Finite Group Databasebeta

← C31×#12183 #21468942

Subgroups — C31×#12183 #21468942

Order 7936.

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

normal subgroups only

This group has 362 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×#2843C211yesmaximal, characteristic, complemented
2C31×#2039C211yesmaximal, characteristic
3C31×#1616C211yesmaximal, characteristic
4C31×#1604C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1353C211yesmaximal, characteristic, complemented
7C31×#1325C211yesmaximal, characteristic, complemented
8C31×#537C2211yescharacteristic
9C31×#481—21no—
10C31×#448—21no—
11C2×C31×#104—21no—
12C31×#417—21no—
13C31×#400—21no—
14C31×#400C2211yescharacteristic
15C31×#398—21no—
16C31×#398—21no—
17C31×#398—21no—
18C31×#395—21no—
19C31×#395C2211yescharacteristic
20C31×#384C2211yescharacteristic
21C31×#383—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C2×SD16×C31—41no—
30C2×M16×C31—21no—
31C31×C22⋊Q8—41no—
32C31×#121—21no—
33C31×#121—21no—
34C31×#121—41no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31C2311yescharacteristic
44C2×C42×C31D411yescharacteristic
45C31×#106—41no—
46C31×#104—42no—
47C31×#103—41no—
48C31×#103—41no—
49C31×#102—41no—
50C31×#102—41no—
51C31×C22⋊C8—21no—
52C31×C22⋊C8—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95D411yescharacteristic
62C31×#95D411yescharacteristic
63#41×C31—41no—
64#41×C31—41no—
65#41×C31—41no—
66#41×C31—41no—
67C2×Q8×C31—21no—
68C2×Q8×C31—41no—
69C2×Q8×C31—81no—
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—41no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—41no—
81C22×C4×C31C2×D411yescharacteristic
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D411yescharacteristic
84SD16×C31—82no—
85M16×C31—41no—
86M16×C31—41no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—81no—
96C4⋊C4×C31—81no—
97C4⋊C4×C31—81no—
98C4⋊C4×C31—81no—
99C4⋊C4×C31—81no—
100C4⋊C4×C31—81no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C42×C31—21no—
104C42×C31SD1612yescomplemented
105#12183C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—41no—
107C23×C31#12011yescharacteristic
108Q8×C31—41no—
109Q8×C31—41no—
110Q8×C31—42no—
111Q8×C31—81no—
112D4×C31—41no—
113D4×C31—41no—
114D4×C31—81no—
115D4×C31—81no—
116D4×C31—81no—
117D4×C31—81no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
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×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—42no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31C2×SD1611yescharacteristic
145C2×C4×C31#13711yescharacteristic
146C8×C31—81no—
147C8×C31—81no—
148C8×C31—81no—
149#2843C2×C3111yescharacteristic, complemented
150#2039C2×C3111yescharacteristic
151#1616C2×C3111yescharacteristic
152#1604C2×C3111yescharacteristic
153#1354C2×C3111yescharacteristic, complemented
154#1353C2×C3111yescharacteristic, complemented
155#1325C2×C3111yescharacteristic, complemented
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31#44811yescharacteristic
162C22×C31#45411yescharacteristic
163C22×C31B(4,𝔽2)11yescharacteristic
164C4×C31—21no—
165C4×C31—21no—
166C4×C31—42no—
167C4×C31—41no—
168C4×C31—41no—
169C4×C31—81no—
170C4×C31—81no—
171C4×C31—81no—
172C4×C31—81no—
173C4×C31—81no—
174C4×C31—81no—
175#537C22×C3111yescharacteristic
176#481—21no—
177#448—21no—
178C2×#104—21no—
179#417—21no—
180#400—21no—
181#400C22×C3111yescharacteristic
182#398—21no—
183#398—21no—
184#398—21no—
185#395—21no—
186#395C22×C3111yescharacteristic
187#384C22×C3111yescharacteristic
188#383—21no—
189#343C22×C3111yescharacteristic
190#327—21no—
191#327—21no—
192#324C22×C3111yescharacteristic
193#324C22×C3111yescharacteristic
194C2×C31—21no—
195C2×C31—21no—
196C2×C31—161no—
197C2×C31#159911yescharacteristic
198C2×C31#220011yescharacteristic
199C2×C31#220311yescharacteristic
200C2×#41—21no—
201C22×Q8D4×C3111yescharacteristic
202C2×SD16—41no—
203C2×M16—21no—
204C22⋊Q8—41no—
205#121—21no—
206#121—21no—
207#121—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4D4×C3111yescharacteristic
215C2×C4⋊C4D4×C3111yescharacteristic
216C2×C4⋊C4C23×C3111yescharacteristic
217C2×C42D4×C3111yescharacteristic
218#106—41no—
219#104—42no—
220#103—41no—
221#103—41no—
222#102—41no—
223#102—41no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95D4×C3111yescharacteristic
235#95D4×C3111yescharacteristic
236C31#1218311yescharacteristic, direct product factor, Sylow
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×Q8—81no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4C2×D4×C3111yescharacteristic
256C22×C4C2×D4×C3111yescharacteristic
257C22×C4C2×D4×C3111yescharacteristic
258SD16—82no—
259M16—41no—
260M16—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—21no—
278C42SD16×C3112yescomplemented
279C23—41no—
280C23C31×#12011yescharacteristic
281Q8—41no—
282Q8—41no—
283Q8—42no—
284Q8—81no—
285D4—41no—
286D4—41no—
287D4—81no—
288D4—81no—
289D4—81no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C2×SD16×C3111yescharacteristic
318C2×C4C31×#13711yescharacteristic
319C8—81no—
320C8—81no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C31×#44811yescharacteristic
328C22C31×#45411yescharacteristic
329C22C31×B(4,𝔽2)11yescharacteristic
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C2—21no—
342C2—21no—
343C2—161no—
344C2C31×#159911yescharacteristic
345C2C31×#220011yescharacteristic
346C2C31×#220311yescharacteristic
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×#2843C2semidirect
5. C31×#1354C2semidirect
6. C31×#1353C2semidirect
7. C31×#1325C2semidirect
104. C42×C31SD16semidirect
105. #12183C31direct (N × H)
149. #2843C2×C31semidirect
153. #1354C2×C31semidirect
154. #1353C2×C31semidirect
155. #1325C2×C31semidirect
236. C31#12183direct (N × H)
278. C42SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28432
C31×#20392
C31×#16162
C31×#16042
C31×#13542
C31×#13532
C31×#13252
#1218331