Finite Group Databasebeta

← C31×#12144 #21468903

Subgroups — C31×#12144 #21468903

Order 7936.

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

normal subgroups only

This group has 398 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×#2795C211yesmaximal, characteristic, complemented
2C31×#2039C211yesmaximal, characteristic, complemented
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1598C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic, complemented
7C31×#1325C211yesmaximal, characteristic, complemented
8C31×#537C2211yescharacteristic
9C31×#481—21no—
10C31×#447—21no—
11C2×C31×#104—21no—
12C31×#417—21no—
13C31×#400C2211yescharacteristic
14C31×#396—21no—
15C31×#394—21no—
16C31×#394C2211yescharacteristic
17C31×#388—21no—
18C31×#388—21no—
19C31×#388—21no—
20C31×#387—21no—
21C31×#384C2211yescharacteristic
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×D4×C31D411yescharacteristic
29C2×D8×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×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×C22⋊C4×C31—21no—
47C2×C42×C31—21no—
48C2×C42×C31D411yescharacteristic
49C31×#106—41no—
50C31×#104—42no—
51C31×#103—41no—
52C31×#102—41no—
53C31×#102—41no—
54C31×#102—41no—
55C31×C22⋊C8—21no—
56C31×C22⋊C8—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—41no—
74C2×D4×C31—81no—
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
84D8×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—42no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—81no—
97C4⋊C4×C31—81no—
98C22⋊C4×C31—42no—
99C22⋊C4×C31—42no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—81no—
106C22⋊C4×C31—81no—
107C42×C31—21no—
108C42×C31—21no—
109C42×C31—81no—
110#12144C3111yesmaximal, characteristic, direct product factor, Sylow
111C23×C31—41no—
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—22no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
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#13511yescharacteristic
155C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
156C8×C31—81no—
157C8×C31—81no—
158C8×C31—81no—
159#2795C2×C3111yescharacteristic, complemented
160#2039C2×C3111yescharacteristic, complemented
161#1616C2×C3111yescharacteristic, complemented
162#1598C2×C3111yescharacteristic
163#1353C2×C3111yescharacteristic, complemented
164#1345C2×C3111yescharacteristic, complemented
165#1325C2×C3111yescharacteristic, complemented
166C22×C31—21no—
167C22×C31—21no—
168C22×C31—81no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31#44911yescharacteristic
179C22×C31#45411yescharacteristic
180C22×C31B(4,𝔽2)11yescharacteristic
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—42no—
184C4×C31—41no—
185C4×C31—41no—
186C4×C31—81no—
187C4×C31—81no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191#537C22×C3111yescharacteristic
192#481—21no—
193#447—21no—
194C2×#104—21no—
195#417—21no—
196#400C22×C3111yescharacteristic
197#396—21no—
198#394—21no—
199#394C22×C3111yescharacteristic
200#388—21no—
201#388—21no—
202#388—21no—
203#387—21no—
204#384C22×C3111yescharacteristic
205#343C22×C3111yescharacteristic
206#327—21no—
207#327—21no—
208#324C22×C3111yescharacteristic
209#324C22×C3111yescharacteristic
210C2×C31—21no—
211C2×C31—21no—
212C2×C31—161no—
213C2×C31—161no—
214C2×C31#159711yescharacteristic
215C2×C31#220011yescharacteristic
216C2×C31#220411yescharacteristic
217C2×#41—21no—
218C22×D4D4×C3111yescharacteristic
219C2×D8—41no—
220C2×M16—21no—
221C22⋊Q8—41no—
222#121—21no—
223#121—21no—
224#121—41no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4D4×C3111yescharacteristic
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C4⋊C4C23×C3111yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42D4×C3111yescharacteristic
239#106—41no—
240#104—42no—
241#103—41no—
242#102—41no—
243#102—41no—
244#102—41no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95D4×C3111yescharacteristic
252#95D4×C3111yescharacteristic
253C31#1214411yescharacteristic, direct product factor, Sylow
254C24—21no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259C2×Q8—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—81no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4C2×D4×C3111yescharacteristic
273C22×C4C2×D4×C3111yescharacteristic
274C22×C4C2×D4×C3111yescharacteristic
275D8—82no—
276M16—41no—
277M16—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C42—21no—
299C42—21no—
300C42—81no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23C31×#12011yescharacteristic
310Q8—41no—
311Q8—41no—
312D4—41no—
313D4—41no—
314D4—42no—
315D4—81no—
316D4—81no—
317D4—81no—
318D4—81no—
319D4—81no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
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—81no—
343C2×C4—81no—
344C2×C4C31×#13511yescharacteristic
345C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
346C8—81no—
347C8—81no—
348C8—81no—
349C22—21no—
350C22—21no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22C31×#44911yescharacteristic
362C22C31×#45411yescharacteristic
363C22C31×B(4,𝔽2)11yescharacteristic
364C4—21no—
365C4—21no—
366C4—42no—
367C4—41no—
368C4—41no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C2—21no—
375C2—21no—
376C2—161no—
377C2—161no—
378C2C31×#159711yescharacteristic
379C2C31×#220011yescharacteristic
380C2C31×#220411yescharacteristic
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×#2795C2semidirect
2. C31×#2039C2semidirect
3. C31×#1616C2semidirect
5. C31×#1353C2semidirect
6. C31×#1345C2semidirect
7. C31×#1325C2semidirect
110. #12144C31direct (N × H)
159. #2795C2×C31semidirect
160. #2039C2×C31semidirect
161. #1616C2×C31semidirect
163. #1353C2×C31semidirect
164. #1345C2×C31semidirect
165. #1325C2×C31semidirect
253. C31#12144direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27952
C31×#20392
C31×#16162
C31×#15982
C31×#13532
C31×#13452
C31×#13252
#1214431