Finite Group Databasebeta

← C31×#19527 #21476286

Subgroups — C31×#19527 #21476286

Order 7936.

574 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3255C211yesmaximal, characteristic, complemented
2C31×#3246C211yesmaximal, characteristic
3C31×#3210C211yesmaximal, characteristic
4C31×#2985C211yesmaximal, characteristic, complemented
5C31×#1700C212yesmaximal, complemented
6C31×#1686C212yesmaximal, complemented
7C31×#1682C212yesmaximal, complemented
8C31×#1670C212yesmaximal, complemented
9C31×#1571C211yesmaximal, characteristic, complemented
10C31×#1557C211yesmaximal, characteristic
11C31×#1552C211yesmaximal, characteristic, complemented
12C31×#533C2211yescharacteristic
13C31×#533C2211yescharacteristic
14C31×#532C2211yescharacteristic
15C31×#494C2211yescharacteristic
16C31×#492C2211yescharacteristic
17C31×#487C2211yescharacteristic
18C31×#486C2211yescharacteristic
19C31×#476—21no—
20C31×#475—21no—
21C31×#475C2212yes—
22C31×#475C2212yes—
23C31×#470—22no—
24C31×#465—22no—
25C31×#464—22no—
26C31×#462—21no—
27C31×#462C2212yes—
28C31×#461—21no—
29C31×#460C2212yes—
30C31×#452—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×#332C2212yes—
42C31×#332C2212yes—
43C31×#332C2212yes—
44C31×#331C2212yes—
45C2×#41×C31—21no—
46C2×Q16×C31—22no—
47C2×Q16×C31—22no—
48C2×SD16×C31—22no—
49C2×SD16×C31—22no—
50C2×M16×C31—21no—
51C22×C8×C31—21no—
52C31×C4⋊Q8D411yescharacteristic
53C31×C4⋊Q8D411yescharacteristic
54C31×C4⋊Q8C2311yescharacteristic
55C31×C4⋊Q8C2311yescharacteristic
56C31×C4⋊Q8C2311yescharacteristic
57C31×#127C2311yescharacteristic
58C31×#125—21no—
59C31×#125—21no—
60C31×#124—21no—
61C31×C22⋊Q8—21no—
62C31×C22⋊Q8—21no—
63C31×C22⋊Q8—21no—
64C31×C22⋊Q8D412yes—
65C31×#121—21no—
66C31×#121—21no—
67C4×Q8×C31—21no—
68C4×Q8×C31—21no—
69C4×Q8×C31—21no—
70C4×Q8×C31C2312yes—
71C4×D4×C31—21no—
72C4×D4×C31—21no—
73C31×#117—21no—
74C31×#117—21no—
75C2×C42×C31C2311yescharacteristic
76C31×#106—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×#103—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—
100C2×Q8×C31—21no—
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—41no—
109C22×C4×C31—41no—
110C22×C4×C31C2×D411yescharacteristic
111C22×C4×C31C2×D411yescharacteristic
112C22×C4×C31#4111yescharacteristic
113Q16×C31—82no—
114Q16×C31—82no—
115SD16×C31—82no—
116SD16×C31—82no—
117M16×C31—81no—
118C2×C8×C31—22no—
119C2×C8×C31—22no—
120C2×C8×C31—22no—
121C2×C8×C31—81no—
122C2×C8×C31C2×D412yes—
123C2×C8×C31C2×D412yes—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—21no—
126C4⋊C4×C31—21no—
127C4⋊C4×C31—22no—
128C4⋊C4×C31—22no—
129C4⋊C4×C31—21no—
130C4⋊C4×C31—21no—
131C4⋊C4×C31—21no—
132C4⋊C4×C31—21no—
133C4⋊C4×C31—21no—
134C4⋊C4×C31—21no—
135C4⋊C4×C31—41no—
136C4⋊C4×C31—41no—
137C4⋊C4×C31C2×D412yes—
138C22⋊C4×C31—22no—
139C22⋊C4×C31—41no—
140C22⋊C4×C31—41no—
141C22⋊C4×C31—41no—
142C42×C31—22no—
143C42×C31—41no—
144C42×C31C2×D411yescharacteristic
145C42×C31C2×D411yescharacteristic
146C42×C31#4111yescharacteristic
147C42×C31C2411yescharacteristic
148#19527C3111yesmaximal, characteristic, direct product factor, Sylow
149C23×C31—41no—
150C23×C31#12111yescharacteristic
151Q8×C31—22no—
152Q8×C31—22no—
153Q8×C31—41no—
154Q8×C31—42no—
155Q8×C31—42no—
156Q8×C31—41no—
157Q8×C31—81no—
158D4×C31—41no—
159D4×C31—42no—
160D4×C31—41no—
161D4×C31—81no—
162D4×C31—82no—
163C2×C4×C31—21no—
164C2×C4×C31—22no—
165C2×C4×C31—21no—
166C2×C4×C31—22no—
167C2×C4×C31—22no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—82no—
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12111yescharacteristic
176C2×C4×C31C22×D411yescharacteristic
177C2×C4×C31C22×D411yescharacteristic
178C2×C4×C31C2×#4111yescharacteristic
179C8×C31—22no—
180C8×C31—22no—
181C8×C31—42no—
182C8×C31—42no—
183C8×C31—42no—
184#3255C2×C3111yescharacteristic, complemented
185#3246C2×C3111yescharacteristic
186#3210C2×C3111yescharacteristic
187#2985C2×C3111yescharacteristic, complemented
188#1700C2×C3112yescomplemented
189#1686C2×C3112yescomplemented
190#1682C2×C3112yescomplemented
191#1670C2×C3112yescomplemented
192#1571C2×C3111yescharacteristic, complemented
193#1557C2×C3111yescharacteristic
194#1552C2×C3111yescharacteristic, complemented
195C22×C31—21no—
196C22×C31—21no—
197C22×C31—21no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31C2×#12111yescharacteristic
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—42no—
207C4×C31—42no—
208C4×C31—81no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31#57611yescharacteristic
212C4×C31#57712yes—
213C4×C31#57811yescharacteristic
214#533C22×C3111yescharacteristic
215#533C22×C3111yescharacteristic
216#532C22×C3111yescharacteristic
217#494C22×C3111yescharacteristic
218#492C22×C3111yescharacteristic
219#487C22×C3111yescharacteristic
220#486C22×C3111yescharacteristic
221#476—21no—
222#475—21no—
223#475C22×C3112yes—
224#475C22×C3112yes—
225#470—22no—
226#465—22no—
227#464—22no—
228#462—21no—
229#462C22×C3112yes—
230#461—21no—
231#460C22×C3112yes—
232#452—22no—
233#446—22no—
234#446C22×C3112yes—
235C8×Q8C22×C3112yes—
236#433—22no—
237#432C22×C3111yescharacteristic
238#431C22×C3111yescharacteristic
239#423C22×C3111yescharacteristic
240C2×C4⋊C8C22×C3111yescharacteristic
241#335C22×C3112yes—
242#335C22×C3112yes—
243#332C22×C3112yes—
244#332C22×C3112yes—
245#332C22×C3112yes—
246#331C22×C3112yes—
247C2×C31—41no—
248C2×C31—161no—
249C2×C31#304311yescharacteristic
250C2×C31#306111yescharacteristic
251C2×C31#306911yescharacteristic
252C2×#41—21no—
253C2×Q16—22no—
254C2×Q16—22no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×M16—21no—
258C22×C8—21no—
259C4⋊Q8D4×C3111yescharacteristic
260C4⋊Q8D4×C3111yescharacteristic
261C4⋊Q8C23×C3111yescharacteristic
262C4⋊Q8C23×C3111yescharacteristic
263C4⋊Q8C23×C3111yescharacteristic
264#127C23×C3111yescharacteristic
265#125—21no—
266#125—21no—
267#124—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8D4×C3112yes—
272#121—21no—
273#121—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8C23×C3112yes—
278C4×D4—21no—
279C4×D4—21no—
280#117—21no—
281#117—21no—
282C2×C42C23×C3111yescharacteristic
283#106—22no—
284#106—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8D4×C3112yes—
290C4⋊C8C23×C3112yes—
291C4⋊C8C23×C3112yes—
292C4⋊C8C23×C3112yes—
293#103—22no—
294#103—22no—
295#103—22no—
296#103—22no—
297#102—22no—
298#102—22no—
299C22⋊C8—21no—
300C22⋊C8D4×C3112yes—
301#97—22no—
302#97—21no—
303C4×C8—22no—
304C4×C8C23×C3111yescharacteristic
305C4×C8C23×C3111yescharacteristic
306C31#1952711yescharacteristic, direct product factor, Sylow
307#41—82no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8C2×D4×C3112yes—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C22×C4—41no—
318C22×C4C2×D4×C3111yescharacteristic
319C22×C4C2×D4×C3111yescharacteristic
320C22×C4#41×C3111yescharacteristic
321Q16—82no—
322Q16—82no—
323SD16—82no—
324SD16—82no—
325M16—81no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—81no—
330C2×C8C2×D4×C3112yes—
331C2×C8C2×D4×C3112yes—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4C2×D4×C3112yes—
346C22⋊C4—22no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—22no—
351C42—41no—
352C42C2×D4×C3111yescharacteristic
353C42C2×D4×C3111yescharacteristic
354C42#41×C3111yescharacteristic
355C42C24×C3111yescharacteristic
356C23—41no—
357C23C31×#12111yescharacteristic
358Q8—22no—
359Q8—22no—
360Q8—41no—
361Q8—42no—
362Q8—42no—
363Q8—41no—
364Q8—81no—
365D4—41no—
366D4—42no—
367D4—41no—
368D4—81no—
369D4—82no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4C31×#12111yescharacteristic
381C2×C4C31×#12111yescharacteristic
382C2×C4C31×#12111yescharacteristic
383C2×C4C22×D4×C3111yescharacteristic
384C2×C4C22×D4×C3111yescharacteristic
385C2×C4C2×#41×C3111yescharacteristic
386C8—22no—
387C8—22no—
388C8—42no—
389C8—42no—
390C8—42no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×C31×#12111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4C31×#57611yescharacteristic
408C4C31×#57712yes—
409C4C31×#57811yescharacteristic
410C2—41no—
411C2—161no—
412C2C31×#304311yescharacteristic
413C2C31×#306111yescharacteristic
414C2C31×#306911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#121
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#532C22
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×#3255C2semidirect
4. C31×#2985C2semidirect
5. C31×#1700C2semidirect
6. C31×#1686C2semidirect
7. C31×#1682C2semidirect
8. C31×#1670C2semidirect
9. C31×#1571C2semidirect
11. C31×#1552C2semidirect
148. #19527C31direct (N × H)
184. #3255C2×C31semidirect
187. #2985C2×C31semidirect
188. #1700C2×C31semidirect
189. #1686C2×C31semidirect
190. #1682C2×C31semidirect
191. #1670C2×C31semidirect
192. #1571C2×C31semidirect
194. #1552C2×C31semidirect
306. C31#19527direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32552
C31×#32462
C31×#32102
C31×#29852
C31×#17002
C31×#16862
C31×#16822
C31×#16702
C31×#15712
C31×#15572
C31×#15522
#1952731