Finite Group Databasebeta

← C31×#12127 #21468886

Subgroups — C31×#12127 #21468886

Order 7936.

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

normal subgroups only

This group has 454 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×#2804C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic
3C31×#1616C212yesmaximal, complemented
4C31×#1353C212yesmaximal, complemented
5C31×#1318C211yesmaximal, characteristic, complemented
6C2×C31×AGL(1,ℤ/8)—21no—
7C31×#537C2211yescharacteristic
8C31×#481—22no—
9C2×C31×#104—21no—
10C31×#400C2212yes—
11C31×#393—22no—
12C31×#391—21no—
13C31×#390C2211yescharacteristic
14C31×#386—21no—
15C31×#385—22no—
16C31×#343C2211yescharacteristic
17C31×#327—22no—
18C31×#324C2212yes—
19C2×#41×C31—21no—
20C22×Q8×C31—21no—
21C22×D4×C31D411yescharacteristic
22C23×C4×C31—21no—
23C31×AGL(1,ℤ/8)—42no—
24C31×AGL(1,ℤ/8)—42no—
25C2×SD16×C31—41no—
26C2×D8×C31—41no—
27C2×M16×C31—21no—
28C31×C22⋊Q8—41no—
29C31×#121—22no—
30C31×#121—41no—
31C2×C4⋊C4×C31—21no—
32C2×C4⋊C4×C31D412yes—
33C2×C4⋊C4×C31C2311yescharacteristic
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—21no—
37C2×C42×C31D411yescharacteristic
38C31×#106—42no—
39C31×#104—42no—
40C31×#102—42no—
41C31×C22⋊C8—22no—
42C31×#95—22no—
43C31×#95—22no—
44C31×#95—22no—
45C31×#95D412yes—
46C24×C31—21no—
47#41×C31—41no—
48#41×C31—41no—
49#41×C31—41no—
50#41×C31—41no—
51C2×Q8×C31—21no—
52C2×Q8×C31—42no—
53C2×Q8×C31—81no—
54C2×Q8×C31—81no—
55C2×D4×C31—21no—
56C2×D4×C31—22no—
57C2×D4×C31—21no—
58C2×D4×C31—41no—
59C2×D4×C31—41no—
60C2×D4×C31—41no—
61C2×D4×C31SD1612yes—
62C22×C4×C31—21no—
63C22×C4×C31—22no—
64C22×C4×C31—22no—
65C22×C4×C31—41no—
66C22×C4×C31—82no—
67C22×C4×C31—81no—
68C22×C4×C31C2×D411yescharacteristic
69C22×C4×C31C2×D412yes—
70SD16×C31—44no—
71D8×C31—44no—
72M16×C31—22no—
73M16×C31—41no—
74C2×C8×C31—41no—
75C2×C8×C31—42no—
76C4⋊C4×C31—22no—
77C4⋊C4×C31—41no—
78C4⋊C4×C31—42no—
79C4⋊C4×C31—81no—
80C22⋊C4×C31—44no—
81C22⋊C4×C31—42no—
82C22⋊C4×C31—82no—
83C22⋊C4×C31—81no—
84C42×C31—41no—
85#12127C3111yesmaximal, characteristic, direct product factor, Sylow
86C23×C31—22no—
87C23×C31—42no—
88C23×C31—42no—
89C23×C31—41no—
90C23×C31—41no—
91C23×C31—41no—
92C23×C31#12011yescharacteristic
93Q8×C31—41no—
94Q8×C31—41no—
95Q8×C31—84no—
96D4×C31—42no—
97D4×C31—41no—
98D4×C31—41no—
99D4×C31—81no—
100D4×C31—82no—
101D4×C31—81no—
102D4×C31—81no—
103C2×C4×C31—21no—
104C2×C4×C31—21no—
105C2×C4×C31—22no—
106C2×C4×C31—22no—
107C2×C4×C31—41no—
108C2×C4×C31—41no—
109C2×C4×C31—42no—
110C2×C4×C31—42no—
111C2×C4×C31—42no—
112C2×C4×C31—42no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—41no—
116C2×C4×C31—42no—
117C2×C4×C31—81no—
118C2×C4×C31—81no—
119C2×C4×C31—84no—
120C2×C4×C31C2×SD1611yescharacteristic
121C2×C4×C31#13711yescharacteristic
122C8×C31—42no—
123C8×C31—82no—
124#2804C2×C3111yescharacteristic, complemented
125#2006C2×C3111yescharacteristic
126#1616C2×C3112yescomplemented
127#1353C2×C3112yescomplemented
128#1318C2×C3111yescharacteristic, complemented
129C22×C31—21no—
130C22×C31—21no—
131C22×C31—44no—
132C22×C31—42no—
133C22×C31—82no—
134C22×C31—81no—
135C22×C31—81no—
136C22×C31—81no—
137C22×C31—81no—
138C22×C31—81no—
139C22×C31#44811yescharacteristic
140C22×C31#45311yescharacteristic
141C22×C31B(4,𝔽2)11yescharacteristic
142C4×C31—21no—
143C4×C31—21no—
144C4×C31—42no—
145C4×C31—42no—
146C4×C31—82no—
147C4×C31—81no—
148C4×C31—82no—
149C2×AGL(1,ℤ/8)—21no—
150#537C22×C3111yescharacteristic
151#481—22no—
152C2×#104—21no—
153#400C22×C3112yes—
154#393—22no—
155#391—21no—
156#390C22×C3111yescharacteristic
157#386—21no—
158#385—22no—
159#343C22×C3111yescharacteristic
160#327—22no—
161#324C22×C3112yes—
162C2×C31—21no—
163C2×C31—21no—
164C2×C31—82no—
165C2×C31—161no—
166C2×C31#159611yescharacteristic
167C2×C31#219811yescharacteristic
168C2×C31#220211yescharacteristic
169C2×#41—21no—
170C22×Q8—21no—
171C22×D4D4×C3111yescharacteristic
172C23×C4—21no—
173AGL(1,ℤ/8)—42no—
174AGL(1,ℤ/8)—42no—
175C2×SD16—41no—
176C2×D8—41no—
177C2×M16—21no—
178C22⋊Q8—41no—
179#121—22no—
180#121—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4D4×C3112yes—
183C2×C4⋊C4C23×C3111yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C42D4×C3111yescharacteristic
188#106—42no—
189#104—42no—
190#102—42no—
191C22⋊C8—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95D4×C3112yes—
196C31#1212711yescharacteristic, direct product factor, Sylow
197C24—21no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—81no—
205C2×Q8—81no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4SD16×C3112yes—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4C2×D4×C3111yescharacteristic
220C22×C4C2×D4×C3112yes—
221SD16—44no—
222D8—44no—
223M16—22no—
224M16—41no—
225C2×C8—41no—
226C2×C8—42no—
227C4⋊C4—22no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—81no—
231C22⋊C4—44no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C22⋊C4—81no—
235C42—41no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23C31×#12011yescharacteristic
243Q8—41no—
244Q8—41no—
245Q8—84no—
246D4—42no—
247D4—41no—
248D4—41no—
249D4—81no—
250D4—82no—
251D4—81no—
252D4—81no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4C2×SD16×C3111yescharacteristic
271C2×C4C31×#13711yescharacteristic
272C8—42no—
273C8—82no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—42no—
278C22—82no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22C31×#44811yescharacteristic
285C22C31×#45311yescharacteristic
286C22C31×B(4,𝔽2)11yescharacteristic
287C4—21no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—82no—
292C4—81no—
293C4—82no—
294C2—21no—
295C2—21no—
296C2—82no—
297C2—161no—
298C2C31×#159611yescharacteristic
299C2C31×#219811yescharacteristic
300C2C31×#220211yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2006C2
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×#2804C2semidirect
3. C31×#1616C2semidirect
4. C31×#1353C2semidirect
5. C31×#1318C2semidirect
85. #12127C31direct (N × H)
124. #2804C2×C31semidirect
126. #1616C2×C31semidirect
127. #1353C2×C31semidirect
128. #1318C2×C31semidirect
196. C31#12127direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28042
C31×#20062
C31×#16162
C31×#13532
C31×#13182
#1212731