Finite Group Databasebeta

← #31913467

Subgroups — #31913467

Order 18176.

434 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22589981C212yesmaximal
2#22588390C211yesmaximal, characteristic, complemented
3#22588386C211yesmaximal, characteristic, complemented
4#22586469C212yesmaximal, complemented
5C71×#2929C211yesmaximal, characteristic
6C2×#14556064—21no—
7#17986037—21no—
8#17985983C2212yes—
9#17985973C2212yes—
10#17985789—22no—
11#17985786—22no—
12C71⋊#338—21no—
13#17985756—21no—
14C2×#131×C71C2211yescharacteristic
15C23×C8×C71C411yescharacteristic, complemented
16#525×C71C2211yescharacteristic
17#435×C71—21no—
18#435×C71—21no—
19C8×D4×C71—21no—
20C8×D4×C71—21no—
21#424×C71C411yescharacteristic, complemented
22#408×C71C2211yescharacteristic
23#407×C71C411yescharacteristic, complemented
24#405×C71C411yescharacteristic, complemented
25C2×#11765849—22no—
26#14556087—22no—
27#14556064—41no—
28#14556064—41no—
29#14556062—41no—
30#14556060—41no—
31#14556047—22no—
32#14556045—22no—
33#14556043—22no—
34#14556043—22no—
35C2×#41×C71C2311yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37#131×C71—41no—
38#131×C71—41no—
39C2×M16×C71C2×C411yescharacteristic
40C2×M16×C71C2×C411yescharacteristic
41C2×M16×C71D411yescharacteristic
42C2×M16×C71D411yescharacteristic
43C22×C8×C71—21no—
44C22×C8×C71—21no—
45C22×C8×C71—41no—
46C22×C8×C71—41no—
47C22×C8×C71C2×C411yescharacteristic
48C22×C8×C71C2×C411yescharacteristic
49C22×C8×C71D411yescharacteristic
50C22×C8×C71D411yescharacteristic
51#123×C71D411yescharacteristic
52#123×C71Q811yescharacteristic
53C22⋊Q8×C71—21no—
54#121×C71—21no—
55#120×C71D411yescharacteristic
56#120×C71Q811yescharacteristic
57C4×D4×C71—21no—
58C4×D4×C71—21no—
59#117×C71C2×C411yescharacteristic
60C4⋊C8×C71—21no—
61C4⋊C8×C71—21no—
62C22⋊C8×C71—21no—
63C22⋊C8×C71—21no—
64C22⋊C8×C71—21no—
65C22⋊C8×C71—21no—
66#97×C71—21no—
67C4×C8×C71—21no—
68#11765849—42no—
69#11765849—42no—
70C2×C71⋊C8—42no—
71C4⋊Dic71—42no—
72C22⋊Dic71—42no—
73C4×Dic71—42no—
74C24×C71C4⋊C411yescharacteristic
75#41×C71—41no—
76#41×C71—41no—
77C2×Q8×C71#4111yescharacteristic
78C2×D4×C71—21no—
79C2×D4×C71—21no—
80C2×D4×C71C2×D411yescharacteristic
81C2×D4×C71C2×Q811yescharacteristic
82C2×D4×C71#4111yescharacteristic
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—41no—
86C22×C4×C71—41no—
87C22×C4×C71C4⋊C411yescharacteristic
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×D411yescharacteristic
91M16×C71—41no—
92M16×C71—41no—
93M16×C71—41no—
94M16×C71—41no—
95C2×C8×C71—21no—
96C2×C8×C71—21no—
97C2×C8×C71—21no—
98C2×C8×C71—21no—
99C2×C8×C71—21no—
100C2×C8×C71—21no—
101C2×C8×C71—21no—
102C2×C8×C71—21no—
103C2×C8×C71—42no—
104C2×C8×C71—41no—
105C2×C8×C71—41no—
106C2×C8×C71—41no—
107C2×C8×C71C22⋊C411yescharacteristic
108C2×C8×C71C22⋊C411yescharacteristic
109C2×C8×C71C22⋊C411yescharacteristic
110C2×C8×C71C22⋊C411yescharacteristic
111C4⋊C4×C71—21no—
112C4⋊C4×C71—21no—
113C4⋊C4×C71—21no—
114C22⋊C4×C71—21no—
115C22⋊C4×C71—21no—
116C22⋊C4×C71—21no—
117C22⋊C4×C71—21no—
118C22⋊C4×C71C4⋊C411yescharacteristic
119C22⋊C4×C71C4⋊C411yescharacteristic
120C42×C71—21no—
121C23×C71—21no—
122C23×C71—41no—
123C23×C71—42no—
124C23×C71C2×C4⋊C411yescharacteristic
125C23×C71#12111yescharacteristic
126C23×C71C22⋊Q811yescharacteristic
127Q8×C71—41no—
128D4×C71—41no—
129D4×C71—41no—
130D4×C71—41no—
131D4×C71—42no—
132D4×C71—42no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—42no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71C2×C22⋊C411yescharacteristic
147C2×C4×C71#11711yescharacteristic
148C2×C4×C71#12111yescharacteristic
149C2×C4×C71C22⋊Q811yescharacteristic
150C2×Dic71—82no—
151C8×C71—21no—
152C8×C71—21no—
153C8×C71—21no—
154C8×C71—41no—
155C8×C71—41no—
156C8×C71#9911yescharacteristic
157C8×C71#9911yescharacteristic
158C71⋊C8—82no—
159C22×C71—22no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—42no—
168C22×C71—42no—
169C22×C71—41no—
170C22×C71#38011yescharacteristic
171C4×C71—21no—
172C4×C71—21no—
173C4×C71—21no—
174C4×C71—41no—
175C4×C71—41no—
176C4×C71—41no—
177C4×C71C2×#9911yescharacteristic
178C4×C71#41011yescharacteristic
179Dic71—162no—
180#12805—711nomaximal, Sylow
181C2×C71—21no—
182C2×C71—41no—
183C2×C71—41no—
184C2×C71—42no—
185C2×C71—41no—
186C2×C71#179911yescharacteristic
187#2929D7111yescharacteristic
188#2120—712no—
189#1813—711no—
190#1812—711no—
191#1395—712no—
192C71#1280511yescharacteristic, complemented, Sylow
193C2×#131C2×D7111yescharacteristic
194C23×C8Dic7111yescharacteristic, complemented
195#525C2×D7111yescharacteristic
196#435—21no—
197#435—21no—
198C8×D4—21no—
199C8×D4—21no—
200C2×#108—1421no—
201#427—1421no—
202#424Dic7111yescharacteristic, complemented
203#413—712no—
204#410—712no—
205#408C2×D7111yescharacteristic
206#407Dic7111yescharacteristic, complemented
207#405Dic7111yescharacteristic, complemented
208#352—1422no—
209AΣL(1,𝔽16)—1422no—
210#338—1421no—
211H(3,ℤ/4)—1421no—
212C2×#41C22×D7111yescharacteristic
213C23×C4C2×Dic7111yescharacteristic
214#131—41no—
215#131—41no—
216C2×M16—1422no—
217C2×M16C2×Dic7111yescharacteristic
218C2×M16C2×Dic7111yescharacteristic
219C2×M16C22⋊D7111yescharacteristic
220C2×M16C22⋊D7111yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8C2×Dic7111yescharacteristic
226C22×C8C2×Dic7111yescharacteristic
227C22×C8C22⋊D7111yescharacteristic
228C22×C8C22⋊D7111yescharacteristic
229#123Dic14211yescharacteristic
230#123D28411yescharacteristic
231C22⋊Q8—21no—
232#121—21no—
233#120Dic14211yescharacteristic
234#120D28411yescharacteristic
235C4×D4—21no—
236C4×D4—21no—
237#117—1422no—
238#117C2×Dic7111yescharacteristic
239#108—2841no—
240#108—2841no—
241#107—2841no—
242#106—2841no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245#101—1422no—
246#100—1422no—
247#99—1422no—
248#99—1422no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#97—21no—
254C4×C8—21no—
255C24C4⋊Dic7111yescharacteristic
256#41—41no—
257#41—41no—
258C2×Q8#1176586511yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C2×D28411yescharacteristic
262C2×D4C2×Dic14211yescharacteristic
263C2×D4#1176586511yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C4⋊Dic7111yescharacteristic
269C22×C4C22×Dic7111yescharacteristic
270C22×C4C2×C22⋊D7111yescharacteristic
271C22×C4C2×C22⋊D7111yescharacteristic
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—2842no—
277M16—2842no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—2842no—
291C2×C8C22⋊Dic7111yescharacteristic
292C2×C8C22⋊Dic7111yescharacteristic
293C2×C8C22⋊Dic7111yescharacteristic
294C2×C8C22⋊Dic7111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—2842no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2842no—
304C22⋊C4C4⋊Dic7111yescharacteristic
305C22⋊C4C4⋊Dic7111yescharacteristic
306C42—21no—
307C42—2842no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23C2×C4⋊Dic7111yescharacteristic
312C23#1455610211yescharacteristic
313C23#1455610711yescharacteristic
314Q8—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—42no—
319D4—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—5682no—
334C2×C4C2×C22⋊Dic7111yescharacteristic
335C2×C4#1455608711yescharacteristic
336C2×C4#1455610211yescharacteristic
337C2×C4#1455610711yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—41no—
342C8—41no—
343C8—5682no—
344C8#1455604311yescharacteristic
345C8#1455604311yescharacteristic
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22#1798585911yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—11362no—
365C4C2×#1455604311yescharacteristic
366C4#1798597311yescharacteristic
367C2—21no—
368C2—41no—
369C2—42no—
370C2—41no—
371C2—41no—
372C2#2258832811yescharacteristic
subgroupG/H
centerC4#17985973
commutatorC22×C4×C71C22×C4
FrattiniC2×#41C22×D71
FittingC71×#2929C2
socleC2×C71#1799
prime radical#525×C71C22
cosocleC2×#41×C71C23
hypercenter#2929D71
hypocenterC71#12805
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #22588390C2semidirect
3. #22588386C2semidirect
4. #22586469C2semidirect
15. C23×C8×C71C4semidirect
21. #424×C71C4semidirect
23. #407×C71C4semidirect
24. #405×C71C4semidirect
192. C71#12805semidirect
194. C23×C8Dic71semidirect
202. #424Dic71semidirect
206. #407Dic71semidirect
207. #405Dic71semidirect

Maximal subgroups

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

subgroupindex
#225899812
#225883902
#225883862
#225864692
C71×#29292
#1280571