Finite Group Databasebeta

← C31×#7748 #21464507

Subgroups — C31×#7748 #21464507

Order 7936.

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

normal subgroups only

This group has 536 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×#2806C211yesmaximal, characteristic, complemented
2C31×#2041C211yesmaximal, characteristic
3C31×#2014C211yesmaximal, characteristic
4C31×#2014C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1272C211yesmaximal, characteristic, complemented
7C31×#1271C211yesmaximal, characteristic, complemented
8C22×SD16×C31—21no—
9C22×SD16×C31—21no—
10C2×C31×#127C2211yescharacteristic
11C2×C31×#102—21no—
12C2×C31×#102—21no—
13C2×C31×#102—21no—
14C2×C31×#102—21no—
15C31×#398—21no—
16C31×#398—21no—
17C31×#395—21no—
18C31×#391—21no—
19C31×#391C2211yescharacteristic
20C31×#391C2211yescharacteristic
21C31×#389—21no—
22C31×#389C2211yescharacteristic
23C31×#387—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31—21no—
30C22×D4×C31—21no—
31C22×D4×C31—21no—
32C2×SD16×C31—42no—
33C2×SD16×C31—42no—
34C2×SD16×C31—42no—
35C2×SD16×C31—42no—
36C2×SD16×C31—42no—
37C2×SD16×C31—42no—
38C22×C8×C31—21no—
39C22×C8×C31—21no—
40C22×C8×C31—21no—
41C31×#127—42no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31D411yescharacteristic
48C2×C4⋊C4×C31D411yescharacteristic
49C2×C4⋊C4×C31D411yescharacteristic
50C2×C4⋊C4×C31D411yescharacteristic
51C2×C42×C31—21no—
52C2×C42×C31C2311yescharacteristic
53C31×#102—42no—
54C31×#102—42no—
55C31×#102—42no—
56C31×#102—42no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95—21no—
64C31×#95—21no—
65C24×C31—41no—
66C2×Q8×C31—21no—
67C2×Q8×C31—21no—
68C2×Q8×C31—21no—
69C2×Q8×C31—41no—
70C2×Q8×C31—41no—
71C2×Q8×C31—41no—
72C2×Q8×C31—41no—
73C2×Q8×C31SD1612yescomplemented
74C2×D4×C31—22no—
75C2×D4×C31—22no—
76C2×D4×C31—22no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—41no—
83C2×D4×C31—41no—
84C2×D4×C31—41no—
85C2×D4×C31—41no—
86C2×D4×C31—41no—
87C2×D4×C31—41no—
88C2×D4×C31—41no—
89C2×D4×C31—41no—
90C2×D4×C31—41no—
91C2×D4×C31—41no—
92C22×C4×C31—21no—
93C22×C4×C31—21no—
94C22×C4×C31—21no—
95C22×C4×C31—21no—
96C22×C4×C31—21no—
97C22×C4×C31—21no—
98C22×C4×C31C2×D411yescharacteristic
99C22×C4×C31C2×D411yescharacteristic
100C22×C4×C31C2×D411yescharacteristic
101SD16×C31—48no—
102SD16×C31—48no—
103C2×C8×C31—22no—
104C2×C8×C31—22no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C2×C8×C31—41no—
109C2×C8×C31—41no—
110C2×C8×C31—41no—
111C2×C8×C31—41no—
112C4⋊C4×C31—21no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—81no—
119C4⋊C4×C31—81no—
120C4⋊C4×C31—81no—
121C4⋊C4×C31—81no—
122C4⋊C4×C31SD1612yescomplemented
123C42×C31—41no—
124C42×C31—81no—
125#7748C3111yesmaximal, characteristic, direct product factor, Sylow
126C23×C31—42no—
127C23×C31—81no—
128C23×C31—81no—
129C23×C31—81no—
130C23×C31—81no—
131C23×C31—81no—
132C23×C31—81no—
133C23×C31#12011yescharacteristic
134Q8×C31—42no—
135Q8×C31—42no—
136Q8×C31—81no—
137Q8×C31—81no—
138D4×C31—42no—
139D4×C31—42no—
140D4×C31—42no—
141D4×C31—42no—
142D4×C31—42no—
143D4×C31—42no—
144D4×C31—81no—
145D4×C31—81no—
146D4×C31—81no—
147D4×C31—81no—
148D4×C31—81no—
149D4×C31—81no—
150C2×C4×C31—21no—
151C2×C4×C31—21no—
152C2×C4×C31—21no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—21no—
156C2×C4×C31—22no—
157C2×C4×C31—22no—
158C2×C4×C31—41no—
159C2×C4×C31—41no—
160C2×C4×C31—41no—
161C2×C4×C31—41no—
162C2×C4×C31—41no—
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—41no—
170C2×C4×C31—41no—
171C2×C4×C31—41no—
172C2×C4×C31—41no—
173C2×C4×C31—41no—
174C2×C4×C31C2×SD1611yescharacteristic
175C2×C4×C31C2×SD1611yescharacteristic
176C2×C4×C31#13511yescharacteristic
177C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C31#13711yescharacteristic
179C2×C4×C31#13711yescharacteristic
180C8×C31—42no—
181C8×C31—42no—
182C8×C31—81no—
183#2806C2×C3111yescharacteristic, complemented
184#2041C2×C3111yescharacteristic
185#2014C2×C3111yescharacteristic
186#2014C2×C3111yescharacteristic
187#1272C2×C3111yescharacteristic, complemented
188#1272C2×C3111yescharacteristic, complemented
189#1271C2×C3111yescharacteristic, complemented
190C22×C31—82no—
191C22×C31—82no—
192C22×C31—82no—
193C22×C31—161no—
194C22×C31—161no—
195C22×C31—161no—
196C22×C31—161no—
197C22×C31#44811yescharacteristic
198C22×C31#44811yescharacteristic
199C22×C31#44911yescharacteristic
200C22×C31#45311yescharacteristic
201C22×C31#45311yescharacteristic
202C22×C31#45411yescharacteristic
203C22×C31B(4,𝔽2)11yescharacteristic
204C4×C31—21no—
205C4×C31—21no—
206C4×C31—21no—
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—42no—
211C4×C31—42no—
212C4×C31—81no—
213C4×C31—81no—
214C4×C31—81no—
215C4×C31—81no—
216C22×SD16—21no—
217C22×SD16—21no—
218C2×#127C22×C3111yescharacteristic
219C2×#102—21no—
220C2×#102—21no—
221C2×#102—21no—
222C2×#102—21no—
223#398—21no—
224#398—21no—
225#395—21no—
226#391—21no—
227#391C22×C3111yescharacteristic
228#391C22×C3111yescharacteristic
229#389—21no—
230#389C22×C3111yescharacteristic
231#387—21no—
232#336C22×C3111yescharacteristic
233#336C22×C3111yescharacteristic
234#336C22×C3111yescharacteristic
235C2×C31—162no—
236C2×C31#159611yescharacteristic
237C2×C31#159911yescharacteristic
238C2×C31#160011yescharacteristic
239C2×C31#162211yescharacteristic
240C2×C31#162311yescharacteristic
241C2×C31#162611yescharacteristic
242C2×C31#165611yescharacteristic
243C22×Q8D4×C3111yescharacteristic
244C22×Q8D4×C3111yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C2×SD16—42no—
249C2×SD16—42no—
250C2×SD16—42no—
251C2×SD16—42no—
252C2×SD16—42no—
253C2×SD16—42no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257#127—42no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4D4×C3111yescharacteristic
264C2×C4⋊C4D4×C3111yescharacteristic
265C2×C4⋊C4D4×C3111yescharacteristic
266C2×C4⋊C4D4×C3111yescharacteristic
267C2×C42—21no—
268C2×C42C23×C3111yescharacteristic
269#102—42no—
270#102—42no—
271#102—42no—
272#102—42no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281C31#774811yescharacteristic, direct product factor, Sylow
282C24—41no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8SD16×C3112yescomplemented
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4C2×D4×C3111yescharacteristic
316C22×C4C2×D4×C3111yescharacteristic
317C22×C4C2×D4×C3111yescharacteristic
318SD16—48no—
319SD16—48no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4SD16×C3112yescomplemented
340C42—41no—
341C42—81no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23C31×#12011yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—42no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4C2×SD16×C3111yescharacteristic
391C2×C4C2×SD16×C3111yescharacteristic
392C2×C4C31×#13511yescharacteristic
393C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C31×#13711yescharacteristic
395C2×C4C31×#13711yescharacteristic
396C8—42no—
397C8—42no—
398C8—81no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22C31×#44811yescharacteristic
407C22C31×#44811yescharacteristic
408C22C31×#44911yescharacteristic
409C22C31×#45311yescharacteristic
410C22C31×#45311yescharacteristic
411C22C31×#45411yescharacteristic
412C22C31×B(4,𝔽2)11yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—42no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C2—162no—
426C2C31×#159611yescharacteristic
427C2C31×#159911yescharacteristic
428C2C31×#160011yescharacteristic
429C2C31×#162211yescharacteristic
430C2C31×#162311yescharacteristic
431C2C31×#162611yescharacteristic
432C2C31×#165611yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C31×#2806C2semidirect
5. C31×#1272C2semidirect
6. C31×#1272C2semidirect
7. C31×#1271C2semidirect
73. C2×Q8×C31SD16semidirect
122. C4⋊C4×C31SD16semidirect
125. #7748C31direct (N × H)
183. #2806C2×C31semidirect
187. #1272C2×C31semidirect
188. #1272C2×C31semidirect
189. #1271C2×C31semidirect
281. C31#7748direct (N × H)
290. C2×Q8SD16×C31semidirect
339. C4⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#28062
C31×#20412
C31×#20142
C31×#12722
C31×#12712
#774831