Finite Group Databasebeta

← C31×#15986 #21472745

Subgroups — C31×#15986 #21472745

Order 7936.

2938 conjugacy classes of subgroups; only the 374 normal subgroups are shown.

normal subgroups only

This group has 2938 conjugacy classes of subgroups — too many to draw. Only its 374 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
1C22×C31×#126C211yesmaximal, characteristic, complemented
2C2×C31×#379C213yesmaximal
3C31×#1440C213yesmaximal, complemented
4C31×#1439C216yesmaximal, complemented
5C31×#1437C212yesmaximal, complemented
6C26×C31C2211yescharacteristic
7C2×C31×#126—212no—
8C22×C4⋊C4×C31C2213yescomplemented
9C22×C22⋊C4×C31C2216yes—
10C22×C22⋊C4×C31C2216yes—
11C22×C22⋊C4×C31C2213yes—
12C22×C22⋊C4×C31C2213yes—
13C22×C42×C31C2211yescharacteristic, complemented
14C31×#379—212no—
15C2×C31×#95C2216yescomplemented
16C2×C31×#95C2216yescomplemented
17C25×C31—26no—
18C25×C31—212no—
19C25×C31—46no—
20C25×C31C2311yescharacteristic
21C25×C31C2312yes—
22C23×C4×C31C2316yes—
23C23×C4×C31C2313yes—
24C23×C4×C31C2313yes—
25C31×#126—232no—
26C2×C4⋊C4×C31—26no—
27C2×C4⋊C4×C31—26no—
28C2×C4⋊C4×C31—26no—
29C2×C22⋊C4×C31—26no—
30C2×C22⋊C4×C31—26no—
31C2×C22⋊C4×C31—26no—
32C2×C22⋊C4×C31—212no—
33C2×C22⋊C4×C31—212no—
34C2×C22⋊C4×C31—212no—
35C2×C22⋊C4×C31—212no—
36C2×C22⋊C4×C31—46no—
37C2×C22⋊C4×C31—46no—
38C2×C22⋊C4×C31—46no—
39C2×C22⋊C4×C31D4112yes—
40C2×C42×C31—26no—
41C31×#95—46no—
42C31×#95—46no—
43C24×C31—26no—
44C24×C31—26no—
45C24×C31—26no—
46C24×C31—26no—
47C24×C31—26no—
48C24×C31—26no—
49C24×C31—224no—
50C24×C31—424no—
51C24×C31—424no—
52C24×C31—424no—
53C24×C31—412no—
54C24×C31—424no—
55C24×C31—48no—
56C24×C31—44no—
57C24×C31—48no—
58C24×C31C2×D416yes—
59C24×C31#41112yes—
60C24×C31C2411yescharacteristic
61C22×C4×C31—23no—
62C22×C4×C31—23no—
63C22×C4×C31—23no—
64C22×C4×C31—23no—
65C22×C4×C31—26no—
66C22×C4×C31—26no—
67C22×C4×C31—26no—
68C22×C4×C31—26no—
69C22×C4×C31—26no—
70C22×C4×C31—26no—
71C22×C4×C31—26no—
72C22×C4×C31—26no—
73C22×C4×C31—26no—
74C22×C4×C31—26no—
75C22×C4×C31—26no—
76C22×C4×C31C2×D416yes—
77C22×C4×C31C2×D416yes—
78C4⋊C4×C31—224no—
79C22⋊C4×C31—224no—
80C22⋊C4×C31—424no—
81C22⋊C4×C31—424no—
82C22⋊C4×C31—412no—
83C42×C31—28no—
84#15986C3111yesmaximal, characteristic, direct product factor, Sylow
85C23×C31—26no—
86C23×C31—212no—
87C23×C31—26no—
88C23×C31—212no—
89C23×C31—212no—
90C23×C31—212no—
91C23×C31—212no—
92C23×C31—212no—
93C23×C31—212no—
94C23×C31—212no—
95C23×C31—448no—
96C23×C31—496no—
97C23×C31—496no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—46no—
102C23×C31—46no—
103C23×C31—46no—
104C23×C31—46no—
105C23×C31—46no—
106C23×C31—46no—
107C23×C31—42no—
108C23×C31—42no—
109C23×C31—42no—
110C23×C31#12014yes—
111C23×C31#12618yescomplemented
112C23×C31C22×D413yes—
113C23×C31C2×#4116yes—
114C23×C31H(5,𝔽2)16yes—
115C2×C4×C31—26no—
116C2×C4×C31—26no—
117C2×C4×C31—212no—
118C2×C4×C31—212no—
119C2×C4×C31—43no—
120C2×C4×C31—43no—
121C2×C4×C31—43no—
122C2×C4×C31—43no—
123C2×C4×C31—46no—
124C2×C4×C31—46no—
125C2×C4×C31—46no—
126C2×C4×C31—46no—
127C2×C4×C31—46no—
128C2×C4×C31—46no—
129C2×C4×C31—46no—
130C2×C4×C31—46no—
131C2×C4×C31—46no—
132C22×#126C2×C3111yescharacteristic, complemented
133C2×#379C2×C3113yes—
134#1440C2×C3113yescomplemented
135#1439C2×C3116yescomplemented
136#1437C2×C3112yescomplemented
137C22×C31—212no—
138C22×C31—224no—
139C22×C31—464no—
140C22×C31—412no—
141C22×C31—412no—
142C22×C31—412no—
143C22×C31—412no—
144C22×C31—412no—
145C22×C31—412no—
146C22×C31C2×#12011yescharacteristic
147C22×C31#52513yes—
148C22×C31C2×#12612yes—
149C22×C31#53513yes—
150C22×C31#53816yes—
151C22×C31#54616yes—
152C22×C31#54616yes—
153C22×C31#55116yes—
154C22×C31#56012yes—
155C4×C31—46no—
156C4×C31—46no—
157C4×C31—86no—
158C26C22×C3111yescharacteristic
159C2×#126—212no—
160C22×C4⋊C4C22×C3113yescomplemented
161C22×C22⋊C4C22×C3116yes—
162C22×C22⋊C4C22×C3116yes—
163C22×C22⋊C4C22×C3113yes—
164C22×C22⋊C4C22×C3113yes—
165C22×C42C22×C3111yescharacteristic, complemented
166#379—212no—
167C2×#95C22×C3116yescomplemented
168C2×#95C22×C3116yescomplemented
169C2×C31—44no—
170C2×C31—48no—
171C2×C31#240413yes—
172C2×C31#248016yes—
173C2×C31#268516yes—
174C25—26no—
175C25—212no—
176C25—46no—
177C25C23×C3111yescharacteristic
178C25C23×C3112yes—
179C23×C4C23×C3116yes—
180C23×C4C23×C3113yes—
181C23×C4C23×C3113yes—
182#126—232no—
183C2×C4⋊C4—26no—
184C2×C4⋊C4—26no—
185C2×C4⋊C4—26no—
186C2×C22⋊C4—26no—
187C2×C22⋊C4—26no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—212no—
190C2×C22⋊C4—212no—
191C2×C22⋊C4—212no—
192C2×C22⋊C4—212no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4D4×C31112yes—
197C2×C42—26no—
198#95—46no—
199#95—46no—
200C31#1598611yescharacteristic, direct product factor, Sylow
201C24—26no—
202C24—26no—
203C24—26no—
204C24—26no—
205C24—26no—
206C24—26no—
207C24—224no—
208C24—424no—
209C24—424no—
210C24—424no—
211C24—412no—
212C24—424no—
213C24—48no—
214C24—44no—
215C24—48no—
216C24C2×D4×C3116yes—
217C24#41×C31112yes—
218C24C24×C3111yescharacteristic
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4C2×D4×C3116yes—
235C22×C4C2×D4×C3116yes—
236C4⋊C4—224no—
237C22⋊C4—224no—
238C22⋊C4—424no—
239C22⋊C4—424no—
240C22⋊C4—412no—
241C42—28no—
242C23—26no—
243C23—212no—
244C23—26no—
245C23—212no—
246C23—212no—
247C23—212no—
248C23—212no—
249C23—212no—
250C23—212no—
251C23—212no—
252C23—448no—
253C23—496no—
254C23—496no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—46no—
259C23—46no—
260C23—46no—
261C23—46no—
262C23—46no—
263C23—46no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23C31×#12014yes—
268C23C31×#12618yescomplemented
269C23C22×D4×C3113yes—
270C23C2×#41×C3116yes—
271C23C31×H(5,𝔽2)16yes—
272C2×C4—26no—
273C2×C4—26no—
274C2×C4—212no—
275C2×C4—212no—
276C2×C4—43no—
277C2×C4—43no—
278C2×C4—43no—
279C2×C4—43no—
280C2×C4—46no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—46no—
285C2×C4—46no—
286C2×C4—46no—
287C2×C4—46no—
288C2×C4—46no—
289C22—212no—
290C22—224no—
291C22—464no—
292C22—412no—
293C22—412no—
294C22—412no—
295C22—412no—
296C22—412no—
297C22—412no—
298C22C2×C31×#12011yescharacteristic
299C22C31×#52513yes—
300C22C2×C31×#12612yes—
301C22C31×#53513yes—
302C22C31×#53816yes—
303C22C31×#54616yes—
304C22C31×#54616yes—
305C22C31×#55116yes—
306C22C31×#56012yes—
307C4—46no—
308C4—46no—
309C4—86no—
310C2—44no—
311C2—48no—
312C2C31×#240413yes—
313C2C31×#248016yes—
314C2C31×#268516yes—
subgroupG/H
center, socleC24×C31C24
commutator, Frattini, cosocleC24C24×C31
prime radicalC26×C31C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×#126C2semidirect
3. C31×#1440C2semidirect
4. C31×#1439C2semidirect
5. C31×#1437C2semidirect
8. C22×C4⋊C4×C31C22semidirect
13. C22×C42×C31C22semidirect
15. C2×C31×#95C22semidirect
16. C2×C31×#95C22semidirect
84. #15986C31direct (N × H)
111. C23×C31#126semidirect
132. C22×#126C2×C31semidirect
134. #1440C2×C31semidirect
135. #1439C2×C31semidirect
136. #1437C2×C31semidirect
160. C22×C4⋊C4C22×C31semidirect
165. C22×C42C22×C31semidirect
167. C2×#95C22×C31semidirect
168. C2×#95C22×C31semidirect
200. C31#15986direct (N × H)
268. C23C31×#126semidirect

Maximal subgroups

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

subgroupindex
C22×C31×#1262
C2×C31×#3792
C31×#14402
C31×#14392
C31×#14372
#1598631