Finite Group Databasebeta

← C31×#10005 #21466764

Subgroups — C31×#10005 #21466764

Order 7936.

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

normal subgroups only

This group has 566 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×#3234C211yesmaximal, characteristic, complemented
2C31×#3109C211yesmaximal, characteristic
3C31×#2562C211yesmaximal, characteristic, complemented
4C31×#1980C211yesmaximal, characteristic, complemented
5C31×#1528C211yesmaximal, characteristic, complemented
6C31×#1490C211yesmaximal, characteristic, complemented
7C31×#1285C211yesmaximal, characteristic, complemented
8C31×#532C2211yescharacteristic
9C2×C31×#125C2211yescharacteristic
10C2×C4×D4×C31C2211yescharacteristic
11C31×#501—21no—
12C31×#490—21no—
13C31×#481—21no—
14C31×#481—21no—
15C31×#480—21no—
16C31×#478—22no—
17C31×#476—21no—
18C31×#459—21no—
19C31×#447—21no—
20C31×#442—22no—
21C31×#432C2211yescharacteristic
22C31×#432C2211yescharacteristic
23C2×C31×#104—22no—
24C4×M16×C31C2211yescharacteristic
25C31×#400—21no—
26C31×#394—21no—
27C31×#388—22no—
28C31×#380—21no—
29C31×#380—21no—
30C4×C4⋊C4×C31C2211yescharacteristic
31C31×#330—22no—
32C31×D4⋊C8—22no—
33C2×#41×C31—21no—
34C22×D4×C31D411yescharacteristic
35C23×C4×C31—21no—
36C2×D8×C31—22no—
37C2×M16×C31—21no—
38C31×C4⋊Q8D411yescharacteristic
39C31×#127D411yescharacteristic
40C31×#125—21no—
41C31×#125—21no—
42C31×#125—21no—
43C31×#125—21no—
44C31×#124—21no—
45C31×#121—21no—
46C31×#121—21no—
47C4×D4×C31—21no—
48C4×D4×C31—21no—
49C4×D4×C31—21no—
50C4×D4×C31—21no—
51C4×D4×C31—21no—
52C4×D4×C31—21no—
53C2×C4⋊C4×C31—21no—
54C2×C4⋊C4×C31—21no—
55C2×C4⋊C4×C31—21no—
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×C31D412yes—
63C2×C42×C31C2311yescharacteristic
64C31×#107—22no—
65C31×#106—22no—
66C31×#106—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8D412yes—
69C31×#104—44no—
70C31×#102—22no—
71C31×#102—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×D4×C31—21no—
88C2×D4×C31—21no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—21no—
92C2×D4×C31—42no—
93C2×D4×C31—41no—
94C22×C4×C31—21no—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31—42no—
99C22×C4×C31—42no—
100C22×C4×C31—41no—
101C22×C4×C31—41no—
102C22×C4×C31—41no—
103C22×C4×C31C2×D411yescharacteristic
104C22×C4×C31C2×D411yescharacteristic
105C22×C4×C31C2×D411yescharacteristic
106D8×C31—44no—
107M16×C31—22no—
108M16×C31—41no—
109C2×C8×C31—22no—
110C2×C8×C31—22no—
111C2×C8×C31—41no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—21no—
114C4⋊C4×C31—21no—
115C4⋊C4×C31—21no—
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—
123C4⋊C4×C31—41no—
124C4⋊C4×C31—41no—
125C22⋊C4×C31—44no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C22⋊C4×C31—41no—
129C22⋊C4×C31—41no—
130C22⋊C4×C31—82no—
131C42×C31—42no—
132C42×C31C2×D411yescharacteristic
133C42×C31C2×D411yescharacteristic
134C42×C31C2×D411yescharacteristic
135C42×C31#4111yescharacteristic
136#10005C3111yesmaximal, characteristic, direct product factor, Sylow
137C23×C31—22no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31#12011yescharacteristic
146Q8×C31—42no—
147D4×C31—22no—
148D4×C31—22no—
149D4×C31—41no—
150D4×C31—41no—
151D4×C31—42no—
152D4×C31—42no—
153D4×C31—42no—
154D4×C31—81no—
155D4×C31—81no—
156C2×C4×C31—21no—
157C2×C4×C31—21no—
158C2×C4×C31—22no—
159C2×C4×C31—41no—
160C2×C4×C31—42no—
161C2×C4×C31—42no—
162C2×C4×C31—42no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—42no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—82no—
173C2×C4×C31—82no—
174C2×C4×C31—82no—
175C2×C4×C31#12011yescharacteristic
176C2×C4×C31#12011yescharacteristic
177C2×C4×C31#12111yescharacteristic
178C2×C4×C31#12111yescharacteristic
179C2×C4×C31#12111yescharacteristic
180C2×C4×C31#12711yescharacteristic
181C8×C31—42no—
182C8×C31—42no—
183C8×C31—42no—
184#3234C2×C3111yescharacteristic, complemented
185#3109C2×C3111yescharacteristic
186#2562C2×C3111yescharacteristic, complemented
187#1980C2×C3111yescharacteristic, complemented
188#1528C2×C3111yescharacteristic, complemented
189#1490C2×C3111yescharacteristic, complemented
190#1285C2×C3111yescharacteristic, complemented
191C22×C31—21no—
192C22×C31—21no—
193C22×C31—42no—
194C22×C31—42no—
195C22×C31—42no—
196C22×C31—81no—
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—82no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31#45412yes—
217#532C22×C3111yescharacteristic
218C2×#125C22×C3111yescharacteristic
219C2×C4×D4C22×C3111yescharacteristic
220#501—21no—
221#490—21no—
222#481—21no—
223#481—21no—
224#480—21no—
225#478—22no—
226#476—21no—
227#459—21no—
228#447—21no—
229#442—22no—
230#432C22×C3111yescharacteristic
231#432C22×C3111yescharacteristic
232C2×#104—22no—
233C4×M16C22×C3111yescharacteristic
234#400—21no—
235#394—21no—
236#388—22no—
237#380—21no—
238#380—21no—
239C4×C4⋊C4C22×C3111yescharacteristic
240#330—22no—
241D4⋊C8—22no—
242C2×C31—21no—
243C2×C31—21no—
244C2×C31—82no—
245C2×C31—161no—
246C2×C31#200311yescharacteristic
247C2×C31#200611yescharacteristic
248C2×C31#201011yescharacteristic
249C2×#41—21no—
250C22×D4D4×C3111yescharacteristic
251C23×C4—21no—
252C2×D8—22no—
253C2×M16—21no—
254C4⋊Q8D4×C3111yescharacteristic
255#127D4×C3111yescharacteristic
256#125—21no—
257#125—21no—
258#125—21no—
259#125—21no—
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C3111yescharacteristic
273C2×C4⋊C4D4×C3111yescharacteristic
274C2×C4⋊C4D4×C3111yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C42D4×C3112yes—
279C2×C42C23×C3111yescharacteristic
280#107—22no—
281#106—22no—
282#106—22no—
283C4⋊C8—21no—
284C4⋊C8D4×C3112yes—
285#104—44no—
286#102—22no—
287#102—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#1000511yescharacteristic, direct product factor, Sylow
300C24—21no—
301#41—42no—
302#41—42no—
303C2×Q8—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—41no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3111yescharacteristic
322C22×C4C2×D4×C3111yescharacteristic
323D8—44no—
324M16—22no—
325M16—41no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C22⋊C4—44no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—82no—
348C42—42no—
349C42C2×D4×C3111yescharacteristic
350C42C2×D4×C3111yescharacteristic
351C42C2×D4×C3111yescharacteristic
352C42#41×C3111yescharacteristic
353C23—22no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C31×#12011yescharacteristic
362Q8—42no—
363D4—22no—
364D4—22no—
365D4—41no—
366D4—41no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4C31×#12011yescharacteristic
392C2×C4C31×#12011yescharacteristic
393C2×C4C31×#12111yescharacteristic
394C2×C4C31×#12111yescharacteristic
395C2×C4C31×#12111yescharacteristic
396C2×C4C31×#12711yescharacteristic
397C8—42no—
398C8—42no—
399C8—42no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—81no—
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—82no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4C31×#45412yes—
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2C31×#200311yescharacteristic
431C2C31×#200611yescharacteristic
432C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3109C2
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×#3234C2semidirect
3. C31×#2562C2semidirect
4. C31×#1980C2semidirect
5. C31×#1528C2semidirect
6. C31×#1490C2semidirect
7. C31×#1285C2semidirect
136. #10005C31direct (N × H)
184. #3234C2×C31semidirect
186. #2562C2×C31semidirect
187. #1980C2×C31semidirect
188. #1528C2×C31semidirect
189. #1490C2×C31semidirect
190. #1285C2×C31semidirect
299. C31#10005direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32342
C31×#31092
C31×#25622
C31×#19802
C31×#15282
C31×#14902
C31×#12852
#1000531