Finite Group Databasebeta

← C31×#12147 #21468906

Subgroups — C31×#12147 #21468906

Order 7936.

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

normal subgroups only

This group has 354 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×#2769C211yesmaximal, characteristic, complemented
2C31×#2040C211yesmaximal, characteristic
3C31×#1616C211yesmaximal, characteristic
4C31×#1610C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1346C211yesmaximal, characteristic, complemented
7C31×#1324C211yesmaximal, characteristic, complemented
8C31×#537C2211yescharacteristic
9C31×#481—21no—
10C31×#480—21no—
11C2×C31×#104—21no—
12C31×#417—21no—
13C31×#400—21no—
14C31×#400C2211yescharacteristic
15C31×#398—21no—
16C31×#398C2211yescharacteristic
17C31×#389—21no—
18C31×#387—21no—
19C31×#387—21no—
20C31×#383—21no—
21C31×#382C2211yescharacteristic
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C2×M16×C31—21no—
29C31×C22⋊Q8—41no—
30C31×#121—21no—
31C31×#121—21no—
32C31×#121—41no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31D411yescharacteristic
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×#104—42no—
46C31×#103—41no—
47C31×#102—41no—
48C31×#102—41no—
49C31×#102—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×#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×D4×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—41no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—41no—
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31C2×D411yescharacteristic
82M16×C31—41no—
83M16×C31—41no—
84C2×C8×C31—41no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
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—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—81no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C42×C31—21no—
101C42×C31—21no—
102C42×C31—42no—
103#12147C3111yesmaximal, characteristic, direct product factor, Sylow
104C23×C31—41no—
105C23×C31#12011yescharacteristic
106Q8×C31—41no—
107Q8×C31—41no—
108D4×C31—41no—
109D4×C31—41no—
110D4×C31—81no—
111D4×C31—81no—
112D4×C31—81no—
113D4×C31—81no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—22no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
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×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—81no—
140C2×C4×C31—81no—
141C2×C4×C31#13511yescharacteristic
142C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
143C8×C31—81no—
144C8×C31—81no—
145C8×C31—81no—
146#2769C2×C3111yescharacteristic, complemented
147#2040C2×C3111yescharacteristic
148#1616C2×C3111yescharacteristic
149#1610C2×C3111yescharacteristic
150#1353C2×C3111yescharacteristic, complemented
151#1346C2×C3111yescharacteristic, complemented
152#1324C2×C3111yescharacteristic, complemented
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—81no—
156C22×C31—81no—
157C22×C31—81no—
158C22×C31#44911yescharacteristic
159C22×C31#45411yescharacteristic
160C22×C31B(4,𝔽2)11yescharacteristic
161C4×C31—21no—
162C4×C31—21no—
163C4×C31—42no—
164C4×C31—41no—
165C4×C31—41no—
166C4×C31—81no—
167C4×C31—81no—
168C4×C31—81no—
169C4×C31—81no—
170C4×C31—81no—
171C4×C31—81no—
172#537C22×C3111yescharacteristic
173#481—21no—
174#480—21no—
175C2×#104—21no—
176#417—21no—
177#400—21no—
178#400C22×C3111yescharacteristic
179#398—21no—
180#398C22×C3111yescharacteristic
181#389—21no—
182#387—21no—
183#387—21no—
184#383—21no—
185#382C22×C3111yescharacteristic
186#343C22×C3111yescharacteristic
187#327—21no—
188#327—21no—
189#324C22×C3111yescharacteristic
190#324C22×C3111yescharacteristic
191C2×C31—21no—
192C2×C31—21no—
193C2×C31—161no—
194C2×C31#159711yescharacteristic
195C2×C31#220111yescharacteristic
196C2×C31#220411yescharacteristic
197C2×#41—21no—
198C2×M16—21no—
199C22⋊Q8—41no—
200#121—21no—
201#121—21no—
202#121—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4D4×C3111yescharacteristic
208C2×C4⋊C4D4×C3111yescharacteristic
209C2×C4⋊C4D4×C3111yescharacteristic
210C2×C4⋊C4C23×C3111yescharacteristic
211C2×C42—21no—
212C2×C42D4×C3111yescharacteristic
213#107—41no—
214#106—41no—
215#104—42no—
216#103—41no—
217#102—41no—
218#102—41no—
219#102—41no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95D4×C3111yescharacteristic
232#95D4×C3111yescharacteristic
233C31#1214711yescharacteristic, direct product factor, Sylow
234#41—41no—
235#41—41no—
236#41—41no—
237#41—41no—
238C2×Q8—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4C2×D4×C3111yescharacteristic
251C22×C4C2×D4×C3111yescharacteristic
252C22×C4C2×D4×C3111yescharacteristic
253M16—41no—
254M16—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C42—21no—
272C42—21no—
273C42—42no—
274C23—41no—
275C23C31×#12011yescharacteristic
276Q8—41no—
277Q8—41no—
278D4—41no—
279D4—41no—
280D4—81no—
281D4—81no—
282D4—81no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
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—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4C31×#13511yescharacteristic
312C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
313C8—81no—
314C8—81no—
315C8—81no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C31×#44911yescharacteristic
322C22C31×#45411yescharacteristic
323C22C31×B(4,𝔽2)11yescharacteristic
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2C31×#159711yescharacteristic
339C2C31×#220111yescharacteristic
340C2C31×#220411yescharacteristic
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×#2769C2semidirect
5. C31×#1353C2semidirect
6. C31×#1346C2semidirect
7. C31×#1324C2semidirect
103. #12147C31direct (N × H)
146. #2769C2×C31semidirect
150. #1353C2×C31semidirect
151. #1346C2×C31semidirect
152. #1324C2×C31semidirect
233. C31#12147direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27692
C31×#20402
C31×#16162
C31×#16102
C31×#13532
C31×#13462
C31×#13242
#1214731