Finite Group Databasebeta

← #31902173

Subgroups — #31902173

Order 18176.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 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×#1844C211yesmaximal, 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×#106×C71—21no—
19C2×#106×C71—21no—
20C2×C4×C8×C71C2211yescharacteristic
21#384×C71C2211yescharacteristic
22#384×C71C2211yescharacteristic
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
53#106×C71—42no—
54#106×C71—42no—
55C4×C8×C71—21no—
56C4×C8×C71—21no—
57C23×D71—21no—
58C2×D284—21no—
59C2×D284—21no—
60C2×D284—21no—
61C2×D284—21no—
62C2×D284—21no—
63C2×D284SD1612yes—
64C22×Dic71—21no—
65C2×C4×D71—41no—
66C2×C4×D71—41no—
67C2×C4×D71—41no—
68C2×C4×D71—41no—
69C2×C4×D71—41no—
70C2×C4×D71—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—
78C2×C71⋊C8—41no—
79C2×C71⋊C8—41no—
80C4⋊Dic71—21no—
81C4⋊Dic71—21no—
82(C2×D71)⋊C4—41no—
83(C2×D71)⋊C4—41no—
84C22×C4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71C2×D411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71#4111yescharacteristic
89C2×C8×C71—21no—
90C2×C8×C71—21no—
91C2×C8×C71—21no—
92C2×C8×C71—21no—
93C2×C8×C71D812yes—
94C2×C8×C71SD1612yes—
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—41no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—81no—
104C4⋊C4×C71—81no—
105C42×C71#4111yescharacteristic
106C42×C71#4111yescharacteristic
107C42×C71#4111yescharacteristic
108C42×C71#4111yescharacteristic
109C23×C71#12111yescharacteristic
110C22×D71—22no—
111C22×D71—41no—
112C22×D71—41no—
113C22×D71—41no—
114C22×D71—41no—
115C22×D71—41no—
116C22×D71—41no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71—81no—
124C2×C4×C71—81no—
125C2×C4×C71#12311yescharacteristic
126C2×C4×C71#12311yescharacteristic
127C2×C4×C71#12311yescharacteristic
128C2×C4×C71#12411yescharacteristic
129C2×C4×C71#12611yescharacteristic
130C2×C4×C71#12611yescharacteristic
131C2×C4×C71C2×D811yescharacteristic
132C2×C4×C71C2×SD1611yescharacteristic
133C2×C4×C71C2×SD1611yescharacteristic
134C2×C4×C71#13511yescharacteristic
135C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C71#13711yescharacteristic
138C2×C4×C71#13711yescharacteristic
139C2×Dic71—41no—
140C2×Dic71—41no—
141C2×Dic71—41no—
142D284—22no—
143D284—22no—
144D284—41no—
145D284—41no—
146C4×D71—81no—
147C4×D71—81no—
148C4×D71—81no—
149C4×D71—81no—
150C8×C71—22no—
151C8×C71—22no—
152C71⋊C8—81no—
153C71⋊C8—81no—
154C71⋊C8—81no—
155C22×C71#39711yescharacteristic
156C22×C71#45911yescharacteristic
157C22×C71#46111yescharacteristic
158C22×C71#46511yescharacteristic
159C22×C71#46811yescharacteristic
160C22×C71#47111yescharacteristic
161C22×C71#47111yescharacteristic
162C2×D71—42no—
163C2×D71—42no—
164C2×D71—42no—
165C2×D71—81no—
166C2×D71—81no—
167C2×D71—81no—
168C2×D71—81no—
169C4×C71—21no—
170C4×C71—21no—
171C4×C71—161no—
172C4×C71—161no—
173C4×C71#48011yescharacteristic
174C4×C71#48111yescharacteristic
175C4×C71#48111yescharacteristic
176C4×C71#48211yescharacteristic
177C4×C71#48311yescharacteristic
178C4×C71#48511yescharacteristic
179C4×C71#48611yescharacteristic
180C4×C71#48911yescharacteristic
181Dic71—81no—
182#10619—711nomaximal, Sylow
183C2×C71#166511yescharacteristic
184C2×C71#167111yescharacteristic
185C2×C71#168511yescharacteristic
186C2×C71#169111yescharacteristic
187C2×C71#207111yescharacteristic
188C2×C71#207711yescharacteristic
189C2×C71#207911yescharacteristic
190D71—82no—
191#1954—711no—
192#1844D7111yescharacteristic, complemented
193#1802—711no—
194#1802—711no—
195#1296—711no—
196#1296—711no—
197#1277—711no—
198C71#1061911yescharacteristic, complemented, Sylow
199C2×C4×D4—711no—
200C2×#106—21no—
201C2×#106—21no—
202C2×C4⋊C8—711no—
203C2×C4⋊C8—711no—
204C2×C4⋊C8—711no—
205C2×#102—1421no—
206C2×#102—1421no—
207C2×C22⋊C8—1421no—
208C2×C22⋊C8—1421no—
209C2×C4×C8C2×D7111yescharacteristic
210#384C2×D7111yescharacteristic
211#384C2×D7111yescharacteristic
212#340—1421no—
213#340—1421no—
214C22×D4—711no—
215C23×C4—1421no—
216C22×C8—1421no—
217C22×C8—1421no—
218C22×C8—1421no—
219C22×C8C22⋊D7111yescharacteristic
220C22×C8C22⋊D7111yescharacteristic
221C4×D4—1421no—
222C4×D4—1421no—
223C4×D4—1421no—
224C4×D4—1421no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—711no—
232C2×C22⋊C4—1421no—
233C2×C42C22×D7111yescharacteristic
234#106—42no—
235#106—42no—
236C4⋊C8—1421no—
237C4⋊C8—1421no—
238C4⋊C8—1421no—
239C4⋊C8—1421no—
240C4⋊C8—1421no—
241C4⋊C8—1421no—
242#102—2842no—
243#102—2842no—
244C22⋊C8—2842no—
245C22⋊C8—2842no—
246C4×C8—21no—
247C4×C8—21no—
248C24—1421no—
249C2×D4—712no—
250C2×D4—1421no—
251C2×D4—1421no—
252C2×D4—1421no—
253C2×D4—1421no—
254C2×D4—1421no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—2841no—
258C22×C4—2841no—
259C22×C4—2841no—
260C22×C4—2841no—
261C22×C4—2841no—
262C22×C4—2841no—
263C22×C4—1421no—
264C22×C4D4×D7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266C22×C4#1176586411yescharacteristic
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—2841no—
272C2×C8—2841no—
273C2×C8—2841no—
274C2×C8—2841no—
275C2×C8—2841no—
276C2×C8—2841no—
277C2×C8—2841no—
278C2×C8—2841no—
279C2×C8—2841no—
280C2×C8#1176585012yes—
281C2×C8#1176585212yes—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C22⋊C4—2841no—
295C22⋊C4—2841no—
296C42#1176586511yescharacteristic
297C42#1176586511yescharacteristic
298C42#1176586611yescharacteristic
299C42#1176586611yescharacteristic
300C23—2841no—
301C23—2841no—
302C23—2841no—
303C23—2841no—
304C23—2841no—
305C23—2841no—
306C23—1422no—
307C23#1455609911yescharacteristic
308D4—2841no—
309D4—2841no—
310D4—1422no—
311D4—1422no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—2841no—
321C2×C4—2841no—
322C2×C4—2841no—
323C2×C4—5681no—
324C2×C4—5681no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4#1455611111yescharacteristic
328C2×C4#1455611311yescharacteristic
329C2×C4#1455611311yescharacteristic
330C2×C4#1455611711yescharacteristic
331C2×C4#1455612311yescharacteristic
332C2×C4#1455612311yescharacteristic
333C2×C4C2×#1176585011yescharacteristic
334C2×C4SD16×D7111yescharacteristic
335C2×C4C2×#1176585211yescharacteristic
336C2×C4#1455614911yescharacteristic
337C2×C4#1455615411yescharacteristic
338C2×C4#1455615511yescharacteristic
339C2×C4#1455615711yescharacteristic
340C2×C4#1455615911yescharacteristic
341C8—22no—
342C8—22no—
343C8—5681no—
344C8—5681no—
345C8—5681no—
346C22—2842no—
347C22—2842no—
348C22—2842no—
349C22—5681no—
350C22—5681no—
351C22—5681no—
352C22—5681no—
353C22#1798592511yescharacteristic
354C22#1798619111yescharacteristic
355C22#1798620111yescharacteristic
356C22#1798622511yescharacteristic
357C22#1798624211yescharacteristic
358C22#1798625911yescharacteristic
359C22#1798625911yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—161no—
364C4—5681no—
365C4#1798631211yescharacteristic
366C4#1798631711yescharacteristic
367C4#1798631911yescharacteristic
368C4#1798632511yescharacteristic
369C4#1798633111yescharacteristic
370C4#1798634311yescharacteristic
371C4#1798634711yescharacteristic
372C4#1798636111yescharacteristic
373C2—5682no—
374C2#2258767811yescharacteristic
375C2#2258771211yescharacteristic
376C2#2258780211yescharacteristic
377C2#2258783611yescharacteristic
378C2#2258974411yescharacteristic
379C2#2258977411yescharacteristic
380C2#2258978611yescharacteristic
subgroupG/H
centerC23#14556099
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1844C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenter#1844D71
hypocenterC71#10619
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
4. #22586223C2semidirect
5. #22586223C2semidirect
6. #22586175C2semidirect
7. C71×#1844C2semidirect
192. #1844D71semidirect
198. C71#10619semidirect

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×#18442
#1061971