Finite Group Databasebeta

← C31×#9168 #21465927

Subgroups — C31×#9168 #21465927

Order 7936.

432 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3079C211yesmaximal, characteristic
2C31×#2925C211yesmaximal, characteristic
3C31×#1548C211yesmaximal, characteristic, complemented
4C2×C31×#332C211yesmaximal, characteristic, complemented
5C31×#1477C211yesmaximal, characteristic
6C31×#1301C211yesmaximal, characteristic, complemented
7C31×#1292C211yesmaximal, characteristic, complemented
8C2×C31×C4⋊Q8C2211yescharacteristic
9C4×#41×C31C2211yescharacteristic
10C31×#477C412yes—
11C31×#476C412yes—
12C31×#475—21no—
13C31×#474—21no—
14C31×#433—22no—
15C31×#432—21no—
16C31×#431—21no—
17C31×#431C2211yescharacteristic
18C31×#428—21no—
19C31×#423C2211yescharacteristic
20C31×#423C2211yescharacteristic
21C2×C31×C4⋊C8C2211yescharacteristic
22C2×C31×C4⋊C8C2211yescharacteristic
23C31×#417—21no—
24C31×#408—21no—
25C31×#405—21no—
26C31×#333—22no—
27C31×#332—21no—
28C31×#332—21no—
29C31×Q8⋊C8—21no—
30C31×D4⋊C8—21no—
31C2×#41×C31D411yescharacteristic
32C22×Q8×C31—21no—
33C2×M16×C31—21no—
34C2×M16×C31—21no—
35C22×C8×C31—21no—
36C22×C8×C31—21no—
37C31×C4⋊Q8—21no—
38C31×C4⋊Q8—21no—
39C31×C4⋊Q8—21no—
40C31×C4⋊Q8C2×C412yes—
41C4×Q8×C31C2×C411yescharacteristic
42C4×Q8×C31D411yescharacteristic
43C4×D4×C31—21no—
44C4×D4×C31—21no—
45C4×D4×C31C2×C411yescharacteristic
46C4×D4×C31D411yescharacteristic
47C31×#117—21no—
48C31×#117D411yescharacteristic
49C2×C4⋊C4×C31—21no—
50C2×C4⋊C4×C31—21no—
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×#107—22no—
54C31×#106—22no—
55C31×C4⋊C8—21no—
56C31×C4⋊C8—21no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×C4⋊C8—21no—
60C31×C4⋊C8—21no—
61C31×C4⋊C8C2×C412yes—
62C31×C4⋊C8D411yescharacteristic
63C31×C4⋊C8D411yescharacteristic
64C31×C4⋊C8D412yes—
65C31×#103—22no—
66C31×#102—22no—
67C31×C22⋊C8—22no—
68C31×C22⋊C8—22no—
69C31×C22⋊C8—21no—
70C31×#97—22no—
71C4×C8×C31—22no—
72C4×C8×C31—21no—
73#41×C31—21no—
74#41×C31—21no—
75#41×C31—21no—
76#41×C31—21no—
77C2×Q8×C31—22no—
78C2×Q8×C31—41no—
79C2×Q8×C31—81no—
80C2×Q8×C31—81no—
81C2×Q8×C31C22⋊C411yescharacteristic
82C2×D4×C31—21no—
83C2×D4×C31C22⋊C411yescharacteristic
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—21no—
87C22×C4×C31—41no—
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31#4111yescharacteristic
90C22×C4×C31#4111yescharacteristic
91M16×C31—41no—
92M16×C31—41no—
93M16×C31—41no—
94M16×C31—41no—
95C2×C8×C31—22no—
96C2×C8×C31—21no—
97C2×C8×C31—22no—
98C2×C8×C31—21no—
99C2×C8×C31—41no—
100C2×C8×C31—41no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—81no—
104C4⋊C4×C31—21no—
105C4⋊C4×C31—21no—
106C4⋊C4×C31—22no—
107C4⋊C4×C31—22no—
108C4⋊C4×C31—41no—
109C4⋊C4×C31—41no—
110C4⋊C4×C31C22⋊C411yescharacteristic
111C4⋊C4×C31C22⋊C411yescharacteristic
112C22⋊C4×C31—21no—
113C22⋊C4×C31—21no—
114C22⋊C4×C31—21no—
115C42×C31—21no—
116C42×C31—21no—
117C42×C31—21no—
118C42×C31C22×C411yescharacteristic
119C42×C31C2×D411yescharacteristic
120C42×C31C2×D411yescharacteristic
121C42×C31C2×D411yescharacteristic
122#9168C3111yesmaximal, characteristic, direct product factor, Sylow
123C23×C31—21no—
124C23×C31#12311yescharacteristic
125Q8×C31—21no—
126Q8×C31—21no—
127Q8×C31—82no—
128Q8×C31—81no—
129Q8×C31—81no—
130D4×C31—21no—
131D4×C31—21no—
132D4×C31—41no—
133D4×C31—41no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—42no—
145C2×C4×C31—81no—
146C2×C4×C31—82no—
147C2×C4×C31C2×C22⋊C411yescharacteristic
148C2×C4×C31C4×D411yescharacteristic
149C2×C4×C31C4×D411yescharacteristic
150C2×C4×C31#12011yescharacteristic
151C2×C4×C31#12111yescharacteristic
152C2×C4×C31#12111yescharacteristic
153C2×C4×C31#13511yescharacteristic
154C2×C4×C31#13511yescharacteristic
155C2×C4×C31#13711yescharacteristic
156C2×C4×C31#13711yescharacteristic
157C8×C31—42no—
158C8×C31—41no—
159C8×C31—41no—
160C8×C31—42no—
161C8×C31—81no—
162C8×C31—81no—
163#3079C2×C3111yescharacteristic
164#2925C2×C3111yescharacteristic
165#1548C2×C3111yescharacteristic, complemented
166C2×#332C2×C3111yescharacteristic, complemented
167#1477C2×C3111yescharacteristic
168#1301C2×C3111yescharacteristic, complemented
169#1292C2×C3111yescharacteristic, complemented
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—41no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31#38611yescharacteristic
176C22×C31#48511yescharacteristic
177C22×C31#48511yescharacteristic
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—21no—
181C4×C31—41no—
182C4×C31—41no—
183C4×C31—41no—
184C4×C31—82no—
185C4×C31#41611yescharacteristic
186C4×C31#41911yescharacteristic
187C4×C31#45211yescharacteristic
188C4×C31#45211yescharacteristic
189C4×C31#47111yescharacteristic
190C4×C31#47211yescharacteristic
191C2×C4⋊Q8C22×C3111yescharacteristic
192C4×#41C22×C3111yescharacteristic
193#477C4×C3112yes—
194#476C4×C3112yes—
195#475—21no—
196#474—21no—
197#433—22no—
198#432—21no—
199#431—21no—
200#431C22×C3111yescharacteristic
201#428—21no—
202#423C22×C3111yescharacteristic
203#423C22×C3111yescharacteristic
204C2×C4⋊C8C22×C3111yescharacteristic
205C2×C4⋊C8C22×C3111yescharacteristic
206#417—21no—
207#408—21no—
208#405—21no—
209#333—22no—
210#332—21no—
211#332—21no—
212Q8⋊C8—21no—
213D4⋊C8—21no—
214C2×C31—21no—
215C2×C31—21no—
216C2×C31—81no—
217C2×C31#187311yescharacteristic
218C2×C31#190211yescharacteristic
219C2×C31#190711yescharacteristic
220C2×#41D4×C3111yescharacteristic
221C22×Q8—21no—
222C2×M16—21no—
223C2×M16—21no—
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C2×C4×C3112yes—
230C4×Q8C2×C4×C3111yescharacteristic
231C4×Q8D4×C3111yescharacteristic
232C4×D4—21no—
233C4×D4—21no—
234C4×D4C2×C4×C3111yescharacteristic
235C4×D4D4×C3111yescharacteristic
236#117—21no—
237#117D4×C3111yescharacteristic
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C42—21no—
241C2×C42C23×C3111yescharacteristic
242#107—22no—
243#106—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C2×C4×C3112yes—
251C4⋊C8D4×C3111yescharacteristic
252C4⋊C8D4×C3111yescharacteristic
253C4⋊C8D4×C3112yes—
254#103—22no—
255#102—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—21no—
259#97—22no—
260C4×C8—22no—
261C4×C8—21no—
262C31#916811yescharacteristic, direct product factor, Sylow
263#41—21no—
264#41—21no—
265#41—21no—
266#41—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8C22⋊C4×C3111yescharacteristic
272C2×D4—21no—
273C2×D4C22⋊C4×C3111yescharacteristic
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4C2×D4×C3111yescharacteristic
279C22×C4#41×C3111yescharacteristic
280C22×C4#41×C3111yescharacteristic
281M16—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—81no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4C22⋊C4×C3111yescharacteristic
301C4⋊C4C22⋊C4×C3111yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42C22×C4×C3111yescharacteristic
309C42C2×D4×C3111yescharacteristic
310C42C2×D4×C3111yescharacteristic
311C42C2×D4×C3111yescharacteristic
312C23—21no—
313C23C31×#12311yescharacteristic
314Q8—21no—
315Q8—21no—
316Q8—82no—
317Q8—81no—
318Q8—81no—
319D4—21no—
320D4—21no—
321D4—41no—
322D4—41no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4C2×C22⋊C4×C3111yescharacteristic
337C2×C4C4×D4×C3111yescharacteristic
338C2×C4C4×D4×C3111yescharacteristic
339C2×C4C31×#12011yescharacteristic
340C2×C4C31×#12111yescharacteristic
341C2×C4C31×#12111yescharacteristic
342C2×C4C31×#13511yescharacteristic
343C2×C4C31×#13511yescharacteristic
344C2×C4C31×#13711yescharacteristic
345C2×C4C31×#13711yescharacteristic
346C8—42no—
347C8—41no—
348C8—41no—
349C8—42no—
350C8—81no—
351C8—81no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22C31×#38611yescharacteristic
358C22C31×#48511yescharacteristic
359C22C31×#48511yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—82no—
367C4C31×#41611yescharacteristic
368C4C31×#41911yescharacteristic
369C4C31×#45211yescharacteristic
370C4C31×#45211yescharacteristic
371C4C31×#47111yescharacteristic
372C4C31×#47211yescharacteristic
373C2—21no—
374C2—21no—
375C2—81no—
376C2C31×#187311yescharacteristic
377C2C31×#190211yescharacteristic
378C2C31×#190711yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×#41×C31D4
LCSG ▷ 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×#1548C2semidirect
4. C2×C31×#332C2semidirect
6. C31×#1301C2semidirect
7. C31×#1292C2semidirect
122. #9168C31direct (N × H)
165. #1548C2×C31semidirect
166. C2×#332C2×C31semidirect
168. #1301C2×C31semidirect
169. #1292C2×C31semidirect
262. C31#9168direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#30792
C31×#29252
C31×#15482
C2×C31×#3322
C31×#14772
C31×#13012
C31×#12922
#916831