Finite Group Databasebeta

← #31917051

Subgroups — #31917051

Order 18176.

446 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556071C211yesmaximal, characteristic
2C2×C4⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
3C2×#17985803C211yesmaximal, characteristic, complemented
4C2×#17985799C211yesmaximal, characteristic
5#22589206C211yesmaximal, characteristic
6#22586431C211yesmaximal, characteristic, complemented
7C71×#1969C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C2×#14556071—22no—
10C2×#14556071—22no—
11C2×#14556071—22no—
12C22×C71⋊C16C2211yescharacteristic
13C22×C71⋊C16C2211yescharacteristic
14C2×#14556062C2211yescharacteristic
15C2×#14556050—21no—
16#17985901—21no—
17C4⋊(C71⋊C16)—22no—
18#17985803—22no—
19#17985799—22no—
20C22×Q16×C71C2211yescharacteristic
21C2×#107×C71C2211yescharacteristic
22C2×#103×C71—21no—
23C2×C4×C8×C71C2211yescharacteristic
24#391×C71—21no—
25C22×D284—21no—
26C2×D568—21no—
27C2×D568—21no—
28C2×D568—21no—
29C2×D568—21no—
30C2×D568—21no—
31C2×D568D412yes—
32C2×C4⋊Dic71—21no—
33C2×(C2×D71)⋊C4—41no—
34C2×(C2×D71)⋊C4—41no—
35#14556071—28no—
36C2×C71⋊C16—21no—
37C2×C71⋊C16—21no—
38C2×C71⋊C16—21no—
39C2×C71⋊C16—21no—
40C2×C71⋊C16—21no—
41C2×C71⋊C16D412yes—
42#14556062—21no—
43#14556062—21no—
44#14556050—42no—
45C22×Q8×C71—21no—
46C2×Q16×C71—21no—
47C2×Q16×C71—21no—
48C2×Q16×C71—21no—
49C2×Q16×C71—21no—
50C2×Q16×C71—21no—
51C2×Q16×C71D412yes—
52C22×C8×C71D411yescharacteristic
53C22×C8×C71C2311yescharacteristic
54C2×C4⋊C4×C71—21no—
55C2×C42×C71D411yescharacteristic
56#107×C71—21no—
57#107×C71—21no—
58#103×C71—42no—
59C4×C8×C71—21no—
60C4×C8×C71—21no—
61#95×C71—41no—
62#95×C71—41no—
63C23×D71—41no—
64C2×D284—22no—
65C2×D284—41no—
66C2×D284—41no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284—41no—
70C22×Dic71—41no—
71D568—22no—
72D568—22no—
73D568—41no—
74D568—41no—
75C4⋊Dic71—41no—
76C4⋊Dic71—41no—
77(C2×D71)⋊C4—82no—
78(C2×D71)⋊C4—82no—
79C71⋊C16—22no—
80C71⋊C16—41no—
81C2×Q8×C71—22no—
82C2×Q8×C71—41no—
83C2×Q8×C71—41no—
84C2×Q8×C71—41no—
85C2×Q8×C71—41no—
86C2×Q8×C71—41no—
87C22×C4×C71—41no—
88C22×C4×C71—41no—
89C22×C4×C71D811yescharacteristic
90C22×C4×C71D811yescharacteristic
91C22×C4×C71C2×D411yescharacteristic
92Q16×C71—22no—
93Q16×C71—22no—
94Q16×C71—41no—
95Q16×C71—41no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—21no—
99C2×C8×C71C2×D411yescharacteristic
100C2×C8×C71C2×D411yescharacteristic
101C2×C8×C71C2×D411yescharacteristic
102C2×C8×C71#4111yescharacteristic
103C2×C8×C71#4111yescharacteristic
104C2×C8×C71#4111yescharacteristic
105C4⋊C4×C71—41no—
106C4⋊C4×C71—41no—
107C42×C71—21no—
108C42×C71—21no—
109C23×C71C2×D811yescharacteristic
110C22×D71—42no—
111C22×D71—81no—
112C22×D71—81no—
113C22×D71—81no—
114C22×D71—81no—
115C22×D71—81no—
116C22×D71—81no—
117Q8×C71—42no—
118Q8×C71—42no—
119Q8×C71—81no—
120Q8×C71—81no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—42no—
127C2×C4×C71—81no—
128C2×C4×C71—81no—
129C2×C4×C71—81no—
130C2×C4×C71—81no—
131C2×C4×C71—81no—
132C2×C4×C71SD3212yescomplemented
133C2×C4×C71#12111yescharacteristic
134C2×C4×C71#12111yescharacteristic
135C2×C4×C71#12111yescharacteristic
136C2×C4×C71#13511yescharacteristic
137C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
139C2×Dic71—81no—
140C2×Dic71—81no—
141C2×Dic71—81no—
142D284—42no—
143D284—42no—
144D284—81no—
145D284—81no—
146C8×C71—41no—
147C8×C71#12011yescharacteristic
148C8×C71#12311yescharacteristic
149C8×C71#12311yescharacteristic
150C8×C71#12411yescharacteristic
151C22×C71#45911yescharacteristic
152C22×C71#45911yescharacteristic
153C22×C71#46611yescharacteristic
154C22×C71C2×SD3211yescharacteristic
155C22×C71#50811yescharacteristic
156C22×C71#50911yescharacteristic
157C22×C71#51011yescharacteristic
158C2×D71—82no—
159C2×D71—82no—
160C2×D71—82no—
161C2×D71—161no—
162C2×D71—161no—
163C2×D71—161no—
164C2×D71—161no—
165C4×C71—22no—
166C4×C71—41no—
167C4×C71—82no—
168C4×C71—161no—
169C4×C71#39411yescharacteristic
170C4×C71#46311yescharacteristic
171C4×C71#46311yescharacteristic
172C4×C71#46911yescharacteristic
173Dic71—161no—
174#13449—711nomaximal, Sylow
175C2×C71#205411yescharacteristic
176C2×C71#220711yescharacteristic
177C2×C71#220811yescharacteristic
178C2×C71#221011yescharacteristic
179C2×C71#221111yescharacteristic
180C2×C71#221711yescharacteristic
181C2×C71#222011yescharacteristic
182D71—162no—
183C22×SD32—711no—
184C2×C4⋊C16—711no—
185C2×#358—711no—
186C2×#357—711no—
187#1970—711no—
188#1969D7111yescharacteristic, complemented
189#1380—711no—
190C71#1344911yescharacteristic, complemented, Sylow
191C22×Q16C2×D7111yescharacteristic
192C22×D8—711no—
193C2×SD32—1422no—
194C2×SD32—1422no—
195C2×SD32—1422no—
196C22×C16—711no—
197C22×C16—711no—
198C2×#107—711no—
199C2×#107C2×D7111yescharacteristic
200C2×#103—21no—
201C2×#102—1421no—
202C2×C4×C8C2×D7111yescharacteristic
203#391—21no—
204#390—1421no—
205C4⋊C16—1422no—
206#358—1422no—
207#357—1422no—
208C22×Q8—21no—
209C22×D4—1421no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16D28412yes—
216C2×D8—712no—
217C2×D8—1421no—
218C2×D8—1421no—
219C2×D8—1421no—
220C2×D8—1421no—
221C2×D8—1421no—
222C22×C8C22⋊D7111yescharacteristic
223C22×C8C22×D7111yescharacteristic
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1421no—
226C2×C22⋊C4—2841no—
227C2×C22⋊C4—2841no—
228C2×C42C22⋊D7111yescharacteristic
229SD32—1428no—
230C2×C16—712no—
231C2×C16—1421no—
232C2×C16—1421no—
233C2×C16—1421no—
234C2×C16—1421no—
235C2×C16—1421no—
236#107—21no—
237#107—21no—
238#107—1421no—
239#107—1421no—
240#103—42no—
241#102—2842no—
242C4×C8—21no—
243C4×C8—21no—
244#95—41no—
245#95—41no—
246C24—2841no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—2841no—
254C2×D4—2841no—
255C2×D4—2841no—
256C2×D4—2841no—
257C2×D4—2841no—
258C2×D4—1422no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—2841no—
262C22×C4#1176585011yescharacteristic
263C22×C4#1176585011yescharacteristic
264C22×C4C2×C22⋊D7111yescharacteristic
265Q16—22no—
266Q16—22no—
267Q16—41no—
268Q16—41no—
269D8—2841no—
270D8—2841no—
271D8—1422no—
272D8—1422no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8D4×D7111yescharacteristic
277C2×C8D4×D7111yescharacteristic
278C2×C8C2×D28411yescharacteristic
279C2×C8#1176586411yescharacteristic
280C2×C8#1176586411yescharacteristic
281C2×C8#1176586511yescharacteristic
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—2841no—
285C4⋊C4—2841no—
286C22⋊C4—5682no—
287C22⋊C4—5682no—
288C42—21no—
289C42—21no—
290C16—2841no—
291C16—1422no—
292C23—2842no—
293C23—5681no—
294C23—5681no—
295C23—5681no—
296C23—5681no—
297C23—5681no—
298C23—5681no—
299C23C2×#1176585011yescharacteristic
300Q8—42no—
301Q8—42no—
302Q8—81no—
303Q8—81no—
304D4—2842no—
305D4—2842no—
306D4—5681no—
307D4—5681no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—5681no—
320C2×C4—5681no—
321C2×C4—5681no—
322C2×C4#1455607112yescomplemented
323C2×C4#1455609911yescharacteristic
324C2×C4#1455610011yescharacteristic
325C2×C4#1455610211yescharacteristic
326C2×C4#1455614811yescharacteristic
327C2×C4#1455615311yescharacteristic
328C2×C4#1455615411yescharacteristic
329C8—41no—
330C8#1455609811yescharacteristic
331C8#1455611011yescharacteristic
332C8#1455611211yescharacteristic
333C8#1455611411yescharacteristic
334C22—5682no—
335C22—5682no—
336C22—5682no—
337C22—11361no—
338C22—11361no—
339C22—11361no—
340C22—11361no—
341C22#1798619111yescharacteristic
342C22#1798619411yescharacteristic
343C22#1798623311yescharacteristic
344C22C2×#1455607111yescharacteristic
345C22#1798642911yescharacteristic
346C22#1798643411yescharacteristic
347C22#1798644011yescharacteristic
348C4—22no—
349C4—41no—
350C4—82no—
351C4—161no—
352C4—11361no—
353C4#1798591511yescharacteristic
354C4#1798621111yescharacteristic
355C4#1798621411yescharacteristic
356C4#1798625011yescharacteristic
357C2—11362no—
358C2#2258965011yescharacteristic
359C2#2259038011yescharacteristic
360C2#2259038211yescharacteristic
361C2#2259039211yescharacteristic
362C2#2259040211yescharacteristic
363C2#2259043811yescharacteristic
364C2#2259045511yescharacteristic
subgroupG/H
centerC23C2×#11765850
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#1969C2
socleC23×C71C2×D8
prime radicalC22×D568C22
hypercenter#1969D71
hypocenterC71#13449
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C71⋊C16)C2semidirect
3. C2×#17985803C2semidirect
6. #22586431C2semidirect
7. C71×#1969C2semidirect
132. C2×C4×C71SD32semidirect
188. #1969D71semidirect
190. C71#13449semidirect
322. C2×C4#14556071semidirect

Maximal subgroups

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

subgroupindex
C22×#145560712
C2×C4⋊(C71⋊C16)2
C2×#179858032
C2×#179857992
#225892062
#225864312
C71×#19692
#1344971