Finite Group Databasebeta

← C31×#10006 #21466765

Subgroups — C31×#10006 #21466765

Order 7936.

572 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3213C211yesmaximal, characteristic
2C31×#3112C211yesmaximal, characteristic, complemented
3C31×#2592C211yesmaximal, characteristic, complemented
4C31×#1980C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic, complemented
6C31×#1491C211yesmaximal, characteristic, complemented
7C31×#1285C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C31×#496—21no—
12C31×#489—21no—
13C31×#480—21no—
14C31×#477—21no—
15C31×#463—22no—
16C31×#460—21no—
17C31×#448—21no—
18C31×#448—21no—
19C31×#447—21no—
20C31×#440—22no—
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C2×C31×#104—22no—
24C4×M16×C31C2211yescharacteristic
25C31×#397—21no—
26C31×#393—21no—
27C31×#391—21no—
28C31×#390—21no—
29C31×#388—22no—
30C4×C4⋊C4×C31C2211yescharacteristic
31C31×#329—22no—
32C31×Q8⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C22×Q8×C31D411yescharacteristic
36C22×D4×C31D411yescharacteristic
37C2×SD16×C31—22no—
38C2×SD16×C31—22no—
39C2×D8×C31—22no—
40C2×M16×C31—21no—
41C31×C4⋊Q8D411yescharacteristic
42C31×#127D411yescharacteristic
43C31×#124—21no—
44C31×#124—21no—
45C31×#124—21no—
46C31×#124—21no—
47C31×#124—21no—
48C31×#121—21no—
49C31×#121—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×D4×C31—21no—
55C4×D4×C31—21no—
56C2×C4⋊C4×C31—21no—
57C2×C4⋊C4×C31—21no—
58C2×C4⋊C4×C31D411yescharacteristic
59C2×C22⋊C4×C31—22no—
60C2×C22⋊C4×C31—21no—
61C2×C22⋊C4×C31—21no—
62C2×C22⋊C4×C31—22no—
63C2×C42×C31—21no—
64C2×C42×C31D412yes—
65C2×C42×C31C2311yescharacteristic
66C31×#107—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8D412yes—
69C31×#104—44no—
70C31×#103—22no—
71C31×#103—22no—
72C31×#102—22no—
73C31×#102—22no—
74C31×C22⋊C8—21no—
75C31×C22⋊C8—21no—
76C31×#97—21no—
77C31×#97—21no—
78C31×#97D412yes—
79C4×C8×C31—21no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—21no—
83C24×C31—21no—
84#41×C31—42no—
85#41×C31—42no—
86C2×Q8×C31—21no—
87C2×Q8×C31—21no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—42no—
92C2×Q8×C31—42no—
93C2×D4×C31—21no—
94C2×D4×C31—21no—
95C2×D4×C31—21no—
96C2×D4×C31—21no—
97C2×D4×C31—21no—
98C2×D4×C31—42no—
99C2×D4×C31—41no—
100C22×C4×C31—21no—
101C22×C4×C31—21no—
102C22×C4×C31—22no—
103C22×C4×C31—21no—
104C22×C4×C31—41no—
105C22×C4×C31C2×D411yescharacteristic
106C22×C4×C31C2×D411yescharacteristic
107C22×C4×C31C2×D411yescharacteristic
108SD16×C31—44no—
109SD16×C31—82no—
110D8×C31—82no—
111M16×C31—22no—
112M16×C31—41no—
113C2×C8×C31—22no—
114C2×C8×C31—22no—
115C2×C8×C31—41no—
116C4⋊C4×C31—21no—
117C4⋊C4×C31—21no—
118C4⋊C4×C31—21no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—42no—
121C4⋊C4×C31—41no—
122C4⋊C4×C31—41no—
123C22⋊C4×C31—44no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—82no—
131C42×C31—42no—
132C42×C31—41no—
133C42×C31—41no—
134C42×C31C2×D411yescharacteristic
135C42×C31C2×D411yescharacteristic
136C42×C31C2×D411yescharacteristic
137C42×C31#4111yescharacteristic
138#10006C3111yesmaximal, characteristic, direct product factor, Sylow
139C23×C31—42no—
140C23×C31—42no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31#12011yescharacteristic
146Q8×C31—22no—
147Q8×C31—22no—
148Q8×C31—41no—
149Q8×C31—41no—
150Q8×C31—42no—
151Q8×C31—42no—
152Q8×C31—81no—
153D4×C31—42no—
154D4×C31—42no—
155D4×C31—42no—
156D4×C31—42no—
157D4×C31—81no—
158D4×C31—81no—
159D4×C31—81no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—22no—
163C2×C4×C31—22no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—42no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—82no—
177C2×C4×C31#12011yescharacteristic
178C2×C4×C31#12011yescharacteristic
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12111yescharacteristic
181C2×C4×C31#12111yescharacteristic
182C2×C4×C31#12711yescharacteristic
183C8×C31—42no—
184C8×C31—42no—
185C8×C31—42no—
186#3213C2×C3111yescharacteristic
187#3112C2×C3111yescharacteristic, complemented
188#2592C2×C3111yescharacteristic, complemented
189#1980C2×C3111yescharacteristic, complemented
190#1527C2×C3111yescharacteristic, complemented
191#1491C2×C3111yescharacteristic, complemented
192#1285C2×C3111yescharacteristic, complemented
193C22×C31—21no—
194C22×C31—21no—
195C22×C31—82no—
196C22×C31—82no—
197C22×C31—81no—
198C22×C31—81no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31#39211yescharacteristic
204C22×C31#45311yescharacteristic
205C22×C31#45411yescharacteristic
206C4×C31—21no—
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—42no—
212C4×C31—82no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31#45512yes—
217#532C22×C3111yescharacteristic
218C2×#124C22×C3111yescharacteristic
219C2×C4×Q8C22×C3111yescharacteristic
220#496—21no—
221#489—21no—
222#480—21no—
223#477—21no—
224#463—22no—
225#460—21no—
226#448—21no—
227#448—21no—
228#447—21no—
229#440—22no—
230#432C22×C3111yescharacteristic
231#432C22×C3111yescharacteristic
232C2×#104—22no—
233C4×M16C22×C3111yescharacteristic
234#397—21no—
235#393—21no—
236#391—21no—
237#390—21no—
238#388—22no—
239C4×C4⋊C4C22×C3111yescharacteristic
240#329—22no—
241Q8⋊C8—22no—
242C2×C31—21no—
243C2×C31—21no—
244C2×C31—161no—
245C2×C31—161no—
246C2×C31#200411yescharacteristic
247C2×C31#200611yescharacteristic
248C2×C31#201011yescharacteristic
249C2×#41—21no—
250C22×Q8D4×C3111yescharacteristic
251C22×Q8D4×C3111yescharacteristic
252C22×D4D4×C3111yescharacteristic
253C2×SD16—22no—
254C2×SD16—22no—
255C2×D8—22no—
256C2×M16—21no—
257C4⋊Q8D4×C3111yescharacteristic
258#127D4×C3111yescharacteristic
259#124—21no—
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#121—21no—
265#121—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×D4—21no—
271C4×D4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4D4×C3111yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C42—21no—
280C2×C42D4×C3112yes—
281C2×C42C23×C3111yescharacteristic
282#107—22no—
283C4⋊C8—21no—
284C4⋊C8D4×C3112yes—
285#104—44no—
286#103—22no—
287#103—22no—
288#102—22no—
289#102—22no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292#97—21no—
293#97—21no—
294#97D4×C3112yes—
295C4×C8—21no—
296#95—22no—
297#95—22no—
298#95—21no—
299C31#1000611yescharacteristic, direct product factor, Sylow
300C24—21no—
301#41—42no—
302#41—42no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—42no—
316C2×D4—41no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4C2×D4×C3111yescharacteristic
323C22×C4C2×D4×C3111yescharacteristic
324C22×C4C2×D4×C3111yescharacteristic
325SD16—44no—
326SD16—82no—
327D8—82no—
328M16—22no—
329M16—41no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—44no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—82no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42C2×D4×C3111yescharacteristic
352C42C2×D4×C3111yescharacteristic
353C42C2×D4×C3111yescharacteristic
354C42#41×C3111yescharacteristic
355C23—42no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C31×#12011yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366Q8—42no—
367Q8—42no—
368Q8—81no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4—81no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—82no—
393C2×C4C31×#12011yescharacteristic
394C2×C4C31×#12011yescharacteristic
395C2×C4C31×#12111yescharacteristic
396C2×C4C31×#12111yescharacteristic
397C2×C4C31×#12111yescharacteristic
398C2×C4C31×#12711yescharacteristic
399C8—42no—
400C8—42no—
401C8—42no—
402C22—21no—
403C22—21no—
404C22—82no—
405C22—82no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C31×#39211yescharacteristic
413C22C31×#45311yescharacteristic
414C22C31×#45411yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—42no—
421C4—82no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4C31×#45512yes—
426C2—21no—
427C2—21no—
428C2—161no—
429C2—161no—
430C2C31×#200411yescharacteristic
431C2C31×#200611yescharacteristic
432C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3213C2
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
2. C31×#3112C2semidirect
3. C31×#2592C2semidirect
4. C31×#1980C2semidirect
5. C31×#1527C2semidirect
6. C31×#1491C2semidirect
7. C31×#1285C2semidirect
138. #10006C31direct (N × H)
187. #3112C2×C31semidirect
188. #2592C2×C31semidirect
189. #1980C2×C31semidirect
190. #1527C2×C31semidirect
191. #1491C2×C31semidirect
192. #1285C2×C31semidirect
299. C31#10006direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32132
C31×#31122
C31×#25922
C31×#19802
C31×#15272
C31×#14912
C31×#12852
#1000631