Finite Group Databasebeta

← #31907683

Subgroups — #31907683

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#22597921C211yesmaximal, characteristic
2#22589951C211yesmaximal, characteristic, complemented
3#22586351C212yesmaximal, complemented
4#22586348C212yesmaximal, complemented
5C2×#340×C71C211yesmaximal, characteristic, complemented
6C2×#14556150C411yescharacteristic, complemented
7#17986510—21no—
8C2×#14556068—21no—
9#17986240C411yescharacteristic, complemented
10#17986240C2211yescharacteristic
11#17986235C411yescharacteristic, complemented
12#17986235C2211yescharacteristic
13#17986228—21no—
14#17986228—21no—
15#17986037C411yescharacteristic, complemented
16#17985996—21no—
17C22⋊(C71⋊C16)—21no—
18C22⋊(C71⋊C16)C2212yes—
19C23×C8×C71C2211yescharacteristic
20C22×C4⋊C4×C71—21no—
21#340×C71—21no—
22#340×C71C2212yes—
23C2×#11765865—21no—
24#14556150—42no—
25C2×Dic284C2×C411yescharacteristic, complemented
26C2×#11765854—21no—
27C2×D568C2×C411yescharacteristic, complemented
28#14556111—41no—
29#14556107—21no—
30#14556107—21no—
31#14556102—21no—
32#14556102—21no—
33#14556097—41no—
34C4×C22⋊D71—41no—
35C4×C22⋊D71—41no—
36#14556087—21no—
37#14556068—42no—
38C2×C71⋊C16—22no—
39#14556062C2×C411yescharacteristic, complemented
40#14556062C2×C411yescharacteristic, complemented
41#14556060—21no—
42#14556053—21no—
43#14556053—21no—
44#14556050—21no—
45#14556050—21no—
46C23×C4×C71—41no—
47C23×C4×C71D411yescharacteristic
48C22×C8×C71—42no—
49C22×C8×C71C2×C411yescharacteristic
50C22×C8×C71C2×C411yescharacteristic
51C22×C8×C71D411yescharacteristic
52C22×C8×C71D411yescharacteristic
53C22×C8×C71D411yescharacteristic
54C22×C8×C71C2311yescharacteristic
55C2×C4⋊C4×C71—24no—
56C2×C4⋊C4×C71—22no—
57C2×C4⋊C4×C71—42no—
58C2×C4⋊C4×C71—41no—
59#11765865—42no—
60#11765865—42no—
61C2×Dic142—21no—
62C2×D284—21no—
63C2×C22⋊D71—41no—
64C2×C22⋊D71—41no—
65C2×C22⋊D71—41no—
66C2×C4×D71—41no—
67Dic284—41no—
68#11765854—42no—
69D568—41no—
70C4⋊Dic71—21no—
71C4⋊Dic71—21no—
72Dic71⋊C4—41no—
73Dic71⋊C4—41no—
74C22⋊Dic71—41no—
75C22⋊Dic71—41no—
76C22⋊Dic71—41no—
77(C2×D71)⋊C4—41no—
78(C2×D71)⋊C4—41no—
79C4×Dic71—41no—
80C71⋊C16—42no—
81C24×C71D811yescharacteristic
82C22×C4×C71—22no—
83C22×C4×C71—22no—
84C22×C4×C71—44no—
85C22×C4×C71—42no—
86C22×C4×C71—82no—
87C22×C4×C71—82no—
88C22×C4×C71C22⋊C411yescharacteristic
89C22×C4×C71C22⋊C411yescharacteristic
90C22×C4×C71D811yescharacteristic
91C22×C4×C71SD1611yescharacteristic
92C22×C4×C71SD1611yescharacteristic
93C22×C4×C71C2×D411yescharacteristic
94C2×C8×C71—22no—
95C2×C8×C71—22no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C2×C8×C71—41no—
99C2×C8×C71—41no—
100C2×C8×C71C22⋊C411yescharacteristic
101C2×C8×C71C22⋊C411yescharacteristic
102C2×C8×C71C22×C411yescharacteristic
103C2×C8×C71C2×D411yescharacteristic
104C4⋊C4×C71—24no—
105C4⋊C4×C71—44no—
106C4⋊C4×C71—42no—
107C23×C71—22no—
108C23×C71—22no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71#10211yescharacteristic
112C23×C71#10211yescharacteristic
113C23×C71C2×D811yescharacteristic
114C22×D71—41no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—44no—
123C2×C4×C71—88no—
124C2×C4×C71—82no—
125C2×C4×C71—82no—
126C2×C4×C71#9912yes—
127C2×C4×C71#10211yescharacteristic
128C2×C4×C71#10211yescharacteristic
129C2×C4×C71#10412yes—
130C2×C4×C71C2×C22⋊C411yescharacteristic
131C2×C4×C71C2×SD1611yescharacteristic
132C22⋊D71—82no—
133C22⋊D71—82no—
134C22⋊D71—81no—
135C22⋊D71—81no—
136C2×Dic71—41no—
137C2×Dic71—41no—
138C2×Dic71—41no—
139D284—41no—
140D284—41no—
141C4×D71—82no—
142Dic142—41no—
143Dic142—41no—
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#32712yes—
157C22×C71#36012yes—
158C22×C71C2×#10211yescharacteristic
159C22×C71#50911yescharacteristic
160C22×C71#51011yescharacteristic
161C2×D71—81no—
162C2×D71—81no—
163C2×D71—81no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—84no—
167C4×C71#32712yes—
168C4×C71C2×#9911yescharacteristic
169C4×C71C2×#10411yescharacteristic
170Dic71—81no—
171Dic71—81no—
172Dic71—81no—
173#11704—711nomaximal, Sylow
174C2×C71—21no—
175C2×C71—22no—
176C2×C71—22no—
177C2×C71—21no—
178C2×C71C2×#32711yescharacteristic
179C2×C71#213911yescharacteristic
180C2×C71C2×#36011yescharacteristic
181D71—161no—
182#3050—711no—
183#2112—711no—
184C2×#340D7111yescharacteristic, complemented
185#1348—712no—
186#1347—712no—
187C71#1170411yescharacteristic, complemented, Sylow
188C2×#135—711no—
189C23×C8C2×D7111yescharacteristic
190#525—1421no—
191C22×C4⋊C4—21no—
192C2×M32—1421no—
193#467—711no—
194#467—711no—
195#466—711no—
196#466—711no—
197#465—1421no—
198#465—1421no—
199#427—711no—
200#416—1421no—
201C22⋊C16—712no—
202C22⋊C16—1421no—
203#340—21no—
204#340C2×D7112yes—
205C2×#41—1421no—
206C23×C4—41no—
207C23×C4C22⋊D7111yescharacteristic
208#135—2842no—
209C2×Q16—711no—
210C2×SD16—1421no—
211C2×D8—711no—
212C22×C8—42no—
213C22×C8C4×D7111yescharacteristic
214C22×C8C4×D7111yescharacteristic
215C22×C8D28411yescharacteristic
216C22×C8D28411yescharacteristic
217C22×C8C22⋊D7111yescharacteristic
218C22×C8C22×D7111yescharacteristic
219#123—2841no—
220C22⋊Q8—1421no—
221C22⋊Q8—1421no—
222#121—1421no—
223#121—1421no—
224#120—2841no—
225C4×D4—2841no—
226C4×D4—2841no—
227#117—1421no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232M32—2842no—
233C2×C16—1422no—
234#107—711no—
235#107—711no—
236#106—1421no—
237#103—1421no—
238#103—1421no—
239#102—1421no—
240#102—1421no—
241C24#1176585011yescharacteristic
242#41—2842no—
243#41—2842no—
244C2×Q8—1421no—
245C2×D4—2841no—
246C2×D4—2841no—
247C2×D4—2841no—
248C2×D4—1421no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—2841no—
256C22×C4(C2×D71)⋊C411yescharacteristic
257C22×C4(C2×D71)⋊C411yescharacteristic
258C22×C4#1176585011yescharacteristic
259C22×C4#1176585211yescharacteristic
260C22×C4#1176585211yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262Q16—2841no—
263SD16—2842no—
264D8—2841no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8(C2×D71)⋊C411yescharacteristic
272C2×C8(C2×D71)⋊C411yescharacteristic
273C2×C8C2×C4×D7111yescharacteristic
274C2×C8C2×D28411yescharacteristic
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—2841no—
279C4⋊C4—2841no—
280C4⋊C4—1421no—
281C4⋊C4—1421no—
282C22⋊C4—2841no—
283C22⋊C4—2841no—
284C22⋊C4—2841no—
285C22⋊C4—2841no—
286C22⋊C4—2841no—
287C42—2841no—
288C16—2842no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—2841no—
294C23#1455604811yescharacteristic
295C23#1455604811yescharacteristic
296C23C2×#1176585011yescharacteristic
297Q8—2841no—
298Q8—2841no—
299D4—2841no—
300D4—2841no—
301D4—5682no—
302D4—5681no—
303D4—5681no—
304D4—5682no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—2841no—
317C2×C4—2841no—
318C2×C4—2841no—
319C2×C4—5682no—
320C2×C4#1455604212yes—
321C2×C4#1455604811yescharacteristic
322C2×C4#1455604811yescharacteristic
323C2×C4#1455605612yes—
324C2×C4C2×(C2×D71)⋊C411yescharacteristic
325C2×C4C2×#1176585211yescharacteristic
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—5681no—
339C22—5681no—
340C22—5681no—
341C22#1798573212yes—
342C22#1798580812yes—
343C22C2×#1455604811yescharacteristic
344C22#1798643511yescharacteristic
345C22#1798644011yescharacteristic
346C4—21no—
347C4—21no—
348C4—84no—
349C4—5681no—
350C4—5681no—
351C4—5681no—
352C4#1798573212yes—
353C4C2×#1455604211yescharacteristic
354C4C2×#1455605611yescharacteristic
355C2—21no—
356C2—22no—
357C2—21no—
358C2—22no—
359C2—11361no—
360C2C2×#1798573211yescharacteristic
361C2#2259005011yescharacteristic
362C2C2×#1798580811yescharacteristic
subgroupG/H
centerC22C2×#14556048
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC2×#340×C71C2
socleC22×C71C2×#102
prime radical#22597921C2
cosocleC22×C8×C71C23
hypercenterC2×#340D71
hypocenterC71#11704
LCSG ▷ C2×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. #22589951C2semidirect
3. #22586351C2semidirect
4. #22586348C2semidirect
5. C2×#340×C71C2semidirect
6. C2×#14556150C4semidirect
9. #17986240C4semidirect
11. #17986235C4semidirect
15. #17986037C4semidirect
25. C2×Dic284C2×C4semidirect
27. C2×D568C2×C4semidirect
39. #14556062C2×C4semidirect
40. #14556062C2×C4semidirect
184. C2×#340D71semidirect
187. C71#11704semidirect

Maximal subgroups

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

subgroupindex
#225979212
#225899512
#225863512
#225863482
C2×#340×C712
#1170471