Finite Group Databasebeta

← #31887496

Subgroups — #31887496

Order 18176.

742 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986052C211yesmaximal, characteristic
2#22590747C211yesmaximal, characteristic
3#22586219C214yesmaximal
4C2×#382×C71C211yesmaximal, characteristic
5C2×#14556087—21no—
6#17986052—22no—
7#17986052C414yescomplemented
8C2×C4⋊(C71⋊C8)—21no—
9C2×C22⋊(C71⋊C8)C2212yes—
10C2×C4×C71⋊C8—21no—
11#17985869C412yescomplemented
12#17985865—21no—
13#17985859—21no—
14#17985859C2212yes—
15C4×C4⋊Dic71—21no—
16C4×C4⋊Dic71C412yescomplemented
17C22×C4⋊C4×C71—21no—
18C22×C42×C71C2211yescharacteristic
19#382×C71—22no—
20#382×C71C414yescomplemented
21C2×#95×C71—21no—
22C2×#95×C71C2212yes—
23C22×C71⋊C8—22no—
24#14556087—42no—
25#14556087—42no—
26C2×C4⋊Dic71—22no—
27C2×C4⋊Dic71—22no—
28C2×C4⋊Dic71C2×C414yes—
29C2×C22⋊Dic71—22no—
30C2×C4×Dic71—22no—
31C4⋊(C71⋊C8)—24no—
32C22⋊(C71⋊C8)—22no—
33C22⋊(C71⋊C8)C2×C414yes—
34C4×C71⋊C8—24no—
35C71⋊#95—22no—
36C71⋊#95—22no—
37C23×C4×C71—22no—
38C23×C4×C71D411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71C2311yescharacteristic
41C2×C4⋊C4×C71—24no—
42C2×C4⋊C4×C71—42no—
43C2×C4⋊C4×C71—42no—
44C2×C42×C71C2×C412yes—
45C2×C42×C71C2×C411yescharacteristic
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71C2×C412yes—
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71D412yes—
51C2×C42×C71Q812yes—
52#95×C71—24no—
53#95×C71—22no—
54#95×C71C2×C414yes—
55C22×Dic71—24no—
56C2×C71⋊C8—24no—
57C2×C71⋊C8—44no—
58C4⋊Dic71—44no—
59C4⋊Dic71—44no—
60C4⋊Dic71—44no—
61C22⋊Dic71—42no—
62C22⋊Dic71—42no—
63C4×Dic71—44no—
64C24×C71C2×D411yescharacteristic
65C22×C4×C71—24no—
66C22×C4×C71—42no—
67C22×C4×C71—42no—
68C22×C4×C71—44no—
69C22×C4×C71—44no—
70C22×C4×C71—42no—
71C22×C4×C71—42no—
72C22×C4×C71C22⋊C411yescharacteristic
73C22×C4×C71C22⋊C411yescharacteristic
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C412yes—
77C22×C4×C71C22⋊C412yes—
78C22×C4×C71C4⋊C412yes—
79C22×C4×C71C4⋊C412yes—
80C22×C4×C71D812yes—
81C22×C4×C71SD1612yes—
82C22×C4×C71SD1612yes—
83C22×C4×C71Q1612yes—
84C22×C4×C71C22×C411yescharacteristic
85C22×C4×C71C22×C411yescharacteristic
86C22×C4×C71C22×C411yescharacteristic
87C22×C4×C71C2×D411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×Q811yescharacteristic
90C4⋊C4×C71—48no—
91C42×C71C4212yes—
92C42×C71C4212yes—
93C42×C71C22⋊C412yes—
94C42×C71C22⋊C412yes—
95C42×C71C4⋊C412yes—
96C42×C71C4⋊C412yes—
97C42×C71C4⋊C412yes—
98C42×C71C4⋊C412yes—
99C23×C71—22no—
100C23×C71—22no—
101C23×C71C2×C22⋊C411yescharacteristic
102C23×C71C2×C22⋊C411yescharacteristic
103C23×C71C2×C4⋊C411yescharacteristic
104C23×C71C2×D811yescharacteristic
105C23×C71C2×SD1611yescharacteristic
106C23×C71C2×SD1611yescharacteristic
107C23×C71C2×Q1611yescharacteristic
108C2×C4×C71—22no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—42no—
113C2×C4×C71—42no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—41no—
118C2×C4×C71—44no—
119C2×C4×C71—44no—
120C2×C4×C71—44no—
121C2×C4×C71—84no—
122C2×C4×C71—84no—
123C2×C4×C71#9512yes—
124C2×C4×C71#9512yes—
125C2×C4×C71#9512yes—
126C2×C4×C71#9512yes—
127C2×C4×C71#10212yes—
128C2×C4×C71#10212yes—
129C2×C4×C71#10214yes—
130C2×C4×C71#10312yes—
131C2×C4×C71#10312yes—
132C2×C4×C71#10314yes—
133C2×C4×C71#10614yes—
134C2×C4×C71#10714yes—
135C2×C4×C71C2×C4211yescharacteristic
136C2×C4×C71C2×C22⋊C411yescharacteristic
137C2×C4×C71C2×C4⋊C411yescharacteristic
138C2×C4×C71C2×C4⋊C411yescharacteristic
139C2×Dic71—44no—
140C2×Dic71—44no—
141C2×Dic71—44no—
142C71⋊C8—44no—
143C22×C71—21no—
144C22×C71—21no—
145C22×C71—21no—
146C22×C71—21no—
147C22×C71—22no—
148C22×C71—22no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71C2×#9511yescharacteristic
152C22×C71#41012yes—
153C22×C71C2×#10211yescharacteristic
154C22×C71C2×#10211yescharacteristic
155C22×C71C2×#10311yescharacteristic
156C22×C71C2×#10311yescharacteristic
157C22×C71#42112yes—
158C22×C71C2×#10611yescharacteristic
159C22×C71C2×#10711yescharacteristic
160C4×C71—22no—
161C4×C71—22no—
162C4×C71—21no—
163C4×C71—21no—
164C4×C71—21no—
165C4×C71—21no—
166C4×C71—84no—
167C4×C71#34014yes—
168C4×C71#34014yes—
169Dic71—84no—
170#8026—711nomaximal, Sylow
171C2×C71—21no—
172C2×C71—21no—
173C2×C71—21no—
174C2×C71—21no—
175C2×C71#151712yes—
176C2×C71#151812yes—
177C2×C71#173311yescharacteristic
178C2×C71C2×#34011yescharacteristic
179C2×C71#174011yescharacteristic
180C2×#431—711no—
181#2285—711no—
182C2×#382D7111yescharacteristic
183#1294—714no—
184C71#802611yescharacteristic, complemented, Sylow
185C2×#117—1421no—
186C22×C4⋊C4—21no—
187C22×C42C2×D7111yescharacteristic
188#431—714no—
189#431—1422no—
190C2×C4⋊C8—1421no—
191C2×C22⋊C8—712no—
192C2×C4×C8—1421no—
193#384—712no—
194#382—22no—
195#382—1421no—
196#382Dic7114yescomplemented
197#380—712no—
198#380—1421no—
199C4×C4⋊C4—712no—
200C4×C4⋊C4—1421no—
201C2×#95—21no—
202C2×#95C2×D7112yes—
203C23×C4—22no—
204C23×C4C22⋊D7111yescharacteristic
205C23×C4C22⋊D7111yescharacteristic
206C23×C4C22×D7111yescharacteristic
207C22×C8—1422no—
208#117—2842no—
209#117—2842no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—714no—
214C2×C4⋊C4—1422no—
215C2×C4⋊C4—1422no—
216C2×C22⋊C4—1422no—
217C2×C42—1422no—
218C2×C42Dic14212yes—
219C2×C42C4×D7112yes—
220C2×C42C4×D7111yescharacteristic
221C2×C42C4×D7111yescharacteristic
222C2×C42D28411yescharacteristic
223C2×C42D28411yescharacteristic
224C2×C42C2×Dic7112yes—
225C2×C42C22⋊D7112yes—
226C4⋊C8—1424no—
227C22⋊C8—714no—
228C22⋊C8—1422no—
229C4×C8—1424no—
230#95—24no—
231#95—22no—
232#95—1422no—
233#95—1422no—
234#95C2×Dic7114yes—
235C24C2×C22⋊D7111yescharacteristic
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—1424no—
244C22×C4(C2×D71)⋊C411yescharacteristic
245C22×C4(C2×D71)⋊C411yescharacteristic
246C22×C4(C2×D71)⋊C411yescharacteristic
247C22×C4(C2×D71)⋊C411yescharacteristic
248C22×C4C22⋊Dic7112yes—
249C22×C4C22⋊Dic7112yes—
250C22×C4Dic71⋊C412yes—
251C22×C4Dic71⋊C412yes—
252C22×C4#1176585012yes—
253C22×C4#1176585212yes—
254C22×C4#1176585312yes—
255C22×C4#1176585512yes—
256C22×C4C2×C4×D7111yescharacteristic
257C22×C4C2×C4×D7111yescharacteristic
258C22×C4C22×Dic7111yescharacteristic
259C22×C4C2×C22⋊D7111yescharacteristic
260C22×C4C2×D28411yescharacteristic
261C22×C4C2×Dic14211yescharacteristic
262C2×C8—2844no—
263C2×C8—1424no—
264C4⋊C4—48no—
265C4⋊C4—2844no—
266C4⋊C4—2844no—
267C4⋊C4—2844no—
268C22⋊C4—2842no—
269C22⋊C4—2842no—
270C42—2844no—
271C42C4×Dic7112yes—
272C42C4×Dic7112yes—
273C42(C2×D71)⋊C412yes—
274C42(C2×D71)⋊C412yes—
275C42Dic71⋊C412yes—
276C42Dic71⋊C412yes—
277C42C4⋊Dic7112yes—
278C42C4⋊Dic7112yes—
279C23—22no—
280C23—22no—
281C23C2×(C2×D71)⋊C411yescharacteristic
282C23C2×C22⋊Dic7111yescharacteristic
283C23C2×Dic71⋊C411yescharacteristic
284C23C2×#1176585011yescharacteristic
285C23C2×#1176585211yescharacteristic
286C23C2×#1176585311yescharacteristic
287C23C2×#1176585511yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—2844no—
304C2×C4—2844no—
305C2×C4—2844no—
306C2×C4C71⋊#9512yes—
307C2×C4C71⋊#9512yes—
308C2×C4C71⋊#9512yes—
309C2×C4C71⋊#9512yes—
310C2×C4#1455604812yes—
311C2×C4#1455604812yes—
312C2×C4#1455604914yes—
313C2×C4#1455605112yes—
314C2×C4#1455605112yes—
315C2×C4#1455605214yes—
316C2×C4#1455605914yes—
317C2×C4#1455606114yes—
318C2×C4C2×C4×Dic7111yescharacteristic
319C2×C4C2×(C2×D71)⋊C411yescharacteristic
320C2×C4C2×Dic71⋊C411yescharacteristic
321C2×C4C2×C4⋊Dic7111yescharacteristic
322C8—2844no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22C2×C71⋊#9511yescharacteristic
332C22#1798597212yes—
333C22C2×#1455604811yescharacteristic
334C22C2×#1455604911yescharacteristic
335C22C2×#1455605111yescharacteristic
336C22C2×#1455605211yescharacteristic
337C22#1798601712yes—
338C22C2×#1455605911yescharacteristic
339C22C2×#1455606111yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—84no—
347C4—5684no—
348C4#1798576114yes—
349C4#1798576114yes—
350C2—21no—
351C2—21no—
352C2—21no—
353C2—21no—
354C2#2258689912yes—
355C2#2258690312yes—
356C2#2258805011yescharacteristic
357C2C2×#1798576111yescharacteristic
358C2#2258807211yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C2×C42
FrattiniC23×C4C22×D71
FittingC2×#382×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenterC2×#382D71
hypocenterC71#8026
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
7. #17986052C4semidirect
11. #17985869C4semidirect
16. C4×C4⋊Dic71C4semidirect
20. #382×C71C4semidirect
184. C71#8026semidirect
196. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179860522
#225907472
#225862192
C2×#382×C712
#802671