Finite Group Databasebeta

← C31×#12258 #21469017

Subgroups — C31×#12258 #21469017

Order 7936.

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

normal subgroups only

This group has 352 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×#2612C211yesmaximal, characteristic
2C31×#2061C211yesmaximal, characteristic
3C31×#1617C211yesmaximal, characteristic
4C31×#1605C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic
6C31×#1353C211yesmaximal, characteristic
7C31×#1324C211yesmaximal, characteristic
8C31×#544C2211yescharacteristic
9C31×#484—21no—
10C31×#451—21no—
11C31×#428—21no—
12C31×#400C2211yescharacteristic
13C31×#398—21no—
14C31×#395C2211yescharacteristic
15C31×#391—21no—
16C31×#389—21no—
17C31×#387—21no—
18C31×#387—21no—
19C31×#382C2211yescharacteristic
20C31×#376—21no—
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×C31—21no—
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31C2311yescharacteristic
42C2×C42×C31—21no—
43C2×C42×C31—21no—
44C2×C42×C31D411yescharacteristic
45C31×#107—41no—
46C31×#107—41no—
47C31×#106—41no—
48C31×#103—41no—
49C31×#103—41no—
50C31×C22⋊C8—21no—
51C31×C22⋊C8—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—42no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—81no—
94C22⋊C4×C31—41no—
95C42×C31—21no—
96C42×C31—42no—
97C42×C31—42no—
98C42×C31—41no—
99C42×C31Q1612yescomplemented
100#12258C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31#12011yescharacteristic
102Q8×C31—42no—
103Q8×C31—81no—
104Q8×C31—81no—
105Q8×C31—81no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—22no—
109C2×C4×C31—22no—
110C2×C4×C31—22no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
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—41no—
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—41no—
131C2×C4×C31—41no—
132C2×C4×C31C2×Q1611yescharacteristic
133C2×C4×C31#13711yescharacteristic
134C8×C31—42no—
135C8×C31—81no—
136C8×C31—81no—
137#2612C2×C3111yescharacteristic
138#2061C2×C3111yescharacteristic
139#1617C2×C3111yescharacteristic
140#1605C2×C3111yescharacteristic
141#1354C2×C3111yescharacteristic
142#1353C2×C3111yescharacteristic
143#1324C2×C3111yescharacteristic
144C22×C31—21no—
145C22×C31—21no—
146C22×C31#45111yescharacteristic
147C22×C31#45611yescharacteristic
148C22×C31B(4,𝔽2)11yescharacteristic
149C4×C31—21no—
150C4×C31—21no—
151C4×C31—42no—
152C4×C31—42no—
153C4×C31—41no—
154C4×C31—41no—
155C4×C31—42no—
156C4×C31—81no—
157C4×C31—81no—
158C4×C31—81no—
159C4×C31—81no—
160C4×C31—81no—
161#544C22×C3111yescharacteristic
162#484—21no—
163#451—21no—
164#428—21no—
165#400C22×C3111yescharacteristic
166#398—21no—
167#395C22×C3111yescharacteristic
168#391—21no—
169#389—21no—
170#387—21no—
171#387—21no—
172#382C22×C3111yescharacteristic
173#376—21no—
174#343C22×C3111yescharacteristic
175#339—21no—
176#328—21no—
177#328—21no—
178#324C22×C3111yescharacteristic
179#324C22×C3111yescharacteristic
180C2×C31—21no—
181C2×C31—21no—
182C2×C31#160511yescharacteristic
183C2×C31#219711yescharacteristic
184C2×C31#220411yescharacteristic
185C22×Q8D4×C3111yescharacteristic
186C2×Q16—41no—
187C2×M16—21no—
188C4⋊Q8—41no—
189#125—41no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#117—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4D4×C3111yescharacteristic
198C2×C4⋊C4D4×C3111yescharacteristic
199C2×C4⋊C4C23×C3111yescharacteristic
200C2×C42—21no—
201C2×C42—21no—
202C2×C42D4×C3111yescharacteristic
203#107—41no—
204#107—41no—
205#106—41no—
206#103—41no—
207#103—41no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95D4×C3111yescharacteristic
219#95D4×C3111yescharacteristic
220C31#1225811yescharacteristic, direct product factor, Sylow
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×D4×C3111yescharacteristic
232C22×C4C2×D4×C3111yescharacteristic
233C22×C4C2×D4×C3111yescharacteristic
234Q16—82no—
235M16—22no—
236M16—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42—41no—
258C42Q16×C3112yescomplemented
259C23C31×#12011yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—81no—
263Q8—81no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
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×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4C2×Q16×C3111yescharacteristic
291C2×C4C31×#13711yescharacteristic
292C8—42no—
293C8—81no—
294C8—81no—
295C22—21no—
296C22—21no—
297C22C31×#45111yescharacteristic
298C22C31×#45611yescharacteristic
299C22C31×B(4,𝔽2)11yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C2—21no—
313C2—21no—
314C2C31×#160511yescharacteristic
315C2C31×#219711yescharacteristic
316C2C31×#220411yescharacteristic
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. C42×C31Q16semidirect
100. #12258C31direct (N × H)
220. C31#12258direct (N × H)
258. C42Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#26122
C31×#20612
C31×#16172
C31×#16052
C31×#13542
C31×#13532
C31×#13242
#1225831