Finite Group Databasebeta

← C31×#12160 #21468919

Subgroups — C31×#12160 #21468919

Order 7936.

396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3250C211yesmaximal, characteristic, complemented
2C31×#2187C211yesmaximal, characteristic, complemented
3C31×#2184C211yesmaximal, characteristic
4C31×#2054C211yesmaximal, characteristic, complemented
5C31×#1348C211yesmaximal, characteristic, complemented
6C31×#1347C211yesmaximal, characteristic, complemented
7C31×#1322C211yesmaximal, characteristic, complemented
8C22×D8×C31C2211yescharacteristic
9C31×#529—21no—
10C2×C31×SD32—21no—
11C2×C31×D16—21no—
12C31×#479—21no—
13C31×#478—21no—
14C31×#467C2211yescharacteristic
15C31×#466C2211yescharacteristic
16C31×#465—21no—
17C31×#464—21no—
18C31×#463—21no—
19C2×C31×C4⋊C8C2211yescharacteristic
20C2×C31×#102—21no—
21C31×#390—21no—
22C31×#357—21no—
23C31×#357—21no—
24C31×C22⋊C16C2211yescharacteristic
25C31×C22⋊C16C2211yescharacteristic
26C31×#340C2211yescharacteristic
27C22×D4×C31—21no—
28C2×Q16×C31D411yescharacteristic
29C2×SD16×C31—21no—
30C2×D8×C31—21no—
31C2×D8×C31—21no—
32C2×D8×C31—42no—
33C2×D8×C31D411yescharacteristic
34C22×C8×C31D411yescharacteristic
35C22×C8×C31C2311yescharacteristic
36C31×#126—41no—
37C31×#125—21no—
38C31×#124—21no—
39C31×#123—41no—
40C31×C22⋊Q8—21no—
41C31×#121—21no—
42C4×Q8×C31—21no—
43C4×D4×C31—21no—
44C4×D4×C31—41no—
45C31×#117—41no—
46C2×C4⋊C4×C31—21no—
47C2×C22⋊C4×C31—41no—
48C2×C22⋊C4×C31—41no—
49C2×C42×C31D411yescharacteristic
50C31×SD32—42no—
51C31×D16—42no—
52C2×C16×C31—21no—
53C2×C16×C31—21no—
54C31×#107D411yescharacteristic
55C31×#107D411yescharacteristic
56C31×#106—21no—
57C31×C4⋊C8—21no—
58C31×C4⋊C8—21no—
59C31×#103—21no—
60C31×#103—21no—
61C31×#102—21no—
62C31×#102—21no—
63C31×#102—42no—
64C24×C31—41no—
65C2×Q8×C31—21no—
66C2×D4×C31—22no—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31—41no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C22×C4×C31—41no—
75C22×C4×C31—41no—
76C22×C4×C31D811yescharacteristic
77C22×C4×C31D811yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79Q16×C31—41no—
80SD16×C31—81no—
81D8×C31—22no—
82D8×C31—22no—
83D8×C31—41no—
84D8×C31—41no—
85D8×C31—41no—
86C2×C8×C31—21no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31C2×D411yescharacteristic
90C2×C8×C31C2×D411yescharacteristic
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—41no—
103C22⋊C4×C31—41no—
104C22⋊C4×C31—82no—
105C22⋊C4×C31—82no—
106C42×C31—21no—
107C42×C31—21no—
108C42×C31—41no—
109C16×C31—41no—
110C16×C31—41no—
111#12160C3111yesmaximal, characteristic, direct product factor, Sylow
112C23×C31—42no—
113C23×C31—41no—
114C23×C31—81no—
115C23×C31—81no—
116C23×C31—81no—
117C23×C31—81no—
118C23×C31—81no—
119C23×C31—81no—
120C23×C31C2×D811yescharacteristic
121Q8×C31—41no—
122Q8×C31—41no—
123D4×C31—42no—
124D4×C31—41no—
125D4×C31—42no—
126D4×C31—41no—
127D4×C31—81no—
128D4×C31—81no—
129D4×C31—81no—
130D4×C31—81no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—82no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31—81no—
145C2×C4×C31#12011yescharacteristic
146C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
147C8×C31—21no—
148C8×C31—21no—
149C8×C31—41no—
150#3250C2×C3111yescharacteristic, complemented
151#2187C2×C3111yescharacteristic, complemented
152#2184C2×C3111yescharacteristic
153#2054C2×C3111yescharacteristic, complemented
154#1348C2×C3111yescharacteristic, complemented
155#1347C2×C3111yescharacteristic, complemented
156#1322C2×C3111yescharacteristic, complemented
157C22×C31—21no—
158C22×C31—21no—
159C22×C31—82no—
160C22×C31—82no—
161C22×C31—82no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—161no—
166C22×C31—161no—
167C22×C31—161no—
168C22×C31—161no—
169C22×C31#44711yescharacteristic
170C22×C31#50811yescharacteristic
171C22×C31#50911yescharacteristic
172C4×C31—21no—
173C4×C31—41no—
174C4×C31—41no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C4×C31—161no—
179C4×C31#45411yescharacteristic
180C4×C31B(4,𝔽2)11yescharacteristic
181C22×D8C22×C3111yescharacteristic
182#529—21no—
183C2×SD32—21no—
184C2×D16—21no—
185#479—21no—
186#478—21no—
187#467C22×C3111yescharacteristic
188#466C22×C3111yescharacteristic
189#465—21no—
190#464—21no—
191#463—21no—
192C2×C4⋊C8C22×C3111yescharacteristic
193C2×#102—21no—
194#390—21no—
195#357—21no—
196#357—21no—
197C22⋊C16C22×C3111yescharacteristic
198C22⋊C16C22×C3111yescharacteristic
199#340C22×C3111yescharacteristic
200C2×C31—21no—
201C2×C31—21no—
202C2×C31—161no—
203C2×C31—162no—
204C2×C31#159811yescharacteristic
205C2×C31#218611yescharacteristic
206C2×C31#219011yescharacteristic
207C22×D4—21no—
208C2×Q16D4×C3111yescharacteristic
209C2×SD16—21no—
210C2×D8—21no—
211C2×D8—21no—
212C2×D8—42no—
213C2×D8D4×C3111yescharacteristic
214C22×C8D4×C3111yescharacteristic
215C22×C8C23×C3111yescharacteristic
216#126—41no—
217#125—21no—
218#124—21no—
219#123—41no—
220C22⋊Q8—21no—
221#121—21no—
222C4×Q8—21no—
223C4×D4—21no—
224C4×D4—41no—
225#117—41no—
226C2×C4⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42D4×C3111yescharacteristic
230SD32—42no—
231D16—42no—
232C2×C16—21no—
233C2×C16—21no—
234#107D4×C3111yescharacteristic
235#107D4×C3111yescharacteristic
236#106—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239#103—21no—
240#103—21no—
241#102—21no—
242#102—21no—
243#102—42no—
244C31#1216011yescharacteristic, direct product factor, Sylow
245C24—41no—
246C2×Q8—21no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4D8×C3111yescharacteristic
258C22×C4D8×C3111yescharacteristic
259C22×C4C2×D4×C3111yescharacteristic
260Q16—41no—
261SD16—81no—
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266D8—41no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8C2×D4×C3111yescharacteristic
271C2×C8C2×D4×C3111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—21no—
288C42—21no—
289C42—41no—
290C16—41no—
291C16—41no—
292C23—42no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23C2×D8×C3111yescharacteristic
301Q8—41no—
302Q8—41no—
303D4—42no—
304D4—41no—
305D4—42no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—81no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4C31×#12011yescharacteristic
326C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
327C8—21no—
328C8—21no—
329C8—41no—
330C22—21no—
331C22—21no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—161no—
339C22—161no—
340C22—161no—
341C22—161no—
342C22C31×#44711yescharacteristic
343C22C31×#50811yescharacteristic
344C22C31×#50911yescharacteristic
345C4—21no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—161no—
352C4C31×#45411yescharacteristic
353C4C31×B(4,𝔽2)11yescharacteristic
354C2—21no—
355C2—21no—
356C2—162no—
357C2—161no—
358C2C31×#159811yescharacteristic
359C2C31×#218611yescharacteristic
360C2C31×#219011yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC31×#2184C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×C4⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3250C2semidirect
2. C31×#2187C2semidirect
4. C31×#2054C2semidirect
5. C31×#1348C2semidirect
6. C31×#1347C2semidirect
7. C31×#1322C2semidirect
111. #12160C31direct (N × H)
150. #3250C2×C31semidirect
151. #2187C2×C31semidirect
153. #2054C2×C31semidirect
154. #1348C2×C31semidirect
155. #1347C2×C31semidirect
156. #1322C2×C31semidirect
244. C31#12160direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32502
C31×#21872
C31×#21842
C31×#20542
C31×#13482
C31×#13472
C31×#13222
#1216031