Finite Group Databasebeta

← C31×#19526 #21476285

Subgroups — C31×#19526 #21476285

Order 7936.

596 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3254C211yesmaximal, characteristic, complemented
2C31×#3246C211yesmaximal, characteristic, complemented
3C31×#3207C211yesmaximal, characteristic
4C31×#2985C211yesmaximal, characteristic, complemented
5C31×#1698C212yesmaximal, complemented
6C31×#1686C212yesmaximal, complemented
7C31×#1684C212yesmaximal, complemented
8C31×#1668C212yesmaximal, complemented
9C31×#1571C211yesmaximal, characteristic, complemented
10C31×#1557C211yesmaximal, characteristic, complemented
11C31×#1551C211yesmaximal, characteristic, complemented
12C31×#533C2211yescharacteristic
13C31×#532C2211yescharacteristic
14C31×#532C2211yescharacteristic
15C31×#493C2211yescharacteristic
16C31×#492C2211yescharacteristic
17C31×#487C2211yescharacteristic
18C31×#486C2211yescharacteristic
19C31×#477C2212yes—
20C31×#476—21no—
21C31×#475C2212yes—
22C31×#474—21no—
23C31×#468—22no—
24C31×#465—22no—
25C31×#463—22no—
26C31×#461—21no—
27C31×#461—21no—
28C31×#460C2212yes—
29C31×#460C2212yes—
30C31×#449—22no—
31C31×#446—22no—
32C31×#446C2212yes—
33C8×Q8×C31C2212yes—
34C31×#433—22no—
35C31×#432C2211yescharacteristic
36C31×#431C2211yescharacteristic
37C31×#423C2211yescharacteristic
38C2×C31×C4⋊C8C2211yescharacteristic
39C31×#335C2212yes—
40C31×#335C2212yes—
41C31×#333C2212yes—
42C31×#332C2212yes—
43C31×#331C2212yes—
44C31×#331C2212yes—
45C2×#41×C31—21no—
46C2×#41×C31—21no—
47C2×SD16×C31—22no—
48C2×SD16×C31—22no—
49C2×SD16×C31—22no—
50C2×D8×C31—22no—
51C2×M16×C31—21no—
52C22×C8×C31—21no—
53C31×C4⋊Q8D411yescharacteristic
54C31×C4⋊Q8D411yescharacteristic
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2311yescharacteristic
57C31×#127C2311yescharacteristic
58C31×#127C2311yescharacteristic
59C31×#125—21no—
60C31×#124—21no—
61C31×#124—21no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8D412yes—
64C31×#121—21no—
65C31×#121—21no—
66C31×#121—21no—
67C31×#121—21no—
68C4×Q8×C31—21no—
69C4×Q8×C31C2312yes—
70C4×D4×C31—21no—
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C4×D4×C31—21no—
74C31×#117—21no—
75C2×C42×C31C2311yescharacteristic
76C31×#107—22no—
77C31×#106—22no—
78C31×C4⋊C8—22no—
79C31×C4⋊C8—22no—
80C31×C4⋊C8—21no—
81C31×C4⋊C8—21no—
82C31×C4⋊C8D412yes—
83C31×C4⋊C8C2312yes—
84C31×C4⋊C8C2312yes—
85C31×C4⋊C8C2312yes—
86C31×#103—22no—
87C31×#103—22no—
88C31×#103—22no—
89C31×#102—22no—
90C31×#102—22no—
91C31×#102—22no—
92C31×C22⋊C8—21no—
93C31×C22⋊C8D412yes—
94C31×#97—22no—
95C31×#97—21no—
96C4×C8×C31—22no—
97C4×C8×C31C2311yescharacteristic
98C4×C8×C31C2311yescharacteristic
99#41×C31—82no—
100#41×C31—82no—
101C2×Q8×C31—21no—
102C2×Q8×C31—21no—
103C2×Q8×C31—21no—
104C2×Q8×C31C2×D412yes—
105C2×D4×C31—21no—
106C2×D4×C31—21no—
107C2×D4×C31—21no—
108C2×D4×C31—21no—
109C2×D4×C31—21no—
110C2×D4×C31—21no—
111C2×D4×C31—41no—
112C2×D4×C31—41no—
113C22×C4×C31—41no—
114C22×C4×C31—41no—
115C22×C4×C31C2×D411yescharacteristic
116C22×C4×C31C2×D411yescharacteristic
117C22×C4×C31#4111yescharacteristic
118SD16×C31—82no—
119SD16×C31—82no—
120SD16×C31—82no—
121D8×C31—82no—
122M16×C31—81no—
123C2×C8×C31—22no—
124C2×C8×C31—22no—
125C2×C8×C31—22no—
126C2×C8×C31—81no—
127C2×C8×C31C2×D412yes—
128C2×C8×C31C2×D412yes—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—22no—
133C4⋊C4×C31—22no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—21no—
136C4⋊C4×C31—21no—
137C4⋊C4×C31—21no—
138C4⋊C4×C31C2×D412yes—
139C22⋊C4×C31—22no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C22⋊C4×C31—41no—
143C22⋊C4×C31—41no—
144C42×C31—22no—
145C42×C31C2×D411yescharacteristic
146C42×C31C2×D411yescharacteristic
147C42×C31#4111yescharacteristic
148C42×C31C2411yescharacteristic
149#19526C3111yesmaximal, characteristic, direct product factor, Sylow
150C23×C31—41no—
151C23×C31—41no—
152C23×C31#12111yescharacteristic
153Q8×C31—22no—
154Q8×C31—22no—
155Q8×C31—42no—
156Q8×C31—81no—
157Q8×C31—81no—
158D4×C31—41no—
159D4×C31—41no—
160D4×C31—42no—
161D4×C31—42no—
162D4×C31—41no—
163D4×C31—41no—
164D4×C31—81no—
165D4×C31—81no—
166D4×C31—82no—
167D4×C31—82no—
168C2×C4×C31—21no—
169C2×C4×C31—22no—
170C2×C4×C31—21no—
171C2×C4×C31—22no—
172C2×C4×C31—22no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—82no—
177C2×C4×C31—82no—
178C2×C4×C31#12111yescharacteristic
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31C22×D411yescharacteristic
182C2×C4×C31C22×D411yescharacteristic
183C2×C4×C31C2×#4111yescharacteristic
184C8×C31—22no—
185C8×C31—22no—
186C8×C31—42no—
187C8×C31—42no—
188C8×C31—42no—
189#3254C2×C3111yescharacteristic, complemented
190#3246C2×C3111yescharacteristic, complemented
191#3207C2×C3111yescharacteristic
192#2985C2×C3111yescharacteristic, complemented
193#1698C2×C3112yescomplemented
194#1686C2×C3112yescomplemented
195#1684C2×C3112yescomplemented
196#1668C2×C3112yescomplemented
197#1571C2×C3111yescharacteristic, complemented
198#1557C2×C3111yescharacteristic, complemented
199#1551C2×C3111yescharacteristic, complemented
200C22×C31—21no—
201C22×C31—21no—
202C22×C31—21no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31C2×#12111yescharacteristic
210C4×C31—21no—
211C4×C31—21no—
212C4×C31—21no—
213C4×C31—21no—
214C4×C31—42no—
215C4×C31—42no—
216C4×C31—81no—
217C4×C31—81no—
218C4×C31#57611yescharacteristic
219C4×C31#57712yes—
220C4×C31#57811yescharacteristic
221#533C22×C3111yescharacteristic
222#532C22×C3111yescharacteristic
223#532C22×C3111yescharacteristic
224#493C22×C3111yescharacteristic
225#492C22×C3111yescharacteristic
226#487C22×C3111yescharacteristic
227#486C22×C3111yescharacteristic
228#477C22×C3112yes—
229#476—21no—
230#475C22×C3112yes—
231#474—21no—
232#468—22no—
233#465—22no—
234#463—22no—
235#461—21no—
236#461—21no—
237#460C22×C3112yes—
238#460C22×C3112yes—
239#449—22no—
240#446—22no—
241#446C22×C3112yes—
242C8×Q8C22×C3112yes—
243#433—22no—
244#432C22×C3111yescharacteristic
245#431C22×C3111yescharacteristic
246#423C22×C3111yescharacteristic
247C2×C4⋊C8C22×C3111yescharacteristic
248#335C22×C3112yes—
249#335C22×C3112yes—
250#333C22×C3112yes—
251#332C22×C3112yes—
252#331C22×C3112yes—
253#331C22×C3112yes—
254C2×C31—41no—
255C2×C31—161no—
256C2×C31—161no—
257C2×C31#304311yescharacteristic
258C2×C31#306111yescharacteristic
259C2×C31#306811yescharacteristic
260C2×#41—21no—
261C2×#41—21no—
262C2×SD16—22no—
263C2×SD16—22no—
264C2×SD16—22no—
265C2×D8—22no—
266C2×M16—21no—
267C22×C8—21no—
268C4⋊Q8D4×C3111yescharacteristic
269C4⋊Q8D4×C3111yescharacteristic
270C4⋊Q8C23×C3111yescharacteristic
271C4⋊Q8C23×C3111yescharacteristic
272#127C23×C3111yescharacteristic
273#127C23×C3111yescharacteristic
274#125—21no—
275#124—21no—
276#124—21no—
277C22⋊Q8—21no—
278C22⋊Q8D4×C3112yes—
279#121—21no—
280#121—21no—
281#121—21no—
282#121—21no—
283C4×Q8—21no—
284C4×Q8C23×C3112yes—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289#117—21no—
290C2×C42C23×C3111yescharacteristic
291#107—22no—
292#106—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8D4×C3112yes—
298C4⋊C8C23×C3112yes—
299C4⋊C8C23×C3112yes—
300C4⋊C8C23×C3112yes—
301#103—22no—
302#103—22no—
303#103—22no—
304#102—22no—
305#102—22no—
306#102—22no—
307C22⋊C8—21no—
308C22⋊C8D4×C3112yes—
309#97—22no—
310#97—21no—
311C4×C8—22no—
312C4×C8C23×C3111yescharacteristic
313C4×C8C23×C3111yescharacteristic
314C31#1952611yescharacteristic, direct product factor, Sylow
315#41—82no—
316#41—82no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C2×D4×C3112yes—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—41no—
328C2×D4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×D4×C3111yescharacteristic
332C22×C4C2×D4×C3111yescharacteristic
333C22×C4#41×C3111yescharacteristic
334SD16—82no—
335SD16—82no—
336SD16—82no—
337D8—82no—
338M16—81no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—81no—
343C2×C8C2×D4×C3112yes—
344C2×C8C2×D4×C3112yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C2×D4×C3112yes—
355C22⋊C4—22no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C42—22no—
361C42C2×D4×C3111yescharacteristic
362C42C2×D4×C3111yescharacteristic
363C42#41×C3111yescharacteristic
364C42C24×C3111yescharacteristic
365C23—41no—
366C23—41no—
367C23C31×#12111yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—42no—
371Q8—81no—
372Q8—81no—
373D4—41no—
374D4—41no—
375D4—42no—
376D4—42no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381D4—82no—
382D4—82no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4C31×#12111yescharacteristic
394C2×C4C31×#12111yescharacteristic
395C2×C4C31×#12111yescharacteristic
396C2×C4C22×D4×C3111yescharacteristic
397C2×C4C22×D4×C3111yescharacteristic
398C2×C4C2×#41×C3111yescharacteristic
399C8—22no—
400C8—22no—
401C8—42no—
402C8—42no—
403C8—42no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22C2×C31×#12111yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4C31×#57611yescharacteristic
423C4C31×#57712yes—
424C4C31×#57811yescharacteristic
425C2—41no—
426C2—161no—
427C2—161no—
428C2C31×#304311yescharacteristic
429C2C31×#306111yescharacteristic
430C2C31×#306811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3207C2
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
1. C31×#3254C2semidirect
2. C31×#3246C2semidirect
4. C31×#2985C2semidirect
5. C31×#1698C2semidirect
6. C31×#1686C2semidirect
7. C31×#1684C2semidirect
8. C31×#1668C2semidirect
9. C31×#1571C2semidirect
10. C31×#1557C2semidirect
11. C31×#1551C2semidirect
149. #19526C31direct (N × H)
189. #3254C2×C31semidirect
190. #3246C2×C31semidirect
192. #2985C2×C31semidirect
193. #1698C2×C31semidirect
194. #1686C2×C31semidirect
195. #1684C2×C31semidirect
196. #1668C2×C31semidirect
197. #1571C2×C31semidirect
198. #1557C2×C31semidirect
199. #1551C2×C31semidirect
314. C31#19526direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32542
C31×#32462
C31×#32072
C31×#29852
C31×#16982
C31×#16862
C31×#16842
C31×#16682
C31×#15712
C31×#15572
C31×#15512
#1952631