Finite Group Databasebeta

← #31900493

Subgroups — #31900493

Order 18176.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589267C211yesmaximal, characteristic
2#22589267C211yesmaximal, characteristic
3#22589145C211yesmaximal, characteristic
4#22586223C211yesmaximal, characteristic, complemented
5#22586181C211yesmaximal, characteristic, complemented
6#22586181C211yesmaximal, characteristic, complemented
7C71×#2552C211yesmaximal, characteristic, complemented
8C2×#14556125C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×#14556048—21no—
12C2×#14556048—21no—
13C2×#14556048—21no—
14C2×#14556048—21no—
15C2×#14556048—22no—
16C2×C4×C71⋊C8C2211yescharacteristic
17C2×#125×C71—21no—
18#398×C71—22no—
19#389×C71—21no—
20#389×C71—21no—
21#384×C71—21no—
22#384×C71C2211yescharacteristic
23C4×C4⋊C4×C71C2211yescharacteristic
24C4×C4⋊C4×C71C2211yescharacteristic
25C22×D284—21no—
26C22×D284—21no—
27C22×D284—21no—
28C22×C71⋊C8—21no—
29C22×C71⋊C8—21no—
30C22×C71⋊C8—21no—
31#14556125—21no—
32#14556125—21no—
33#14556125—21no—
34#14556125Q812yes—
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39#14556048—42no—
40#14556048—42no—
41#14556048—44no—
42#14556048—42no—
43#14556048—42no—
44C4×C71⋊C8—21no—
45C4×C71⋊C8—21no—
46#125×C71—42no—
47#125×C71—42no—
48C2×C4⋊C4×C71—22no—
49C2×C4⋊C4×C71—22no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71—21no—
52C2×C4⋊C4×C71—22no—
53C2×C4⋊C4×C71D411yescharacteristic
54C2×C4⋊C4×C71D411yescharacteristic
55C2×C4⋊C4×C71D411yescharacteristic
56C2×C4⋊C4×C71D411yescharacteristic
57C2×C4⋊C4×C71D412yes—
58C2×C42×C71—21no—
59C2×C42×C71—21no—
60C2×C42×C71C2311yescharacteristic
61#95×C71—22no—
62#95×C71—21no—
63#95×C71—21no—
64C23×D71—41no—
65C2×D284—22no—
66C2×D284—22no—
67C2×D284—22no—
68C2×D284—42no—
69C2×D284—42no—
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—41no—
80C2×D284—41no—
81C2×C71⋊C8—22no—
82C2×C71⋊C8—41no—
83C2×C71⋊C8—41no—
84C2×C71⋊C8—41no—
85C2×C71⋊C8—41no—
86C2×C71⋊C8—41no—
87C2×C71⋊C8—41no—
88C2×C71⋊C8—41no—
89C2×C71⋊C8—41no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71—22no—
95C22×C4×C71C2×D411yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C22×C4×C71C2×D411yescharacteristic
98C4⋊C4×C71—21no—
99C4⋊C4×C71—21no—
100C4⋊C4×C71—21no—
101C4⋊C4×C71—21no—
102C4⋊C4×C71—21no—
103C4⋊C4×C71—21no—
104C4⋊C4×C71—42no—
105C4⋊C4×C71—44no—
106C4⋊C4×C71—44no—
107C4⋊C4×C71—41no—
108C4⋊C4×C71—41no—
109C4⋊C4×C71—41no—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—82no—
112C4⋊C4×C71D812yes—
113C4⋊C4×C71SD1612yes—
114C42×C71—41no—
115C42×C71—41no—
116C42×C71—41no—
117C42×C71—41no—
118C42×C71C2×Q811yescharacteristic
119C42×C71#4111yescharacteristic
120C42×C71#4111yescharacteristic
121C42×C71#4111yescharacteristic
122C23×C71#12011yescharacteristic
123C22×D71—42no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71—81no—
127C22×D71—81no—
128C22×D71—81no—
129C22×D71—81no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—42no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71C22⋊Q811yescharacteristic
148C2×C4×C71C22⋊Q811yescharacteristic
149C2×C4×C71C22⋊Q811yescharacteristic
150C2×C4×C71#12411yescharacteristic
151C2×C4×C71#12411yescharacteristic
152C2×C4×C71#12411yescharacteristic
153C2×C4×C71C2×D811yescharacteristic
154C2×C4×C71C2×SD1611yescharacteristic
155C2×C4×C71#13511yescharacteristic
156C2×C4×C71#13511yescharacteristic
157C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C71#13711yescharacteristic
161D284—44no—
162D284—42no—
163D284—42no—
164D284—42no—
165D284—42no—
166D284—82no—
167D284—81no—
168D284—81no—
169D284—81no—
170D284—81no—
171C71⋊C8—42no—
172C71⋊C8—81no—
173C71⋊C8—81no—
174C22×C71#39311yescharacteristic
175C22×C71#44711yescharacteristic
176C22×C71#44811yescharacteristic
177C22×C71#44911yescharacteristic
178C22×C71#44911yescharacteristic
179C22×C71#45311yescharacteristic
180C22×C71#45411yescharacteristic
181C2×D71—82no—
182C2×D71—82no—
183C2×D71—82no—
184C2×D71—161no—
185C2×D71—161no—
186C2×D71—161no—
187C2×D71—161no—
188C4×C71—21no—
189C4×C71—21no—
190C4×C71—42no—
191C4×C71—42no—
192C4×C71—81no—
193C4×C71—82no—
194C4×C71—81no—
195C4×C71#47411yescharacteristic
196C4×C71#47611yescharacteristic
197C4×C71#47811yescharacteristic
198C4×C71#47811yescharacteristic
199C4×C71#48611yescharacteristic
200C4×C71#48811yescharacteristic
201C4×C71#48911yescharacteristic
202C4×C71#49011yescharacteristic
203#10354—711nomaximal, Sylow
204C2×C71#162111yescharacteristic
205C2×C71#162211yescharacteristic
206C2×C71#162511yescharacteristic
207C2×C71#162611yescharacteristic
208C2×C71#202311yescharacteristic
209C2×C71#202511yescharacteristic
210C2×C71#202711yescharacteristic
211D71—162no—
212#2552D7111yescharacteristic, complemented
213#1982—711no—
214#1982—711no—
215#1957—711no—
216#1296—711no—
217#1279—711no—
218#1279—711no—
219C71#1035411yescharacteristic, complemented, Sylow
220C2×#127—711no—
221C2×#125—21no—
222C2×C4⋊C8—711no—
223C2×C4⋊C8—711no—
224C2×#102—1422no—
225C2×#102—1421no—
226C2×#102—1421no—
227C2×#102—1421no—
228C2×#102—1421no—
229C2×C4×C8—711no—
230#398—22no—
231#389—21no—
232#389—21no—
233#384—21no—
234#384C2×D7111yescharacteristic
235C4×C4⋊C4C2×D7111yescharacteristic
236C4×C4⋊C4C2×D7111yescharacteristic
237C22×D4—1421no—
238C22×D4—1421no—
239C22×D4—1421no—
240C22×C8—1421no—
241C22×C8—1421no—
242C22×C8—1421no—
243#127—712no—
244#127—1421no—
245#127—1421no—
246#127—1421no—
247#125—42no—
248#125—42no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C22⋊D7111yescharacteristic
255C2×C4⋊C4C22⋊D7111yescharacteristic
256C2×C4⋊C4C22⋊D7111yescharacteristic
257C2×C4⋊C4C22⋊D7111yescharacteristic
258C2×C4⋊C4C22⋊D7112yes—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C22×D7111yescharacteristic
262C4⋊C8—1421no—
263C4⋊C8—1421no—
264C4⋊C8—1421no—
265C4⋊C8—1421no—
266#102—2842no—
267#102—2842no—
268#102—2844no—
269#102—2842no—
270#102—2842no—
271C4×C8—1421no—
272C4×C8—1421no—
273#95—22no—
274#95—21no—
275#95—21no—
276C24—2841no—
277C2×D4—2842no—
278C2×D4—2842no—
279C2×D4—2841no—
280C2×D4—2841no—
281C2×D4—2841no—
282C2×D4—2841no—
283C2×D4—2841no—
284C2×D4—2841no—
285C2×D4—2841no—
286C2×D4—2841no—
287C2×D4—2841no—
288C2×D4—2841no—
289C2×D4—2841no—
290C2×D4—1422no—
291C2×D4—1422no—
292C2×D4—1422no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4C2×C22⋊D7111yescharacteristic
299C22×C4C2×C22⋊D7111yescharacteristic
300C22×C4C2×C22⋊D7111yescharacteristic
301C2×C8—2841no—
302C2×C8—2841no—
303C2×C8—2841no—
304C2×C8—2841no—
305C2×C8—2841no—
306C2×C8—2841no—
307C2×C8—2841no—
308C2×C8—2841no—
309C2×C8—1422no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—82no—
324C4⋊C4#1176585012yes—
325C4⋊C4#1176585212yes—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42Q8×D7111yescharacteristic
331C42#1176586611yescharacteristic
332C42#1176586611yescharacteristic
333C42#1176586611yescharacteristic
334C23—2842no—
335C23—5681no—
336C23—5681no—
337C23—5681no—
338C23—5681no—
339C23—5681no—
340C23—5681no—
341C23#1455609711yescharacteristic
342D4—2844no—
343D4—2842no—
344D4—2842no—
345D4—2842no—
346D4—2842no—
347D4—5682no—
348D4—5681no—
349D4—5681no—
350D4—5681no—
351D4—5681no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4#1455610411yescharacteristic
370C2×C4#1455610411yescharacteristic
371C2×C4#1455610411yescharacteristic
372C2×C4#1455611711yescharacteristic
373C2×C4#1455611711yescharacteristic
374C2×C4#1455611711yescharacteristic
375C2×C4C2×#1176585011yescharacteristic
376C2×C4C2×#1176585211yescharacteristic
377C2×C4#1455614811yescharacteristic
378C2×C4#1455614811yescharacteristic
379C2×C4#1455615311yescharacteristic
380C2×C4#1455615311yescharacteristic
381C2×C4#1455615411yescharacteristic
382C2×C4#1455615911yescharacteristic
383C8—2842no—
384C8—5681no—
385C8—5681no—
386C22—5682no—
387C22—5682no—
388C22—5682no—
389C22—11361no—
390C22—11361no—
391C22—11361no—
392C22—11361no—
393C22#1798591011yescharacteristic
394C22#1798613311yescharacteristic
395C22#1798613811yescharacteristic
396C22#1798614411yescharacteristic
397C22#1798614411yescharacteristic
398C22#1798616711yescharacteristic
399C22#1798617211yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—81no—
405C4—82no—
406C4—81no—
407C4#1798627611yescharacteristic
408C4#1798628611yescharacteristic
409C4#1798629611yescharacteristic
410C4#1798629611yescharacteristic
411C4#1798634711yescharacteristic
412C4#1798635611yescharacteristic
413C4#1798636111yescharacteristic
414C4#1798636511yescharacteristic
415C2—11362no—
416C2#2258740511yescharacteristic
417C2#2258741211yescharacteristic
418C2#2258742911yescharacteristic
419C2#2258743611yescharacteristic
420C2#2258947311yescharacteristic
421C2#2258948311yescharacteristic
422C2#2258949311yescharacteristic
subgroupG/H
centerC23#14556097
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#2552C2
socleC23×C71#120
prime radicalC2×#14556125C22
hypercenter#2552D71
hypocenterC71#10354
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2552

Complements

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

NH ≅ G/Ntype
4. #22586223C2semidirect
5. #22586181C2semidirect
6. #22586181C2semidirect
7. C71×#2552C2semidirect
212. #2552D71semidirect
219. C71#10354semidirect

Maximal subgroups

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

subgroupindex
#225892672
#225891452
#225862232
#225861812
C71×#25522
#1035471