Finite Group Databasebeta

← #31902207

Subgroups — #31902207

Order 18176.

402 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589134C211yesmaximal, characteristic
2#22588337C211yesmaximal, characteristic
3#22588337C211yesmaximal, characteristic
4#22586223C211yesmaximal, characteristic, complemented
5#22586223C211yesmaximal, characteristic, complemented
6#22586175C211yesmaximal, characteristic, complemented
7C71×#1834C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×C4⋊(C71⋊C8)C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556048—21no—
13C2×#14556048—21no—
14C2×(C2×D71)⋊C8—21no—
15C2×(C2×D71)⋊C8—21no—
16#17985761—21no—
17#17985761—21no—
18C2×C4×C8×C71C2211yescharacteristic
19#384×C71C2211yescharacteristic
20#384×C71C2211yescharacteristic
21#340×C71—21no—
22#340×C71—21no—
23C22×D284D411yescharacteristic
24C22×C4×D71—21no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8—21no—
27C22×C71⋊C8—21no—
28C4×D284—21no—
29C4×D284—21no—
30C4×D284—21no—
31C4×D284—21no—
32C2×C4⋊Dic71D411yescharacteristic
33C2×(C2×D71)⋊C4—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36C4⋊(C71⋊C8)—21no—
37C4⋊(C71⋊C8)—21no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40#14556048—42no—
41#14556048—42no—
42(C2×D71)⋊C8—42no—
43(C2×D71)⋊C8—42no—
44C22×C8×C71D411yescharacteristic
45C22×C8×C71D411yescharacteristic
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C4⋊C4×C71—21no—
50C2×C4⋊C4×C71—21no—
51C2×C4⋊C4×C71—21no—
52C2×C42×C71C2311yescharacteristic
53C4×C8×C71—21no—
54C4×C8×C71—21no—
55C23×D71—21no—
56C2×D284—21no—
57C2×D284—21no—
58C2×D284—21no—
59C2×D284—21no—
60C2×D284—21no—
61C2×D284SD1612yes—
62C22×Dic71—21no—
63C2×C4×D71—41no—
64C2×C4×D71—41no—
65C2×C4×D71—41no—
66C2×C4×D71—41no—
67C2×C4×D71—41no—
68C2×C4×D71—41no—
69C2×C71⋊C8—41no—
70C2×C71⋊C8—41no—
71C2×C71⋊C8—41no—
72C2×C71⋊C8—41no—
73C2×C71⋊C8—41no—
74C2×C71⋊C8—41no—
75C2×C71⋊C8—41no—
76C2×C71⋊C8—41no—
77C2×C71⋊C8—41no—
78C4⋊Dic71—21no—
79C4⋊Dic71—21no—
80(C2×D71)⋊C4—41no—
81(C2×D71)⋊C4—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×D411yescharacteristic
86C22×C4×C71#4111yescharacteristic
87C2×C8×C71—21no—
88C2×C8×C71—21no—
89C2×C8×C71—21no—
90C2×C8×C71—21no—
91C2×C8×C71—21no—
92C2×C8×C71—21no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—81no—
102C4⋊C4×C71—81no—
103C42×C71#4111yescharacteristic
104C42×C71#4111yescharacteristic
105C42×C71#4111yescharacteristic
106C42×C71#4111yescharacteristic
107C23×C71#12111yescharacteristic
108C22×D71—22no—
109C22×D71—41no—
110C22×D71—41no—
111C22×D71—41no—
112C22×D71—41no—
113C22×D71—41no—
114C22×D71—41no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—81no—
118C2×C4×C71—81no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71#12311yescharacteristic
124C2×C4×C71#12311yescharacteristic
125C2×C4×C71#12311yescharacteristic
126C2×C4×C71#12411yescharacteristic
127C2×C4×C71#12611yescharacteristic
128C2×C4×C71#12611yescharacteristic
129C2×C4×C71C2×SD1611yescharacteristic
130C2×C4×C71#13511yescharacteristic
131C2×C4×C71#13511yescharacteristic
132C2×C4×C71#13511yescharacteristic
133C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C71#13711yescharacteristic
137C2×Dic71—41no—
138C2×Dic71—41no—
139C2×Dic71—41no—
140D284—22no—
141D284—22no—
142D284—41no—
143D284—41no—
144C4×D71—81no—
145C4×D71—81no—
146C4×D71—81no—
147C4×D71—81no—
148C8×C71—41no—
149C8×C71—41no—
150C71⋊C8—81no—
151C71⋊C8—81no—
152C71⋊C8—81no—
153C22×C71#39711yescharacteristic
154C22×C71#46311yescharacteristic
155C22×C71#46311yescharacteristic
156C22×C71#46511yescharacteristic
157C22×C71#46811yescharacteristic
158C22×C71#47211yescharacteristic
159C22×C71#47311yescharacteristic
160C2×D71—42no—
161C2×D71—42no—
162C2×D71—42no—
163C2×D71—81no—
164C2×D71—81no—
165C2×D71—81no—
166C2×D71—81no—
167C4×C71—21no—
168C4×C71—21no—
169C4×C71—161no—
170C4×C71—161no—
171C4×C71#48111yescharacteristic
172C4×C71#48211yescharacteristic
173C4×C71#48211yescharacteristic
174C4×C71#48211yescharacteristic
175C4×C71#48311yescharacteristic
176C4×C71#48511yescharacteristic
177C4×C71#48811yescharacteristic
178C4×C71#49011yescharacteristic
179Dic71—81no—
180#10625—711nomaximal, Sylow
181C2×C71#167711yescharacteristic
182C2×C71#167911yescharacteristic
183C2×C71#169711yescharacteristic
184C2×C71#169911yescharacteristic
185C2×C71#207311yescharacteristic
186C2×C71#207711yescharacteristic
187C2×C71#207811yescharacteristic
188D71—82no—
189#1954—711no—
190#1834D7111yescharacteristic, complemented
191#1802—711no—
192#1802—711no—
193#1296—711no—
194#1296—711no—
195#1277—711no—
196C71#1062511yescharacteristic, complemented, Sylow
197C2×C4×D4—711no—
198C2×C4⋊C8—711no—
199C2×C4⋊C8—711no—
200C2×C4⋊C8—711no—
201C2×#102—1421no—
202C2×#102—1421no—
203C2×C22⋊C8—1421no—
204C2×C22⋊C8—1421no—
205C2×C4×C8C2×D7111yescharacteristic
206#384C2×D7111yescharacteristic
207#384C2×D7111yescharacteristic
208#340—21no—
209#340—21no—
210#340—1421no—
211#340—1421no—
212C22×D4—711no—
213C23×C4—1421no—
214C22×C8—1421no—
215C22×C8—1421no—
216C22×C8—1421no—
217C22×C8C22⋊D7111yescharacteristic
218C22×C8C22⋊D7111yescharacteristic
219C4×D4—1421no—
220C4×D4—1421no—
221C4×D4—1421no—
222C4×D4—1421no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—711no—
230C2×C22⋊C4—1421no—
231C2×C42C22×D7111yescharacteristic
232C4⋊C8—1421no—
233C4⋊C8—1421no—
234C4⋊C8—1421no—
235C4⋊C8—1421no—
236C4⋊C8—1421no—
237C4⋊C8—1421no—
238#102—2842no—
239#102—2842no—
240C22⋊C8—2842no—
241C22⋊C8—2842no—
242C4×C8—21no—
243C4×C8—21no—
244C24—1421no—
245C2×D4—712no—
246C2×D4—1421no—
247C2×D4—1421no—
248C2×D4—1421no—
249C2×D4—1421no—
250C2×D4—1421no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—2841no—
254C22×C4—2841no—
255C22×C4—2841no—
256C22×C4—2841no—
257C22×C4—2841no—
258C22×C4—2841no—
259C22×C4—1421no—
260C22×C4D4×D7111yescharacteristic
261C22×C4C2×C22⋊D7111yescharacteristic
262C22×C4#1176586411yescharacteristic
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—2841no—
270C2×C8—2841no—
271C2×C8—2841no—
272C2×C8—2841no—
273C2×C8—2841no—
274C2×C8—2841no—
275C2×C8—2841no—
276C2×C8—2841no—
277C2×C8—2841no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—1421no—
289C4⋊C4—1421no—
290C22⋊C4—2841no—
291C22⋊C4—2841no—
292C42#1176586511yescharacteristic
293C42#1176586511yescharacteristic
294C42#1176586611yescharacteristic
295C42#1176586611yescharacteristic
296C23—2841no—
297C23—2841no—
298C23—2841no—
299C23—2841no—
300C23—2841no—
301C23—2841no—
302C23—1422no—
303C23#1455609911yescharacteristic
304D4—2841no—
305D4—2841no—
306D4—1422no—
307D4—1422no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—2841no—
317C2×C4—2841no—
318C2×C4—2841no—
319C2×C4—5681no—
320C2×C4—5681no—
321C2×C4—5681no—
322C2×C4—5681no—
323C2×C4#1455611111yescharacteristic
324C2×C4#1455611311yescharacteristic
325C2×C4#1455611311yescharacteristic
326C2×C4#1455611711yescharacteristic
327C2×C4#1455612311yescharacteristic
328C2×C4#1455612311yescharacteristic
329C2×C4SD16×D7111yescharacteristic
330C2×C4#1455614811yescharacteristic
331C2×C4#1455614811yescharacteristic
332C2×C4#1455614911yescharacteristic
333C2×C4#1455615311yescharacteristic
334C2×C4#1455615311yescharacteristic
335C2×C4#1455615511yescharacteristic
336C2×C4#1455615711yescharacteristic
337C8—41no—
338C8—41no—
339C8—5681no—
340C8—5681no—
341C8—5681no—
342C22—2842no—
343C22—2842no—
344C22—2842no—
345C22—5681no—
346C22—5681no—
347C22—5681no—
348C22—5681no—
349C22#1798592511yescharacteristic
350C22#1798621111yescharacteristic
351C22#1798621111yescharacteristic
352C22#1798622511yescharacteristic
353C22#1798624211yescharacteristic
354C22#1798626611yescharacteristic
355C22#1798627111yescharacteristic
356C4—21no—
357C4—21no—
358C4—161no—
359C4—161no—
360C4—5681no—
361C4#1798631911yescharacteristic
362C4#1798632311yescharacteristic
363C4#1798632311yescharacteristic
364C4#1798632511yescharacteristic
365C4#1798633111yescharacteristic
366C4#1798634311yescharacteristic
367C4#1798635611yescharacteristic
368C4#1798636511yescharacteristic
369C2—5682no—
370C2#2258774611yescharacteristic
371C2#2258776011yescharacteristic
372C2#2258787011yescharacteristic
373C2#2258788411yescharacteristic
374C2#2258975411yescharacteristic
375C2#2258977411yescharacteristic
376C2#2258978111yescharacteristic
subgroupG/H
centerC23#14556099
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1834C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenter#1834D71
hypocenterC71#10625
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
4. #22586223C2semidirect
5. #22586223C2semidirect
6. #22586175C2semidirect
7. C71×#1834C2semidirect
190. #1834D71semidirect
196. C71#10625semidirect

Maximal subgroups

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

subgroupindex
#225891342
#225883372
#225862232
#225861752
C71×#18342
#1062571