Finite Group Databasebeta

← C31×#12200 #21468959

Subgroups — C31×#12200 #21468959

Order 7936.

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

normal subgroups only

This group has 424 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×#2830C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic
3C31×#1610C211yesmaximal, characteristic, complemented
4C31×#1604C211yesmaximal, characteristic, complemented
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1346C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#537C2211yescharacteristic
10C31×#480—21no—
11C31×#448—21no—
12C2×C31×#104—21no—
13C31×#398C2211yescharacteristic
14C31×#397—21no—
15C31×#397—21no—
16C31×#396—21no—
17C31×#395C2211yescharacteristic
18C31×#394—21no—
19C31×#393—21no—
20C31×#390C2211yescharacteristic
21C31×#387—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C31×AGL(1,ℤ/8)—42no—
31C31×AGL(1,ℤ/8)—42no—
32C2×SD16×C31—41no—
33C2×SD16×C31—41no—
34C2×D8×C31—41no—
35C2×M16×C31—21no—
36C31×C22⋊Q8—41no—
37C31×#121—21no—
38C31×#121—21no—
39C31×#121—41no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31C2311yescharacteristic
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31D411yescharacteristic
52C31×#107—41no—
53C31×#104—42no—
54C31×#103—41no—
55C31×#102—41no—
56C31×C22⋊C8—21no—
57C31×C22⋊C8—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×Q8×C31—41no—
72C2×Q8×C31—81no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90SD16×C31—44no—
91SD16×C31—82no—
92D8×C31—44no—
93M16×C31—22no—
94M16×C31—41no—
95C2×C8×C31—41no—
96C2×C8×C31—41no—
97C2×C8×C31—41no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—81no—
105C4⋊C4×C31—81no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—81no—
109C22⋊C4×C31—81no—
110C22⋊C4×C31—81no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C22⋊C4×C31—81no—
114C42×C31—41no—
115#12200C3111yesmaximal, characteristic, direct product factor, Sylow
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#12011yescharacteristic
125Q8×C31—41no—
126Q8×C31—41no—
127Q8×C31—42no—
128Q8×C31—81no—
129D4×C31—41no—
130D4×C31—42no—
131D4×C31—41no—
132D4×C31—81no—
133D4×C31—81no—
134D4×C31—81no—
135D4×C31—81no—
136D4×C31—81no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—21no—
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—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—41no—
158C2×C4×C31—41no—
159C2×C4×C31—81no—
160C2×C4×C31—81no—
161C2×C4×C31#13511yescharacteristic
162C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
163C8×C31—42no—
164C8×C31—81no—
165C8×C31—81no—
166#2830C2×C3111yescharacteristic, complemented
167#2006C2×C3111yescharacteristic
168#1610C2×C3111yescharacteristic, complemented
169#1604C2×C3111yescharacteristic, complemented
170#1354C2×C3111yescharacteristic, complemented
171#1346C2×C3111yescharacteristic, complemented
172#1318C2×C3111yescharacteristic, complemented
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31—81no—
184C22×C31—81no—
185C22×C31#44911yescharacteristic
186C22×C31#45311yescharacteristic
187C22×C31B(4,𝔽2)11yescharacteristic
188C4×C31—21no—
189C4×C31—21no—
190C4×C31—41no—
191C4×C31—41no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C2×AGL(1,ℤ/8)—21no—
199#537C22×C3111yescharacteristic
200#480—21no—
201#448—21no—
202C2×#104—21no—
203#398C22×C3111yescharacteristic
204#397—21no—
205#397—21no—
206#396—21no—
207#395C22×C3111yescharacteristic
208#394—21no—
209#393—21no—
210#390C22×C3111yescharacteristic
211#387—21no—
212#343C22×C3111yescharacteristic
213#327—21no—
214#327—21no—
215#324C22×C3111yescharacteristic
216#324C22×C3111yescharacteristic
217C2×C31—21no—
218C2×C31—21no—
219C2×C31—161no—
220C2×C31—161no—
221C2×C31#160011yescharacteristic
222C2×C31#220011yescharacteristic
223C2×C31#220311yescharacteristic
224C2×#41—21no—
225C22×Q8D4×C3111yescharacteristic
226C22×D4D4×C3111yescharacteristic
227AGL(1,ℤ/8)—42no—
228AGL(1,ℤ/8)—42no—
229C2×SD16—41no—
230C2×SD16—41no—
231C2×D8—41no—
232C2×M16—21no—
233C22⋊Q8—41no—
234#121—21no—
235#121—21no—
236#121—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4D4×C3111yescharacteristic
241C2×C4⋊C4C23×C3111yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C42D4×C3111yescharacteristic
249#107—41no—
250#104—42no—
251#103—41no—
252#102—41no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95D4×C3111yescharacteristic
261#95D4×C3111yescharacteristic
262C31#1220011yescharacteristic, direct product factor, Sylow
263C24—21no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C2×D4×C3111yescharacteristic
286C22×C4C2×D4×C3111yescharacteristic
287C22×C4C2×D4×C3111yescharacteristic
288SD16—44no—
289SD16—82no—
290D8—44no—
291M16—22no—
292M16—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C31×#12011yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—42no—
325Q8—81no—
326D4—41no—
327D4—42no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C31×#13511yescharacteristic
359C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
360C8—42no—
361C8—81no—
362C8—81no—
363C22—21no—
364C22—21no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22C31×#44911yescharacteristic
376C22C31×#45311yescharacteristic
377C22C31×B(4,𝔽2)11yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C2—21no—
389C2—21no—
390C2—161no—
391C2—161no—
392C2C31×#160011yescharacteristic
393C2C31×#220011yescharacteristic
394C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2006C2
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×#2830C2semidirect
3. C31×#1610C2semidirect
4. C31×#1604C2semidirect
5. C31×#1354C2semidirect
6. C31×#1346C2semidirect
7. C31×#1318C2semidirect
115. #12200C31direct (N × H)
166. #2830C2×C31semidirect
168. #1610C2×C31semidirect
169. #1604C2×C31semidirect
170. #1354C2×C31semidirect
171. #1346C2×C31semidirect
172. #1318C2×C31semidirect
262. C31#12200direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28302
C31×#20062
C31×#16102
C31×#16042
C31×#13542
C31×#13462
C31×#13182
#1220031