—
206#336—1421no—
207#336—1421no—
208C22×Q8C2×Dic7111yescharacteristic
209C23×C4C2×Dic7111yescharacteristic
210C23×C4C22⋊D7111yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C2×M16—2842no—
213C2×M16—2842no—
214C2×M16—1424no—
215C22×C8—2842no—
216C22×C8—2842no—
217C22×C8—1424no—
218C22×C8—1422no—
219C22×C8—1422no—
220C22⋊Q8—42no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C4⋊C4C2×Dic7111yescharacteristic
226C2×C4⋊C4C2×Dic7111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—42no—
230C2×C42C2×Dic7111yescharacteristic
231C2×C42C22⋊D7111yescharacteristic
232C2×C42C22⋊D7111yescharacteristic
233C2×C42C22×D7111yescharacteristic
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24C22⋊Dic7111yescharacteristic
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C22⋊Dic7111yescharacteristic
259C22×C4C22⋊Dic7111yescharacteristic
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4#1176585212yes—
262C22×C4#1176585512yes—
263C22×C4C22×Dic7111yescharacteristic
264C22×C4C2×C22⋊D7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266M16—2848no—
267C2×C8—2848no—
268C2×C8—2842no—
269C2×C8—2842no—
270C2×C8—2842no—
271C2×C8—2842no—
272C2×C8—1424no—
273C2×C8—1424no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23#1455604312yes—
292C23#1455605212yes—
293C23#1455605512yes—
294C23C2×C22⋊Dic7111yescharacteristic
295Q8—82no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—44no—
314C2×C4—41no—
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#1455605212yes—
327C2×C4#1455605512yes—
328C2×C4#1455613512yes—
329C2×C4C2×#1176585211yescharacteristic
330C2×C4C2×#1176585511yescharacteristic
331C2×C4#1455615311yescharacteristic
332C2×C4#1455615811yescharacteristic
333C8—2844no—
334C8—2844no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#1798573614yes—
342C22#1798596611yescharacteristic
343C22C2×#1455604311yescharacteristic
344C22#1798598311yescharacteristic
345C22C2×#1455605211yescharacteristic
346C22#1798600011yescharacteristic
347C22C2×#1455605511yescharacteristic
348C22C2×#1455605511yescharacteristic
349C4—22no—
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C2—24no—
362C2#2258668811yescharacteristic
363C2#2258675311yescharacteristic
364C2#2258679511yescharacteristic
365C2C2×#1798573611yescharacteristic
366C2#2258684411yescharacteristic
367C2#2258692811yescharacteristic
368C2#2258703011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2855C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#2855D71
hypocenterC71#7681
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
3. #22586166C2semidirect
4. #22586163C2semidirect
190. C71#7681semidirect

Maximal subgroups

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

subgroupindex
#225880342
C2×#179857542
#225861662
#225861632
C71×#28552
#768171