Finite Group Databasebeta

← C31×#10088 #21466847

Subgroups — C31×#10088 #21466847

Order 7936.

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

normal subgroups only

This group has 770 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, complemented
2C31×#3151C211yesmaximal, characteristic, complemented
3C31×#2657C211yesmaximal, characteristic, complemented
4C31×#1555C211yesmaximal, characteristic, complemented
5C31×#1527C211yesmaximal, characteristic, complemented
6C31×#1505C211yesmaximal, characteristic, complemented
7C31×#1300C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—22no—
9C31×#532C2211yescharacteristic
10C2×C31×#127—21no—
11C2×C31×#127C2211yescharacteristic
12C2×C31×#124—21no—
13C2×C31×#124C2211yescharacteristic
14C31×#496—21no—
15C31×#496—22no—
16C31×#493—21no—
17C31×#492—21no—
18C31×#489—21no—
19C31×#463—22no—
20C31×#448—21no—
21C31×#447—21no—
22C31×#432C2211yescharacteristic
23C31×#432C2211yescharacteristic
24C4×M16×C31C2211yescharacteristic
25C31×#392—22no—
26C31×#392—22no—
27C31×#376C2211yescharacteristic
28C31×#331—22no—
29C31×#329—22no—
30C31×#327—22no—
31C2×#41×C31—21no—
32C22×Q8×C31D411yescharacteristic
33C22×D4×C31—24no—
34C22×D4×C31—21no—
35C22×D4×C31—21no—
36C22×D4×C31D411yescharacteristic
37C22×D4×C31D412yes—
38C31×AGL(1,ℤ/8)—48no—
39C2×SD16×C31—24no—
40C2×SD16×C31—22no—
41C2×D8×C31—24no—
42C2×D8×C31—22no—
43C2×M16×C31—21no—
44C31×C4⋊Q8D411yescharacteristic, complemented
45C31×#127—21no—
46C31×#127—21no—
47C31×#127—21no—
48C31×#127—21no—
49C31×#127—44no—
50C31×#127D411yescharacteristic, complemented
51C31×#124—21no—
52C31×#124—21no—
53C31×#124—21no—
54C31×#124—21no—
55C31×#124—21no—
56C31×#124—44no—
57C31×#121—21no—
58C31×#121—21no—
59C4×D4×C31—21no—
60C4×D4×C31—21no—
61C2×C22⋊C4×C31—24no—
62C2×C22⋊C4×C31—21no—
63C2×C22⋊C4×C31—21no—
64C2×C42×C31—21no—
65C2×C42×C31C2311yescharacteristic
66C31×C4⋊C8—21no—
67C31×C4⋊C8D412yescomplemented
68C31×#103—22no—
69C31×#102—22no—
70C31×C22⋊C8—21no—
71C31×C22⋊C8—21no—
72C31×#97—21no—
73C31×#97—21no—
74C31×#97D412yescomplemented
75C4×C8×C31—21no—
76C31×#95—21no—
77C31×#95D412yes—
78C24×C31—22no—
79C24×C31—21no—
80#41×C31—44no—
81C2×Q8×C31—21no—
82C2×Q8×C31—21no—
83C2×Q8×C31—21no—
84C2×Q8×C31—42no—
85C2×D4×C31—21no—
86C2×D4×C31—21no—
87C2×D4×C31—21no—
88C2×D4×C31—22no—
89C2×D4×C31—21no—
90C2×D4×C31—21no—
91C2×D4×C31—22no—
92C2×D4×C31—44no—
93C2×D4×C31—42no—
94C2×D4×C31—42no—
95C2×D4×C31—42no—
96C2×D4×C31—48no—
97C2×D4×C31—41no—
98C2×D4×C31—41no—
99C2×D4×C31—41no—
100C2×D4×C31—41no—
101C2×D4×C31—41no—
102C2×D4×C31—82no—
103C2×D4×C31—84no—
104C2×D4×C31—84no—
105C22×C4×C31—22no—
106C22×C4×C31—21no—
107C22×C4×C31—41no—
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111SD16×C31—48no—
112SD16×C31—82no—
113D8×C31—48no—
114D8×C31—82no—
115M16×C31—24no—
116C2×C8×C31—22no—
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C22⋊C4×C31—48no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
124C22⋊C4×C31—41no—
125C22⋊C4×C31—41no—
126C22⋊C4×C31—41no—
127C22⋊C4×C31—41no—
128C42×C31—24no—
129C42×C31C2×D411yescharacteristic
130C42×C31C2×D411yescharacteristic
131C42×C31C2×D411yescharacteristic
132C42×C31#4111yescharacteristic
133#10088C3111yesmaximal, characteristic, direct product factor, Sylow
134C23×C31—42no—
135C23×C31—42no—
136C23×C31—42no—
137C23×C31—42no—
138C23×C31—44no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—42no—
142C23×C31—44no—
143C23×C31—42no—
144C23×C31#12011yescharacteristic
145Q8×C31—42no—
146Q8×C31—42no—
147Q8×C31—81no—
148D4×C31—42no—
149D4×C31—44no—
150D4×C31—42no—
151D4×C31—42no—
152D4×C31—81no—
153D4×C31—82no—
154D4×C31—81no—
155D4×C31—82no—
156D4×C31—82no—
157D4×C31—82no—
158D4×C31—816no—
159D4×C31—82no—
160D4×C31—82no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—24no—
164C2×C4×C31—41no—
165C2×C4×C31—42no—
166C2×C4×C31—42no—
167C2×C4×C31—42no—
168C2×C4×C31—41no—
169C2×C4×C31—41no—
170C2×C4×C31—82no—
171C2×C4×C31#12011yescharacteristic
172C2×C4×C31#12011yescharacteristic
173C2×C4×C31#12111yescharacteristic
174C2×C4×C31#12111yescharacteristic
175C2×C4×C31#12111yescharacteristic
176C2×C4×C31#12711yescharacteristic
177C8×C31—24no—
178C8×C31—42no—
179C8×C31—81no—
180#3213C2×C3111yescharacteristic, complemented
181#3151C2×C3111yescharacteristic, complemented
182#2657C2×C3111yescharacteristic, complemented
183#1555C2×C3111yescharacteristic, complemented
184#1527C2×C3111yescharacteristic, complemented
185#1505C2×C3111yescharacteristic, complemented
186#1300C2×C3111yescharacteristic, complemented
187C22×C31—21no—
188C22×C31—21no—
189C22×C31—84no—
190C22×C31—84no—
191C22×C31—84no—
192C22×C31—81no—
193C22×C31—82no—
194C22×C31—81no—
195C22×C31—81no—
196C22×C31—82no—
197C22×C31—81no—
198C22×C31—82no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31#39211yescharacteristic
202C22×C31#45312yes—
203C4×C31—21no—
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—21no—
208C4×C31—44no—
209C4×C31—81no—
210C4×C31—81no—
211C4×C31#45312yes—
212C2×AGL(1,ℤ/8)—22no—
213#532C22×C3111yescharacteristic
214C2×#127—21no—
215C2×#127C22×C3111yescharacteristic
216C2×#124—21no—
217C2×#124C22×C3111yescharacteristic
218#496—21no—
219#496—22no—
220#493—21no—
221#492—21no—
222#489—21no—
223#463—22no—
224#448—21no—
225#447—21no—
226#432C22×C3111yescharacteristic
227#432C22×C3111yescharacteristic
228C4×M16C22×C3111yescharacteristic
229#392—22no—
230#392—22no—
231#376C22×C3111yescharacteristic
232#331—22no—
233#329—22no—
234#327—22no—
235C2×C31—22no—
236C2×C31—161no—
237C2×C31—161no—
238C2×C31—162no—
239C2×C31#200211yescharacteristic
240C2×C31#200211yescharacteristic
241C2×C31#200811yescharacteristic
242C2×#41—21no—
243C22×Q8D4×C3111yescharacteristic
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4D4×C3111yescharacteristic
248C22×D4D4×C3112yes—
249AGL(1,ℤ/8)—48no—
250C2×SD16—24no—
251C2×SD16—22no—
252C2×D8—24no—
253C2×D8—22no—
254C2×M16—21no—
255C4⋊Q8D4×C3111yescharacteristic, complemented
256#127—21no—
257#127—21no—
258#127—21no—
259#127—21no—
260#127—44no—
261#127D4×C3111yescharacteristic, complemented
262#124—21no—
263#124—21no—
264#124—21no—
265#124—21no—
266#124—21no—
267#124—44no—
268#121—21no—
269#121—21no—
270C4×D4—21no—
271C4×D4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—21no—
276C2×C42C23×C3111yescharacteristic
277C4⋊C8—21no—
278C4⋊C8D4×C3112yescomplemented
279#103—22no—
280#102—22no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283#97—21no—
284#97—21no—
285#97D4×C3112yescomplemented
286C4×C8—21no—
287#95—21no—
288#95D4×C3112yes—
289C31#1008811yescharacteristic, direct product factor, Sylow
290C24—22no—
291C24—21no—
292#41—44no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—48no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—82no—
315C2×D4—84no—
316C2×D4—84no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4C2×D4×C3111yescharacteristic
321C22×C4C2×D4×C3111yescharacteristic
322C22×C4C2×D4×C3111yescharacteristic
323SD16—48no—
324SD16—82no—
325D8—48no—
326D8—82no—
327M16—24no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C22⋊C4—48no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C42—24no—
341C42C2×D4×C3111yescharacteristic
342C42C2×D4×C3111yescharacteristic
343C42C2×D4×C3111yescharacteristic
344C42#41×C3111yescharacteristic
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23—44no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23C31×#12011yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—81no—
359D4—42no—
360D4—44no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—82no—
365D4—81no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—816no—
370D4—82no—
371D4—82no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—82no—
382C2×C4C31×#12011yescharacteristic
383C2×C4C31×#12011yescharacteristic
384C2×C4C31×#12111yescharacteristic
385C2×C4C31×#12111yescharacteristic
386C2×C4C31×#12111yescharacteristic
387C2×C4C31×#12711yescharacteristic
388C8—24no—
389C8—42no—
390C8—81no—
391C22—21no—
392C22—21no—
393C22—84no—
394C22—84no—
395C22—84no—
396C22—81no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—82no—
401C22—81no—
402C22—82no—
403C22—81no—
404C22—81no—
405C22C31×#39211yescharacteristic
406C22C31×#45312yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—44no—
413C4—81no—
414C4—81no—
415C4C31×#45312yes—
416C2—22no—
417C2—161no—
418C2—162no—
419C2—161no—
420C2C31×#200211yescharacteristic
421C2C31×#200211yescharacteristic
422C2C31×#200811yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
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×#3213C2semidirect
2. C31×#3151C2semidirect
3. C31×#2657C2semidirect
4. C31×#1555C2semidirect
5. C31×#1527C2semidirect
6. C31×#1505C2semidirect
7. C31×#1300C2semidirect
44. C31×C4⋊Q8D4semidirect
50. C31×#127D4semidirect
67. C31×C4⋊C8D4semidirect
74. C31×#97D4semidirect
133. #10088C31direct (N × H)
180. #3213C2×C31semidirect
181. #3151C2×C31semidirect
182. #2657C2×C31semidirect
183. #1555C2×C31semidirect
184. #1527C2×C31semidirect
185. #1505C2×C31semidirect
186. #1300C2×C31semidirect
255. C4⋊Q8D4×C31semidirect
261. #127D4×C31semidirect
278. C4⋊C8D4×C31semidirect
285. #97D4×C31semidirect
289. C31#10088direct (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×#31512
C31×#26572
C31×#15552
C31×#15272
C31×#15052
C31×#13002
#1008831