Finite Group Databasebeta

← #31902052

Subgroups — #31902052

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589267C211yesmaximal, characteristic
2#22589146C211yesmaximal, characteristic
3#22589103C211yesmaximal, characteristic, complemented
4C2×#17985752C211yesmaximal, characteristic, complemented
5C2×#17985742C211yesmaximal, characteristic
6#22586224C211yesmaximal, characteristic, complemented
7C71×#1939C211yesmaximal, characteristic, complemented
8C2×#14556125C2211yescharacteristic
9C2×#14556059—21no—
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556050—21no—
13C2×#14556050—21no—
14C2×#14556048—21no—
15C2×#14556048—21no—
16#17985869C2211yescharacteristic
17#17985761—21no—
18#17985752—22no—
19#17985742—22no—
20C2×C4⋊C8×C71—21no—
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×C4×C8×C71C2211yescharacteristic
23C4×C4⋊C4×C71C2211yescharacteristic
24#336×C71—21no—
25C22×D284—21no—
26C22×D284—21no—
27C22×D284—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30#14556125—21no—
31#14556125—21no—
32#14556125—21no—
33#14556125Q812yes—
34C2×C4⋊Dic71—21no—
35C2×C4⋊Dic71—21no—
36C2×C4⋊Dic71—21no—
37#14556059—42no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)D412yes—
42#14556050—42no—
43#14556050—42no—
44#14556048—42no—
45#14556048—42no—
46C22×C8×C71—21no—
47C22×C8×C71D411yescharacteristic
48C22×C8×C71D411yescharacteristic
49C2×C4⋊C4×C71D411yescharacteristic
50C2×C4⋊C4×C71D411yescharacteristic
51C2×C42×C71—21no—
52C2×C42×C71C2311yescharacteristic
53C4⋊C8×C71—21no—
54C4⋊C8×C71—21no—
55C4⋊C8×C71—42no—
56C4×C8×C71—21no—
57C4×C8×C71—21no—
58#95×C71—21no—
59C23×D71—41no—
60C2×D284—22no—
61C2×D284—22no—
62C2×D284—22no—
63C2×D284—41no—
64C2×D284—41no—
65C2×D284—41no—
66C2×D284—41no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284—41no—
70C2×D284—41no—
71C2×D284—41no—
72C2×D284—41no—
73C2×D284—41no—
74C2×D284—81no—
75C2×D284—81no—
76C22×Dic71—41no—
77C2×C71⋊C8—22no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—41no—
80C2×C71⋊C8—41no—
81C2×C71⋊C8—41no—
82C2×C71⋊C8—41no—
83C4⋊Dic71—41no—
84C4⋊Dic71—41no—
85C4⋊Dic71—41no—
86C4⋊Dic71—41no—
87C4⋊Dic71—81no—
88C22×C4×C71—21no—
89C22×C4×C71—21no—
90C22×C4×C71C2×D411yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71#4111yescharacteristic
93C2×C8×C71—21no—
94C2×C8×C71—21no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—41no—
99C2×C8×C71—41no—
100C2×C8×C71—41no—
101C2×C8×C71SD1612yes—
102C4⋊C4×C71—21no—
103C4⋊C4×C71—21no—
104C4⋊C4×C71—21no—
105C4⋊C4×C71D812yes—
106C42×C71—41no—
107C42×C71—41no—
108C42×C71C2×D411yescharacteristic
109C42×C71C2×Q811yescharacteristic
110C42×C71#4111yescharacteristic
111C42×C71#4111yescharacteristic
112C23×C71#12111yescharacteristic
113C22×D71—42no—
114C22×D71—81no—
115C22×D71—81no—
116C22×D71—81no—
117C22×D71—81no—
118C22×D71—81no—
119C22×D71—81no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—22no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71#12111yescharacteristic
129C2×C4×C71#12111yescharacteristic
130C2×C4×C71C22⋊Q811yescharacteristic
131C2×C4×C71C22⋊Q811yescharacteristic
132C2×C4×C71C22⋊Q811yescharacteristic
133C2×C4×C71#12611yescharacteristic
134C2×C4×C71C2×D811yescharacteristic
135C2×C4×C71C2×SD1611yescharacteristic
136C2×C4×C71#13511yescharacteristic
137C2×C4×C71#13511yescharacteristic
138C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C71#13711yescharacteristic
142C2×Dic71—81no—
143C2×Dic71—81no—
144C2×Dic71—81no—
145D284—42no—
146D284—42no—
147D284—42no—
148D284—42no—
149D284—82no—
150D284—82no—
151D284—81no—
152D284—81no—
153D284—81no—
154D284—81no—
155C8×C71—22no—
156C8×C71—41no—
157C8×C71—81no—
158C71⋊C8—42no—
159C71⋊C8—81no—
160C22×C71#39611yescharacteristic
161C22×C71#46011yescharacteristic
162C22×C71#46311yescharacteristic
163C22×C71#46611yescharacteristic
164C22×C71#46811yescharacteristic
165C22×C71#47111yescharacteristic
166C22×C71#47211yescharacteristic
167C2×D71—82no—
168C2×D71—82no—
169C2×D71—82no—
170C2×D71—161no—
171C2×D71—161no—
172C2×D71—161no—
173C2×D71—161no—
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—42no—
177C4×C71—81no—
178C4×C71#45911yescharacteristic
179C4×C71#46411yescharacteristic
180C4×C71#46511yescharacteristic
181C4×C71#46911yescharacteristic
182C4×C71#47411yescharacteristic
183C4×C71#47611yescharacteristic
184C4×C71#47811yescharacteristic
185C4×C71#47811yescharacteristic
186Dic71—161no—
187#10599—711nomaximal, Sylow
188C2×C71#166811yescharacteristic
189C2×C71#168111yescharacteristic
190C2×C71#168611yescharacteristic
191C2×C71#169711yescharacteristic
192C2×C71#206311yescharacteristic
193C2×C71#206511yescharacteristic
194C2×C71#206711yescharacteristic
195D71—162no—
196#1982—711no—
197#1957—711no—
198#1948—711no—
199#1939D7111yescharacteristic, complemented
200C2×#335—711no—
201C2×#331—711no—
202#1296—711no—
203C71#1059911yescharacteristic, complemented, Sylow
204C2×#127—711no—
205C2×#106—1421no—
206C2×C4⋊C8—21no—
207C2×C4⋊C8—711no—
208C2×C4⋊C8—711no—
209C2×C4⋊C8C2×D7111yescharacteristic
210C2×#102—1421no—
211C2×#102—1421no—
212C2×#102—1421no—
213C2×#102—1421no—
214C2×C4×C8C2×D7111yescharacteristic
215#384—711no—
216C4×C4⋊C4C2×D7111yescharacteristic
217#340—1421no—
218#336—21no—
219#335—1422no—
220#331—1422no—
221C22×D4—1421no—
222C22×D4—1421no—
223C22×D4—1421no—
224C22×C8—21no—
225C22×C8—1421no—
226C22×C8—1421no—
227C22×C8C22⋊D7111yescharacteristic
228C22×C8C22⋊D7111yescharacteristic
229#127—712no—
230#127—1421no—
231#127—1421no—
232#127—1421no—
233C2×C4⋊C4—1421no—
234C2×C4⋊C4—1421no—
235C2×C4⋊C4—1421no—
236C2×C4⋊C4D28411yescharacteristic
237C2×C4⋊C4D28411yescharacteristic
238C2×C42—21no—
239C2×C42C22×D7111yescharacteristic
240#106—2842no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8—712no—
245C4⋊C8—1421no—
246C4⋊C8—1421no—
247C4⋊C8—1421no—
248#102—2842no—
249#102—2842no—
250#102—2842no—
251#102—2842no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C24—2841no—
256C2×D4—2841no—
257C2×D4—2841no—
258C2×D4—2841no—
259C2×D4—2841no—
260C2×D4—2841no—
261C2×D4—2841no—
262C2×D4—2841no—
263C2×D4—2841no—
264C2×D4—2841no—
265C2×D4—2841no—
266C2×D4—2841no—
267C2×D4—5681no—
268C2×D4—5681no—
269C2×D4—1422no—
270C2×D4—1422no—
271C2×D4—1422no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—2841no—
275C22×C4C2×C22⋊D7111yescharacteristic
276C22×C4C2×D28411yescharacteristic
277C22×C4#1176586511yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—2841no—
287C2×C8—2841no—
288C2×C8—2841no—
289C2×C8—2841no—
290C2×C8—2841no—
291C2×C8—1422no—
292C2×C8#1176585212yes—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—2841no—
297C4⋊C4—2841no—
298C4⋊C4—2841no—
299C4⋊C4—2841no—
300C4⋊C4—5681no—
301C4⋊C4D56812yes—
302C42—41no—
303C42—41no—
304C42D4×D7111yescharacteristic
305C42Q8×D7111yescharacteristic
306C42#1176586411yescharacteristic
307C42#1176586611yescharacteristic
308C23—2842no—
309C23—5681no—
310C23—5681no—
311C23—5681no—
312C23—5681no—
313C23—5681no—
314C23—5681no—
315C23#1455610211yescharacteristic
316D4—2842no—
317D4—2842no—
318D4—2842no—
319D4—2842no—
320D4—5682no—
321D4—5682no—
322D4—5681no—
323D4—5681no—
324D4—5681no—
325D4—5681no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—5681no—
335C2×C4—5681no—
336C2×C4—5681no—
337C2×C4#1455610011yescharacteristic
338C2×C4#1455610311yescharacteristic
339C2×C4#1455610411yescharacteristic
340C2×C4#1455610611yescharacteristic
341C2×C4#1455610811yescharacteristic
342C2×C4#1455612311yescharacteristic
343C2×C4C2×D56811yescharacteristic
344C2×C4C2×#1176585211yescharacteristic
345C2×C4#1455614811yescharacteristic
346C2×C4#1455615011yescharacteristic
347C2×C4#1455615311yescharacteristic
348C2×C4#1455615411yescharacteristic
349C2×C4#1455615611yescharacteristic
350C2×C4#1455616111yescharacteristic
351C8—22no—
352C8—41no—
353C8—81no—
354C8—2842no—
355C8—5681no—
356C22—5682no—
357C22—5682no—
358C22—5682no—
359C22—11361no—
360C22—11361no—
361C22—11361no—
362C22—11361no—
363C22#1798592111yescharacteristic
364C22#1798619911yescharacteristic
365C22#1798621411yescharacteristic
366C22#1798623511yescharacteristic
367C22#1798624511yescharacteristic
368C22#1798626211yescharacteristic
369C22#1798626911yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—11361no—
375C4#1798619511yescharacteristic
376C4#1798622211yescharacteristic
377C4#1798623011yescharacteristic
378C4#1798625011yescharacteristic
379C4#1798628011yescharacteristic
380C4#1798628611yescharacteristic
381C4#1798629611yescharacteristic
382C4#1798630011yescharacteristic
383C2—11362no—
384C2#2258770011yescharacteristic
385C2#2258777711yescharacteristic
386C2#2258781211yescharacteristic
387C2#2258787311yescharacteristic
388C2#2258969911yescharacteristic
389C2#2258971111yescharacteristic
390C2#2258972711yescharacteristic
subgroupG/H
centerC23#14556102
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1939C2
socleC23×C71#121
prime radicalC2×#14556125C22
hypercenter#1939D71
hypocenterC71#10599
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939

Complements

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

NH ≅ G/Ntype
3. #22589103C2semidirect
4. C2×#17985752C2semidirect
6. #22586224C2semidirect
7. C71×#1939C2semidirect
199. #1939D71semidirect
203. C71#10599semidirect

Maximal subgroups

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

subgroupindex
#225892672
#225891462
#225891032
C2×#179857522
C2×#179857422
#225862242
C71×#19392
#1059971