Finite Group Databasebeta

← C31×#7752 #21464511

Subgroups — C31×#7752 #21464511

Order 7936.

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

normal subgroups only

This group has 574 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×#2682C211yesmaximal, characteristic
2C31×#2057C211yesmaximal, characteristic, complemented
3C31×#2023C211yesmaximal, characteristic, complemented
4C31×#2013C211yesmaximal, characteristic, complemented
5C31×#1271C211yesmaximal, characteristic, complemented
6C31×#1270C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic, complemented
8C22×SD16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#123—21no—
11C2×C31×#121—21no—
12C2×C31×#107—21no—
13C2×C31×#103—21no—
14C2×C31×#102—21no—
15C31×#396—21no—
16C31×#394—21no—
17C31×#392—21no—
18C31×#390C2211yescharacteristic
19C31×#390C2211yescharacteristic
20C31×#389C2211yescharacteristic
21C31×#385—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31—21no—
28C22×D4×C31—21no—
29C22×D4×C31—21no—
30C22×D4×C31D411yescharacteristic
31C22×D4×C31D411yescharacteristic
32C23×C4×C31—21no—
33C2×SD16×C31—42no—
34C2×SD16×C31—42no—
35C2×SD16×C31—42no—
36C22×C8×C31—21no—
37C22×C8×C31—21no—
38C22×C8×C31—21no—
39C31×C4⋊Q8—42no—
40C31×#123—42no—
41C31×#123—42no—
42C31×#121—42no—
43C31×#121—42no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31—21no—
46C2×C4⋊C4×C31—21no—
47C2×C4⋊C4×C31—21no—
48C2×C4⋊C4×C31D411yescharacteristic
49C2×C4⋊C4×C31D411yescharacteristic
50C2×C4⋊C4×C31D411yescharacteristic
51C2×C4⋊C4×C31D411yescharacteristic
52C2×C22⋊C4×C31—21no—
53C2×C22⋊C4×C31—21no—
54C2×C22⋊C4×C31—21no—
55C2×C22⋊C4×C31—21no—
56C2×C22⋊C4×C31—21no—
57C2×C22⋊C4×C31—21no—
58C2×C22⋊C4×C31—21no—
59C2×C42×C31C2311yescharacteristic
60C31×#107—42no—
61C31×#103—42no—
62C31×#102—42no—
63C31×#95—21no—
64C31×#95—21no—
65C24×C31—21no—
66C24×C31—21no—
67C2×Q8×C31—22no—
68C2×Q8×C31—41no—
69C2×Q8×C31—41no—
70C2×Q8×C31—41no—
71C2×Q8×C31—41no—
72C2×Q8×C31—41no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—42no—
81C2×D4×C31—42no—
82C2×D4×C31—42no—
83C2×D4×C31—81no—
84C2×D4×C31—81no—
85C2×D4×C31—81no—
86C2×D4×C31D812yes—
87C22×C4×C31—21no—
88C22×C4×C31—21no—
89C22×C4×C31—21no—
90C22×C4×C31—21no—
91C22×C4×C31—42no—
92C22×C4×C31—42no—
93C22×C4×C31—41no—
94C22×C4×C31—81no—
95C22×C4×C31C2×D411yescharacteristic
96C22×C4×C31C2×D411yescharacteristic
97C22×C4×C31C2×D411yescharacteristic
98SD16×C31—48no—
99C2×C8×C31—22no—
100C2×C8×C31—22no—
101C2×C8×C31—41no—
102C2×C8×C31—41no—
103C2×C8×C31—41no—
104C2×C8×C31—41no—
105C2×C8×C31—41no—
106C2×C8×C31—41no—
107C2×C8×C31—41no—
108C4⋊C4×C31—21no—
109C4⋊C4×C31—22no—
110C4⋊C4×C31—22no—
111C4⋊C4×C31—42no—
112C4⋊C4×C31—41no—
113C4⋊C4×C31—41no—
114C4⋊C4×C31—41no—
115C4⋊C4×C31—41no—
116C4⋊C4×C31—41no—
117C4⋊C4×C31—41no—
118C4⋊C4×C31—41no—
119C4⋊C4×C31SD1612yescomplemented
120C22⋊C4×C31—42no—
121C22⋊C4×C31—42no—
122C22⋊C4×C31—41no—
123C22⋊C4×C31—41no—
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—81no—
131C42×C31—41no—
132#7752C3111yesmaximal, characteristic, direct product factor, Sylow
133C23×C31—22no—
134C23×C31—41no—
135C23×C31—41no—
136C23×C31—41no—
137C23×C31—41no—
138C23×C31—41no—
139C23×C31—41no—
140C23×C31—41no—
141C23×C31—41no—
142C23×C31—41no—
143C23×C31—41no—
144C23×C31—41no—
145C23×C31—41no—
146C23×C31—41no—
147C23×C31#12011yescharacteristic
148Q8×C31—42no—
149Q8×C31—42no—
150Q8×C31—81no—
151Q8×C31—81no—
152D4×C31—42no—
153D4×C31—42no—
154D4×C31—81no—
155D4×C31—81no—
156D4×C31—84no—
157D4×C31—84no—
158C2×C4×C31—21no—
159C2×C4×C31—21no—
160C2×C4×C31—21no—
161C2×C4×C31—21no—
162C2×C4×C31—21no—
163C2×C4×C31—21no—
164C2×C4×C31—22no—
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×C31—41no—
175C2×C4×C31—42no—
176C2×C4×C31—41no—
177C2×C4×C31—84no—
178C2×C4×C31—81no—
179C2×C4×C31—81no—
180C2×C4×C31C2×D811yescharacteristic
181C2×C4×C31C2×SD1611yescharacteristic
182C2×C4×C31#13511yescharacteristic
183C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C31#13711yescharacteristic
186C8×C31—42no—
187C8×C31—42no—
188C8×C31—81no—
189#2682C2×C3111yescharacteristic
190#2057C2×C3111yescharacteristic, complemented
191#2023C2×C3111yescharacteristic, complemented
192#2013C2×C3111yescharacteristic, complemented
193#1271C2×C3111yescharacteristic, complemented
194#1270C2×C3111yescharacteristic, complemented
195#1270C2×C3111yescharacteristic, complemented
196C22×C31—42no—
197C22×C31—42no—
198C22×C31—42no—
199C22×C31—81no—
200C22×C31—81no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31—81no—
205C22×C31—81no—
206C22×C31—81no—
207C22×C31—81no—
208C22×C31—81no—
209C22×C31—81no—
210C22×C31#44711yescharacteristic
211C22×C31#45011yescharacteristic
212C22×C31#45211yescharacteristic
213C22×C31#45311yescharacteristic
214C22×C31#45511yescharacteristic
215C22×C31#45611yescharacteristic
216C22×C31B(4,𝔽2)11yescharacteristic
217C4×C31—21no—
218C4×C31—21no—
219C4×C31—21no—
220C4×C31—21no—
221C4×C31—21no—
222C4×C31—21no—
223C4×C31—42no—
224C4×C31—81no—
225C4×C31—82no—
226C4×C31—81no—
227C4×C31—81no—
228C22×SD16—21no—
229C2×C4⋊Q8C22×C3111yescharacteristic
230C2×#123—21no—
231C2×#121—21no—
232C2×#107—21no—
233C2×#103—21no—
234C2×#102—21no—
235#396—21no—
236#394—21no—
237#392—21no—
238#390C22×C3111yescharacteristic
239#390C22×C3111yescharacteristic
240#389C22×C3111yescharacteristic
241#385—21no—
242#340—21no—
243#340—21no—
244#336C22×C3111yescharacteristic
245#336C22×C3111yescharacteristic
246#336C22×C3111yescharacteristic
247C2×C31—82no—
248C2×C31—161no—
249C2×C31#159511yescharacteristic
250C2×C31#160411yescharacteristic
251C2×C31#160611yescharacteristic
252C2×C31#163111yescharacteristic
253C2×C31#163911yescharacteristic
254C2×C31#164711yescharacteristic
255C2×C31#166111yescharacteristic
256C22×Q8—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4D4×C3111yescharacteristic
260C22×D4D4×C3111yescharacteristic
261C23×C4—21no—
262C2×SD16—42no—
263C2×SD16—42no—
264C2×SD16—42no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C4⋊Q8—42no—
269#123—42no—
270#123—42no—
271#121—42no—
272#121—42no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4D4×C3111yescharacteristic
278C2×C4⋊C4D4×C3111yescharacteristic
279C2×C4⋊C4D4×C3111yescharacteristic
280C2×C4⋊C4D4×C3111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C42C23×C3111yescharacteristic
289#107—42no—
290#103—42no—
291#102—42no—
292#95—21no—
293#95—21no—
294C31#775211yescharacteristic, direct product factor, Sylow
295C24—21no—
296C24—21no—
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—81no—
314C2×D4—81no—
315C2×D4—81no—
316C2×D4D8×C3112yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—81no—
325C22×C4C2×D4×C3111yescharacteristic
326C22×C4C2×D4×C3111yescharacteristic
327C22×C4C2×D4×C3111yescharacteristic
328SD16—48no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4SD16×C3112yescomplemented
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—81no—
361C42—41no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23C31×#12011yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—42no—
382D4—42no—
383D4—81no—
384D4—81no—
385D4—84no—
386D4—84no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4C2×D8×C3111yescharacteristic
410C2×C4C2×SD16×C3111yescharacteristic
411C2×C4C31×#13511yescharacteristic
412C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
414C2×C4C31×#13711yescharacteristic
415C8—42no—
416C8—42no—
417C8—81no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22C31×#44711yescharacteristic
433C22C31×#45011yescharacteristic
434C22C31×#45211yescharacteristic
435C22C31×#45311yescharacteristic
436C22C31×#45511yescharacteristic
437C22C31×#45611yescharacteristic
438C22C31×B(4,𝔽2)11yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4—81no—
450C2—82no—
451C2—161no—
452C2C31×#159511yescharacteristic
453C2C31×#160411yescharacteristic
454C2C31×#160611yescharacteristic
455C2C31×#163111yescharacteristic
456C2C31×#163911yescharacteristic
457C2C31×#164711yescharacteristic
458C2C31×#166111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2682C2
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
2. C31×#2057C2semidirect
3. C31×#2023C2semidirect
4. C31×#2013C2semidirect
5. C31×#1271C2semidirect
6. C31×#1270C2semidirect
7. C31×#1270C2semidirect
119. C4⋊C4×C31SD16semidirect
132. #7752C31direct (N × H)
190. #2057C2×C31semidirect
191. #2023C2×C31semidirect
192. #2013C2×C31semidirect
193. #1271C2×C31semidirect
194. #1270C2×C31semidirect
195. #1270C2×C31semidirect
294. C31#7752direct (N × H)
349. C4⋊C4SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#26822
C31×#20572
C31×#20232
C31×#20132
C31×#12712
C31×#12702
#775231