Finite Group Databasebeta

← #31919069

Subgroups — #31919069

Order 18176.

2878 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 427 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985854C213yesmaximal
2#22586578C212yesmaximal, complemented
3#22586577C213yesmaximal, complemented
4#22586576C216yesmaximal, complemented
5C71×#1437C211yesmaximal, characteristic, complemented
6C25×D71C2211yescharacteristic
7C22×C22⋊Dic71C2213yes—
8C22×(C2×D71)⋊C4C2216yes—
9C22×(C2×D71)⋊C4C2216yes—
10C22×C4×Dic71C2213yescomplemented
11#17985854C4124yescomplemented
12C2×C71⋊#95C2213yescomplemented
13C2×C71⋊#95C2216yescomplemented
14C22×C22⋊C4×C71C2213yes—
15C22×C42×C71C2211yescharacteristic, complemented
16C2×#95×C71C2213yescomplemented
17C24×D71—43no—
18C24×D71—43no—
19C24×D71—46no—
20C24×D71C2×C4112yes—
21C24×D71C2312yes—
22C23×Dic71C2316yes—
23C2×C22⋊Dic71—46no—
24C2×C22⋊Dic71C2×C4112yes—
25C2×(C2×D71)⋊C4—46no—
26C2×(C2×D71)⋊C4—46no—
27C2×(C2×D71)⋊C4—46no—
28C2×(C2×D71)⋊C4—46no—
29C2×(C2×D71)⋊C4C2×C4124yes—
30C2×(C2×D71)⋊C4C2×C4124yes—
31C2×C4×Dic71—43no—
32C2×C4×Dic71—46no—
33C71⋊#95—43no—
34C71⋊#95—46no—
35C25×C71C2311yescharacteristic
36C23×C4×C71C2313yes—
37C23×C4×C71C2313yes—
38C2×C22⋊C4×C71—43no—
39C2×C22⋊C4×C71—43no—
40C2×C22⋊C4×C71C2×C4112yes—
41C2×C42×C71—43no—
42#95×C71—43no—
43C23×D71—26no—
44C23×D71—26no—
45C23×D71—26no—
46C23×D71—26no—
47C23×D71—424no—
48C23×D71—424no—
49C23×D71—412no—
50C23×D71—412no—
51C23×D71—44no—
52C23×D71—424no—
53C23×D71—412no—
54C23×D71—48no—
55C23×D71—412no—
56C23×D71—44no—
57C23×D71C42116yescomplemented
58C23×D71C22×C4112yes—
59C22×Dic71—26no—
60C22×Dic71—26no—
61C22×Dic71—26no—
62C22×Dic71—26no—
63C22×Dic71—26no—
64C22×Dic71—26no—
65C22×Dic71C22×C4112yes—
66C22⋊Dic71—412no—
67(C2×D71)⋊C4—424no—
68(C2×D71)⋊C4—424no—
69C4×Dic71—412no—
70C24×C71—23no—
71C24×C71—23no—
72C24×C71—26no—
73C24×C71C22×C416yes—
74C24×C71C2411yescharacteristic
75C22×C4×C71—23no—
76C22×C4×C71—26no—
77C22×C4×C71—23no—
78C22×C4×C71—26no—
79C22×C4×C71—23no—
80C22×C4×C71—23no—
81C22×C4×C71—26no—
82C22×C4×C71—26no—
83C22×C4×C71C22×C416yes—
84C22×C4×C71C22×C416yes—
85C22⋊C4×C71—412no—
86C42×C71—44no—
87C23×C71—26no—
88C23×C71—26no—
89C23×C71—212no—
90C23×C71—212no—
91C23×C71—41no—
92C23×C71—46no—
93C23×C71—43no—
94C23×C71—42no—
95C23×C71—43no—
96C23×C71—41no—
97C23×C71C2×C4214yes—
98C23×C71C23×C413yes—
99C23×C71H(5,𝔽2)13yes—
100C23×C71H(5,𝔽2)13yes—
101C23×C71H(5,𝔽2)16yes—
102C22×D71—212no—
103C22×D71—212no—
104C22×D71—212no—
105C22×D71—212no—
106C22×D71—212no—
107C22×D71—212no—
108C22×D71—448no—
109C22×D71—448no—
110C22×D71—448no—
111C22×D71—496no—
112C22×D71—42no—
113C22×D71—46no—
114C22×D71—46no—
115C22×D71—46no—
116C22×D71—42no—
117C22×D71—42no—
118C22×D71—46no—
119C22×D71—42no—
120C22×D71C2×C4218yes—
121C2×C4×C71—26no—
122C2×C4×C71—26no—
123C2×C4×C71—212no—
124C2×C4×C71—212no—
125C2×C4×C71—43no—
126C2×C4×C71—46no—
127C2×C4×C71—46no—
128C2×C4×C71—43no—
129C2×C4×C71—43no—
130C2×C4×C71—43no—
131C2×Dic71—212no—
132C2×Dic71—212no—
133C2×Dic71—212no—
134C2×Dic71—46no—
135C2×Dic71—46no—
136C2×Dic71—46no—
137C2×Dic71—46no—
138C22×C71—212no—
139C22×C71—46no—
140C22×C71—46no—
141C22×C71—412no—
142C22×C71C22×C4211yescharacteristic
143C22×C71#51816yes—
144C22×C71#51816yes—
145C22×C71#51813yes—
146C22×C71#51813yes—
147C22×C71#56011yescharacteristic
148C22×C71#56016yes—
149C22×C71#56013yes—
150C22×C71#56012yes—
151C22×C71H(3,𝔽4)13yes—
152C22×C71H(3,𝔽4)11yescharacteristic
153C2×D71—224no—
154C2×D71—464no—
155C2×D71—412no—
156C2×D71—412no—
157C2×D71—412no—
158C2×D71—412no—
159C4×C71—46no—
160C4×C71—46no—
161Dic71—412no—
162#13827—711nomaximal, Sylow
163C2×C71—44no—
164C2×C71#227313yes—
165C2×C71#237513yes—
166C2×C71#237513yes—
167C2×C71#237516yes—
168D71—48no—
169C2×#379—713no—
170#1437—712no—
171#1437—716no—
172#1437—713no—
173#1437D7111yescharacteristic, complemented
174C71#1382711yescharacteristic, complemented, Sylow
175C26—711no—
176C22×C22⋊C4—713no—
177C22×C22⋊C4—716no—
178C22×C22⋊C4—716no—
179C22×C22⋊C4C2×D7113yes—
180C22×C42—713no—
181C22×C42C2×D7111yescharacteristic, complemented
182#379—7124no—
183C2×#95—716no—
184C2×#95—713no—
185C2×#95C2×D7113yescomplemented
186C25—2843no—
187C25—2843no—
188C25—2846no—
189C25—7112no—
190C25—712no—
191C25C22×D7111yescharacteristic
192C23×C4—716no—
193C23×C4C22×D7113yes—
194C23×C4C22×D7113yes—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—2846no—
198C2×C22⋊C4—2846no—
199C2×C22⋊C4—2846no—
200C2×C22⋊C4—2846no—
201C2×C22⋊C4—2846no—
202C2×C22⋊C4—7112no—
203C2×C22⋊C4—7124no—
204C2×C22⋊C4—7124no—
205C2×C22⋊C4C4×D71112yes—
206C2×C42—43no—
207C2×C42—2843no—
208C2×C42—2846no—
209#95—43no—
210#95—2843no—
211#95—2846no—
212C24—23no—
213C24—23no—
214C24—26no—
215C24—28424no—
216C24—28424no—
217C24—28412no—
218C24—28412no—
219C24—2844no—
220C24—28424no—
221C24—28412no—
222C24—2848no—
223C24—28412no—
224C24—2844no—
225C24—7116no—
226C24—7112no—
227C24—1426no—
228C24—1426no—
229C24—1426no—
230C24—1426no—
231C24C2×C4×D7116yes—
232C24C23×D7111yescharacteristic
233C22×C4—23no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—7112no—
242C22×C4—1426no—
243C22×C4—1426no—
244C22×C4—1426no—
245C22×C4—1426no—
246C22×C4—1426no—
247C22×C4—1426no—
248C22×C4C2×C4×D7116yes—
249C22×C4C2×C4×D7116yes—
250C22⋊C4—412no—
251C22⋊C4—28424no—
252C22⋊C4—28424no—
253C22⋊C4—28412no—
254C42—44no—
255C42—28412no—
256C23—212no—
257C23—212no—
258C23—26no—
259C23—26no—
260C23—41no—
261C23—46no—
262C23—43no—
263C23—42no—
264C23—43no—
265C23—41no—
266C23—28448no—
267C23—28448no—
268C23—28448no—
269C23—28496no—
270C23—2842no—
271C23—2846no—
272C23—2846no—
273C23—2846no—
274C23—2842no—
275C23—2842no—
276C23—2846no—
277C23—2842no—
278C23—718no—
279C23—14212no—
280C23—14212no—
281C23—14212no—
282C23—14212no—
283C23—14212no—
284C23—14212no—
285C23C42×D7114yes—
286C23C22×C4×D7113yes—
287C23#1455617313yes—
288C23#1455617313yes—
289C23#1455617416yes—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—43no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—2846no—
301C2×C4—2846no—
302C2×C4—2846no—
303C2×C4—2846no—
304C2×C4—14212no—
305C2×C4—14212no—
306C2×C4—14212no—
307C22—212no—
308C22—46no—
309C22—46no—
310C22—412no—
311C22—28464no—
312C22—28412no—
313C22—28412no—
314C22—28412no—
315C22—28412no—
316C22—14224no—
317C22C2×C42×D7111yescharacteristic
318C22#1798647116yes—
319C22#1798647116yes—
320C22#1798647213yes—
321C22#1798647313yes—
322C22#1798673611yescharacteristic
323C22#1798673716yes—
324C22#1798673913yes—
325C22#1798674012yes—
326C22#1798674113yes—
327C22#1798674211yescharacteristic
328C4—46no—
329C4—46no—
330C4—28412no—
331C2—44no—
332C2—2848no—
333C2#2259069213yes—
334C2#2259147713yes—
335C2#2259147813yes—
336C2#2259148016yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC71×#1437C2
socle, cosocleC24×C71C24
prime radicalC25×D71C22
hypercenter#1437D71
hypocenterC71#13827
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #22586578C2semidirect
3. #22586577C2semidirect
4. #22586576C2semidirect
5. C71×#1437C2semidirect
10. C22×C4×Dic71C22semidirect
11. #17985854C4semidirect
12. C2×C71⋊#95C22semidirect
13. C2×C71⋊#95C22semidirect
15. C22×C42×C71C22semidirect
16. C2×#95×C71C22semidirect
57. C23×D71C42semidirect
173. #1437D71semidirect
174. C71#13827semidirect
181. C22×C42C2×D71semidirect
185. C2×#95C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858542
#225865782
#225865772
#225865762
C71×#14372
#1382771