Finite Group Databasebeta

← C31×#12237 #21468996

Subgroups — C31×#12237 #21468996

Order 7936.

334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2789C211yesmaximal, characteristic
2C31×#2061C211yesmaximal, characteristic
3C31×#1611C211yesmaximal, characteristic
4C31×#1605C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic
6C31×#1346C211yesmaximal, characteristic
7C31×#1324C211yesmaximal, characteristic
8C31×#544C2211yescharacteristic
9C31×#483—21no—
10C31×#451—21no—
11C31×#428—21no—
12C31×#401—21no—
13C31×#400—21no—
14C31×#398C2211yescharacteristic
15C31×#395C2211yescharacteristic
16C31×#391—21no—
17C31×#387—21no—
18C31×#387—21no—
19C31×#387—21no—
20C31×#382C2211yescharacteristic
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C2×Q16×C31—41no—
29C2×M16×C31—21no—
30C31×C4⋊Q8—41no—
31C31×#125—41no—
32C31×C22⋊Q8—21no—
33C31×C22⋊Q8—21no—
34C31×#117—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31C2311yescharacteristic
41C2×C42×C31—21no—
42C2×C42×C31D411yescharacteristic
43C31×#107—41no—
44C31×#106—41no—
45C31×#106—41no—
46C31×#103—41no—
47C31×#103—41no—
48C31×C22⋊C8—21no—
49C31×C22⋊C8—21no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C2×Q8×C31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—42no—
65C2×Q8×C31—41no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31C2×D411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75Q16×C31—82no—
76M16×C31—22no—
77M16×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—42no—
87C4⋊C4×C31—42no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C22⋊C4×C31—41no—
95C42×C31—21no—
96C42×C31—21no—
97C42×C31—41no—
98C42×C31—81no—
99#12237C3111yesmaximal, characteristic, direct product factor, Sylow
100C23×C31#12011yescharacteristic
101Q8×C31—42no—
102Q8×C31—81no—
103Q8×C31—81no—
104Q8×C31—81no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—22no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—41no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—42no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31#13511yescharacteristic
131C2×C4×C31#13711yescharacteristic
132C8×C31—42no—
133C8×C31—81no—
134C8×C31—81no—
135#2789C2×C3111yescharacteristic
136#2061C2×C3111yescharacteristic
137#1611C2×C3111yescharacteristic
138#1605C2×C3111yescharacteristic
139#1354C2×C3111yescharacteristic
140#1346C2×C3111yescharacteristic
141#1324C2×C3111yescharacteristic
142C22×C31—21no—
143C22×C31—21no—
144C22×C31#45211yescharacteristic
145C22×C31#45611yescharacteristic
146C22×C31B(4,𝔽2)11yescharacteristic
147C4×C31—21no—
148C4×C31—21no—
149C4×C31—41no—
150C4×C31—41no—
151C4×C31—42no—
152C4×C31—81no—
153C4×C31—81no—
154C4×C31—81no—
155C4×C31—81no—
156C4×C31—81no—
157C4×C31—81no—
158C4×C31—81no—
159#544C22×C3111yescharacteristic
160#483—21no—
161#451—21no—
162#428—21no—
163#401—21no—
164#400—21no—
165#398C22×C3111yescharacteristic
166#395C22×C3111yescharacteristic
167#391—21no—
168#387—21no—
169#387—21no—
170#387—21no—
171#382C22×C3111yescharacteristic
172#343C22×C3111yescharacteristic
173#339—21no—
174#328—21no—
175#328—21no—
176#324C22×C3111yescharacteristic
177#324C22×C3111yescharacteristic
178C2×C31—21no—
179C2×C31—21no—
180C2×C31#160311yescharacteristic
181C2×C31#219711yescharacteristic
182C2×C31#220311yescharacteristic
183C22×Q8D4×C3111yescharacteristic
184C2×Q16—41no—
185C2×M16—21no—
186C4⋊Q8—41no—
187#125—41no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4D4×C3111yescharacteristic
195C2×C4⋊C4D4×C3111yescharacteristic
196C2×C4⋊C4C23×C3111yescharacteristic
197C2×C42—21no—
198C2×C42D4×C3111yescharacteristic
199#107—41no—
200#106—41no—
201#106—41no—
202#103—41no—
203#103—41no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95D4×C3111yescharacteristic
217#95D4×C3111yescharacteristic
218C31#1223711yescharacteristic, direct product factor, Sylow
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—42no—
222C2×Q8—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4C2×D4×C3111yescharacteristic
230C22×C4C2×D4×C3111yescharacteristic
231C22×C4C2×D4×C3111yescharacteristic
232Q16—82no—
233M16—22no—
234M16—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—41no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—81no—
256C23C31×#12011yescharacteristic
257Q8—42no—
258Q8—81no—
259Q8—81no—
260Q8—81no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4C31×#13511yescharacteristic
287C2×C4C31×#13711yescharacteristic
288C8—42no—
289C8—81no—
290C8—81no—
291C22—21no—
292C22—21no—
293C22C31×#45211yescharacteristic
294C22C31×#45611yescharacteristic
295C22C31×B(4,𝔽2)11yescharacteristic
296C4—21no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—42no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C2—21no—
309C2—21no—
310C2C31×#160311yescharacteristic
311C2C31×#219711yescharacteristic
312C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC23×C31#120
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
99. #12237C31direct (N × H)
218. C31#12237direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27892
C31×#20612
C31×#16112
C31×#16052
C31×#13542
C31×#13462
C31×#13242
#1223731