Finite Group Databasebeta

← #31901674

Subgroups — #31901674

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986347C211yesmaximal, characteristic
2#22589267C211yesmaximal, characteristic
3#22589059C211yesmaximal, characteristic, complemented
4C2×#17985748C211yesmaximal, characteristic, complemented
5C2×#17985742C211yesmaximal, characteristic
6#22586174C211yesmaximal, characteristic, complemented
7C71×#1987C211yesmaximal, characteristic, complemented
8C2×#14556125C2211yescharacteristic
9#17986347—22no—
10#17986347—22no—
11C2×C4⋊(C71⋊C8)—21no—
12C2×C4⋊(C71⋊C8)C2211yescharacteristic
13C2×C4⋊(C71⋊C8)C2211yescharacteristic
14C2×#14556048—21no—
15C2×#14556048—21no—
16C2×#14556048—21no—
17C2×#14556048—21no—
18C2×C4×C71⋊C8C2211yescharacteristic
19#17985754—21no—
20#17985748—22no—
21#17985742—22no—
22C2×C4⋊Q8×C71C2211yescharacteristic
23C2×C4⋊C8×C71C2211yescharacteristic
24C4×C4⋊C4×C71C2211yescharacteristic
25#340×C71—21no—
26#340×C71—21no—
27C22×D284—21no—
28C22×D284—21no—
29C22×D284—21no—
30C22×C71⋊C8—21no—
31C22×C71⋊C8—21no—
32C22×C71⋊C8D411yescharacteristic
33C22×C71⋊C8D411yescharacteristic
34#14556125—21no—
35#14556125—21no—
36#14556125—21no—
37#14556125D412yes—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)—21no—
42C4⋊(C71⋊C8)—42no—
43#14556048—24no—
44#14556048—24no—
45#14556048—42no—
46#14556048—42no—
47C4×C71⋊C8—21no—
48C4×C71⋊C8—21no—
49C22×Q8×C71—21no—
50C22×C8×C71—21no—
51C4⋊Q8×C71—21no—
52C4⋊Q8×C71—21no—
53C4⋊Q8×C71—21no—
54C4⋊Q8×C71D412yes—
55C2×C4⋊C4×C71—21no—
56C2×C4⋊C4×C71—21no—
57C2×C4⋊C4×C71D411yescharacteristic
58C2×C4⋊C4×C71D411yescharacteristic
59C2×C42×C71—21no—
60C2×C42×C71C2311yescharacteristic
61C4⋊C8×C71—21no—
62C4⋊C8×C71—21no—
63#95×C71—21no—
64C23×D71—41no—
65C2×D284—22no—
66C2×D284—22no—
67C2×D284—22no—
68C2×D284—41no—
69C2×D284—41no—
70C2×D284—41no—
71C2×D284—41no—
72C2×D284—41no—
73C2×D284—41no—
74C2×D284—41no—
75C2×D284—41no—
76C2×D284—41no—
77C2×D284—41no—
78C2×D284—41no—
79C2×D284—81no—
80C2×D284—81no—
81C2×C71⋊C8—21no—
82C2×C71⋊C8—21no—
83C2×C71⋊C8—21no—
84C2×C71⋊C8—21no—
85C2×C71⋊C8—21no—
86C2×C71⋊C8—21no—
87C2×C71⋊C8—41no—
88C2×C71⋊C8—41no—
89C2×C71⋊C8—41no—
90C2×C71⋊C8—41no—
91C2×C71⋊C8—41no—
92C2×C71⋊C8—41no—
93C2×Q8×C71—22no—
94C2×Q8×C71—41no—
95C2×Q8×C71—81no—
96C2×Q8×C71—81no—
97C22×C4×C71—21no—
98C22×C4×C71—21no—
99C22×C4×C71—41no—
100C22×C4×C71C2×D411yescharacteristic
101C22×C4×C71C2×D411yescharacteristic
102C22×C4×C71#4111yescharacteristic
103C2×C8×C71—41no—
104C2×C8×C71—41no—
105C2×C8×C71—41no—
106C4⋊C4×C71—21no—
107C4⋊C4×C71—22no—
108C4⋊C4×C71—22no—
109C4⋊C4×C71—21no—
110C4⋊C4×C71—21no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C4⋊C4×C71SD1612yes—
114C42×C71—41no—
115C42×C71—41no—
116C42×C71C2×D411yescharacteristic
117C42×C71C2×D411yescharacteristic
118C42×C71#4111yescharacteristic
119C42×C71#4111yescharacteristic
120C23×C71#12111yescharacteristic
121C22×D71—42no—
122C22×D71—81no—
123C22×D71—81no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71—81no—
127C22×D71—81no—
128Q8×C71—82no—
129Q8×C71—82no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—22no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—42no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—81no—
140C2×C4×C71—81no—
141C2×C4×C71#12011yescharacteristic
142C2×C4×C71#12111yescharacteristic
143C2×C4×C71#12111yescharacteristic
144C2×C4×C71#12311yescharacteristic
145C2×C4×C71#12311yescharacteristic
146C2×C4×C71#12411yescharacteristic
147C2×C4×C71C2×SD1611yescharacteristic
148C2×C4×C71#13511yescharacteristic
149C2×C4×C71#13511yescharacteristic
150C2×C4×C71#13511yescharacteristic
151C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C71#13711yescharacteristic
154C2×C4×C71#13711yescharacteristic
155D284—42no—
156D284—42no—
157D284—42no—
158D284—42no—
159D284—82no—
160D284—82no—
161D284—81no—
162D284—81no—
163D284—81no—
164D284—81no—
165C8×C71—81no—
166C71⋊C8—41no—
167C71⋊C8—41no—
168C71⋊C8—81no—
169C71⋊C8—81no—
170C22×C71#39411yescharacteristic
171C22×C71#46311yescharacteristic
172C22×C71#46411yescharacteristic
173C22×C71#46511yescharacteristic
174C22×C71#46811yescharacteristic
175C22×C71#47211yescharacteristic
176C22×C71#47211yescharacteristic
177C2×D71—82no—
178C2×D71—82no—
179C2×D71—82no—
180C2×D71—161no—
181C2×D71—161no—
182C2×D71—161no—
183C2×D71—161no—
184C4×C71—21no—
185C4×C71—21no—
186C4×C71—42no—
187C4×C71—81no—
188C4×C71—82no—
189C4×C71#44911yescharacteristic
190C4×C71#45211yescharacteristic
191C4×C71#46011yescharacteristic
192C4×C71#46111yescharacteristic
193C4×C71#46311yescharacteristic
194C4×C71#46411yescharacteristic
195C4×C71#48811yescharacteristic
196C4×C71#48911yescharacteristic
197#10539—711nomaximal, Sylow
198C2×C71#167711yescharacteristic
199C2×C71#167811yescharacteristic
200C2×C71#169711yescharacteristic
201C2×C71#169811yescharacteristic
202C2×C71#204211yescharacteristic
203C2×C71#204511yescharacteristic
204C2×C71#205511yescharacteristic
205D71—162no—
206C2×#486—711no—
207#1987D7111yescharacteristic, complemented
208#1982—711no—
209#1939—711no—
210C2×#333—711no—
211C2×#331—711no—
212#1277—711no—
213C71#1053911yescharacteristic, complemented, Sylow
214C2×C4⋊Q8C2×D7111yescharacteristic
215C2×#127—711no—
216#486—1422no—
217#486—1422no—
218C2×C4⋊C8—711no—
219C2×C4⋊C8—711no—
220C2×C4⋊C8—1421no—
221C2×C4⋊C8C2×D7111yescharacteristic
222C2×#102—1421no—
223C2×#102—1421no—
224C2×#102—1421no—
225C2×#102—1421no—
226C2×C4×C8—711no—
227C4×C4⋊C4C2×D7111yescharacteristic
228#340—21no—
229#340—21no—
230#336—1421no—
231#333—1422no—
232#331—1422no—
233C22×Q8—21no—
234C22×D4—1421no—
235C22×D4—1421no—
236C22×D4—1421no—
237C22×C8—21no—
238C22×C8—711no—
239C22×C8—711no—
240C22×C8—1421no—
241C22×C8—1421no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8D28412yes—
246#127—712no—
247#127—1421no—
248#127—1421no—
249#127—1421no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊D7111yescharacteristic
253C2×C4⋊C4C22⋊D7111yescharacteristic
254C2×C42—21no—
255C2×C42C22×D7111yescharacteristic
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—2842no—
259C4⋊C8—1421no—
260C4⋊C8—1421no—
261C4⋊C8—1421no—
262C4⋊C8—1421no—
263#102—2842no—
264#102—2842no—
265#102—1424no—
266#102—1424no—
267C4×C8—1421no—
268C4×C8—1421no—
269#95—21no—
270C24—2841no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—81no—
274C2×Q8—81no—
275C2×D4—2841no—
276C2×D4—2841no—
277C2×D4—2841no—
278C2×D4—2841no—
279C2×D4—2841no—
280C2×D4—2841no—
281C2×D4—2841no—
282C2×D4—2841no—
283C2×D4—2841no—
284C2×D4—2841no—
285C2×D4—2841no—
286C2×D4—5681no—
287C2×D4—5681no—
288C2×D4—1422no—
289C2×D4—1422no—
290C2×D4—1422no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4D4×D7111yescharacteristic
295C22×C4C2×C22⋊D7111yescharacteristic
296C22×C4#1176586411yescharacteristic
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—2841no—
301C2×C8—2841no—
302C2×C8—2841no—
303C2×C8—2841no—
304C2×C8—2841no—
305C2×C8—2841no—
306C2×C8—1421no—
307C2×C8—1421no—
308C2×C8—1421no—
309C2×C8—1421no—
310C2×C8—1421no—
311C2×C8—1421no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4#1176585212yes—
320C42—41no—
321C42—41no—
322C42D4×D7111yescharacteristic
323C42C2×D28411yescharacteristic
324C42#1176586411yescharacteristic
325C42#1176586511yescharacteristic
326C23—2842no—
327C23—5681no—
328C23—5681no—
329C23—5681no—
330C23—5681no—
331C23—5681no—
332C23—5681no—
333C23#1455610011yescharacteristic
334Q8—82no—
335Q8—82no—
336D4—2842no—
337D4—2842no—
338D4—2842no—
339D4—2842no—
340D4—5682no—
341D4—5682no—
342D4—5681no—
343D4—5681no—
344D4—5681no—
345D4—5681no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#1455609811yescharacteristic
358C2×C4#1455609911yescharacteristic
359C2×C4#1455610211yescharacteristic
360C2×C4#1455611011yescharacteristic
361C2×C4#1455611211yescharacteristic
362C2×C4#1455611411yescharacteristic
363C2×C4C2×#1176585211yescharacteristic
364C2×C4#1455614811yescharacteristic
365C2×C4#1455614911yescharacteristic
366C2×C4#1455615111yescharacteristic
367C2×C4#1455615311yescharacteristic
368C2×C4#1455615511yescharacteristic
369C2×C4#1455615911yescharacteristic
370C2×C4#1455616011yescharacteristic
371C8—81no—
372C8—2841no—
373C8—2841no—
374C8—5681no—
375C8—5681no—
376C22—5682no—
377C22—5682no—
378C22—5682no—
379C22—11361no—
380C22—11361no—
381C22—11361no—
382C22—11361no—
383C22#1798591511yescharacteristic
384C22#1798621611yescharacteristic
385C22#1798622311yescharacteristic
386C22#1798623011yescharacteristic
387C22#1798624711yescharacteristic
388C22#1798626711yescharacteristic
389C22#1798626711yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4#1798614711yescharacteristic
396C4#1798616411yescharacteristic
397C4#1798619911yescharacteristic
398C4#1798620111yescharacteristic
399C4#1798621111yescharacteristic
400C4#1798622111yescharacteristic
401C4#1798635511yescharacteristic
402C4#1798636011yescharacteristic
403C2—11362no—
404C2#2258775111yescharacteristic
405C2#2258775811yescharacteristic
406C2#2258787511yescharacteristic
407C2#2258788211yescharacteristic
408C2#2258957811yescharacteristic
409C2#2258959511yescharacteristic
410C2#2258965511yescharacteristic
subgroupG/H
centerC23#14556100
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1987C2
socleC23×C71#121
prime radicalC2×#14556125C22
hypercenter#1987D71
hypocenterC71#10539
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. #22589059C2semidirect
4. C2×#17985748C2semidirect
6. #22586174C2semidirect
7. C71×#1987C2semidirect
207. #1987D71semidirect
213. C71#10539semidirect

Maximal subgroups

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

subgroupindex
C2×#179863472
#225892672
#225890592
C2×#179857482
C2×#179857422
#225861742
C71×#19872
#1053971