Finite Group Databasebeta

← #31907685

Subgroups — #31907685

Order 18176.

504 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589951C211yesmaximal, characteristic, complemented
2C2×#17985762C211yesmaximal, characteristic, complemented
3#22586352C212yesmaximal, complemented
4#22586349C212yesmaximal, complemented
5C71×#3050C211yesmaximal, characteristic
6C22×C4⋊Dic71—21no—
7C2×#14556068—21no—
8C22⋊(C71⋊C16)—21no—
9C22⋊(C71⋊C16)C2212yes—
10#17985762—21no—
11#17985762C2212yes—
12C2×#135×C71C411yescharacteristic, complemented
13C23×C8×C71C2211yescharacteristic
14#525×C71—21no—
15#467×C71C411yescharacteristic, complemented
16#467×C71C2211yescharacteristic
17#466×C71C411yescharacteristic, complemented
18#466×C71C2211yescharacteristic
19#465×C71—21no—
20#465×C71—21no—
21#427×C71C411yescharacteristic, complemented
22#416×C71—21no—
23C23×Dic71—41no—
24C2×C4⋊Dic71—24no—
25C2×C4⋊Dic71—22no—
26C2×C4⋊Dic71—42no—
27C2×C4⋊Dic71—41no—
28#14556068—42no—
29C2×C71⋊C16—22no—
30C2×#41×C71—21no—
31C23×C4×C71D411yescharacteristic
32#135×C71—42no—
33C2×Q16×C71C2×C411yescharacteristic, complemented
34C2×SD16×C71—21no—
35C2×D8×C71C2×C411yescharacteristic, complemented
36C22×C8×C71—42no—
37C22×C8×C71C2×C411yescharacteristic
38C22×C8×C71C2×C411yescharacteristic
39C22×C8×C71D411yescharacteristic
40C22×C8×C71D411yescharacteristic
41C22×C8×C71D411yescharacteristic
42C22×C8×C71C2311yescharacteristic
43#123×C71—41no—
44C22⋊Q8×C71—21no—
45C22⋊Q8×C71—21no—
46#121×C71—21no—
47#121×C71—21no—
48#120×C71—41no—
49C4×D4×C71—41no—
50C4×D4×C71—41no—
51#117×C71—21no—
52#107×C71C2×C411yescharacteristic, complemented
53#107×C71C2×C411yescharacteristic, complemented
54#106×C71—21no—
55#103×C71—21no—
56#103×C71—21no—
57#102×C71—21no—
58#102×C71—21no—
59C22×Dic71—44no—
60C22×Dic71—42no—
61C22×Dic71—82no—
62C22×Dic71—82no—
63C4⋊Dic71—24no—
64C4⋊Dic71—44no—
65C4⋊Dic71—42no—
66C71⋊C16—42no—
67C24×C71D811yescharacteristic
68#41×C71—42no—
69#41×C71—42no—
70C2×Q8×C71—21no—
71C2×D4×C71—21no—
72C2×D4×C71—41no—
73C2×D4×C71—41no—
74C2×D4×C71—41no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—41no—
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71D811yescharacteristic
81C22×C4×C71SD1611yescharacteristic
82C22×C4×C71SD1611yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84Q16×C71—41no—
85SD16×C71—42no—
86D8×C71—41no—
87C2×C8×C71—22no—
88C2×C8×C71—22no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C2×C8×C71C22⋊C411yescharacteristic
94C2×C8×C71C22⋊C411yescharacteristic
95C2×C8×C71C22×C411yescharacteristic
96C2×C8×C71C2×D411yescharacteristic
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C22⋊C4×C71—41no—
105C22⋊C4×C71—41no—
106C42×C71—41no—
107C23×C71—22no—
108C23×C71—22no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—41no—
112C23×C71#10211yescharacteristic
113C23×C71#10211yescharacteristic
114C23×C71C2×D811yescharacteristic
115Q8×C71—41no—
116Q8×C71—41no—
117D4×C71—41no—
118D4×C71—41no—
119D4×C71—82no—
120D4×C71—82no—
121D4×C71—81no—
122D4×C71—81no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—22no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—82no—
134C2×C4×C71#9912yes—
135C2×C4×C71#10211yescharacteristic
136C2×C4×C71#10211yescharacteristic
137C2×C4×C71#10412yes—
138C2×C4×C71C2×C22⋊C411yescharacteristic
139C2×C4×C71C2×SD1611yescharacteristic
140C2×Dic71—44no—
141C2×Dic71—88no—
142C2×Dic71—82no—
143C2×Dic71—82no—
144C8×C71—21no—
145C8×C71—21no—
146C8×C71—21no—
147C8×C71—21no—
148C22×C71—24no—
149C22×C71—24no—
150C22×C71—21no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—81no—
157C22×C71—81no—
158C22×C71—81no—
159C22×C71#32712yes—
160C22×C71#36012yes—
161C22×C71C2×#10211yescharacteristic
162C22×C71#50911yescharacteristic
163C22×C71#51011yescharacteristic
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—81no—
167C4×C71—81no—
168C4×C71—81no—
169C4×C71#32712yes—
170C4×C71C2×#9911yescharacteristic
171C4×C71C2×#10411yescharacteristic
172Dic71—84no—
173#11704—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—22no—
176C2×C71—22no—
177C2×C71—21no—
178C2×C71—161no—
179C2×C71C2×#32711yescharacteristic
180C2×C71#213911yescharacteristic
181C2×C71C2×#36011yescharacteristic
182#3050D7111yescharacteristic
183#2112—711no—
184C2×#340—711no—
185#1348—712no—
186#1347—712no—
187C71#1170411yescharacteristic, complemented, Sylow
188C2×#135Dic7111yescharacteristic, complemented
189C23×C8C2×D7111yescharacteristic
190#525—21no—
191C22×C4⋊C4—1421no—
192C2×M32—1421no—
193#467Dic7111yescharacteristic, complemented
194#467C2×D7111yescharacteristic
195#466Dic7111yescharacteristic, complemented
196#466C2×D7111yescharacteristic
197#465—21no—
198#465—21no—
199#427Dic7111yescharacteristic, complemented
200#416—21no—
201C22⋊C16—712no—
202C22⋊C16—1421no—
203#340—712no—
204#340—1421no—
205C2×#41—21no—
206C23×C4—2841no—
207C23×C4C22⋊D7111yescharacteristic
208#135—42no—
209C2×Q16C2×Dic7111yescharacteristic, complemented
210C2×SD16—21no—
211C2×D8C2×Dic7111yescharacteristic, complemented
212C22×C8—42no—
213C22×C8C2×Dic7111yescharacteristic
214C22×C8C2×Dic7111yescharacteristic
215C22×C8C22⋊D7111yescharacteristic
216C22×C8C22⋊D7111yescharacteristic
217C22×C8C22⋊D7111yescharacteristic
218C22×C8C22×D7111yescharacteristic
219#123—41no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222#121—21no—
223#121—21no—
224#120—41no—
225C4×D4—41no—
226C4×D4—41no—
227#117—21no—
228C2×C4⋊C4—2842no—
229C2×C4⋊C4—2841no—
230C2×C4⋊C4—1424no—
231C2×C4⋊C4—1422no—
232M32—2842no—
233C2×C16—1422no—
234#107C2×Dic7111yescharacteristic, complemented
235#107C2×Dic7111yescharacteristic, complemented
236#106—21no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C24#1176585011yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—2844no—
253C22×C4—2842no—
254C22×C4—5682no—
255C22×C4—5682no—
256C22×C4C22⋊Dic7111yescharacteristic
257C22×C4C22⋊Dic7111yescharacteristic
258C22×C4#1176585011yescharacteristic
259C22×C4#1176585311yescharacteristic
260C22×C4#1176585311yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262Q16—41no—
263SD16—42no—
264D8—41no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8C22⋊Dic7111yescharacteristic
272C2×C8C22⋊Dic7111yescharacteristic
273C2×C8C22×Dic7111yescharacteristic
274C2×C8C2×C22⋊D7111yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—2842no—
280C4⋊C4—2844no—
281C4⋊C4—1424no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—41no—
288C16—2842no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—41no—
294C23#1455604911yescharacteristic
295C23#1455604911yescharacteristic
296C23C2×#1176585011yescharacteristic
297Q8—41no—
298Q8—41no—
299D4—41no—
300D4—41no—
301D4—82no—
302D4—81no—
303D4—81no—
304D4—82no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—2844no—
317C2×C4—5688no—
318C2×C4—5682no—
319C2×C4—5682no—
320C2×C4#1455604312yes—
321C2×C4#1455604911yescharacteristic
322C2×C4#1455604911yescharacteristic
323C2×C4#1455605512yes—
324C2×C4C2×C22⋊Dic7111yescharacteristic
325C2×C4C2×#1176585311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1798573312yes—
342C22#1798580912yes—
343C22C2×#1455604911yescharacteristic
344C22#1798643411yescharacteristic
345C22#1798643911yescharacteristic
346C4—21no—
347C4—21no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—5684no—
352C4#1798573312yes—
353C4C2×#1455604311yescharacteristic
354C4C2×#1455605511yescharacteristic
355C2—21no—
356C2—22no—
357C2—21no—
358C2—22no—
359C2—161no—
360C2C2×#1798573311yescharacteristic
361C2#2259005211yescharacteristic
362C2C2×#1798580911yescharacteristic
subgroupG/H
centerC22C2×#14556049
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
Fitting, prime radicalC71×#3050C2
socleC22×C71C2×#102
cosocleC22×C8×C71C23
hypercenter#3050D71
hypocenterC71#11704
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #22589951C2semidirect
2. C2×#17985762C2semidirect
3. #22586352C2semidirect
4. #22586349C2semidirect
12. C2×#135×C71C4semidirect
15. #467×C71C4semidirect
17. #466×C71C4semidirect
21. #427×C71C4semidirect
33. C2×Q16×C71C2×C4semidirect
35. C2×D8×C71C2×C4semidirect
52. #107×C71C2×C4semidirect
53. #107×C71C2×C4semidirect
187. C71#11704semidirect
188. C2×#135Dic71semidirect
193. #467Dic71semidirect
195. #466Dic71semidirect
199. #427Dic71semidirect
209. C2×Q16C2×Dic71semidirect
211. C2×D8C2×Dic71semidirect
234. #107C2×Dic71semidirect
235. #107C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
#225899512
C2×#179857622
#225863522
#225863492
C71×#30502
#1170471