Finite Group Databasebeta

← C31×#12238 #21468997

Subgroups — C31×#12238 #21468997

Order 7936.

368 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2795C211yesmaximal, characteristic
2C31×#2060C211yesmaximal, characteristic, complemented
3C31×#1617C211yesmaximal, characteristic, complemented
4C31×#1599C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1345C211yesmaximal, characteristic
7C31×#1325C211yesmaximal, characteristic, complemented
8C31×#544C2211yescharacteristic
9C31×#484—21no—
10C31×#450—21no—
11C31×#428—21no—
12C31×#400C2211yescharacteristic
13C31×#396—21no—
14C31×#394—21no—
15C31×#394C2211yescharacteristic
16C31×#388—21no—
17C31×#388—21no—
18C31×#388—21no—
19C31×#387—21no—
20C31×#384C2211yescharacteristic
21C31×#343C2211yescharacteristic
22C31×#339—21no—
23C31×#328—21no—
24C31×#328—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C22×D4×C31D411yescharacteristic
28C2×SD16×C31—41no—
29C2×M16×C31—21no—
30C31×C4⋊Q8—41no—
31C31×#125—41no—
32C31×C22⋊Q8—21no—
33C31×C22⋊Q8—21no—
34C31×#117—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31C2311yescharacteristic
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C42×C31—21no—
48C2×C42×C31D411yescharacteristic
49C31×#107—41no—
50C31×#107—41no—
51C31×#106—41no—
52C31×#103—41no—
53C31×#102—41no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95D411yescharacteristic
61C31×#95D411yescharacteristic
62C24×C31—21no—
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C2×D4×C31—41no—
66C2×D4×C31—81no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31C2×D411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75SD16×C31—82no—
76M16×C31—22no—
77M16×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C4⋊C4×C31—21no—
82C4⋊C4×C31—21no—
83C4⋊C4×C31—21no—
84C4⋊C4×C31—21no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—42no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—81no—
92C4⋊C4×C31—81no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—41no—
96C22⋊C4×C31—41no—
97C22⋊C4×C31—41no—
98C22⋊C4×C31—81no—
99C22⋊C4×C31—81no—
100C22⋊C4×C31—81no—
101C42×C31—21no—
102C42×C31—21no—
103C42×C31—41no—
104C42×C31—81no—
105#12238C3111yesmaximal, characteristic, direct product factor, Sylow
106C23×C31—41no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31#12011yescharacteristic
114Q8×C31—81no—
115Q8×C31—81no—
116D4×C31—42no—
117D4×C31—81no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—22no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31#13511yescharacteristic
142C2×C4×C31#13711yescharacteristic
143C8×C31—42no—
144C8×C31—81no—
145C8×C31—81no—
146#2795C2×C3111yescharacteristic
147#2060C2×C3111yescharacteristic, complemented
148#1617C2×C3111yescharacteristic, complemented
149#1599C2×C3111yescharacteristic
150#1353C2×C3111yescharacteristic, complemented
151#1345C2×C3111yescharacteristic
152#1325C2×C3111yescharacteristic, complemented
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—81no—
156C22×C31—81no—
157C22×C31—81no—
158C22×C31—81no—
159C22×C31—81no—
160C22×C31—81no—
161C22×C31—81no—
162C22×C31#45211yescharacteristic
163C22×C31#45611yescharacteristic
164C22×C31B(4,𝔽2)11yescharacteristic
165C4×C31—21no—
166C4×C31—21no—
167C4×C31—42no—
168C4×C31—41no—
169C4×C31—41no—
170C4×C31—81no—
171C4×C31—81no—
172C4×C31—81no—
173C4×C31—81no—
174C4×C31—81no—
175C4×C31—81no—
176#544C22×C3111yescharacteristic
177#484—21no—
178#450—21no—
179#428—21no—
180#400C22×C3111yescharacteristic
181#396—21no—
182#394—21no—
183#394C22×C3111yescharacteristic
184#388—21no—
185#388—21no—
186#388—21no—
187#387—21no—
188#384C22×C3111yescharacteristic
189#343C22×C3111yescharacteristic
190#339—21no—
191#328—21no—
192#328—21no—
193#324C22×C3111yescharacteristic
194#324C22×C3111yescharacteristic
195C2×C31—21no—
196C2×C31—21no—
197C2×C31—161no—
198C2×C31#160311yescharacteristic
199C2×C31#220011yescharacteristic
200C2×C31#220411yescharacteristic
201C22×D4D4×C3111yescharacteristic
202C2×SD16—41no—
203C2×M16—21no—
204C4⋊Q8—41no—
205#125—41no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4D4×C3111yescharacteristic
213C2×C4⋊C4D4×C3111yescharacteristic
214C2×C4⋊C4C23×C3111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42—21no—
222C2×C42D4×C3111yescharacteristic
223#107—41no—
224#107—41no—
225#106—41no—
226#103—41no—
227#102—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95D4×C3111yescharacteristic
235#95D4×C3111yescharacteristic
236C31#1223811yescharacteristic, direct product factor, Sylow
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×D4—41no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C2×D4×C3111yescharacteristic
248C22×C4C2×D4×C3111yescharacteristic
249C22×C4C2×D4×C3111yescharacteristic
250SD16—82no—
251M16—22no—
252M16—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C42—41no—
279C42—81no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C31×#12011yescharacteristic
288Q8—81no—
289Q8—81no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4C31×#13511yescharacteristic
316C2×C4C31×#13711yescharacteristic
317C8—42no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C31×#45211yescharacteristic
330C22C31×#45611yescharacteristic
331C22C31×B(4,𝔽2)11yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2C31×#160311yescharacteristic
347C2C31×#220011yescharacteristic
348C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2060C2semidirect
3. C31×#1617C2semidirect
5. C31×#1353C2semidirect
7. C31×#1325C2semidirect
105. #12238C31direct (N × H)
147. #2060C2×C31semidirect
148. #1617C2×C31semidirect
150. #1353C2×C31semidirect
152. #1325C2×C31semidirect
236. C31#12238direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27952
C31×#20602
C31×#16172
C31×#15992
C31×#13532
C31×#13452
C31×#13252
#1223831