Finite Group Databasebeta

← #31916608

Subgroups — #31916608

Order 18176.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986235C211yesmaximal, characteristic
2C2×#17985825C211yesmaximal, characteristic, complemented
3C2×#17985801C211yesmaximal, characteristic, complemented
4C2×#17985801C211yesmaximal, characteristic, complemented
5#22588369C211yesmaximal, characteristic, complemented
6#22586432C211yesmaximal, characteristic, complemented
7C2×C22⋊C16×C71C211yesmaximal, characteristic, complemented
8C22×D568C2211yescharacteristic
9C2×#14556102—21no—
10#17986235—22no—
11#17986235—22no—
12C2×#14556062C2211yescharacteristic
13C2×#14556062C2211yescharacteristic
14C2×#14556062C2211yescharacteristic
15C2×#14556050—21no—
16#17985859—21no—
17#17985825—22no—
18#17985801—22no—
19#17985801—22no—
20#17985762—21no—
21C23×C8×C71C2211yescharacteristic
22C22×C16×C71C2211yescharacteristic
23C22×C16×C71C2211yescharacteristic
24C22⋊C16×C71—22no—
25C22×D284—21no—
26C22×C22⋊D71—41no—
27C2×D568—21no—
28C2×D568—21no—
29C2×D568—21no—
30C2×D568—21no—
31C2×D568—21no—
32C2×D568D412yes—
33#14556102—24no—
34#14556102—42no—
35C2×C4⋊Dic71—21no—
36C2×C4⋊Dic71—21no—
37C2×C4⋊Dic71—21no—
38C2×C22⋊Dic71—41no—
39C2×(C2×D71)⋊C4—41no—
40#14556062—21no—
41#14556062—21no—
42#14556062—21no—
43#14556062—21no—
44#14556062—21no—
45#14556062D412yes—
46#14556050—24no—
47C71⋊#95—41no—
48C23×C4×C71D411yescharacteristic
49C22×C8×C71—21no—
50C22×C8×C71—21no—
51C22×C8×C71—21no—
52C22×C8×C71—21no—
53C22×C8×C71—21no—
54C22×C8×C71—21no—
55C22×C8×C71D411yescharacteristic
56C22×C8×C71C2311yescharacteristic
57C2×C16×C71—21no—
58C2×C16×C71—21no—
59C2×C16×C71—21no—
60C2×C16×C71—21no—
61C2×C16×C71—21no—
62C2×C16×C71D412yescomplemented
63C23×D71—41no—
64C2×D284—22no—
65C2×D284—41no—
66C2×D284—41no—
67C2×D284—41no—
68C2×D284—41no—
69C2×D284—41no—
70C2×C22⋊D71—44no—
71C2×C22⋊D71—82no—
72C2×C22⋊D71—82no—
73C22×Dic71—41no—
74C22×Dic71—41no—
75C22×Dic71—41no—
76D568—22no—
77D568—22no—
78D568—41no—
79D568—41no—
80C4⋊Dic71—22no—
81C4⋊Dic71—41no—
82C4⋊Dic71—41no—
83C4⋊Dic71—41no—
84C4⋊Dic71—41no—
85C4⋊Dic71—41no—
86C22⋊Dic71—82no—
87(C2×D71)⋊C4—44no—
88C24×C71D811yescharacteristic
89C22×C4×C71—21no—
90C22×C4×C71—21no—
91C22×C4×C71—21no—
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71D811yescharacteristic
96C22×C4×C71C2×D411yescharacteristic
97C2×C8×C71—21no—
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—41no—
101C2×C8×C71—41no—
102C2×C8×C71—41no—
103C2×C8×C71—41no—
104C2×C8×C71C2×D411yescharacteristic
105C2×C8×C71C2×D411yescharacteristic
106C2×C8×C71C2×D411yescharacteristic
107C2×C8×C71#4111yescharacteristic
108C2×C8×C71#4111yescharacteristic
109C2×C8×C71#4111yescharacteristic
110C16×C71—22no—
111C16×C71—41no—
112C23×C71—21no—
113C23×C71—21no—
114C23×C71—21no—
115C23×C71—21no—
116C23×C71—21no—
117C23×C71—21no—
118C23×C71D1612yes—
119C23×C71C2×D811yescharacteristic
120C22×D71—42no—
121C22×D71—81no—
122C22×D71—81no—
123C22×D71—81no—
124C22×D71—81no—
125C22×D71—81no—
126C22×D71—81no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71#12011yescharacteristic
135C2×C4×C71#12111yescharacteristic
136C2×C4×C71#12311yescharacteristic
137C2×C4×C71#13511yescharacteristic
138C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C71#13711yescharacteristic
140C22⋊D71—88no—
141C2×Dic71—42no—
142C2×Dic71—81no—
143C2×Dic71—81no—
144C2×Dic71—81no—
145C2×Dic71—81no—
146C2×Dic71—81no—
147C2×Dic71—81no—
148C2×Dic71—81no—
149C2×Dic71—81no—
150D284—42no—
151D284—42no—
152D284—81no—
153D284—81no—
154C8×C71—41no—
155C8×C71#12111yescharacteristic
156C8×C71#12111yescharacteristic
157C8×C71#12311yescharacteristic
158C8×C71#12411yescharacteristic
159C22×C71—22no—
160C22×C71—22no—
161C22×C71—22no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71#44711yescharacteristic
167C22×C71#46611yescharacteristic
168C22×C71#48011yescharacteristic
169C22×C71C2×D1611yescharacteristic
170C22×C71#50811yescharacteristic
171C22×C71#50911yescharacteristic
172C22×C71#51011yescharacteristic
173C2×D71—82no—
174C2×D71—82no—
175C2×D71—82no—
176C2×D71—161no—
177C2×D71—161no—
178C2×D71—161no—
179C2×D71—161no—
180C4×C71—41no—
181C4×C71#39411yescharacteristic
182C4×C71#45211yescharacteristic
183C4×C71#46811yescharacteristic
184C4×C71#48211yescharacteristic
185Dic71—82no—
186Dic71—161no—
187Dic71—161no—
188#13380—711nomaximal, Sylow
189C2×C71—42no—
190C2×C71#203311yescharacteristic
191C2×C71#218411yescharacteristic
192C2×C71#218911yescharacteristic
193C2×C71#221511yescharacteristic
194C2×C71#222011yescharacteristic
195C2×C71#223211yescharacteristic
196C2×C71#223411yescharacteristic
197D71—162no—
198C2×#466—711no—
199C2×#366—711no—
200C2×#357—711no—
201C2×#357—711no—
202C2×C22⋊C16D7111yescharacteristic, complemented
203#1808—711no—
204#1380—711no—
205C71#1338011yescharacteristic, complemented, Sylow
206C22×D8—711no—
207C23×C8C2×D7111yescharacteristic
208C2×#121—1421no—
209C22×C16C2×D7111yescharacteristic
210C22×C16C2×D7111yescharacteristic
211#466—1422no—
212#466—1422no—
213C2×#107—711no—
214C2×#107—711no—
215C2×#107—711no—
216C2×#102—1421no—
217#380—1421no—
218#366—1422no—
219#357—1422no—
220#357—1422no—
221C22⋊C16—22no—
222#340—1421no—
223C22×D4—2841no—
224C22×D4—1421no—
225C23×C4D28411yescharacteristic
226C2×D8—712no—
227C2×D8—1421no—
228C2×D8—1421no—
229C2×D8—1421no—
230C2×D8—1421no—
231C2×D8—1421no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8D28411yescharacteristic
239C22×C8C22×D7111yescharacteristic
240#121—2842no—
241#121—1424no—
242C2×C4⋊C4—1421no—
243C2×C4⋊C4—1421no—
244C2×C4⋊C4—1421no—
245C2×C22⋊C4—2841no—
246C2×C22⋊C4—2841no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16C22⋊D7112yescomplemented
253#107—712no—
254#107—1421no—
255#107—1421no—
256#107—1421no—
257#107—1421no—
258#107—1421no—
259#102—1424no—
260#95—2841no—
261C24—2841no—
262C24D56811yescharacteristic
263C2×D4—2844no—
264C2×D4—2841no—
265C2×D4—2841no—
266C2×D4—2841no—
267C2×D4—2841no—
268C2×D4—2841no—
269C2×D4—5682no—
270C2×D4—5682no—
271C2×D4—1422no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—2841no—
279C22×C4—2841no—
280C22×C4—2841no—
281C22×C4D56811yescharacteristic
282C22×C4C2×D28411yescharacteristic
283D8—2841no—
284D8—2841no—
285D8—1422no—
286D8—1422no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D7111yescharacteristic
295C2×C8D4×D7111yescharacteristic
296C2×C8C2×C22⋊D7111yescharacteristic
297C2×C8#1176586411yescharacteristic
298C2×C8#1176586411yescharacteristic
299C2×C8#1176586511yescharacteristic
300C4⋊C4—2841no—
301C4⋊C4—2841no—
302C4⋊C4—2841no—
303C4⋊C4—2841no—
304C4⋊C4—2841no—
305C4⋊C4—1422no—
306C22⋊C4—2844no—
307C22⋊C4—5682no—
308C16—22no—
309C16—41no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—2842no—
317C23—5681no—
318C23—5681no—
319C23—5681no—
320C23—5681no—
321C23—5681no—
322C23—5681no—
323C23D113612yes—
324C23C2×D56811yescharacteristic
325D4—2842no—
326D4—2842no—
327D4—5688no—
328D4—5681no—
329D4—5681no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—2842no—
338C2×C4—5681no—
339C2×C4—5681no—
340C2×C4—5681no—
341C2×C4—5681no—
342C2×C4—5681no—
343C2×C4—5681no—
344C2×C4—5681no—
345C2×C4—5681no—
346C2×C4#1455609811yescharacteristic
347C2×C4#1455610211yescharacteristic
348C2×C4#1455611211yescharacteristic
349C2×C4#1455615011yescharacteristic
350C2×C4#1455615611yescharacteristic
351C2×C4#1455616111yescharacteristic
352C8—41no—
353C8#1455609911yescharacteristic
354C8#1455610011yescharacteristic
355C8#1455611011yescharacteristic
356C8#1455611411yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—5682no—
365C22—5682no—
366C22—5682no—
367C22—11361no—
368C22—11361no—
369C22—11361no—
370C22—11361no—
371C22#1798613511yescharacteristic
372C22#1798623511yescharacteristic
373C22#1798631311yescharacteristic
374C22C2×D113611yescharacteristic
375C22#1798643111yescharacteristic
376C22#1798643711yescharacteristic
377C22#1798644211yescharacteristic
378C4—41no—
379C4—5682no—
380C4—11361no—
381C4—11361no—
382C4#1798591511yescharacteristic
383C4#1798616311yescharacteristic
384C4#1798624511yescharacteristic
385C4#1798632411yescharacteristic
386C2—42no—
387C2—11362no—
388C2#2258952311yescharacteristic
389C2#2259025411yescharacteristic
390C2#2259028211yescharacteristic
391C2#2259042611yescharacteristic
392C2#2259045311yescharacteristic
393C2#2259052111yescharacteristic
394C2#2259053211yescharacteristic
subgroupG/H
centerC23C2×D568
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC2×C22⋊C16×C71C2
socleC23×C71C2×D8
prime radicalC2×#17986235C2
hypercenterC2×C22⋊C16D71
hypocenterC71#13380
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#17985825C2semidirect
3. C2×#17985801C2semidirect
4. C2×#17985801C2semidirect
5. #22588369C2semidirect
6. #22586432C2semidirect
7. C2×C22⋊C16×C71C2semidirect
62. C2×C16×C71D4semidirect
202. C2×C22⋊C16D71semidirect
205. C71#13380semidirect
252. C2×C16C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862352
C2×#179858252
C2×#179858012
#225883692
#225864322
C2×C22⋊C16×C712
#1338071