Finite Group Databasebeta

← C31×#7736 #21464495

Subgroups — C31×#7736 #21464495

Order 7936.

560 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C31×#2724C211yesmaximal, characteristic, complemented
2C31×#2041C212yesmaximal, complemented
3C31×#2011C211yesmaximal, characteristic
4C31×#1271C212yesmaximal, complemented
5C31×#1270C211yesmaximal, characteristic, complemented
6C22×D8×C31—21no—
7C2×C31×#127C2211yescharacteristic
8C2×C31×#123—21no—
9C2×C31×#102—21no—
10C2×C31×#102—22no—
11C2×C31×#102—22no—
12C31×#398—21no—
13C31×#397—22no—
14C31×#390C2211yescharacteristic
15C31×#389C2212yes—
16C31×#385—22no—
17C31×#336C2211yescharacteristic
18C31×#336C2212yes—
19C22×D4×C31—22no—
20C22×D4×C31—21no—
21C22×D4×C31D411yescharacteristic
22C23×C4×C31—21no—
23C2×D8×C31—42no—
24C2×D8×C31—42no—
25C2×D8×C31—42no—
26C22×C8×C31—21no—
27C22×C8×C31—22no—
28C31×#127—42no—
29C31×#123—42no—
30C31×#123—42no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31D412yes—
34C2×C4⋊C4×C31D411yescharacteristic
35C2×C4⋊C4×C31D412yes—
36C2×C22⋊C4×C31—21no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C42×C31C2311yescharacteristic
40C31×#102—44no—
41C31×#102—42no—
42C31×#102—44no—
43C31×#95—21no—
44C31×#95—22no—
45C31×#95—22no—
46C24×C31—21no—
47C24×C31—41no—
48C2×D4×C31—24no—
49C2×D4×C31—22no—
50C2×D4×C31—21no—
51C2×D4×C31—41no—
52C2×D4×C31—41no—
53C2×D4×C31—41no—
54C2×D4×C31—41no—
55C2×D4×C31—42no—
56C2×D4×C31—42no—
57C2×D4×C31—42no—
58C2×D4×C31—41no—
59C2×D4×C31—41no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—41no—
63C2×D4×C31D812yes—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—22no—
67C22×C4×C31—42no—
68C22×C4×C31—81no—
69C22×C4×C31—81no—
70C22×C4×C31C2×D411yescharacteristic
71C22×C4×C31C2×D412yes—
72D8×C31—48no—
73C2×C8×C31—22no—
74C2×C8×C31—41no—
75C2×C8×C31—41no—
76C2×C8×C31—42no—
77C2×C8×C31—42no—
78C2×C8×C31—42no—
79C4⋊C4×C31—22no—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—81no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—44no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—42no—
90C42×C31—41no—
91#7736C3111yesmaximal, characteristic, direct product factor, Sylow
92C23×C31—22no—
93C23×C31—42no—
94C23×C31—41no—
95C23×C31—42no—
96C23×C31—41no—
97C23×C31—42no—
98C23×C31—81no—
99C23×C31—82no—
100C23×C31—81no—
101C23×C31—82no—
102C23×C31#12011yescharacteristic
103D4×C31—42no—
104D4×C31—42no—
105D4×C31—42no—
106D4×C31—44no—
107D4×C31—44no—
108D4×C31—81no—
109D4×C31—81no—
110D4×C31—81no—
111D4×C31—82no—
112D4×C31—82no—
113C2×C4×C31—21no—
114C2×C4×C31—22no—
115C2×C4×C31—22no—
116C2×C4×C31—21no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—41no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—84no—
126C2×C4×C31C2×D811yescharacteristic
127C2×C4×C31#13512yes—
128C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C31AGL(1,ℤ/8)12yes—
130C8×C31—42no—
131C8×C31—82no—
132#2724C2×C3111yescharacteristic, complemented
133#2041C2×C3112yescomplemented
134#2011C2×C3111yescharacteristic
135#1271C2×C3112yescomplemented
136#1270C2×C3111yescharacteristic, complemented
137C22×C31—42no—
138C22×C31—44no—
139C22×C31—82no—
140C22×C31—81no—
141C22×C31—81no—
142C22×C31—82no—
143C22×C31—84no—
144C22×C31—161no—
145C22×C31—162no—
146C22×C31—161no—
147C22×C31#44711yescharacteristic
148C22×C31#44912yes—
149C22×C31#45311yescharacteristic
150C22×C31#45412yes—
151C22×C31B(4,𝔽2)11yescharacteristic
152C4×C31—21no—
153C4×C31—21no—
154C4×C31—22no—
155C4×C31—22no—
156C4×C31—82no—
157C4×C31—81no—
158C4×C31—82no—
159C22×D8—21no—
160C2×#127C22×C3111yescharacteristic
161C2×#123—21no—
162C2×#102—21no—
163C2×#102—22no—
164C2×#102—22no—
165#398—21no—
166#397—22no—
167#390C22×C3111yescharacteristic
168#389C22×C3112yes—
169#385—22no—
170#336C22×C3111yescharacteristic
171#336C22×C3112yes—
172C2×C31—82no—
173C2×C31—162no—
174C2×C31#159511yescharacteristic
175C2×C31#159712yes—
176C2×C31#162011yescharacteristic
177C2×C31#162512yes—
178C2×C31#165711yescharacteristic
179C22×D4—21no—
180C22×D4—22no—
181C22×D4D4×C3111yescharacteristic
182C23×C4—21no—
183C2×D8—42no—
184C2×D8—42no—
185C2×D8—42no—
186C22×C8—21no—
187C22×C8—22no—
188#127—42no—
189#123—42no—
190#123—42no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4D4×C3112yes—
194C2×C4⋊C4D4×C3112yes—
195C2×C4⋊C4D4×C3111yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C42C23×C3111yescharacteristic
200#102—44no—
201#102—42no—
202#102—44no—
203#95—22no—
204#95—21no—
205#95—22no—
206C31#773611yescharacteristic, direct product factor, Sylow
207C24—21no—
208C24—41no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4D8×C3112yes—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C2×D4×C3111yescharacteristic
232C22×C4C2×D4×C3112yes—
233D8—48no—
234C2×C8—22no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C42—41no—
252C23—22no—
253C23—42no—
254C23—41no—
255C23—42no—
256C23—41no—
257C23—42no—
258C23—81no—
259C23—82no—
260C23—81no—
261C23—82no—
262C23C31×#12011yescharacteristic
263D4—42no—
264D4—42no—
265D4—42no—
266D4—44no—
267D4—44no—
268D4—81no—
269D4—81no—
270D4—81no—
271D4—82no—
272D4—82no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—84no—
286C2×C4C2×D8×C3111yescharacteristic
287C2×C4C31×#13512yes—
288C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
289C2×C4C31×AGL(1,ℤ/8)12yes—
290C8—42no—
291C8—82no—
292C22—42no—
293C22—44no—
294C22—82no—
295C22—81no—
296C22—81no—
297C22—82no—
298C22—84no—
299C22—161no—
300C22—162no—
301C22—161no—
302C22C31×#44711yescharacteristic
303C22C31×#44912yes—
304C22C31×#45311yescharacteristic
305C22C31×#45412yes—
306C22C31×B(4,𝔽2)11yescharacteristic
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—82no—
312C4—81no—
313C4—82no—
314C2—82no—
315C2—162no—
316C2C31×#159511yescharacteristic
317C2C31×#159712yes—
318C2C31×#162011yescharacteristic
319C2C31×#162512yes—
320C2C31×#165711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2011C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2724C2semidirect
2. C31×#2041C2semidirect
4. C31×#1271C2semidirect
5. C31×#1270C2semidirect
91. #7736C31direct (N × H)
132. #2724C2×C31semidirect
133. #2041C2×C31semidirect
135. #1271C2×C31semidirect
136. #1270C2×C31semidirect
206. C31#7736direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27242
C31×#20412
C31×#20112
C31×#12712
C31×#12702
#773631