Finite Group Databasebeta

← #31914844

Subgroups — #31914844

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
1#22597921C211yesmaximal, characteristic
2C2×#17985810C212yesmaximal, complemented
3#22588365C211yesmaximal, characteristic, complemented
4#22586446C212yesmaximal, complemented
5C22×M32×C71C211yesmaximal, characteristic, complemented
6C2×#14556150C2211yescharacteristic
7#17986510—21no—
8#17986240C411yescharacteristic, complemented
9#17986240C411yescharacteristic, complemented
10#17986235C411yescharacteristic, complemented
11#17986235C411yescharacteristic, complemented
12#17986228—21no—
13#17986228—21no—
14#17986037C2211yescharacteristic
15C2×#14556060C2212yes—
16#17985996—21no—
17#17985859—21no—
18#17985810—24no—
19#17985762—21no—
20C23×C8×C71C2211yescharacteristic
21C2×M32×C71—21no—
22C2×M32×C71—24no—
23C2×M32×C71C2212yes—
24C22×C16×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—
40C2×C4⋊Dic71—22no—
41C2×C22⋊Dic71—41no—
42#14556062C2×C411yescharacteristic
43#14556062C2×C411yescharacteristic
44#14556060—21no—
45#14556060—22no—
46#14556060—22no—
47#14556053—21no—
48#14556053—21no—
49#14556050—21no—
50#14556050—21no—
51C71⋊#95—41no—
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
61M32×C71—24no—
62M32×C71—22no—
63M32×C71D414yescomplemented
64C2×C16×C71—24no—
65C2×C16×C71—22no—
66#11765865—42no—
67#11765865—42no—
68C2×Dic142—21no—
69C2×D284—21no—
70C2×C22⋊D71—41no—
71C2×C22⋊D71—41no—
72C2×C22⋊D71—41no—
73C22×Dic71—42no—
74C2×C4×D71—41no—
75Dic284—22no—
76#11765854—41no—
77#11765854—41no—
78D568—22no—
79C4⋊Dic71—21no—
80C4⋊Dic71—21no—
81C4⋊Dic71—42no—
82C4⋊Dic71—42no—
83Dic71⋊C4—41no—
84Dic71⋊C4—41no—
85C22⋊Dic71—41no—
86C22⋊Dic71—41no—
87C22⋊Dic71—41no—
88C22⋊Dic71—82no—
89(C2×D71)⋊C4—41no—
90(C2×D71)⋊C4—41no—
91C4×Dic71—41no—
92C24×C71D811yescharacteristic
93C22×C4×C71—22no—
94C22×C4×C71—22no—
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71D811yescharacteristic
98C22×C4×C71SD1611yescharacteristic
99C22×C4×C71SD1611yescharacteristic
100C22×C4×C71C2×D411yescharacteristic
101C2×C8×C71—22no—
102C2×C8×C71—22no—
103C2×C8×C71—22no—
104C2×C8×C71—21no—
105C2×C8×C71—21no—
106C2×C8×C71—21no—
107C2×C8×C71—21no—
108C2×C8×C71C22⋊C411yescharacteristic
109C2×C8×C71C22⋊C411yescharacteristic
110C2×C8×C71C22×C411yescharacteristic
111C2×C8×C71C2×D412yes—
112C2×C8×C71C2×D411yescharacteristic
113C2×C8×C71#4112yes—
114C16×C71—24no—
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#10211yescharacteristic
130C2×C4×C71#10211yescharacteristic
131C2×C4×C71C2×C22⋊C411yescharacteristic
132C2×C4×C71#12112yes—
133C2×C4×C71C2×SD1611yescharacteristic
134C2×C4×C71#13512yes—
135C22⋊D71—82no—
136C22⋊D71—82no—
137C22⋊D71—82no—
138C2×Dic71—41no—
139C2×Dic71—41no—
140C2×Dic71—41no—
141C2×Dic71—82no—
142C2×Dic71—82no—
143C2×Dic71—82no—
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#38611yescharacteristic
171C4×C71#41611yescharacteristic
172C4×C71#46512yes—
173Dic71—81no—
174Dic71—81no—
175Dic71—81no—
176Dic71—162no—
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×M32D7111yescharacteristic, complemented
186#3050—711no—
187C2×#360—712no—
188#1807—711no—
189#1386—712no—
190C71#1306511yescharacteristic, complemented, Sylow
191C2×#135—711no—
192C23×C8C2×D7111yescharacteristic
193#525—1421no—
194C2×M32—21no—
195C2×M32—24no—
196C2×M32C2×D7112yes—
197C22×C16—21no—
198#467—711no—
199#467—711no—
200#466—711no—
201#466—711no—
202#465—1421no—
203#465—1421no—
204#427—711no—
205C2×#106—712no—
206#416—1421no—
207#380—1421no—
208#360—1424no—
209#340—1421no—
210C2×#41—1421no—
211C23×C4D28411yescharacteristic
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×C8C22⋊D7111yescharacteristic
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—1422no—
235C2×C22⋊C4—2841no—
236M32—24no—
237M32—22no—
238M32C22⋊D7114yescomplemented
239C2×C16—24no—
240C2×C16—22no—
241#107—711no—
242#107—711no—
243#106—1422no—
244#106—1422no—
245#106—1421no—
246#103—1421no—
247#103—1421no—
248#102—1421no—
249#102—1421no—
250#95—2841no—
251C24D56811yescharacteristic
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—2841no—
262C22×C4—2842no—
263C22×C4(C2×D71)⋊C411yescharacteristic
264C22×C4(C2×D71)⋊C411yescharacteristic
265C22×C4D56811yescharacteristic
266C22×C4#1176585411yescharacteristic
267C22×C4#1176585411yescharacteristic
268C22×C4C2×D28411yescharacteristic
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×C8C2×C22⋊D7112yes—
284C2×C8C2×C22⋊D7111yescharacteristic
285C2×C8#1176586512yes—
286C4⋊C4—2841no—
287C4⋊C4—2841no—
288C4⋊C4—2842no—
289C4⋊C4—2842no—
290C4⋊C4—1421no—
291C4⋊C4—1421no—
292C22⋊C4—2841no—
293C22⋊C4—2841no—
294C22⋊C4—2841no—
295C22⋊C4—2841no—
296C22⋊C4—2841no—
297C22⋊C4—5682no—
298C42—2841no—
299C16—24no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2841no—
304C23#1455605011yescharacteristic
305C23#1455605011yescharacteristic
306C23C2×D56811yescharacteristic
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—2841no—
322C2×C4—2841no—
323C2×C4—2841no—
324C2×C4—5682no—
325C2×C4—5682no—
326C2×C4—5682no—
327C2×C4—5682no—
328C2×C4#1455605011yescharacteristic
329C2×C4#1455605011yescharacteristic
330C2×C4C2×(C2×D71)⋊C411yescharacteristic
331C2×C4#1455610212yes—
332C2×C4C2×#1176585411yescharacteristic
333C2×C4#1455615012yes—
334C8—21no—
335C8—21no—
336C8C4×C22⋊D7112yes—
337C8#1455609711yescharacteristic
338C8#1455611111yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—5681no—
348C22—5681no—
349C22—5681no—
350C22#1798581012yescomplemented
351C22C2×#1455605011yescharacteristic
352C22#1798623512yes—
353C4—21no—
354C4—21no—
355C4—5681no—
356C4—5681no—
357C4—5681no—
358C4—11362no—
359C4#1798587811yescharacteristic
360C4#1798599611yescharacteristic
361C4#1798622812yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—11361no—
366C2#2258876811yescharacteristic
367C2C2×#1798581011yescharacteristic
368C2#2259007711yescharacteristic
subgroupG/H
centerC22C2×#14556050
commutatorC2×C8×C71C22×C4
FrattiniC22×C8C22×D71
FittingC22×M32×C71C2
socleC22×C71C2×#102
prime radical#22597921C2
cosocleC22×C8×C71C23
hypercenterC22×M32D71
hypocenterC71#13065
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
2. C2×#17985810C2semidirect
3. #22588365C2semidirect
4. #22586446C2semidirect
5. C22×M32×C71C2semidirect
8. #17986240C4semidirect
9. #17986240C4semidirect
10. #17986235C4semidirect
11. #17986235C4semidirect
63. M32×C71D4semidirect
162. C22×C71#360semidirect
185. C22×M32D71semidirect
190. C71#13065semidirect
238. M32C22⋊D71semidirect
350. C22#17985810semidirect

Maximal subgroups

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

subgroupindex
#225979212
C2×#179858102
#225883652
#225864462
C22×M32×C712
#1306571