Finite Group Databasebeta

← C31×#7741 #21464500

Subgroups — C31×#7741 #21464500

Order 7936.

536 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2734C211yesmaximal, characteristic, complemented
2C31×#2041C212yesmaximal
3C31×#2014C211yesmaximal, characteristic
4C31×#1272C211yesmaximal, characteristic, complemented
5C31×#1271C212yesmaximal, complemented
6C22×SD16×C31—21no—
7C2×C31×C4⋊Q8—21no—
8C2×C31×#127C2211yescharacteristic
9C2×C31×#102—21no—
10C2×C31×#102—22no—
11C2×C31×#102—22no—
12C31×#398—21no—
13C31×#395—22no—
14C31×#391C2211yescharacteristic
15C31×#389C2212yes—
16C31×#387—22no—
17C31×#336C2211yescharacteristic
18C31×#336C2212yes—
19C22×Q8×C31—21no—
20C22×Q8×C31D411yescharacteristic
21C22×D4×C31—22no—
22C22×D4×C31—21no—
23C2×SD16×C31—42no—
24C2×SD16×C31—42no—
25C2×SD16×C31—42no—
26C22×C8×C31—21no—
27C22×C8×C31—22no—
28C31×C4⋊Q8—42no—
29C31×C4⋊Q8—42no—
30C31×#127—42no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—22no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31D412yes—
35C2×C4⋊C4×C31D412yes—
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C42×C31—21no—
38C2×C42×C31C2311yescharacteristic
39C31×#102—42no—
40C31×#102—44no—
41C31×#102—44no—
42C31×#95—22no—
43C31×#95—21no—
44C31×#95—22no—
45C31×#95—22no—
46C24×C31—41no—
47C2×Q8×C31—21no—
48C2×Q8×C31—42no—
49C2×Q8×C31—41no—
50C2×Q8×C31—41no—
51C2×Q8×C31—81no—
52C2×Q8×C31—81no—
53C2×Q8×C31D812yes—
54C2×D4×C31—24no—
55C2×D4×C31—22no—
56C2×D4×C31—41no—
57C2×D4×C31—41no—
58C2×D4×C31—42no—
59C2×D4×C31—42no—
60C2×D4×C31—42no—
61C2×D4×C31—41no—
62C2×D4×C31—41no—
63C2×D4×C31—42no—
64C2×D4×C31—41no—
65C2×D4×C31—42no—
66C22×C4×C31—22no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—22no—
70C22×C4×C31C2×D411yescharacteristic
71C22×C4×C31C2×D412yes—
72SD16×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—44no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—42no—
83C4⋊C4×C31—42no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—81no—
86C4⋊C4×C31SD1614yescomplemented
87C42×C31—41no—
88C42×C31—42no—
89#7741C3111yesmaximal, characteristic, direct product factor, Sylow
90C23×C31—42no—
91C23×C31—81no—
92C23×C31—82no—
93C23×C31—81no—
94C23×C31—82no—
95C23×C31#12011yescharacteristic
96Q8×C31—42no—
97Q8×C31—81no—
98Q8×C31—84no—
99D4×C31—42no—
100D4×C31—42no—
101D4×C31—44no—
102D4×C31—44no—
103D4×C31—81no—
104D4×C31—81no—
105D4×C31—82no—
106D4×C31—82no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—21no—
111C2×C4×C31—22no—
112C2×C4×C31—24no—
113C2×C4×C31—42no—
114C2×C4×C31—41no—
115C2×C4×C31—42no—
116C2×C4×C31—41no—
117C2×C4×C31—42no—
118C2×C4×C31—41no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31C2×D811yescharacteristic
123C2×C4×C31C2×SD1612yes—
124C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
125C2×C4×C31#13712yes—
126C8×C31—42no—
127C8×C31—82no—
128#2734C2×C3111yescharacteristic, complemented
129#2041C2×C3112yes—
130#2014C2×C3111yescharacteristic
131#1272C2×C3111yescharacteristic, complemented
132#1271C2×C3112yescomplemented
133C22×C31—82no—
134C22×C31—84no—
135C22×C31—161no—
136C22×C31—162no—
137C22×C31—161no—
138C22×C31#44711yescharacteristic
139C22×C31#44812yes—
140C22×C31#45311yescharacteristic
141C22×C31#45412yes—
142C22×C31B(4,𝔽2)11yescharacteristic
143C4×C31—21no—
144C4×C31—21no—
145C4×C31—22no—
146C4×C31—22no—
147C4×C31—42no—
148C4×C31—44no—
149C4×C31—82no—
150C4×C31—81no—
151C22×SD16—21no—
152C2×C4⋊Q8—21no—
153C2×#127C22×C3111yescharacteristic
154C2×#102—21no—
155C2×#102—22no—
156C2×#102—22no—
157#398—21no—
158#395—22no—
159#391C22×C3111yescharacteristic
160#389C22×C3112yes—
161#387—22no—
162#336C22×C3111yescharacteristic
163#336C22×C3112yes—
164C2×C31—162no—
165C2×C31#159511yescharacteristic
166C2×C31#159912yes—
167C2×C31#162311yescharacteristic
168C2×C31#162712yes—
169C2×C31#165711yescharacteristic
170C22×Q8—21no—
171C22×Q8D4×C3111yescharacteristic
172C22×D4—21no—
173C22×D4—22no—
174C2×SD16—42no—
175C2×SD16—42no—
176C2×SD16—42no—
177C22×C8—21no—
178C22×C8—22no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181#127—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4D4×C3112yes—
186C2×C4⋊C4D4×C3112yes—
187C2×C4⋊C4D4×C3111yescharacteristic
188C2×C42—21no—
189C2×C42C23×C3111yescharacteristic
190#102—42no—
191#102—44no—
192#102—44no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—22no—
197C31#774111yescharacteristic, direct product factor, Sylow
198C24—41no—
199C2×Q8—21no—
200C2×Q8—42no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—81no—
204C2×Q8—81no—
205C2×Q8D8×C3112yes—
206C2×D4—24no—
207C2×D4—22no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—42no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C2×D4×C3111yescharacteristic
223C22×C4C2×D4×C3112yes—
224SD16—48no—
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—22no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—81no—
238C4⋊C4SD16×C3114yescomplemented
239C42—41no—
240C42—42no—
241C23—42no—
242C23—81no—
243C23—82no—
244C23—81no—
245C23—82no—
246C23C31×#12011yescharacteristic
247Q8—42no—
248Q8—81no—
249Q8—84no—
250D4—42no—
251D4—42no—
252D4—44no—
253D4—44no—
254D4—81no—
255D4—81no—
256D4—82no—
257D4—82no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4C2×D8×C3111yescharacteristic
274C2×C4C2×SD16×C3112yes—
275C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
276C2×C4C31×#13712yes—
277C8—42no—
278C8—82no—
279C22—82no—
280C22—84no—
281C22—161no—
282C22—162no—
283C22—161no—
284C22C31×#44711yescharacteristic
285C22C31×#44812yes—
286C22C31×#45311yescharacteristic
287C22C31×#45412yes—
288C22C31×B(4,𝔽2)11yescharacteristic
289C4—21no—
290C4—21no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—44no—
295C4—82no—
296C4—81no—
297C2—162no—
298C2C31×#159511yescharacteristic
299C2C31×#159912yes—
300C2C31×#162311yescharacteristic
301C2C31×#162712yes—
302C2C31×#165711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
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×#2734C2semidirect
4. C31×#1272C2semidirect
5. C31×#1271C2semidirect
86. C4⋊C4×C31SD16semidirect
89. #7741C31direct (N × H)
128. #2734C2×C31semidirect
131. #1272C2×C31semidirect
132. #1271C2×C31semidirect
197. C31#7741direct (N × H)
238. C4⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27342
C31×#20412
C31×#20142
C31×#12722
C31×#12712
#774131