Finite Group Databasebeta

← C31×#12141 #21468900

Subgroups — C31×#12141 #21468900

Order 7936.

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

normal subgroups only

This group has 412 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×#2782C211yesmaximal, characteristic, complemented
2C31×#2040C211yesmaximal, characteristic, complemented
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1324C211yesmaximal, characteristic, complemented
8C31×#537C2211yescharacteristic
9C31×#481—21no—
10C31×#447—21no—
11C2×C31×#104—21no—
12C31×#417—21no—
13C31×#400—21no—
14C31×#400C2211yescharacteristic
15C31×#397—21no—
16C31×#394C2211yescharacteristic
17C31×#388—21no—
18C31×#386—21no—
19C31×#385—21no—
20C31×#385—21no—
21C31×#382C2211yescharacteristic
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×D4×C31D411yescharacteristic
29C23×C4×C31—21no—
30C2×D8×C31—41no—
31C2×M16×C31—21no—
32C31×C22⋊Q8—41no—
33C31×#121—21no—
34C31×#121—21no—
35C31×#121—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31C2311yescharacteristic
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31D411yescharacteristic
47C31×#106—41no—
48C31×#104—42no—
49C31×#103—41no—
50C31×#102—41no—
51C31×#102—41no—
52C31×#102—41no—
53C31×C22⋊C8—21no—
54C31×C22⋊C8—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95D411yescharacteristic
62C31×#95D411yescharacteristic
63C24×C31—21no—
64#41×C31—41no—
65#41×C31—41no—
66#41×C31—41no—
67#41×C31—41no—
68C2×Q8×C31—21no—
69C2×D4×C31—21no—
70C2×D4×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—41no—
73C2×D4×C31—42no—
74C2×D4×C31—41no—
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×C31—81no—
82C22×C4×C31—81no—
83C22×C4×C31—81no—
84C22×C4×C31C2×D411yescharacteristic
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87D8×C31—82no—
88M16×C31—41no—
89M16×C31—41no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C2×C8×C31—41no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—21no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—81no—
99C4⋊C4×C31—81no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—42no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—41no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—81no—
107C22⋊C4×C31—81no—
108C42×C31—21no—
109C42×C31—21no—
110#12141C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—22no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31#12011yescharacteristic
120Q8×C31—41no—
121Q8×C31—41no—
122D4×C31—41no—
123D4×C31—41no—
124D4×C31—42no—
125D4×C31—81no—
126D4×C31—81no—
127D4×C31—81no—
128D4×C31—81no—
129D4×C31—81no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
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—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—81no—
153C2×C4×C31—81no—
154C2×C4×C31—84no—
155C2×C4×C31#13511yescharacteristic
156C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
157C8×C31—81no—
158C8×C31—81no—
159C8×C31—81no—
160#2782C2×C3111yescharacteristic, complemented
161#2040C2×C3111yescharacteristic, complemented
162#1616C2×C3111yescharacteristic, complemented
163#1598C2×C3111yescharacteristic
164#1353C2×C3111yescharacteristic, complemented
165#1345C2×C3111yescharacteristic, complemented
166#1324C2×C3111yescharacteristic, complemented
167C22×C31—21no—
168C22×C31—21no—
169C22×C31—42no—
170C22×C31—42no—
171C22×C31—42no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31#44911yescharacteristic
180C22×C31#45411yescharacteristic
181C22×C31B(4,𝔽2)11yescharacteristic
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—81no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—81no—
192#537C22×C3111yescharacteristic
193#481—21no—
194#447—21no—
195C2×#104—21no—
196#417—21no—
197#400—21no—
198#400C22×C3111yescharacteristic
199#397—21no—
200#394C22×C3111yescharacteristic
201#388—21no—
202#386—21no—
203#385—21no—
204#385—21no—
205#382C22×C3111yescharacteristic
206#343C22×C3111yescharacteristic
207#327—21no—
208#327—21no—
209#324C22×C3111yescharacteristic
210#324C22×C3111yescharacteristic
211C2×C31—21no—
212C2×C31—21no—
213C2×C31—82no—
214C2×C31—161no—
215C2×C31#159711yescharacteristic
216C2×C31#219711yescharacteristic
217C2×C31#220211yescharacteristic
218C2×#41—21no—
219C22×D4D4×C3111yescharacteristic
220C23×C4—21no—
221C2×D8—41no—
222C2×M16—21no—
223C22⋊Q8—41no—
224#121—21no—
225#121—21no—
226#121—41no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C4⋊C4D4×C3111yescharacteristic
231C2×C4⋊C4C23×C3111yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42D4×C3111yescharacteristic
238#106—41no—
239#104—42no—
240#103—41no—
241#102—41no—
242#102—41no—
243#102—41no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95D4×C3111yescharacteristic
253#95D4×C3111yescharacteristic
254C31#1214111yescharacteristic, direct product factor, Sylow
255C24—21no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4C2×D4×C3111yescharacteristic
277C22×C4C2×D4×C3111yescharacteristic
278C22×C4C2×D4×C3111yescharacteristic
279D8—82no—
280M16—41no—
281M16—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—21no—
301C42—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C31×#12011yescharacteristic
311Q8—41no—
312Q8—41no—
313D4—41no—
314D4—41no—
315D4—42no—
316D4—81no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—84no—
346C2×C4C31×#13511yescharacteristic
347C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
348C8—81no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C31×#44911yescharacteristic
364C22C31×#45411yescharacteristic
365C22C31×B(4,𝔽2)11yescharacteristic
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C2—21no—
377C2—21no—
378C2—82no—
379C2—161no—
380C2C31×#159711yescharacteristic
381C2C31×#219711yescharacteristic
382C2C31×#220211yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#1598C2
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×#2782C2semidirect
2. C31×#2040C2semidirect
3. C31×#1616C2semidirect
5. C31×#1353C2semidirect
6. C31×#1345C2semidirect
7. C31×#1324C2semidirect
110. #12141C31direct (N × H)
160. #2782C2×C31semidirect
161. #2040C2×C31semidirect
162. #1616C2×C31semidirect
164. #1353C2×C31semidirect
165. #1345C2×C31semidirect
166. #1324C2×C31semidirect
254. C31#12141direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27822
C31×#20402
C31×#16162
C31×#15982
C31×#13532
C31×#13452
C31×#13242
#1214131