Finite Group Databasebeta

← C31×#10235 #21466994

Subgroups — C31×#10235 #21466994

Order 7936.

712 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 154 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×C31×#476C212yesmaximal, complemented
2C31×#2537C211yesmaximal, characteristic
3C31×#1956C211yesmaximal, characteristic, complemented
4C2×C31×D4⋊C8C212yesmaximal, complemented
5C31×#1296C211yesmaximal, characteristic, complemented
6C2×C31×C4⋊Q8C2211yescharacteristic
7C2×C31×#121—22no—
8C2×C4×D4×C31C2212yes—
9C31×#476—24no—
10C31×#476—24no—
11C2×C31×#106—22no—
12C2×C31×C4⋊C8C2212yes—
13C2×C31×#103—22no—
14C2×C31×#102—22no—
15C2×C4×C8×C31C2211yescharacteristic
16C31×#394—22no—
17C31×#384C2211yescharacteristic
18C31×#380—22no—
19C31×D4⋊C8—24no—
20C22×Q8×C31—21no—
21C22×D4×C31—22no—
22C22×D4×C31D412yes—
23C23×C4×C31—22no—
24C22×C8×C31—21no—
25C22×C8×C31—22no—
26C31×C4⋊Q8—22no—
27C31×C4⋊Q8—21no—
28C31×C4⋊Q8D412yescomplemented
29C31×#121—44no—
30C31×#121—44no—
31C4×D4×C31—22no—
32C4×D4×C31—22no—
33C4×D4×C31—22no—
34C4×D4×C31—22no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31D412yes—
38C2×C4⋊C4×C31D412yes—
39C2×C22⋊C4×C31—24no—
40C2×C22⋊C4×C31—22no—
41C2×C42×C31C2311yescharacteristic
42C31×#106—28no—
43C31×C4⋊C8—22no—
44C31×C4⋊C8D414yescomplemented
45C31×#103—44no—
46C31×#102—28no—
47C4×C8×C31—22no—
48C31×#95—22no—
49C24×C31—22no—
50C2×Q8×C31—22no—
51C2×Q8×C31—42no—
52C2×Q8×C31—42no—
53C2×Q8×C31—41no—
54C2×D4×C31—22no—
55C2×D4×C31—22no—
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31—22no—
59C2×D4×C31—48no—
60C2×D4×C31—82no—
61C2×D4×C31SD1614yes—
62C22×C4×C31—22no—
63C22×C4×C31—22no—
64C22×C4×C31—41no—
65C22×C4×C31—42no—
66C22×C4×C31—42no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—42no—
70C22×C4×C31—42no—
71C22×C4×C31C2×D411yescharacteristic
72C22×C4×C31C2×D412yes—
73C2×C8×C31—22no—
74C2×C8×C31—24no—
75C2×C8×C31—41no—
76C2×C8×C31—41no—
77C2×C8×C31—42no—
78C2×C8×C31—42no—
79C4⋊C4×C31—24no—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—22no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—42no—
86C22⋊C4×C31—48no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C42×C31C2×D411yescharacteristic
90C42×C31C2×D412yes—
91C42×C31#4111yescharacteristic
92#10235C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—24no—
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—42no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31#12011yescharacteristic
101Q8×C31—44no—
102Q8×C31—82no—
103D4×C31—24no—
104D4×C31—24no—
105D4×C31—42no—
106D4×C31—42no—
107D4×C31—88no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—42no—
111C2×C4×C31—42no—
112C2×C4×C31—42no—
113C2×C4×C31—44no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—81no—
117C2×C4×C31—81no—
118C2×C4×C31—82no—
119C2×C4×C31—82no—
120C2×C4×C31—82no—
121C2×C4×C31—82no—
122C2×C4×C31#12012yes—
123C2×C4×C31#12111yescharacteristic
124C2×C4×C31#12112yes—
125C2×C4×C31#12711yescharacteristic
126C2×C4×C31C2×SD1612yes—
127C2×C4×C31#13512yes—
128C2×C4×C31AGL(1,ℤ/8)12yes—
129C2×C4×C31#13712yes—
130C8×C31—42no—
131C8×C31—44no—
132C2×#476C2×C3112yescomplemented
133#2537C2×C3111yescharacteristic
134#1956C2×C3111yescharacteristic, complemented
135C2×D4⋊C8C2×C3112yescomplemented
136#1296C2×C3111yescharacteristic, complemented
137C22×C31—44no—
138C22×C31—44no—
139C22×C31—44no—
140C22×C31—82no—
141C22×C31—82no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31#39211yescharacteristic
145C22×C31#45012yes—
146C22×C31#45212yes—
147C22×C31#45311yescharacteristic
148C22×C31#45411yescharacteristic
149C4×C31—21no—
150C4×C31—21no—
151C4×C31—82no—
152C4×C31—82no—
153C4×C31—82no—
154C4×C31#44812yes—
155C4×C31#44912yes—
156C4×C31#46512yes—
157C4×C31#46812yes—
158C2×C4⋊Q8C22×C3111yescharacteristic
159C2×#121—22no—
160C2×C4×D4C22×C3112yes—
161#476—24no—
162#476—24no—
163C2×#106—22no—
164C2×C4⋊C8C22×C3112yes—
165C2×#103—22no—
166C2×#102—22no—
167C2×C4×C8C22×C3111yescharacteristic
168#394—22no—
169#384C22×C3111yescharacteristic
170#380—22no—
171D4⋊C8—24no—
172C2×C31—84no—
173C2×C31#162911yescharacteristic
174C2×C31#163411yescharacteristic
175C2×C31#163512yes—
176C2×C31#200012yes—
177C2×C31#200611yescharacteristic
178C22×Q8—21no—
179C22×D4—22no—
180C22×D4D4×C3112yes—
181C23×C4—22no—
182C22×C8—21no—
183C22×C8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—21no—
186C4⋊Q8D4×C3112yescomplemented
187#121—44no—
188#121—44no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4D4×C3112yes—
196C2×C4⋊C4D4×C3112yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C42C23×C3111yescharacteristic
200#106—28no—
201C4⋊C8—22no—
202C4⋊C8D4×C3114yescomplemented
203#103—44no—
204#102—28no—
205C4×C8—22no—
206#95—22no—
207C31#1023511yescharacteristic, direct product factor, Sylow
208C24—22no—
209C2×Q8—22no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—48no—
219C2×D4—82no—
220C2×D4SD16×C3114yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C2×D4×C3111yescharacteristic
231C22×C4C2×D4×C3112yes—
232C2×C8—22no—
233C2×C8—24no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—48no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42C2×D4×C3111yescharacteristic
249C42C2×D4×C3112yes—
250C42#41×C3111yescharacteristic
251C23—24no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23C31×#12011yescharacteristic
259Q8—44no—
260Q8—82no—
261D4—24no—
262D4—24no—
263D4—42no—
264D4—42no—
265D4—88no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4C31×#12012yes—
281C2×C4C31×#12111yescharacteristic
282C2×C4C31×#12112yes—
283C2×C4C31×#12711yescharacteristic
284C2×C4C2×SD16×C3112yes—
285C2×C4C31×#13512yes—
286C2×C4C31×AGL(1,ℤ/8)12yes—
287C2×C4C31×#13712yes—
288C8—42no—
289C8—44no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22C31×#39211yescharacteristic
298C22C31×#45012yes—
299C22C31×#45212yes—
300C22C31×#45311yescharacteristic
301C22C31×#45411yescharacteristic
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—82no—
307C4C31×#44812yes—
308C4C31×#44912yes—
309C4C31×#46512yes—
310C4C31×#46812yes—
311C2—84no—
312C2C31×#162911yescharacteristic
313C2C31×#163411yescharacteristic
314C2C31×#163512yes—
315C2C31×#200012yes—
316C2C31×#200611yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2537C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C31×#476C2semidirect
3. C31×#1956C2semidirect
4. C2×C31×D4⋊C8C2semidirect
5. C31×#1296C2semidirect
28. C31×C4⋊Q8D4semidirect
44. C31×C4⋊C8D4semidirect
92. #10235C31direct (N × H)
132. C2×#476C2×C31semidirect
134. #1956C2×C31semidirect
135. C2×D4⋊C8C2×C31semidirect
136. #1296C2×C31semidirect
186. C4⋊Q8D4×C31semidirect
202. C4⋊C8D4×C31semidirect
207. C31#10235direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4762
C31×#25372
C31×#19562
C2×C31×D4⋊C82
C31×#12962
#1023531