Finite Group Databasebeta

← C31×#10023 #21466782

Subgroups — C31×#10023 #21466782

Order 7936.

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

normal subgroups only

This group has 518 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×#3166C211yesmaximal, characteristic, complemented
2C31×#3110C211yesmaximal, characteristic
3C31×#2633C211yesmaximal, characteristic
4C31×#1534C211yesmaximal, characteristic, complemented
5C31×#1515C211yesmaximal, characteristic, complemented
6C31×#1490C211yesmaximal, characteristic
7C31×#1288C211yesmaximal, characteristic, complemented
8C31×#533C2211yescharacteristic
9C2×C31×#125C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#501—21no—
12C31×#501—21no—
13C31×#499—22no—
14C31×#484—21no—
15C31×#474—21no—
16C31×#461—21no—
17C31×#450—21no—
18C31×#440—22no—
19C31×#432C2211yescharacteristic
20C31×#432C2211yescharacteristic
21C31×#428—21no—
22C31×#428—21no—
23C4×M16×C31C2211yescharacteristic
24C31×#399—21no—
25C31×#398—21no—
26C31×#397—21no—
27C31×#394—21no—
28C31×#385—22no—
29C31×#383C2211yescharacteristic
30C31×#330—22no—
31C31×#328—22no—
32C31×D4⋊C8—22no—
33C22×D4×C31D411yescharacteristic
34C23×C4×C31—21no—
35C2×SD16×C31—22no—
36C2×M16×C31—21no—
37C31×C4⋊Q8D411yescharacteristic
38C31×C4⋊Q8D411yescharacteristic
39C31×#125—21no—
40C31×#125—21no—
41C31×#125—21no—
42C31×#125—21no—
43C31×#125—21no—
44C31×C22⋊Q8—21no—
45C31×C22⋊Q8—21no—
46C4×Q8×C31—21no—
47C4×Q8×C31—21no—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C31×#117—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—22no—
56C2×C4⋊C4×C31D411yescharacteristic
57C2×C4⋊C4×C31D411yescharacteristic
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—22no—
62C2×C42×C31C2311yescharacteristic
63C31×#107—22no—
64C31×#107—22no—
65C31×#107—22no—
66C31×#106—22no—
67C31×#106—22no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8—21no—
70C31×#103—22no—
71C31×#102—22no—
72C31×C22⋊C8—21no—
73C31×C22⋊C8—21no—
74C31×#97—21no—
75C31×#97—21no—
76C31×#97D412yes—
77C4×C8×C31D412yes—
78C31×#95—22no—
79C31×#95—21no—
80C31×#95—21no—
81C31×#95D412yes—
82C24×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—21no—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31—42no—
88C22×C4×C31—22no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—21no—
92C22×C4×C31—41no—
93C22×C4×C31—41no—
94C22×C4×C31—82no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98SD16×C31—44no—
99M16×C31—24no—
100C2×C8×C31—22no—
101C2×C8×C31—22no—
102C2×C8×C31—41no—
103C4⋊C4×C31—21no—
104C4⋊C4×C31—21no—
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—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31—82no—
120C22⋊C4×C31—44no—
121C22⋊C4×C31—41no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—82no—
125C42×C31—41no—
126C42×C31C2×D411yescharacteristic
127C42×C31C2×D411yescharacteristic
128C42×C31C2×D411yescharacteristic
129C42×C31#4111yescharacteristic
130#10023C3111yesmaximal, characteristic, direct product factor, Sylow
131C23×C31—22no—
132C23×C31—42no—
133C23×C31—41no—
134C23×C31—41no—
135C23×C31—42no—
136C23×C31#12011yescharacteristic
137Q8×C31—42no—
138Q8×C31—81no—
139D4×C31—22no—
140D4×C31—22no—
141D4×C31—41no—
142D4×C31—41no—
143C2×C4×C31—22no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—41no—
149C2×C4×C31—42no—
150C2×C4×C31—41no—
151C2×C4×C31—42no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—41no—
157C2×C4×C31—42no—
158C2×C4×C31—82no—
159C2×C4×C31—82no—
160C2×C4×C31#12011yescharacteristic
161C2×C4×C31#12011yescharacteristic
162C2×C4×C31#12111yescharacteristic
163C2×C4×C31#12111yescharacteristic
164C2×C4×C31#12111yescharacteristic
165C2×C4×C31#12711yescharacteristic
166C8×C31—24no—
167C8×C31—42no—
168C8×C31—81no—
169#3166C2×C3111yescharacteristic, complemented
170#3110C2×C3111yescharacteristic
171#2633C2×C3111yescharacteristic
172#1534C2×C3111yescharacteristic, complemented
173#1515C2×C3111yescharacteristic, complemented
174#1490C2×C3111yescharacteristic
175#1288C2×C3111yescharacteristic, complemented
176C22×C31—21no—
177C22×C31—21no—
178C22×C31—44no—
179C22×C31—42no—
180C22×C31—82no—
181C22×C31—81no—
182C22×C31—81no—
183C22×C31#39211yescharacteristic
184C22×C31#45612yes—
185C4×C31—21no—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—82no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—81no—
195C4×C31—81no—
196C4×C31—81no—
197#533C22×C3111yescharacteristic
198C2×#125C22×C3111yescharacteristic
199C2×C4×D4C22×C3111yescharacteristic
200#501—21no—
201#501—21no—
202#499—22no—
203#484—21no—
204#474—21no—
205#461—21no—
206#450—21no—
207#440—22no—
208#432C22×C3111yescharacteristic
209#432C22×C3111yescharacteristic
210#428—21no—
211#428—21no—
212C4×M16C22×C3111yescharacteristic
213#399—21no—
214#398—21no—
215#397—21no—
216#394—21no—
217#385—22no—
218#383C22×C3111yescharacteristic
219#330—22no—
220#328—22no—
221D4⋊C8—22no—
222C2×C31—22no—
223C2×C31—82no—
224C2×C31#200511yescharacteristic
225C2×C31#200811yescharacteristic
226C2×C31#201011yescharacteristic
227C22×D4D4×C3111yescharacteristic
228C23×C4—21no—
229C2×SD16—22no—
230C2×M16—21no—
231C4⋊Q8D4×C3111yescharacteristic
232C4⋊Q8D4×C3111yescharacteristic
233#125—21no—
234#125—21no—
235#125—21no—
236#125—21no—
237#125—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246#117—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4D4×C3111yescharacteristic
251C2×C4⋊C4D4×C3111yescharacteristic
252C2×C4⋊C4D4×C3111yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C42C23×C3111yescharacteristic
257#107—22no—
258#107—22no—
259#107—22no—
260#106—22no—
261#106—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264#103—22no—
265#102—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268#97—21no—
269#97—21no—
270#97D4×C3112yes—
271C4×C8D4×C3112yes—
272#95—22no—
273#95—21no—
274#95—21no—
275#95D4×C3112yes—
276C31#1002311yescharacteristic, direct product factor, Sylow
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—42no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4C2×D4×C3111yescharacteristic
291C22×C4C2×D4×C3111yescharacteristic
292C22×C4C2×D4×C3111yescharacteristic
293SD16—44no—
294M16—24no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—82no—
315C22⋊C4—44no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—82no—
320C42—41no—
321C42C2×D4×C3111yescharacteristic
322C42C2×D4×C3111yescharacteristic
323C42C2×D4×C3111yescharacteristic
324C42#41×C3111yescharacteristic
325C23—22no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23C31×#12011yescharacteristic
331Q8—42no—
332Q8—81no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—41no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4C31×#12011yescharacteristic
355C2×C4C31×#12011yescharacteristic
356C2×C4C31×#12111yescharacteristic
357C2×C4C31×#12111yescharacteristic
358C2×C4C31×#12111yescharacteristic
359C2×C4C31×#12711yescharacteristic
360C8—24no—
361C8—42no—
362C8—81no—
363C22—21no—
364C22—21no—
365C22—44no—
366C22—42no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22C31×#39211yescharacteristic
371C22C31×#45612yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C2—22no—
385C2—82no—
386C2C31×#200511yescharacteristic
387C2C31×#200811yescharacteristic
388C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
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
1. C31×#3166C2semidirect
4. C31×#1534C2semidirect
5. C31×#1515C2semidirect
7. C31×#1288C2semidirect
130. #10023C31direct (N × H)
169. #3166C2×C31semidirect
172. #1534C2×C31semidirect
173. #1515C2×C31semidirect
175. #1288C2×C31semidirect
276. C31#10023direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31662
C31×#31102
C31×#26332
C31×#15342
C31×#15152
C31×#14902
C31×#12882
#1002331