Finite Group Databasebeta

← #31902178

Subgroups — #31902178

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#22589133C211yesmaximal, characteristic
2#22588496C211yesmaximal, characteristic, complemented
3#22588341C211yesmaximal, characteristic
4#22588337C211yesmaximal, characteristic
5#22586224C211yesmaximal, characteristic, complemented
6#22586174C211yesmaximal, characteristic, complemented
7C71×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9C2×#14556059—21no—
10C2×#14556059—21no—
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556050—21no—
13C2×#14556048—21no—
14C2×(C2×D71)⋊C8—21no—
15C2×(C2×D71)⋊C8—21no—
16C2×C4×C71⋊C8C2211yescharacteristic
17#17985869C2211yescharacteristic
18#17985762—21no—
19#17985761—21no—
20C2×C4⋊C8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22#384×C71C2211yescharacteristic
23C22×D284D411yescharacteristic
24C22×C4×D71—21no—
25C22×C71⋊C8—21no—
26C22×C71⋊C8D411yescharacteristic
27C22×C71⋊C8D411yescharacteristic
28C4×D284—21no—
29C4×D284—21no—
30C4×D284—21no—
31C4×D284—21no—
32C2×C4⋊Dic71—21no—
33C2×C4⋊Dic71—21no—
34C2×C4⋊Dic71—21no—
35C2×C4⋊Dic71D411yescharacteristic
36C2×(C2×D71)⋊C4—21no—
37#14556059—42no—
38#14556059—42no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41#14556050—42no—
42#14556048—42no—
43(C2×D71)⋊C8—42no—
44(C2×D71)⋊C8—42no—
45C4×C71⋊C8—21no—
46C4×C71⋊C8—21no—
47C22×C8×C71—21no—
48C22×C8×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—
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—
65C22×Dic71—41no—
66C2×C4×D71—41no—
67C2×C4×D71—41no—
68C2×C4×D71—41no—
69C2×C4×D71—41no—
70C2×C4×D71—41no—
71C2×C4×D71—41no—
72C2×C71⋊C8—21no—
73C2×C71⋊C8—21no—
74C2×C71⋊C8—21no—
75C2×C71⋊C8—21no—
76C2×C71⋊C8—41no—
77C2×C71⋊C8—41no—
78C2×C71⋊C8—41no—
79C2×C71⋊C8D812yes—
80C2×C71⋊C8SD1612yes—
81C4⋊Dic71—21no—
82C4⋊Dic71—21no—
83C4⋊Dic71—41no—
84C4⋊Dic71—41no—
85C4⋊Dic71—41no—
86C4⋊Dic71—41no—
87C4⋊Dic71—81no—
88(C2×D71)⋊C4—41no—
89(C2×D71)⋊C4—41no—
90C22×C4×C71—41no—
91C22×C4×C71C2×D411yescharacteristic
92C22×C4×C71C2×D411yescharacteristic
93C22×C4×C71#4111yescharacteristic
94C2×C8×C71—41no—
95C2×C8×C71—41no—
96C2×C8×C71—41no—
97C2×C8×C71—41no—
98C2×C8×C71—41no—
99C2×C8×C71—41no—
100C4⋊C4×C71—41no—
101C4⋊C4×C71—41no—
102C4⋊C4×C71—41no—
103C4⋊C4×C71—41no—
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#12311yescharacteristic
123C2×C4×C71#12311yescharacteristic
124C2×C4×C71#12311yescharacteristic
125C2×C4×C71#12411yescharacteristic
126C2×C4×C71#12611yescharacteristic
127C2×C4×C71#12611yescharacteristic
128C2×C4×C71C2×D811yescharacteristic
129C2×C4×C71C2×SD1611yescharacteristic
130C2×C4×C71C2×SD1611yescharacteristic
131C2×C4×C71#13511yescharacteristic
132C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C71#13711yescharacteristic
135C2×C4×C71#13711yescharacteristic
136C2×Dic71—41no—
137C2×Dic71—41no—
138C2×Dic71—41no—
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71—81no—
142D284—22no—
143D284—22no—
144D284—41no—
145D284—41no—
146C4×D71—81no—
147C4×D71—81no—
148C4×D71—81no—
149C4×D71—81no—
150C8×C71—81no—
151C8×C71—81no—
152C71⋊C8—22no—
153C71⋊C8—22no—
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#48011yescharacteristic
173C4×C71#48111yescharacteristic
174C4×C71#48111yescharacteristic
175C4×C71#48211yescharacteristic
176C4×C71#48311yescharacteristic
177C4×C71#48511yescharacteristic
178C4×C71#48611yescharacteristic
179C4×C71#48911yescharacteristic
180Dic71—81no—
181Dic71—161no—
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#1844—711no—
193#1802—711no—
194#1802—711no—
195#1296—711no—
196#1296D7111yescharacteristic, complemented
197#1277—711no—
198C71#1061911yescharacteristic, complemented, Sylow
199C2×C4×D4—711no—
200C2×#106—1421no—
201C2×#106—1421no—
202C2×C4⋊C8—711no—
203C2×C4⋊C8C2×D7111yescharacteristic
204C2×C4⋊C8C2×D7111yescharacteristic
205C2×#102—1421no—
206C2×#102—1421no—
207C2×C22⋊C8—1421no—
208C2×C22⋊C8—1421no—
209C2×C4×C8—711no—
210#384—711no—
211#384C2×D7111yescharacteristic
212#340—1421no—
213#340—1421no—
214C22×D4—711no—
215C23×C4—1421no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—711no—
219C22×C8—711no—
220C22×C8—1421no—
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—711no—
229C2×C4⋊C4—1421no—
230C2×C4⋊C4—1421no—
231C2×C4⋊C4—1421no—
232C2×C22⋊C4—1421no—
233C2×C42C22×D7111yescharacteristic
234#106—2842no—
235#106—2842no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1421no—
241C4⋊C8—1421no—
242#102—2842no—
243#102—2842no—
244C22⋊C8—2842no—
245C22⋊C8—2842no—
246C4×C8—1421no—
247C4×C8—1421no—
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—2841no—
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×C4D4×D7111yescharacteristic
266C22×C4#1176586511yescharacteristic
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—2841no—
274C2×C8—2841no—
275C2×C8—2841no—
276C2×C8—712no—
277C2×C8—712no—
278C2×C8—1421no—
279C2×C8—1421no—
280C2×C8—1421no—
281C2×C8—1421no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—2841no—
288C4⋊C4—2841no—
289C4⋊C4—2841no—
290C4⋊C4—2841no—
291C4⋊C4—5681no—
292C4⋊C4—1421no—
293C4⋊C4—1421no—
294C22⋊C4—2841no—
295C22⋊C4—2841no—
296C42#1176586411yescharacteristic
297C42#1176586511yescharacteristic
298C42#1176586511yescharacteristic
299C42#1176586611yescharacteristic
300C23—2841no—
301C23—2841no—
302C23—2841no—
303C23—2841no—
304C23—2841no—
305C23—2841no—
306C23—1422no—
307C23#1455610111yescharacteristic
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—2841no—
318C2×C4—2841no—
319C2×C4—2841no—
320C2×C4—5681no—
321C2×C4—5681no—
322C2×C4—5681no—
323C2×C4—5681no—
324C2×C4—5681no—
325C2×C4—5681no—
326C2×C4—5681no—
327C2×C4#1455611011yescharacteristic
328C2×C4#1455611311yescharacteristic
329C2×C4#1455611311yescharacteristic
330C2×C4#1455611411yescharacteristic
331C2×C4#1455612211yescharacteristic
332C2×C4#1455612311yescharacteristic
333C2×C4D8×D7111yescharacteristic
334C2×C4SD16×D7111yescharacteristic
335C2×C4SD16×D7111yescharacteristic
336C2×C4#1455615111yescharacteristic
337C2×C4#1455615211yescharacteristic
338C2×C4#1455615511yescharacteristic
339C2×C4#1455615711yescharacteristic
340C2×C4#1455616011yescharacteristic
341C8—81no—
342C8—81no—
343C8—5681no—
344C8—1422no—
345C8—1422no—
346C22—2842no—
347C22—2842no—
348C22—2842no—
349C22—5681no—
350C22—5681no—
351C22—5681no—
352C22—5681no—
353C22#1798592811yescharacteristic
354C22#1798619311yescharacteristic
355C22#1798620311yescharacteristic
356C22#1798623111yescharacteristic
357C22#1798624411yescharacteristic
358C22#1798626111yescharacteristic
359C22#1798626511yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—5681no—
364C4—11361no—
365C4#1798631411yescharacteristic
366C4#1798631611yescharacteristic
367C4#1798631911yescharacteristic
368C4#1798632611yescharacteristic
369C4#1798633111yescharacteristic
370C4#1798633911yescharacteristic
371C4#1798634411yescharacteristic
372C4#1798636011yescharacteristic
373C2—5682no—
374C2#2258768411yescharacteristic
375C2#2258771811yescharacteristic
376C2#2258780411yescharacteristic
377C2#2258783811yescharacteristic
378C2#2258974711yescharacteristic
379C2#2258977911yescharacteristic
380C2#2258978911yescharacteristic
subgroupG/H
centerC23#14556101
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1296C2
socleC23×C71#121
prime radicalC22×D284D4
hypercenter#1296D71
hypocenterC71#10619
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #22588496C2semidirect
5. #22586224C2semidirect
6. #22586174C2semidirect
7. C71×#1296C2semidirect
196. #1296D71semidirect
198. C71#10619semidirect

Maximal subgroups

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

subgroupindex
#225891332
#225884962
#225883412
#225883372
#225862242
#225861742
C71×#12962
#1061971