Finite Group Databasebeta

← #31914842

Subgroups — #31914842

Order 18176.

510 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
1C22×#14556068C211yesmaximal, characteristic, complemented
2#22597921C211yesmaximal, characteristic
3C2×#17985808C212yesmaximal, complemented
4#22586447C212yesmaximal, complemented
5C71×#1807C211yesmaximal, characteristic, complemented
6C2×#14556150C2211yescharacteristic
7#17986510—21no—
8C2×#14556068—24no—
9C2×#14556068—21no—
10C2×#14556068C2212yes—
11C22×C71⋊C16—21no—
12#17986240C411yescharacteristic, complemented
13#17986240C411yescharacteristic, complemented
14#17986235C411yescharacteristic, complemented
15#17986235C411yescharacteristic, complemented
16#17986228—21no—
17#17986228—21no—
18#17986037C2211yescharacteristic
19#17985996—21no—
20#17985808—24no—
21C23×C8×C71C2211yescharacteristic
22C2×#106×C71C2212yes—
23#380×C71—21no—
24#340×C71—21no—
25C2×#11765865—21no—
26#14556150—22no—
27#14556150—22no—
28C2×Dic284C2×C411yescharacteristic
29C2×#11765854—21no—
30C2×D568C2×C411yescharacteristic
31#14556111—41no—
32#14556107—21no—
33#14556107—21no—
34#14556102—21no—
35#14556102—21no—
36#14556097—41no—
37C4×C22⋊D71—41no—
38C4×C22⋊D71—41no—
39#14556087—21no—
40#14556068—24no—
41#14556068—22no—
42#14556068D414yescomplemented
43C2×C71⋊C16—24no—
44C2×C71⋊C16—22no—
45#14556062C2×C411yescharacteristic
46#14556062C2×C411yescharacteristic
47#14556060—21no—
48#14556053—21no—
49#14556053—21no—
50#14556050—21no—
51#14556050—21no—
52C23×C4×C71D411yescharacteristic
53C22×C8×C71—22no—
54C22×C8×C71—22no—
55C22×C8×C71C2×C411yescharacteristic
56C22×C8×C71C2×C411yescharacteristic
57C22×C8×C71D411yescharacteristic
58C22×C8×C71D411yescharacteristic
59C22×C8×C71D411yescharacteristic
60C22×C8×C71C2311yescharacteristic
61C2×C4⋊C4×C71—22no—
62C2×C22⋊C4×C71—41no—
63#106×C71—22no—
64#106×C71—22no—
65#95×C71—41no—
66#11765865—42no—
67#11765865—42no—
68C2×Dic142—21no—
69C2×D284—21no—
70C2×C22⋊D71—41no—
71C2×C22⋊D71—41no—
72C2×C22⋊D71—41no—
73C2×C4×D71—41no—
74Dic284—22no—
75#11765854—41no—
76#11765854—41no—
77D568—22no—
78C4⋊Dic71—21no—
79C4⋊Dic71—21no—
80Dic71⋊C4—41no—
81Dic71⋊C4—41no—
82C22⋊Dic71—41no—
83C22⋊Dic71—41no—
84C22⋊Dic71—41no—
85(C2×D71)⋊C4—41no—
86(C2×D71)⋊C4—41no—
87C4×Dic71—41no—
88C71⋊C16—24no—
89C24×C71D811yescharacteristic
90C22×C4×C71—22no—
91C22×C4×C71—22no—
92C22×C4×C71—42no—
93C22×C4×C71C22⋊C411yescharacteristic
94C22×C4×C71C22⋊C411yescharacteristic
95C22×C4×C71D811yescharacteristic
96C22×C4×C71SD1611yescharacteristic
97C22×C4×C71SD1611yescharacteristic
98C22×C4×C71C2×D411yescharacteristic
99C2×C8×C71—22no—
100C2×C8×C71—22no—
101C2×C8×C71—22no—
102C2×C8×C71—21no—
103C2×C8×C71—21no—
104C2×C8×C71—21no—
105C2×C8×C71—21no—
106C2×C8×C71C22⋊C411yescharacteristic
107C2×C8×C71C22⋊C411yescharacteristic
108C2×C8×C71C22×C411yescharacteristic
109C2×C8×C71C2×D412yes—
110C2×C8×C71C2×D411yescharacteristic
111C2×C8×C71#4112yes—
112C4⋊C4×C71—42no—
113C4⋊C4×C71—42no—
114C22⋊C4×C71—82no—
115C23×C71—22no—
116C23×C71—22no—
117C23×C71—41no—
118C23×C71#10211yescharacteristic
119C23×C71#10211yescharacteristic
120C23×C71C2×D811yescharacteristic
121C22×D71—41no—
122C2×C4×C71—22no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—22no—
128C2×C4×C71—22no—
129C2×C4×C71—82no—
130C2×C4×C71—82no—
131C2×C4×C71—82no—
132C2×C4×C71#10211yescharacteristic
133C2×C4×C71#10211yescharacteristic
134C2×C4×C71C2×C22⋊C411yescharacteristic
135C2×C4×C71#12112yes—
136C2×C4×C71C2×SD1611yescharacteristic
137C2×C4×C71#13512yes—
138C22⋊D71—82no—
139C22⋊D71—82no—
140C22⋊D71—82no—
141C2×Dic71—41no—
142C2×Dic71—41no—
143C2×Dic71—41no—
144D284—41no—
145D284—41no—
146C4×D71—82no—
147Dic142—41no—
148Dic142—41no—
149C8×C71—21no—
150C8×C71—21no—
151C8×C71C4×D412yes—
152C8×C71#12011yescharacteristic
153C8×C71#12311yescharacteristic
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—44no—
162C22×C71#36012yescomplemented
163C22×C71C2×#10211yescharacteristic
164C22×C71#46612yes—
165C2×D71—81no—
166C2×D71—81no—
167C2×D71—81no—
168C4×C71—21no—
169C4×C71—21no—
170C4×C71—162no—
171C4×C71#38611yescharacteristic
172C4×C71#41611yescharacteristic
173C4×C71#46512yes—
174Dic71—81no—
175Dic71—81no—
176Dic71—81no—
177#13065—711nomaximal, Sylow
178C2×C71—22no—
179C2×C71—22no—
180C2×C71—41no—
181C2×C71#189311yescharacteristic
182C2×C71C2×#36011yescharacteristic
183C2×C71#214511yescharacteristic
184D71—161no—
185C22×M32—711no—
186#3050—711no—
187C2×#360—712no—
188#1807D7111yescharacteristic, complemented
189#1386—712no—
190C71#1306511yescharacteristic, complemented, Sylow
191C2×#135—711no—
192C23×C8C2×D7111yescharacteristic
193#525—1421no—
194C2×M32—712no—
195C2×M32—1424no—
196C2×M32—1421no—
197C22×C16—1421no—
198#467—711no—
199#467—711no—
200#466—711no—
201#466—711no—
202#465—1421no—
203#465—1421no—
204#427—711no—
205C2×#106C2×D7112yes—
206#416—1421no—
207#380—21no—
208#360—1424no—
209#340—21no—
210C2×#41—1421no—
211C23×C4C22⋊D7111yescharacteristic
212#135—1422no—
213#135—1422no—
214C2×Q16—711no—
215C2×SD16—1421no—
216C2×D8—711no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D7111yescharacteristic
220C22×C8C4×D7111yescharacteristic
221C22×C8D28411yescharacteristic
222C22×C8D28411yescharacteristic
223C22×C8C22⋊D7111yescharacteristic
224C22×C8C22×D7111yescharacteristic
225#123—2841no—
226C22⋊Q8—1421no—
227C22⋊Q8—1421no—
228#121—1421no—
229#121—1421no—
230#120—2841no—
231C4×D4—2841no—
232C4×D4—2841no—
233#117—1421no—
234C2×C4⋊C4—22no—
235C2×C22⋊C4—41no—
236M32—714no—
237M32—1424no—
238M32—1422no—
239C2×C16—1424no—
240C2×C16—1422no—
241#107—711no—
242#107—711no—
243#106—22no—
244#106—22no—
245#106—1421no—
246#103—1421no—
247#103—1421no—
248#102—1421no—
249#102—1421no—
250#95—41no—
251C24#1176585011yescharacteristic
252#41—2842no—
253#41—2842no—
254C2×Q8—1421no—
255C2×D4—2841no—
256C2×D4—2841no—
257C2×D4—2841no—
258C2×D4—1421no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—2841no—
263C22×C4(C2×D71)⋊C411yescharacteristic
264C22×C4(C2×D71)⋊C411yescharacteristic
265C22×C4#1176585011yescharacteristic
266C22×C4#1176585211yescharacteristic
267C22×C4#1176585211yescharacteristic
268C22×C4C2×C22⋊D7111yescharacteristic
269Q16—1422no—
270SD16—2841no—
271SD16—2841no—
272D8—1422no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8(C2×D71)⋊C411yescharacteristic
281C2×C8(C2×D71)⋊C411yescharacteristic
282C2×C8C2×C4×D7111yescharacteristic
283C2×C8D4×D7112yes—
284C2×C8C2×D28411yescharacteristic
285C2×C8#1176586412yes—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C4⋊C4—1421no—
291C4⋊C4—1421no—
292C22⋊C4—82no—
293C22⋊C4—2841no—
294C22⋊C4—2841no—
295C22⋊C4—2841no—
296C22⋊C4—2841no—
297C22⋊C4—2841no—
298C42—2841no—
299C16—1424no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2841no—
304C23#1455604811yescharacteristic
305C23#1455604811yescharacteristic
306C23C2×#1176585011yescharacteristic
307Q8—2841no—
308Q8—2841no—
309D4—2841no—
310D4—2841no—
311D4—5682no—
312D4—5682no—
313D4—5682no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—2841no—
325C2×C4—2841no—
326C2×C4—2841no—
327C2×C4—5682no—
328C2×C4#1455604811yescharacteristic
329C2×C4#1455604811yescharacteristic
330C2×C4C2×(C2×D71)⋊C411yescharacteristic
331C2×C4#1455610012yes—
332C2×C4C2×#1176585211yescharacteristic
333C2×C4#1455614812yes—
334C8—21no—
335C8—21no—
336C8(C22⋊D71)⋊C412yes—
337C8#1455609811yescharacteristic
338C8#1455611211yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—5681no—
348C22—5681no—
349C22—5681no—
350C22#1798580812yescomplemented
351C22C2×#1455604811yescharacteristic
352C22#1798623312yes—
353C4—21no—
354C4—21no—
355C4—162no—
356C4—5681no—
357C4—5681no—
358C4—5681no—
359C4#1798587611yescharacteristic
360C4#1798599311yescharacteristic
361C4#1798623012yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—11361no—
366C2#2258876611yescharacteristic
367C2C2×#1798580811yescharacteristic
368C2#2259007511yescharacteristic
subgroupG/H
centerC22C2×#14556048
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC71×#1807C2
socleC22×C71C2×#102
prime radical#22597921C2
cosocleC22×C8×C71C23
hypercenter#1807D71
hypocenterC71#13065
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C22×#14556068C2semidirect
3. C2×#17985808C2semidirect
4. #22586447C2semidirect
5. C71×#1807C2semidirect
12. #17986240C4semidirect
13. #17986240C4semidirect
14. #17986235C4semidirect
15. #17986235C4semidirect
42. #14556068D4semidirect
162. C22×C71#360semidirect
188. #1807D71semidirect
190. C71#13065semidirect
350. C22#17985808semidirect

Maximal subgroups

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

subgroupindex
C22×#145560682
#225979212
C2×#179858082
#225864472
C71×#18072
#1306571