Finite Group Databasebeta

← C31×#9205 #21465964

Subgroups — C31×#9205 #21465964

Order 7936.

690 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2617C211yesmaximal, characteristic
2C31×#2312C211yesmaximal, characteristic, complemented
3C31×#1519C212yesmaximal, complemented
4C2×C31×#323C211yesmaximal, characteristic, complemented
5C31×#1289C212yesmaximal, complemented
6C2×C31×#121C2211yescharacteristic
7C2×C31×#120—21no—
8C22×C22⋊C4×C31C2211yescharacteristic
9C2×C31×C22⋊C8C2212yes—
10C31×#397C412yescomplemented
11C31×#394—22no—
12C31×#392C412yescomplemented
13C31×#388—24no—
14C31×#386—21no—
15C31×#380C2211yescharacteristic
16C31×#380C2212yes—
17C31×#379—21no—
18C4×C22⋊C4×C31—21no—
19C31×#327—24no—
20C31×#323—22no—
21C25×C31D411yescharacteristic
22C22×D4×C31—21no—
23C22×D4×C31—22no—
24C22×D4×C31C2×C411yescharacteristic
25C23×C4×C31D411yescharacteristic
26C23×C4×C31C2311yescharacteristic
27C22×C8×C31—22no—
28C31×#121—22no—
29C31×#121—22no—
30C31×#120—44no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31C2×C411yescharacteristic
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31C2×C412yes—
44C2×C22⋊C4×C31D412yes—
45C2×C42×C31—22no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—22no—
48C31×#95—22no—
49C31×#95—22no—
50C31×#95C2×C412yes—
51C31×#95D414yescomplemented
52C24×C31—21no—
53C24×C31—21no—
54C24×C31—22no—
55C24×C31—22no—
56C24×C31—21no—
57C24×C31—81no—
58C24×C31—81no—
59C24×C31C22⋊C412yes—
60C24×C31C2×D411yescharacteristic
61C2×D4×C31—22no—
62C2×D4×C31—44no—
63C2×D4×C31—41no—
64C2×D4×C31—41no—
65C2×D4×C31—42no—
66C2×D4×C31—42no—
67C2×D4×C31—82no—
68C2×D4×C31—82no—
69C2×D4×C31—81no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—22no—
74C22×C4×C31—24no—
75C22×C4×C31—81no—
76C22×C4×C31—81no—
77C22×C4×C31C22⋊C412yes—
78C22×C4×C31C22×C411yescharacteristic
79C22×C4×C31C2×D412yes—
80C22×C4×C31C2×D411yescharacteristic
81C22×C4×C31#4112yes—
82C2×C8×C31—42no—
83C2×C8×C31—82no—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—82no—
86C4⋊C4×C31—81no—
87C22⋊C4×C31—24no—
88C22⋊C4×C31—24no—
89C22⋊C4×C31—44no—
90C22⋊C4×C31—42no—
91C22⋊C4×C31—42no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—82no—
95C22⋊C4×C31—82no—
96C22⋊C4×C31—82no—
97C42×C31—82no—
98#9205C3111yesmaximal, characteristic, direct product factor, Sylow
99C23×C31—22no—
100C23×C31—22no—
101C23×C31—22no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31—42no—
105C23×C31—42no—
106C23×C31—42no—
107C23×C31—42no—
108C23×C31—42no—
109C23×C31—42no—
110C23×C31—42no—
111C23×C31—42no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—81no—
116C23×C31—81no—
117C23×C31—84no—
118C23×C31—84no—
119C23×C31C2×C22⋊C411yescharacteristic
120C23×C31#12011yescharacteristic
121C23×C31#12311yescharacteristic
122C23×C31#13512yes—
123C23×C31AGL(1,ℤ/8)12yes—
124D4×C31—82no—
125D4×C31—88no—
126D4×C31—82no—
127D4×C31—82no—
128C2×C4×C31—22no—
129C2×C4×C31—41no—
130C2×C4×C31—42no—
131C2×C4×C31—42no—
132C2×C4×C31—41no—
133C2×C4×C31—42no—
134C2×C4×C31—42no—
135C2×C4×C31—44no—
136C2×C4×C31—44no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—42no—
142C2×C4×C31—41no—
143C2×C4×C31—44no—
144C2×C4×C31—81no—
145C2×C4×C31C4×D412yes—
146C2×C4×C31#12112yes—
147C8×C31—82no—
148#2617C2×C3111yescharacteristic
149#2312C2×C3111yescharacteristic, complemented
150#1519C2×C3112yescomplemented
151C2×#323C2×C3111yescharacteristic, complemented
152#1289C2×C3112yescomplemented
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—21no—
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—42no—
160C22×C31—42no—
161C22×C31—44no—
162C22×C31—42no—
163C22×C31—42no—
164C22×C31—42no—
165C22×C31—82no—
166C22×C31—84no—
167C22×C31—84no—
168C22×C31—82no—
169C22×C31—82no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31#38611yescharacteristic
174C22×C31#41611yescharacteristic
175C22×C31#41811yescharacteristic
176C22×C31#44912yes—
177C22×C31#48212yes—
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—42no—
181C4×C31—84no—
182C4×C31—82no—
183C4×C31—81no—
184C2×#121C22×C3111yescharacteristic
185C2×#120—21no—
186C22×C22⋊C4C22×C3111yescharacteristic
187C2×C22⋊C8C22×C3112yes—
188#397C4×C3112yescomplemented
189#394—22no—
190#392C4×C3112yescomplemented
191#388—24no—
192#386—21no—
193#380C22×C3111yescharacteristic
194#380C22×C3112yes—
195#379—21no—
196C4×C22⋊C4—21no—
197#327—24no—
198#323—22no—
199C2×C31—21no—
200C2×C31—22no—
201C2×C31—21no—
202C2×C31—22no—
203C2×C31—82no—
204C2×C31—161no—
205C2×C31#187511yescharacteristic
206C2×C31#191111yescharacteristic
207C2×C31#191311yescharacteristic
208C25D4×C3111yescharacteristic
209C22×D4—21no—
210C22×D4—22no—
211C22×D4C2×C4×C3111yescharacteristic
212C23×C4D4×C3111yescharacteristic
213C23×C4C23×C3111yescharacteristic
214C22×C8—22no—
215#121—22no—
216#121—22no—
217#120—44no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×C4×C3111yescharacteristic
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C2×C4×C3112yes—
231C2×C22⋊C4D4×C3112yes—
232C2×C42—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235#95—22no—
236#95—22no—
237#95C2×C4×C3112yes—
238#95D4×C3114yescomplemented
239C31#920511yescharacteristic, direct product factor, Sylow
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—21no—
245C24—81no—
246C24—81no—
247C24C22⋊C4×C3112yes—
248C24C2×D4×C3111yescharacteristic
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C22⋊C4×C3112yes—
266C22×C4C22×C4×C3111yescharacteristic
267C22×C4C2×D4×C3112yes—
268C22×C4C2×D4×C3111yescharacteristic
269C22×C4#41×C3112yes—
270C2×C8—42no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—82no—
274C4⋊C4—81no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—82no—
285C42—82no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—81no—
303C23—81no—
304C23—84no—
305C23—84no—
306C23C2×C22⋊C4×C3111yescharacteristic
307C23C31×#12011yescharacteristic
308C23C31×#12311yescharacteristic
309C23C31×#13512yes—
310C23C31×AGL(1,ℤ/8)12yes—
311D4—82no—
312D4—88no—
313D4—82no—
314D4—82no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4C4×D4×C3112yes—
333C2×C4C31×#12112yes—
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22C31×#38611yescharacteristic
356C22C31×#41611yescharacteristic
357C22C31×#41811yescharacteristic
358C22C31×#44912yes—
359C22C31×#48212yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—82no—
364C4—84no—
365C4—81no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—161no—
372C2C31×#187511yescharacteristic
373C2C31×#191111yescharacteristic
374C2C31×#191311yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC31×#2617C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#2312C2semidirect
3. C31×#1519C2semidirect
4. C2×C31×#323C2semidirect
5. C31×#1289C2semidirect
10. C31×#397C4semidirect
12. C31×#392C4semidirect
51. C31×#95D4semidirect
98. #9205C31direct (N × H)
149. #2312C2×C31semidirect
150. #1519C2×C31semidirect
151. C2×#323C2×C31semidirect
152. #1289C2×C31semidirect
188. #397C4×C31semidirect
190. #392C4×C31semidirect
238. #95D4×C31semidirect
239. C31#9205direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26172
C31×#23122
C31×#15192
C2×C31×#3232
C31×#12892
#920531