Finite Group Databasebeta

← C31×#12131 #21468890

Subgroups — C31×#12131 #21468890

Order 7936.

398 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2809C211yesmaximal, characteristic, complemented
2C31×#2007C211yesmaximal, characteristic
3C31×#1616C212yesmaximal
4C31×#1353C212yesmaximal, complemented
5C31×#1319C211yesmaximal, characteristic, complemented
6C2×C31×#137—21no—
7C31×#537C2211yescharacteristic
8C31×#481—22no—
9C2×C31×#104—21no—
10C31×#400C2212yes—
11C31×#398—22no—
12C31×#391—21no—
13C31×#391C2211yescharacteristic
14C31×#389—21no—
15C31×#387—22no—
16C31×#343C2211yescharacteristic
17C31×#327—22no—
18C31×#324C2212yes—
19C2×#41×C31—21no—
20C22×Q8×C31D411yescharacteristic
21C31×#137—42no—
22C31×#137—42no—
23C2×Q16×C31—41no—
24C2×SD16×C31—41no—
25C2×M16×C31—21no—
26C31×C22⋊Q8—41no—
27C31×#121—22no—
28C31×#121—41no—
29C2×C4⋊C4×C31—21no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31D412yes—
34C2×C4⋊C4×C31C2311yescharacteristic
35C2×C42×C31—21no—
36C2×C42×C31D411yescharacteristic
37C31×#106—42no—
38C31×#104—42no—
39C31×#102—42no—
40C31×C22⋊C8—22no—
41C31×#95—22no—
42C31×#95—22no—
43C31×#95—22no—
44C31×#95—22no—
45C31×#95D412yes—
46#41×C31—41no—
47#41×C31—41no—
48#41×C31—41no—
49#41×C31—41no—
50C2×Q8×C31—21no—
51C2×Q8×C31—21no—
52C2×Q8×C31—41no—
53C2×Q8×C31—41no—
54C2×Q8×C31SD1612yes—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—41no—
58C22×C4×C31—21no—
59C22×C4×C31—21no—
60C22×C4×C31—22no—
61C22×C4×C31—22no—
62C22×C4×C31—41no—
63C22×C4×C31C2×D411yescharacteristic
64C22×C4×C31C2×D412yes—
65Q16×C31—44no—
66SD16×C31—44no—
67M16×C31—22no—
68M16×C31—41no—
69C2×C8×C31—41no—
70C2×C8×C31—42no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—41no—
73C4⋊C4×C31—42no—
74C4⋊C4×C31—44no—
75C4⋊C4×C31—42no—
76C4⋊C4×C31—81no—
77C4⋊C4×C31—81no—
78C22⋊C4×C31—42no—
79C42×C31—41no—
80C42×C31—81no—
81#12131C3111yesmaximal, characteristic, direct product factor, Sylow
82C23×C31—41no—
83C23×C31#12011yescharacteristic
84Q8×C31—42no—
85Q8×C31—41no—
86Q8×C31—41no—
87Q8×C31—81no—
88D4×C31—41no—
89D4×C31—41no—
90D4×C31—82no—
91D4×C31—81no—
92D4×C31—81no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—22no—
96C2×C4×C31—22no—
97C2×C4×C31—22no—
98C2×C4×C31—41no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31—41no—
102C2×C4×C31—42no—
103C2×C4×C31—42no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—41no—
109C2×C4×C31—42no—
110C2×C4×C31—81no—
111C2×C4×C31—81no—
112C2×C4×C31C2×SD1611yescharacteristic
113C2×C4×C31#13711yescharacteristic
114C8×C31—42no—
115C8×C31—82no—
116#2809C2×C3111yescharacteristic, complemented
117#2007C2×C3111yescharacteristic
118#1616C2×C3112yes—
119#1353C2×C3112yescomplemented
120#1319C2×C3111yescharacteristic, complemented
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—81no—
124C22×C31—81no—
125C22×C31—81no—
126C22×C31#44811yescharacteristic
127C22×C31#45311yescharacteristic
128C22×C31B(4,𝔽2)11yescharacteristic
129C4×C31—21no—
130C4×C31—21no—
131C4×C31—42no—
132C4×C31—42no—
133C4×C31—42no—
134C4×C31—82no—
135C4×C31—81no—
136C4×C31—82no—
137C2×#137—21no—
138#537C22×C3111yescharacteristic
139#481—22no—
140C2×#104—21no—
141#400C22×C3112yes—
142#398—22no—
143#391—21no—
144#391C22×C3111yescharacteristic
145#389—21no—
146#387—22no—
147#343C22×C3111yescharacteristic
148#327—22no—
149#324C22×C3112yes—
150C2×C31—21no—
151C2×C31—21no—
152C2×C31—161no—
153C2×C31#159611yescharacteristic
154C2×C31#220111yescharacteristic
155C2×C31#220411yescharacteristic
156C2×#41—21no—
157C22×Q8D4×C3111yescharacteristic
158#137—42no—
159#137—42no—
160C2×Q16—41no—
161C2×SD16—41no—
162C2×M16—21no—
163C22⋊Q8—41no—
164#121—22no—
165#121—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4D4×C3112yes—
171C2×C4⋊C4C23×C3111yescharacteristic
172C2×C42—21no—
173C2×C42D4×C3111yescharacteristic
174#106—42no—
175#104—42no—
176#102—42no—
177C22⋊C8—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95D4×C3112yes—
183C31#1213111yescharacteristic, direct product factor, Sylow
184#41—41no—
185#41—41no—
186#41—41no—
187#41—41no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8SD16×C3112yes—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—41no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4C2×D4×C3111yescharacteristic
202C22×C4C2×D4×C3112yes—
203Q16—44no—
204SD16—44no—
205M16—22no—
206M16—41no—
207C2×C8—41no—
208C2×C8—42no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C22⋊C4—42no—
217C42—41no—
218C42—81no—
219C23—41no—
220C23C31×#12011yescharacteristic
221Q8—42no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—41no—
226D4—41no—
227D4—82no—
228D4—81no—
229D4—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4C2×SD16×C3111yescharacteristic
250C2×C4C31×#13711yescharacteristic
251C8—42no—
252C8—82no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22C31×#44811yescharacteristic
259C22C31×#45311yescharacteristic
260C22C31×B(4,𝔽2)11yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4—81no—
268C4—82no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2C31×#159611yescharacteristic
273C2C31×#220111yescharacteristic
274C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2809C2semidirect
4. C31×#1353C2semidirect
5. C31×#1319C2semidirect
81. #12131C31direct (N × H)
116. #2809C2×C31semidirect
119. #1353C2×C31semidirect
120. #1319C2×C31semidirect
183. C31#12131direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28092
C31×#20072
C31×#16162
C31×#13532
C31×#13192
#1213131