Finite Group Databasebeta

← #31884575

Subgroups — #31884575

Order 18176.

464 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985725C211yesmaximal, characteristic
2#22586593C211yesmaximal, characteristic
3C2×#343×C71C211yesmaximal, characteristic
4C22×C4⋊Dic71C411yescharacteristic
5C2×C22⋊(C71⋊C8)—21no—
6C2×C71⋊#95—21no—
7C2×C71⋊#95—21no—
8C2×C71⋊#95C411yescharacteristic
9#17985725—21no—
10#17985725C412yescomplemented
11C22×M16×C71—21no—
12C22×C4⋊C4×C71C2211yescharacteristic
13#343×C71—21no—
14#343×C71C412yescomplemented
15C23×Dic71—21no—
16C23×Dic71—21no—
17C22×C71⋊C8—41no—
18C2×C4⋊Dic71—41no—
19C2×C4⋊Dic71—42no—
20C22⋊(C71⋊C8)—22no—
21C22⋊(C71⋊C8)—41no—
22C71⋊#95—41no—
23C71⋊#95—42no—
24C71⋊#95—41no—
25C71⋊#95—41no—
26C23×C4×C71C2×C411yescharacteristic
27C23×C4×C71D411yescharacteristic
28C23×C4×C71D411yescharacteristic
29C2×M16×C71—22no—
30C2×M16×C71—22no—
31C2×M16×C71—22no—
32C2×M16×C71—41no—
33C2×M16×C71—41no—
34C2×M16×C71—41no—
35C22×C8×C71—41no—
36C2×C4⋊C4×C71—41no—
37C2×C4⋊C4×C71—41no—
38C2×C4⋊C4×C71C2×C411yescharacteristic
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71D411yescharacteristic
41C2×C4⋊C4×C71Q811yescharacteristic
42C22×Dic71—22no—
43C22×Dic71—22no—
44C22×Dic71—41no—
45C22×Dic71—41no—
46C22×Dic71—41no—
47C22×Dic71—41no—
48C22×Dic71—41no—
49C22×Dic71—41no—
50C22×Dic71—41no—
51C22×Dic71—41no—
52C22×Dic71—41no—
53C22×Dic71—41no—
54C22×Dic71—41no—
55C22×Dic71—41no—
56C2×C71⋊C8—42no—
57C2×C71⋊C8—81no—
58C2×C71⋊C8—81no—
59C4⋊Dic71—42no—
60C4⋊Dic71—42no—
61C24×C71C22⋊C411yescharacteristic
62C22×C4×C71—21no—
63C22×C4×C71—21no—
64C22×C4×C71—21no—
65C22×C4×C71—21no—
66C22×C4×C71—21no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—21no—
70C22×C4×C71—41no—
71C22×C4×C71—41no—
72C22×C4×C71—41no—
73C22×C4×C71—41no—
74C22×C4×C71C4211yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71C4⋊C411yescharacteristic
78C22×C4×C71C4⋊C411yescharacteristic
79C22×C4×C71C4⋊C411yescharacteristic
80C22×C4×C71D811yescharacteristic
81C22×C4×C71SD1611yescharacteristic
82C22×C4×C71SD1611yescharacteristic
83C22×C4×C71Q1611yescharacteristic
84M16×C71—24no—
85M16×C71—42no—
86M16×C71—42no—
87M16×C71—42no—
88C2×C8×C71—42no—
89C2×C8×C71—42no—
90C2×C8×C71—42no—
91C4⋊C4×C71—41no—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C23×C71—21no—
96C23×C71—21no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71#9511yescharacteristic
100C23×C71#9911yescharacteristic
101C23×C71#9911yescharacteristic
102C23×C71#10211yescharacteristic
103C23×C71#10311yescharacteristic
104C23×C71#10411yescharacteristic
105C23×C71#10411yescharacteristic
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—42no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—81no—
132C2×C4×C71#10211yescharacteristic
133C2×C4×C71#10311yescharacteristic
134C2×C4×C71#10611yescharacteristic
135C2×C4×C71#10711yescharacteristic
136C2×Dic71—42no—
137C2×Dic71—42no—
138C2×Dic71—42no—
139C2×Dic71—42no—
140C2×Dic71—42no—
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—
150C8×C71—44no—
151C71⋊C8—82no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—21no—
163C22×C71—21no—
164C22×C71#32711yescharacteristic
165C22×C71#32711yescharacteristic
166C22×C71#32811yescharacteristic
167C22×C71#32811yescharacteristic
168C22×C71#33911yescharacteristic
169C22×C71#34011yescharacteristic
170C22×C71#34211yescharacteristic
171C22×C71AΣL(1,𝔽16)11yescharacteristic
172C22×C71#35211yescharacteristic
173C22×C71#35511yescharacteristic
174C22×C71#35611yescharacteristic
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71—21no—
179C4×C71—41no—
180C4×C71—41no—
181C4×C71—41no—
182C4×C71—41no—
183Dic71—82no—
184Dic71—82no—
185#7251—711nomaximal, Sylow
186C2×C71—21no—
187C2×C71—21no—
188C2×C71—21no—
189C2×C71—21no—
190C2×C71#129411yescharacteristic
191C2×C71#134511yescharacteristic
192C2×C71#134611yescharacteristic
193C2×C71#135311yescharacteristic
194C2×C71#135411yescharacteristic
195C2×C71#139011yescharacteristic
196C2×C71#139411yescharacteristic
197C2×#343D7111yescharacteristic
198C2×#324—711no—
199#1441—711no—
200C71#725111yescharacteristic, complemented, Sylow
201C22×M16—21no—
202C22×C4⋊C4—711no—
203C22×C4⋊C4C2×D7111yescharacteristic
204C2×C22⋊C8—1421no—
205C2×#95—711no—
206C2×#95—1421no—
207C2×#95—1421no—
208#343—21no—
209#343Dic7112yescomplemented
210#324—712no—
211#324—1421no—
212C23×C4—1421no—
213C23×C4—1421no—
214C23×C4C4×D7111yescharacteristic
215C23×C4D28411yescharacteristic
216C23×C4C22⋊D7111yescharacteristic
217C2×M16—22no—
218C2×M16—22no—
219C2×M16—22no—
220C2×M16—41no—
221C2×M16—41no—
222C2×M16—41no—
223C22×C8—41no—
224C22×C8—2841no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—2841no—
228C2×C4⋊C4—2842no—
229C2×C4⋊C4Dic14211yescharacteristic
230C2×C4⋊C4C4×D7111yescharacteristic
231C2×C4⋊C4C2×Dic7111yescharacteristic
232C2×C4⋊C4C22⋊D7111yescharacteristic
233C22⋊C8—2841no—
234C22⋊C8—1422no—
235#95—2841no—
236#95—2842no—
237#95—2841no—
238#95—2841no—
239C24(C2×D71)⋊C411yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—2841no—
253C22×C4—2841no—
254C22×C4—2841no—
255C22×C4—2841no—
256C22×C4—2841no—
257C22×C4—2841no—
258C22×C4—2841no—
259C22×C4—2841no—
260C22×C4—2841no—
261C22×C4—2841no—
262C22×C4—2841no—
263C22×C4—2841no—
264C22×C4—1422no—
265C22×C4—1422no—
266C22×C4C4×Dic7111yescharacteristic
267C22×C4(C2×D71)⋊C411yescharacteristic
268C22×C4C22⋊Dic7111yescharacteristic
269C22×C4Dic71⋊C411yescharacteristic
270C22×C4Dic71⋊C411yescharacteristic
271C22×C4C4⋊Dic7111yescharacteristic
272C22×C4#1176585011yescharacteristic
273C22×C4#1176585211yescharacteristic
274C22×C4#1176585311yescharacteristic
275C22×C4#1176585511yescharacteristic
276M16—24no—
277M16—42no—
278M16—42no—
279M16—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—2842no—
284C2×C8—5681no—
285C2×C8—5681no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—2842no—
291C4⋊C4—2842no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C71⋊#9511yescharacteristic
297C23#1455604211yescharacteristic
298C23#1455604211yescharacteristic
299C23#1455604811yescharacteristic
300C23#1455605111yescharacteristic
301C23#1455605611yescharacteristic
302C23#1455605611yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—2842no—
330C2×C4—2842no—
331C2×C4—2842no—
332C2×C4—2842no—
333C2×C4—2842no—
334C2×C4—2842no—
335C2×C4—5681no—
336C2×C4—5681no—
337C2×C4—5681no—
338C2×C4—5681no—
339C2×C4—5681no—
340C2×C4—5681no—
341C2×C4—5681no—
342C2×C4—5681no—
343C2×C4#1455604911yescharacteristic
344C2×C4#1455605211yescharacteristic
345C2×C4#1455605911yescharacteristic
346C2×C4#1455606111yescharacteristic
347C8—44no—
348C8—5682no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1798573211yescharacteristic
362C22#1798573211yescharacteristic
363C22#1798573511yescharacteristic
364C22#1798573511yescharacteristic
365C22#1798575911yescharacteristic
366C22#1798576111yescharacteristic
367C22#1798576511yescharacteristic
368C22#1798578711yescharacteristic
369C22#1798579011yescharacteristic
370C22#1798579511yescharacteristic
371C22#1798579711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—5682no—
381C4—5682no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2#2258621911yescharacteristic
387C2#2258634211yescharacteristic
388C2#2258634511yescharacteristic
389C2#2258636611yescharacteristic
390C2#2258636911yescharacteristic
391C2#2258645911yescharacteristic
392C2#2258646611yescharacteristic
subgroupG/H
centerC23#14556042
commutatorC22×C4×C71C42
FrattiniC22×C4⋊C4C2×D71
FittingC2×#343×C71C2
socleC23×C71#99
prime radicalC24×C71C22⋊C4
cosocleC22×C4⋊C4×C71C22
hypercenterC2×#343D71
hypocenterC71#7251
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
10. #17985725C4semidirect
14. #343×C71C4semidirect
200. C71#7251semidirect
209. #343Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179857252
#225865932
C2×#343×C712
#725171