Finite Group Databasebeta

← C31×#10021 #21466780

Subgroups — C31×#10021 #21466780

Order 7936.

492 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3142C211yesmaximal, characteristic
2C31×#3112C211yesmaximal, characteristic
3C31×#2634C211yesmaximal, characteristic, complemented
4C31×#1534C211yesmaximal, characteristic, complemented
5C31×#1514C211yesmaximal, characteristic
6C31×#1491C211yesmaximal, characteristic, complemented
7C31×#1288C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C31×#125C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C31×#491—21no—
12C31×#490—21no—
13C31×#488—22no—
14C31×#480—21no—
15C31×#477—21no—
16C31×#460—21no—
17C31×#448—21no—
18C31×#440—22no—
19C31×#432C2211yescharacteristic
20C31×#432C2211yescharacteristic
21C31×#417—21no—
22C31×#417—21no—
23C4×M16×C31C2211yescharacteristic
24C31×#400—21no—
25C31×#400—21no—
26C31×#398—21no—
27C31×#395—21no—
28C31×#387—22no—
29C31×#383C2211yescharacteristic
30C31×#330—22no—
31C31×#327—22no—
32C31×Q8⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C2×SD16×C31—22no—
36C2×M16×C31—21no—
37C31×C4⋊Q8D411yescharacteristic
38C31×#127D411yescharacteristic
39C31×#125—21no—
40C31×#125—21no—
41C31×#125—21no—
42C31×#125—21no—
43C31×#124—21no—
44C31×#121—21no—
45C31×#121—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C2×C4⋊C4×C31—22no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C42×C31—21no—
60C2×C42×C31C2311yescharacteristic
61C31×#107—22no—
62C31×C4⋊C8—21no—
63C31×C4⋊C8—21no—
64C31×#103—22no—
65C31×#103—22no—
66C31×#103—22no—
67C31×#102—22no—
68C31×#102—22no—
69C31×#102—22no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×#97—21no—
73C31×#97—21no—
74C31×#97D412yes—
75C4×C8×C31D412yes—
76C31×#95—22no—
77C31×#95—22no—
78C31×#95—22no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95D412yes—
82#41×C31—44no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×Q8×C31—21no—
86C2×Q8×C31—42no—
87C2×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—41no—
91C22×C4×C31—21no—
92C22×C4×C31—22no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—41no—
97C22×C4×C31C2×D411yescharacteristic
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100SD16×C31—44no—
101M16×C31—42no—
102C2×C8×C31—22no—
103C2×C8×C31—22no—
104C2×C8×C31—41no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—21no—
107C4⋊C4×C31—21no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—21no—
110C4⋊C4×C31—21no—
111C4⋊C4×C31—21no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—44no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—41no—
120C22⋊C4×C31—41no—
121C22⋊C4×C31—41no—
122C42×C31—41no—
123C42×C31—41no—
124C42×C31C2×D411yescharacteristic
125C42×C31C2×D411yescharacteristic
126C42×C31C2×D411yescharacteristic
127C42×C31#4111yescharacteristic
128#10021C3111yesmaximal, characteristic, direct product factor, Sylow
129C23×C31—41no—
130C23×C31#12011yescharacteristic
131Q8×C31—22no—
132Q8×C31—22no—
133Q8×C31—41no—
134Q8×C31—41no—
135Q8×C31—42no—
136D4×C31—42no—
137D4×C31—42no—
138D4×C31—81no—
139D4×C31—82no—
140C2×C4×C31—22no—
141C2×C4×C31—22no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—42no—
155C2×C4×C31—42no—
156C2×C4×C31—82no—
157C2×C4×C31#12011yescharacteristic
158C2×C4×C31#12011yescharacteristic
159C2×C4×C31#12111yescharacteristic
160C2×C4×C31#12111yescharacteristic
161C2×C4×C31#12111yescharacteristic
162C2×C4×C31#12711yescharacteristic
163C8×C31—42no—
164C8×C31—42no—
165C8×C31—81no—
166#3142C2×C3111yescharacteristic
167#3112C2×C3111yescharacteristic
168#2634C2×C3111yescharacteristic, complemented
169#1534C2×C3111yescharacteristic, complemented
170#1514C2×C3111yescharacteristic
171#1491C2×C3111yescharacteristic, complemented
172#1288C2×C3111yescharacteristic, complemented
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31#39211yescharacteristic
179C22×C31#45412yes—
180C4×C31—21no—
181C4×C31—21no—
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—21no—
185C4×C31—21no—
186C4×C31—42no—
187C4×C31—82no—
188C4×C31—81no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—81no—
192#532C22×C3111yescharacteristic
193C2×#125C22×C3111yescharacteristic
194C2×C4×Q8C22×C3111yescharacteristic
195#491—21no—
196#490—21no—
197#488—22no—
198#480—21no—
199#477—21no—
200#460—21no—
201#448—21no—
202#440—22no—
203#432C22×C3111yescharacteristic
204#432C22×C3111yescharacteristic
205#417—21no—
206#417—21no—
207C4×M16C22×C3111yescharacteristic
208#400—21no—
209#400—21no—
210#398—21no—
211#395—21no—
212#387—22no—
213#383C22×C3111yescharacteristic
214#330—22no—
215#327—22no—
216Q8⋊C8—22no—
217C2×C31—22no—
218C2×C31—161no—
219C2×C31#200311yescharacteristic
220C2×C31#200911yescharacteristic
221C2×C31#201011yescharacteristic
222C2×#41—21no—
223C22×Q8D4×C3111yescharacteristic
224C2×SD16—22no—
225C2×M16—21no—
226C4⋊Q8D4×C3111yescharacteristic
227#127D4×C3111yescharacteristic
228#125—21no—
229#125—21no—
230#125—21no—
231#125—21no—
232#124—21no—
233#121—21no—
234#121—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—21no—
240C4×D4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4D4×C3111yescharacteristic
246C2×C4⋊C4D4×C3111yescharacteristic
247C2×C4⋊C4D4×C3111yescharacteristic
248C2×C42—21no—
249C2×C42C23×C3111yescharacteristic
250#107—22no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253#103—22no—
254#103—22no—
255#103—22no—
256#102—22no—
257#102—22no—
258#102—22no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261#97—21no—
262#97—21no—
263#97D4×C3112yes—
264C4×C8D4×C3112yes—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95D4×C3112yes—
271C31#1002111yescharacteristic, direct product factor, Sylow
272#41—44no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—42no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4C2×D4×C3111yescharacteristic
288C22×C4C2×D4×C3111yescharacteristic
289C22×C4C2×D4×C3111yescharacteristic
290SD16—44no—
291M16—42no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—44no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—41no—
313C42—41no—
314C42C2×D4×C3111yescharacteristic
315C42C2×D4×C3111yescharacteristic
316C42C2×D4×C3111yescharacteristic
317C42#41×C3111yescharacteristic
318C23—41no—
319C23C31×#12011yescharacteristic
320Q8—22no—
321Q8—22no—
322Q8—41no—
323Q8—41no—
324Q8—42no—
325D4—42no—
326D4—42no—
327D4—81no—
328D4—82no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4C31×#12011yescharacteristic
347C2×C4C31×#12011yescharacteristic
348C2×C4C31×#12111yescharacteristic
349C2×C4C31×#12111yescharacteristic
350C2×C4C31×#12111yescharacteristic
351C2×C4C31×#12711yescharacteristic
352C8—42no—
353C8—42no—
354C8—81no—
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C31×#39211yescharacteristic
361C22C31×#45412yes—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—82no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—81no—
374C2—22no—
375C2—161no—
376C2C31×#200311yescharacteristic
377C2C31×#200911yescharacteristic
378C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2634C2semidirect
4. C31×#1534C2semidirect
6. C31×#1491C2semidirect
7. C31×#1288C2semidirect
128. #10021C31direct (N × H)
168. #2634C2×C31semidirect
169. #1534C2×C31semidirect
171. #1491C2×C31semidirect
172. #1288C2×C31semidirect
271. C31#10021direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31422
C31×#31122
C31×#26342
C31×#15342
C31×#15142
C31×#14912
C31×#12882
#1002131