Finite Group Databasebeta

← C31×#12222 #21468981

Subgroups — C31×#12222 #21468981

Order 7936.

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

normal subgroups only

This group has 346 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×#2799C211yesmaximal, characteristic
2C31×#2028C211yesmaximal, characteristic
3C31×#1617C211yesmaximal, characteristic
4C31×#1605C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic
6C31×#1353C211yesmaximal, characteristic
7C31×#1319C211yesmaximal, characteristic
8C31×#544C2211yescharacteristic
9C31×#484—21no—
10C31×#451—21no—
11C31×#419—21no—
12C31×#400—21no—
13C31×#400C2211yescharacteristic
14C31×#398—21no—
15C31×#395C2211yescharacteristic
16C31×#391—21no—
17C31×#391—21no—
18C31×#391C2211yescharacteristic
19C31×#387—21no—
20C31×#387—21no—
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C2×Q16×C31—41no—
30C2×M16×C31—21no—
31C31×C4⋊Q8—41no—
32C31×#125—41no—
33C31×C22⋊Q8—21no—
34C31×C22⋊Q8—21no—
35C31×#117—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31C2311yescharacteristic
42C2×C42×C31—21no—
43C2×C42×C31D411yescharacteristic
44C31×#107—41no—
45C31×#103—41no—
46C31×#103—41no—
47C31×#103—41no—
48C31×#103—41no—
49C31×C22⋊C8—21no—
50C31×C22⋊C8—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—21no—
65C2×Q8×C31—42no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31—41no—
69C2×Q8×C31Q1612yescomplemented
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31C2×D411yescharacteristic
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79Q16×C31—82no—
80M16×C31—41no—
81M16×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31—41no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—21no—
88C4⋊C4×C31—21no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—42no—
91C4⋊C4×C31—42no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—81no—
97C22⋊C4×C31—41no—
98C42×C31—41no—
99C42×C31—41no—
100C42×C31—81no—
101#12222C3111yesmaximal, characteristic, direct product factor, Sylow
102C23×C31#12011yescharacteristic
103Q8×C31—42no—
104Q8×C31—42no—
105Q8×C31—81no—
106Q8×C31—81no—
107Q8×C31—81no—
108Q8×C31—81no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—22no—
112C2×C4×C31—22no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
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—42no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31C2×Q1611yescharacteristic
135C2×C4×C31#13711yescharacteristic
136C8×C31—81no—
137C8×C31—81no—
138C8×C31—81no—
139#2799C2×C3111yescharacteristic
140#2028C2×C3111yescharacteristic
141#1617C2×C3111yescharacteristic
142#1605C2×C3111yescharacteristic
143#1354C2×C3111yescharacteristic
144#1353C2×C3111yescharacteristic
145#1319C2×C3111yescharacteristic
146C22×C31—21no—
147C22×C31—21no—
148C22×C31#45111yescharacteristic
149C22×C31#45511yescharacteristic
150C22×C31B(4,𝔽2)11yescharacteristic
151C4×C31—21no—
152C4×C31—21no—
153C4×C31—42no—
154C4×C31—41no—
155C4×C31—41no—
156C4×C31—42no—
157C4×C31—81no—
158C4×C31—81no—
159C4×C31—81no—
160C4×C31—81no—
161C4×C31—81no—
162C4×C31—81no—
163#544C22×C3111yescharacteristic
164#484—21no—
165#451—21no—
166#419—21no—
167#400—21no—
168#400C22×C3111yescharacteristic
169#398—21no—
170#395C22×C3111yescharacteristic
171#391—21no—
172#391—21no—
173#391C22×C3111yescharacteristic
174#387—21no—
175#387—21no—
176#343C22×C3111yescharacteristic
177#339—21no—
178#328—21no—
179#328—21no—
180#324C22×C3111yescharacteristic
181#324C22×C3111yescharacteristic
182C2×C31—21no—
183C2×C31—21no—
184C2×C31#160211yescharacteristic
185C2×C31#219711yescharacteristic
186C2×C31#220311yescharacteristic
187C22×Q8D4×C3111yescharacteristic
188C22×Q8D4×C3111yescharacteristic
189C2×Q16—41no—
190C2×M16—21no—
191C4⋊Q8—41no—
192#125—41no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195#117—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4D4×C3111yescharacteristic
201C2×C4⋊C4C23×C3111yescharacteristic
202C2×C42—21no—
203C2×C42D4×C3111yescharacteristic
204#107—41no—
205#103—41no—
206#103—41no—
207#103—41no—
208#103—41no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95D4×C3111yescharacteristic
221#95D4×C3111yescharacteristic
222C31#1222211yescharacteristic, direct product factor, Sylow
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8Q16×C3112yescomplemented
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×D4×C3111yescharacteristic
238C22×C4C2×D4×C3111yescharacteristic
239C22×C4C2×D4×C3111yescharacteristic
240Q16—82no—
241M16—41no—
242M16—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—41no—
259C42—41no—
260C42—41no—
261C42—81no—
262C23C31×#12011yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267Q8—81no—
268Q8—81no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
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—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4C2×Q16×C3111yescharacteristic
295C2×C4C31×#13711yescharacteristic
296C8—81no—
297C8—81no—
298C8—81no—
299C22—21no—
300C22—21no—
301C22C31×#45111yescharacteristic
302C22C31×#45511yescharacteristic
303C22C31×B(4,𝔽2)11yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C2—21no—
317C2—21no—
318C2C31×#160211yescharacteristic
319C2C31×#219711yescharacteristic
320C2C31×#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
69. C2×Q8×C31Q16semidirect
101. #12222C31direct (N × H)
222. C31#12222direct (N × H)
230. C2×Q8Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#27992
C31×#20282
C31×#16172
C31×#16052
C31×#13542
C31×#13532
C31×#13192
#1222231