Finite Group Databasebeta

← C31×#12185 #21468944

Subgroups — C31×#12185 #21468944

Order 7936.

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

normal subgroups only

This group has 360 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×#2844C211yesmaximal, characteristic, complemented
2C31×#2040C211yesmaximal, characteristic
3C31×#1616C212yesmaximal
4C31×#1353C212yesmaximal, complemented
5C31×#1324C211yesmaximal, characteristic, complemented
6C31×#537C2211yescharacteristic
7C31×#481—22no—
8C2×C31×#104—21no—
9C31×#417—21no—
10C31×#400—22no—
11C31×#400C2212yes—
12C31×#395—21no—
13C31×#395—22no—
14C31×#382—21no—
15C31×#382C2211yescharacteristic
16C31×#343C2211yescharacteristic
17C31×#327—22no—
18C31×#324C2212yes—
19C2×#41×C31—21no—
20C22×Q8×C31—21no—
21C2×M16×C31—21no—
22C31×C22⋊Q8—41no—
23C31×#121—22no—
24C31×#121—41no—
25C2×C4⋊C4×C31—22no—
26C2×C4⋊C4×C31—21no—
27C2×C4⋊C4×C31—21no—
28C2×C4⋊C4×C31D412yes—
29C2×C4⋊C4×C31D411yescharacteristic
30C2×C4⋊C4×C31C2311yescharacteristic
31C2×C42×C31D411yescharacteristic
32C31×#106—42no—
33C31×#104—42no—
34C31×#103—41no—
35C31×#102—41no—
36C31×#102—42no—
37C31×C22⋊C8—22no—
38C31×#95—21no—
39C31×#95—22no—
40C31×#95—22no—
41C31×#95—22no—
42C31×#95—22no—
43C31×#95D412yes—
44#41×C31—41no—
45#41×C31—41no—
46#41×C31—41no—
47#41×C31—41no—
48C2×Q8×C31—21no—
49C2×Q8×C31—82no—
50C2×Q8×C31—81no—
51C2×D4×C31—21no—
52C2×D4×C31—22no—
53C2×D4×C31—41no—
54C22×C4×C31—21no—
55C22×C4×C31—22no—
56C22×C4×C31—21no—
57C22×C4×C31—22no—
58C22×C4×C31—41no—
59C22×C4×C31C2×D411yescharacteristic
60C22×C4×C31C2×D412yes—
61M16×C31—41no—
62M16×C31—41no—
63C2×C8×C31—42no—
64C2×C8×C31—41no—
65C4⋊C4×C31—22no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—41no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—81no—
70C4⋊C4×C31—81no—
71C4⋊C4×C31—82no—
72C22⋊C4×C31—42no—
73C42×C31—21no—
74C42×C31SD1612yes—
75#12185C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—41no—
77C23×C31#12011yescharacteristic
78Q8×C31—41no—
79Q8×C31—41no—
80Q8×C31—84no—
81D4×C31—41no—
82D4×C31—41no—
83D4×C31—82no—
84D4×C31—81no—
85D4×C31—81no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—41no—
91C2×C4×C31—41no—
92C2×C4×C31—42no—
93C2×C4×C31—42no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—42no—
97C2×C4×C31—42no—
98C2×C4×C31—42no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31—42no—
102C2×C4×C31—81no—
103C2×C4×C31—81no—
104C2×C4×C31C2×SD1611yescharacteristic
105C2×C4×C31#13711yescharacteristic
106C8×C31—81no—
107C8×C31—82no—
108#2844C2×C3111yescharacteristic, complemented
109#2040C2×C3111yescharacteristic
110#1616C2×C3112yes—
111#1353C2×C3112yescomplemented
112#1324C2×C3111yescharacteristic, complemented
113C22×C31—21no—
114C22×C31—21no—
115C22×C31—81no—
116C22×C31—81no—
117C22×C31—81no—
118C22×C31#44811yescharacteristic
119C22×C31#45411yescharacteristic
120C22×C31B(4,𝔽2)11yescharacteristic
121C4×C31—21no—
122C4×C31—21no—
123C4×C31—42no—
124C4×C31—42no—
125C4×C31—82no—
126C4×C31—81no—
127C4×C31—81no—
128C4×C31—82no—
129#537C22×C3111yescharacteristic
130#481—22no—
131C2×#104—21no—
132#417—21no—
133#400—22no—
134#400C22×C3112yes—
135#395—22no—
136#395—21no—
137#382—21no—
138#382C22×C3111yescharacteristic
139#343C22×C3111yescharacteristic
140#327—22no—
141#324C22×C3112yes—
142C2×C31—21no—
143C2×C31—21no—
144C2×C31—161no—
145C2×C31#159911yescharacteristic
146C2×C31#220111yescharacteristic
147C2×C31#220211yescharacteristic
148C2×#41—21no—
149C22×Q8—21no—
150C2×M16—21no—
151C22⋊Q8—41no—
152#121—22no—
153#121—41no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4D4×C3111yescharacteristic
158C2×C4⋊C4D4×C3112yes—
159C2×C4⋊C4C23×C3111yescharacteristic
160C2×C42D4×C3111yescharacteristic
161#106—42no—
162#104—42no—
163#103—41no—
164#102—41no—
165#102—42no—
166C22⋊C8—22no—
167#95—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95D4×C3112yes—
173C31#1218511yescharacteristic, direct product factor, Sylow
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—82no—
180C2×Q8—81no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4C2×D4×C3111yescharacteristic
190C22×C4C2×D4×C3112yes—
191M16—41no—
192M16—41no—
193C2×C8—42no—
194C2×C8—41no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—82no—
202C22⋊C4—42no—
203C42—21no—
204C42SD16×C3112yes—
205C23—41no—
206C23C31×#12011yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—84no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4C2×SD16×C3111yescharacteristic
234C2×C4C31×#13711yescharacteristic
235C8—81no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22C31×#44811yescharacteristic
243C22C31×#45411yescharacteristic
244C22C31×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—81no—
252C4—82no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2C31×#159911yescharacteristic
257C2C31×#220111yescharacteristic
258C2C31×#220211yescharacteristic
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×#2844C2semidirect
4. C31×#1353C2semidirect
5. C31×#1324C2semidirect
75. #12185C31direct (N × H)
108. #2844C2×C31semidirect
111. #1353C2×C31semidirect
112. #1324C2×C31semidirect
173. C31#12185direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28442
C31×#20402
C31×#16162
C31×#13532
C31×#13242
#1218531