Finite Group Databasebeta

← #31895816

Subgroups — #31895816

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589302C211yesmaximal, characteristic
2#22589289C211yesmaximal, characteristic
3#22589099C211yesmaximal, characteristic
4#22588205C211yesmaximal, characteristic
5#22586224C211yesmaximal, characteristic
6#22586174C211yesmaximal, characteristic
7C2×#333×C71C211yesmaximal, characteristic
8C2×#14556062—21no—
9C2×C4⋊(C71⋊C8)C2211yescharacteristic
10C2×C4×C71⋊C8C2211yescharacteristic
11C2×C8×Dic71—21no—
12#17985869C2211yescharacteristic
13C4×C4⋊Dic71C2211yescharacteristic
14#17985762—21no—
15#17985761—21no—
16#17985761—21no—
17#17985761—21no—
18#17985761—21no—
19#17985753—21no—
20C2×C4⋊Q8×C71C2211yescharacteristic
21C2×C4⋊C8×C71C2211yescharacteristic
22C2×C4⋊C8×C71C2211yescharacteristic
23#333×C71C414yescomplemented
24C22×C71⋊C8—21no—
25C22×C71⋊C8D411yescharacteristic
26C22×C71⋊C8D411yescharacteristic
27C2×C4⋊Dic71—21no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71D411yescharacteristic
31C2×C4⋊Dic71D411yescharacteristic
32C2×C4×Dic71—21no—
33#14556062—42no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36C4×C71⋊C8—21no—
37C4×C71⋊C8—21no—
38C8×Dic71—42no—
39C71⋊#95—21no—
40C22×Q8×C71—21no—
41C22×C8×C71—21no—
42C22×C8×C71—21no—
43C4⋊Q8×C71—21no—
44C4⋊Q8×C71—21no—
45C4⋊Q8×C71—21no—
46C4⋊Q8×C71C2×C412yes—
47C2×C4⋊C4×C71—21no—
48C2×C4⋊C4×C71—21no—
49C2×C42×C71C2311yescharacteristic
50C4⋊C8×C71—21no—
51C4⋊C8×C71—21no—
52C4⋊C8×C71C2×C412yes—
53C4⋊C8×C71C2×C412yes—
54C22×Dic71—21no—
55C22×Dic71—21no—
56C22×Dic71—41no—
57C2×C71⋊C8—21no—
58C2×C71⋊C8—21no—
59C2×C71⋊C8—21no—
60C2×C71⋊C8—21no—
61C2×C71⋊C8—21no—
62C2×C71⋊C8—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8Q1612yes—
66C4⋊Dic71—21no—
67C4⋊Dic71—21no—
68C4⋊Dic71—41no—
69C4⋊Dic71—41no—
70C4⋊Dic71—41no—
71C4⋊Dic71—41no—
72C4⋊Dic71—81no—
73C4⋊Dic71SD1612yes—
74C4⋊Dic71Q1612yes—
75C4×Dic71—41no—
76C4×Dic71—41no—
77C2×Q8×C71—22no—
78C2×Q8×C71—41no—
79C2×Q8×C71—81no—
80C2×Q8×C71—81no—
81C22×C4×C71—41no—
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71C2×D411yescharacteristic
84C22×C4×C71#4111yescharacteristic
85C2×C8×C71—22no—
86C2×C8×C71—22no—
87C2×C8×C71—41no—
88C2×C8×C71—41no—
89C2×C8×C71—41no—
90C2×C8×C71—41no—
91C4⋊C4×C71—22no—
92C4⋊C4×C71—22no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C42×C71C22×C411yescharacteristic
96C42×C71#4111yescharacteristic
97C42×C71#4111yescharacteristic
98C42×C71#4111yescharacteristic
99C23×C71#12111yescharacteristic
100Q8×C71—82no—
101Q8×C71—82no—
102C2×C4×C71—21no—
103C2×C4×C71—21no—
104C2×C4×C71—42no—
105C2×C4×C71—81no—
106C2×C4×C71—81no—
107C2×C4×C71#11711yescharacteristic
108C2×C4×C71C4×D411yescharacteristic
109C2×C4×C71C4×D411yescharacteristic
110C2×C4×C71#12311yescharacteristic
111C2×C4×C71#12411yescharacteristic
112C2×C4×C71#12611yescharacteristic
113C2×C4×C71C2×SD1611yescharacteristic
114C2×C4×C71C2×Q1611yescharacteristic
115C2×C4×C71C2×Q1611yescharacteristic
116C2×C4×C71#13511yescharacteristic
117C2×C4×C71#13511yescharacteristic
118C2×C4×C71#13511yescharacteristic
119C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
120C2×C4×C71#13711yescharacteristic
121C2×Dic71—22no—
122C2×Dic71—22no—
123C2×Dic71—41no—
124C2×Dic71—41no—
125C2×Dic71—41no—
126C2×Dic71—41no—
127C2×Dic71—81no—
128C2×Dic71—81no—
129C2×Dic71—81no—
130C8×C71—42no—
131C8×C71—42no—
132C71⋊C8—22no—
133C71⋊C8—41no—
134C71⋊C8—81no—
135C22×C71#38811yescharacteristic
136C22×C71#46211yescharacteristic
137C22×C71#46311yescharacteristic
138C22×C71#46511yescharacteristic
139C22×C71#46711yescharacteristic
140C22×C71#47111yescharacteristic
141C22×C71#47211yescharacteristic
142C4×C71—21no—
143C4×C71—21no—
144C4×C71—82no—
145C4×C71C4×SD1611yescharacteristic
146C4×C71C4×Q1611yescharacteristic
147C4×C71C4×Q1611yescharacteristic
148C4×C71#44011yescharacteristic
149C4×C71#48411yescharacteristic
150C4×C71#48511yescharacteristic
151C4×C71#48711yescharacteristic
152C4×C71#48811yescharacteristic
153Dic71—42no—
154Dic71—42no—
155Dic71—161no—
156#9568—711nomaximal, Sylow
157C2×C71#167211yescharacteristic
158C2×C71#167611yescharacteristic
159C2×C71#167711yescharacteristic
160C2×C71#168311yescharacteristic
161C2×C71#192811yescharacteristic
162C2×C71#193111yescharacteristic
163C2×C71#193311yescharacteristic
164#1990—711no—
165#1987—711no—
166#1947—711no—
167C8×C4⋊C4—711no—
168C2×#333D7111yescharacteristic
169#1296—711no—
170#1277—711no—
171C71#956811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8C2×D7111yescharacteristic
173C2×#107—1421no—
174C2×C4⋊C8—711no—
175C2×C4⋊C8C2×D7111yescharacteristic
176C2×C4⋊C8C2×D7111yescharacteristic
177C2×C4×C8—711no—
178C2×C4×C8—1421no—
179#384—711no—
180C4×C4⋊C4—711no—
181#340—1421no—
182#340—1421no—
183#340—1421no—
184#340—1421no—
185#340—1421no—
186#336—1421no—
187#333Dic7114yescomplemented
188C22×Q8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—711no—
192C22×C8—711no—
193C22×C8—1421no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C2×Dic7112yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—711no—
201C2×C4⋊C4—711no—
202C2×C4⋊C4—1421no—
203C2×C4⋊C4—1421no—
204C2×C4⋊C4—1421no—
205C2×C42—1421no—
206C2×C42C22×D7111yescharacteristic
207#107—2842no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—1421no—
211C4⋊C8—1421no—
212C4⋊C8C2×Dic7112yes—
213C4⋊C8C2×Dic7112yes—
214C4×C8—2842no—
215C4×C8—1421no—
216C4×C8—1421no—
217#95—1421no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—81no—
221C2×Q8—81no—
222C22×C4—41no—
223C22×C4—2841no—
224C22×C4—1421no—
225C22×C4—1421no—
226C22×C4D4×D7111yescharacteristic
227C22×C4D4×D7111yescharacteristic
228C22×C4#1176586511yescharacteristic
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—2841no—
236C2×C8—2841no—
237C2×C8—2841no—
238C2×C8—712no—
239C2×C8—1421no—
240C2×C8—1421no—
241C2×C8—1421no—
242C2×C8—1421no—
243C2×C8—1421no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—2841no—
249C4⋊C4—2841no—
250C4⋊C4—2841no—
251C4⋊C4—2841no—
252C4⋊C4—5681no—
253C4⋊C4—712no—
254C4⋊C4—712no—
255C4⋊C4—1421no—
256C4⋊C4—1421no—
257C42—2841no—
258C42—2841no—
259C42C22×Dic7111yescharacteristic
260C42#1176586411yescharacteristic
261C42#1176586411yescharacteristic
262C42#1176586511yescharacteristic
263C23#1455610111yescharacteristic
264Q8—82no—
265Q8—82no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—2841no—
272C2×C4—2841no—
273C2×C4—2841no—
274C2×C4—2841no—
275C2×C4—5681no—
276C2×C4—5681no—
277C2×C4—5681no—
278C2×C4—1422no—
279C2×C4—1422no—
280C2×C4#1455608711yescharacteristic
281C2×C4D4×Dic7111yescharacteristic
282C2×C4D4×Dic7111yescharacteristic
283C2×C4#1455611011yescharacteristic
284C2×C4#1455611411yescharacteristic
285C2×C4#1455612111yescharacteristic
286C2×C4SD16×D7111yescharacteristic
287C2×C4Q16×D7111yescharacteristic
288C2×C4Q16×D7111yescharacteristic
289C2×C4#1455614911yescharacteristic
290C2×C4#1455615111yescharacteristic
291C2×C4#1455615111yescharacteristic
292C2×C4#1455615511yescharacteristic
293C2×C4#1455615711yescharacteristic
294C8—42no—
295C8—42no—
296C8—2841no—
297C8—5681no—
298C8—1422no—
299C22#1798589211yescharacteristic
300C22#1798620811yescharacteristic
301C22#1798621711yescharacteristic
302C22#1798623111yescharacteristic
303C22#1798623911yescharacteristic
304C22#1798626111yescharacteristic
305C22#1798626811yescharacteristic
306C4—21no—
307C4—21no—
308C4—82no—
309C4—2842no—
310C4—2842no—
311C4—11361no—
312C4SD16×Dic7111yescharacteristic
313C4Q16×Dic7111yescharacteristic
314C4Q16×Dic7111yescharacteristic
315C4#1798610011yescharacteristic
316C4#1798633311yescharacteristic
317C4#1798633911yescharacteristic
318C4#1798634811yescharacteristic
319C4#1798635511yescharacteristic
320C2#2258772511yescharacteristic
321C2#2258774311yescharacteristic
322C2#2258775211yescharacteristic
323C2#2258779411yescharacteristic
324C2#2258898311yescharacteristic
325C2#2258900411yescharacteristic
326C2#2258901811yescharacteristic
subgroupG/H
centerC23#14556101
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC2×#333×C71C2
socle, prime radicalC23×C71#121
cosocleC2×C42×C71C23
hypercenterC2×#333D71
hypocenterC71#9568
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
23. #333×C71C4semidirect
171. C71#9568semidirect
187. #333Dic71semidirect

Maximal subgroups

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

subgroupindex
#225893022
#225892892
#225890992
#225882052
#225862242
#225861742
C2×#333×C712
#956871