Finite Group Databasebeta

← #31916514

Subgroups — #31916514

Order 18176.

450 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986403C211yesmaximal, characteristic, complemented
2C2×#17985800C211yesmaximal, characteristic
3#22589207C211yesmaximal, characteristic
4#22586432C211yesmaximal, characteristic, complemented
5#22586431C211yesmaximal, characteristic, complemented
6#22586417C211yesmaximal, characteristic, complemented
7C2×#357×C71C211yesmaximal, characteristic
8C2×#14556127—21no—
9C22×C71⋊C16C2211yescharacteristic
10#17986403—22no—
11#17986403—22no—
12C2×#14556062C2211yescharacteristic
13C2×#14556062C2211yescharacteristic
14C2×#14556061—21no—
15C2×#14556049—21no—
16C2×C8×Dic71C2211yescharacteristic
17#17985902—21no—
18#17985800—22no—
19C22×D8×C71C2211yescharacteristic
20C22×C16×C71C2211yescharacteristic
21C2×#107×C71C2211yescharacteristic
22#357×C71—22no—
23C22×Dic142—41no—
24C22×C71⋊C8D411yescharacteristic
25#14556127—24no—
26#14556127—42no—
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×Dic71⋊C4—41no—
30C2×C22⋊Dic71—41no—
31C2×C22⋊Dic71—41no—
32C2×C4×Dic71D411yescharacteristic
33C2×C71⋊C16—21no—
34C2×C71⋊C16—21no—
35C2×C71⋊C16—21no—
36#14556062—21no—
37#14556062—21no—
38#14556062—21no—
39#14556062D412yes—
40#14556061—24no—
41#14556049—42no—
42C8×Dic71—21no—
43C8×Dic71—21no—
44C22×D4×C71—21no—
45C2×D8×C71—21no—
46C2×D8×C71—21no—
47C2×D8×C71—21no—
48C2×D8×C71—21no—
49C2×D8×C71—21no—
50C2×D8×C71Q812yescomplemented
51C22×C8×C71C2311yescharacteristic
52C2×C4⋊C4×C71—21no—
53C2×C16×C71—21no—
54C2×C16×C71—21no—
55C2×C16×C71—21no—
56#107×C71—21no—
57#107×C71D412yes—
58C2×Dic142—44no—
59C2×Dic142—82no—
60C2×Dic142—82no—
61C22×Dic71—41no—
62C22×Dic71—41no—
63C22×Dic71D811yescharacteristic
64C22×Dic71D811yescharacteristic
65C2×C71⋊C8—21no—
66C2×C71⋊C8—21no—
67C2×C71⋊C8SD1612yes—
68C4⋊Dic71—22no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72Dic71⋊C4—44no—
73C22⋊Dic71—82no—
74C22⋊Dic71—82no—
75C4×Dic71—21no—
76C4×Dic71—21no—
77C71⋊C16—41no—
78C24×C71—41no—
79C2×D4×C71—22no—
80C2×D4×C71—41no—
81C2×D4×C71—41no—
82C2×D4×C71—41no—
83C2×D4×C71—41no—
84C2×D4×C71—41no—
85C22×C4×C71—41no—
86C22×C4×C71C2×D411yescharacteristic
87D8×C71—22no—
88D8×C71—22no—
89D8×C71—41no—
90D8×C71—41no—
91C2×C8×C71C2×D411yescharacteristic
92C2×C8×C71C2×D411yescharacteristic
93C2×C8×C71C2×Q811yescharacteristic
94C2×C8×C71#4111yescharacteristic
95C2×C8×C71#4111yescharacteristic
96C2×C8×C71#4111yescharacteristic
97C4⋊C4×C71—22no—
98C4⋊C4×C71—41no—
99C16×C71—41no—
100C23×C71—42no—
101C23×C71—81no—
102C23×C71—81no—
103C23×C71—81no—
104C23×C71—81no—
105C23×C71—81no—
106C23×C71—81no—
107C23×C71C2×D811yescharacteristic
108D4×C71—42no—
109D4×C71—42no—
110D4×C71—81no—
111D4×C71—81no—
112C2×C4×C71—42no—
113C2×C4×C71—81no—
114C2×C4×C71—81no—
115C2×C4×C71#12011yescharacteristic
116C2×C4×C71C22⋊Q811yescharacteristic
117C2×C4×C71#12411yescharacteristic
118C2×C4×C71C2×SD1611yescharacteristic
119C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C71#13711yescharacteristic
121C2×Dic71—21no—
122C2×Dic71—21no—
123C2×Dic71—21no—
124C2×Dic71—21no—
125C2×Dic71—42no—
126C2×Dic71—81no—
127C2×Dic71—81no—
128C2×Dic71—81no—
129C2×Dic71—81no—
130C2×Dic71—81no—
131C2×Dic71D1612yes—
132C2×Dic71SD3212yes—
133Dic142—88no—
134C8×C71C22⋊Q811yescharacteristic
135C8×C71C22⋊Q811yescharacteristic
136C8×C71#12411yescharacteristic
137C8×C71#12411yescharacteristic
138C71⋊C8—22no—
139C22×C71—82no—
140C22×C71—82no—
141C22×C71—82no—
142C22×C71—161no—
143C22×C71—161no—
144C22×C71—161no—
145C22×C71—161no—
146C22×C71#44711yescharacteristic
147C22×C71#47411yescharacteristic
148C22×C71#48611yescharacteristic
149C22×C71C2×D1611yescharacteristic
150C22×C71C2×SD3211yescharacteristic
151C22×C71#50911yescharacteristic
152C22×C71#51011yescharacteristic
153C4×C71—82no—
154C4×C71#39311yescharacteristic
155C4×C71#44811yescharacteristic
156C4×C71#47611yescharacteristic
157C4×C71#48911yescharacteristic
158Dic71—22no—
159Dic71—22no—
160Dic71—82no—
161Dic71—161no—
162#13363—711nomaximal, Sylow
163C2×C71—162no—
164C2×C71#202311yescharacteristic
165C2×C71#218411yescharacteristic
166C2×C71#218511yescharacteristic
167C2×C71#222411yescharacteristic
168C2×C71#222611yescharacteristic
169C2×C71#224011yescharacteristic
170C2×C71#224311yescharacteristic
171C2×#500—711no—
172C2×#357—711no—
173C2×#357D7111yescharacteristic
174#1970—711no—
175#1380—711no—
176#1380—711no—
177#1374—711no—
178C71#1336311yescharacteristic, complemented, Sylow
179C22×D8C2×D7111yescharacteristic
180C2×C4⋊Q8—1421no—
181C22×C16—711no—
182C22×C16C2×D7111yescharacteristic
183#500—1422no—
184#500—1422no—
185C2×#107—711no—
186C2×#107—711no—
187C2×#107—1421no—
188C2×#107C2×D7111yescharacteristic
189C2×#102—1421no—
190C2×C4×C8—711no—
191#390—1421no—
192#357—22no—
193#357—1422no—
194C22×Q8—2841no—
195C22×D4—21no—
196C2×D8—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×D8Dic14212yescomplemented
202C22×C8—711no—
203C22×C8C22×D7111yescharacteristic
204C4⋊Q8—2842no—
205C4⋊Q8—1424no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—2841no—
208C2×C4⋊C4—1421no—
209C2×C4⋊C4—1421no—
210C2×C22⋊C4—2841no—
211C2×C22⋊C4—2841no—
212C2×C42—711no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—1421no—
217C2×C16—1421no—
218C2×C16—1421no—
219#107—21no—
220#107—712no—
221#107—1424no—
222#107—1421no—
223#107—1421no—
224#107—1421no—
225#107C22⋊D7112yes—
226#102—2842no—
227C4×C8—1421no—
228C4×C8—1421no—
229C24—41no—
230C2×Q8—2844no—
231C2×Q8—5682no—
232C2×Q8—5682no—
233C2×D4—22no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—2841no—
241C22×C4—2841no—
242C22×C4—711no—
243C22×C4—711no—
244C22×C4D4×D7111yescharacteristic
245D8—22no—
246D8—22no—
247D8—41no—
248D8—41no—
249C2×C8—712no—
250C2×C8—1421no—
251C2×C8—1421no—
252C2×C8D4×D7111yescharacteristic
253C2×C8C2×C22⋊D7111yescharacteristic
254C2×C8C2×Dic14211yescharacteristic
255C2×C8#1176586411yescharacteristic
256C2×C8#1176586411yescharacteristic
257C2×C8#1176586511yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—2844no—
261C4⋊C4—2841no—
262C4⋊C4—2841no—
263C4⋊C4—2841no—
264C4⋊C4—1422no—
265C22⋊C4—5682no—
266C22⋊C4—5682no—
267C42—1421no—
268C42—1421no—
269C16—41no—
270C16—2841no—
271C23—42no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23—81no—
276C23—81no—
277C23—81no—
278C23D8×D7111yescharacteristic
279Q8—5688no—
280D4—42no—
281D4—42no—
282D4—81no—
283D4—81no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—2842no—
288C2×C4—5681no—
289C2×C4—5681no—
290C2×C4—5681no—
291C2×C4—5681no—
292C2×C4—5681no—
293C2×C4—712no—
294C2×C4—712no—
295C2×C4—1421no—
296C2×C4—1421no—
297C2×C4—1421no—
298C2×C4—1421no—
299C2×C4#1455609611yescharacteristic
300C2×C4#1455610511yescharacteristic
301C2×C4#1455611411yescharacteristic
302C2×C4SD16×D7111yescharacteristic
303C2×C4#1455615211yescharacteristic
304C2×C4#1455615711yescharacteristic
305C8—1422no—
306C8#1455610511yescharacteristic
307C8#1455610711yescharacteristic
308C8#1455611411yescharacteristic
309C8#1455611511yescharacteristic
310C22—82no—
311C22—82no—
312C22—82no—
313C22—161no—
314C22—161no—
315C22—161no—
316C22—161no—
317C22#1798613111yescharacteristic
318C22#1798627711yescharacteristic
319C22#1798634411yescharacteristic
320C22D16×D7111yescharacteristic
321C22SD32×D7111yescharacteristic
322C22#1798643311yescharacteristic
323C22#1798643811yescharacteristic
324C4—82no—
325C4—5682no—
326C4—11361no—
327C4—1422no—
328C4—1422no—
329C4#1798591111yescharacteristic
330C4#1798613611yescharacteristic
331C4#1798628711yescharacteristic
332C4#1798635711yescharacteristic
333C2—162no—
334C2#2258947411yescharacteristic
335C2#2259025011yescharacteristic
336C2#2259025511yescharacteristic
337C2#2259047511yescharacteristic
338C2#2259048511yescharacteristic
339C2#2259056211yescharacteristic
340C2#2259057511yescharacteristic
subgroupG/H
centerC23D8×D71
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC2×#357×C71C2
socleC23×C71C2×D8
prime radicalC22×D8×C71C22
hypercenterC2×#357D71
hypocenterC71#13363
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
1. C2×#17986403C2semidirect
4. #22586432C2semidirect
5. #22586431C2semidirect
6. #22586417C2semidirect
50. C2×D8×C71Q8semidirect
178. C71#13363semidirect
201. C2×D8Dic142semidirect

Maximal subgroups

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

subgroupindex
C2×#179864032
C2×#179858002
#225892072
#225864322
#225864312
#225864172
C2×#357×C712
#1336371