Finite Group Databasebeta

← C31×#12124 #21468883

Subgroups — C31×#12124 #21468883

Order 7936.

542 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2842C211yesmaximal, characteristic, complemented
2C31×#2021C211yesmaximal, characteristic
3C31×#1613C212yesmaximal, complemented
4C31×#1352C212yesmaximal, complemented
5C31×#1316C211yesmaximal, characteristic, complemented
6C31×#534C2211yescharacteristic
7C2×C31×#120—21no—
8C31×#481—22no—
9C2×C31×#99—21no—
10C31×#399—21no—
11C31×#399C2212yes—
12C31×#396—22no—
13C31×#396—22no—
14C31×#379—21no—
15C31×#379C2211yescharacteristic
16C31×#342C2211yescharacteristic
17C31×#327—22no—
18C31×#323C2212yes—
19C25×C31D411yescharacteristic
20C22×D4×C31—21no—
21C22×D4×C31—41no—
22C31×#123—41no—
23C31×#121—22no—
24C31×#120—42no—
25C31×#120—41no—
26C31×#120—42no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31D412yescomplemented
31C2×C22⋊C4×C31—21no—
32C2×C22⋊C4×C31—21no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—21no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—41no—
38C2×C22⋊C4×C31D411yescharacteristic
39C2×C22⋊C4×C31C2311yescharacteristic
40C31×#106—42no—
41C31×#102—42no—
42C31×#99—42no—
43C31×C22⋊C8—22no—
44C31×#95—21no—
45C31×#95D412yescomplemented
46C24×C31—21no—
47C24×C31—21no—
48C24×C31—21no—
49C24×C31—41no—
50C24×C31—41no—
51C24×C31—82no—
52C24×C31SD1612yes—
53C24×C31C2×D411yescharacteristic
54C2×D4×C31—22no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31—41no—
58C2×D4×C31—44no—
59C2×D4×C31—41no—
60C2×D4×C31—41no—
61C2×D4×C31—41no—
62C2×D4×C31—84no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31—21no—
66C22×C4×C31—22no—
67C22×C4×C31—41no—
68C22×C4×C31C2×D412yes—
69C2×C8×C31—42no—
70C4⋊C4×C31—22no—
71C4⋊C4×C31—42no—
72C4⋊C4×C31—82no—
73C4⋊C4×C31—82no—
74C4⋊C4×C31—81no—
75C22⋊C4×C31—22no—
76C22⋊C4×C31—41no—
77C22⋊C4×C31—42no—
78C22⋊C4×C31—44no—
79C22⋊C4×C31—41no—
80C22⋊C4×C31—41no—
81C22⋊C4×C31—82no—
82C22⋊C4×C31—82no—
83C22⋊C4×C31—82no—
84C22⋊C4×C31—81no—
85C22⋊C4×C31—81no—
86#12124C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—21no—
88C23×C31—21no—
89C23×C31—22no—
90C23×C31—42no—
91C23×C31—42no—
92C23×C31—42no—
93C23×C31—42no—
94C23×C31—41no—
95C23×C31—41no—
96C23×C31—44no—
97C23×C31—41no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—81no—
101C23×C31—82no—
102C23×C31—82no—
103C23×C31—82no—
104C23×C31—84no—
105C23×C31—82no—
106C23×C31#12011yescharacteristic
107C23×C31C2×SD1611yescharacteristic
108C23×C31#13711yescharacteristic
109D4×C31—82no—
110D4×C31—81no—
111D4×C31—81no—
112D4×C31—88no—
113D4×C31—81no—
114D4×C31—81no—
115C2×C4×C31—22no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—41no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—42no—
127C2×C4×C31—41no—
128C2×C4×C31—42no—
129C2×C4×C31—82no—
130C8×C31—82no—
131#2842C2×C3111yescharacteristic, complemented
132#2021C2×C3111yescharacteristic
133#1613C2×C3112yescomplemented
134#1352C2×C3112yescomplemented
135#1316C2×C3111yescharacteristic, complemented
136C22×C31—22no—
137C22×C31—21no—
138C22×C31—21no—
139C22×C31—41no—
140C22×C31—41no—
141C22×C31—42no—
142C22×C31—42no—
143C22×C31—42no—
144C22×C31—44no—
145C22×C31—42no—
146C22×C31—81no—
147C22×C31—82no—
148C22×C31—82no—
149C22×C31—84no—
150C22×C31—82no—
151C22×C31—82no—
152C22×C31—81no—
153C22×C31—82no—
154C22×C31—161no—
155C22×C31#44811yescharacteristic
156C22×C31#45311yescharacteristic
157C22×C31B(4,𝔽2)11yescharacteristic
158C4×C31—42no—
159C4×C31—81no—
160C4×C31—82no—
161C4×C31—82no—
162C4×C31—81no—
163C4×C31—82no—
164#534C22×C3111yescharacteristic
165C2×#120—21no—
166#481—22no—
167C2×#99—21no—
168#399—21no—
169#399C22×C3112yes—
170#396—22no—
171#396—22no—
172#379—21no—
173#379C22×C3111yescharacteristic
174#342C22×C3111yescharacteristic
175#327—22no—
176#323C22×C3112yes—
177C2×C31—21no—
178C2×C31—21no—
179C2×C31—41no—
180C2×C31—41no—
181C2×C31—82no—
182C2×C31—161no—
183C2×C31#159611yescharacteristic
184C2×C31#220012yes—
185C25D4×C3111yescharacteristic
186C22×D4—21no—
187C22×D4—41no—
188#123—41no—
189#121—22no—
190#120—42no—
191#120—41no—
192#120—42no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4D4×C3112yescomplemented
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4D4×C3111yescharacteristic
205C2×C22⋊C4C23×C3111yescharacteristic
206#106—42no—
207#102—42no—
208#99—42no—
209C22⋊C8—22no—
210#95—21no—
211#95D4×C3112yescomplemented
212C31#1212411yescharacteristic, direct product factor, Sylow
213C24—21no—
214C24—21no—
215C24—21no—
216C24—41no—
217C24—41no—
218C24—82no—
219C24SD16×C3112yes—
220C24C2×D4×C3111yescharacteristic
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—44no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—84no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4C2×D4×C3112yes—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C22⋊C4—22no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—44no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—81no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—84no—
271C23—82no—
272C23C31×#12011yescharacteristic
273C23C2×SD16×C3111yescharacteristic
274C23C31×#13711yescharacteristic
275D4—82no—
276D4—81no—
277D4—81no—
278D4—88no—
279D4—81no—
280D4—81no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—82no—
296C8—82no—
297C22—22no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—42no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—84no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22—82no—
315C22—161no—
316C22C31×#44811yescharacteristic
317C22C31×#45311yescharacteristic
318C22C31×B(4,𝔽2)11yescharacteristic
319C4—42no—
320C4—81no—
321C4—82no—
322C4—82no—
323C4—81no—
324C4—82no—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—82no—
330C2—161no—
331C2C31×#159611yescharacteristic
332C2C31×#220012yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC31×#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2842C2semidirect
3. C31×#1613C2semidirect
4. C31×#1352C2semidirect
5. C31×#1316C2semidirect
30. C2×C4⋊C4×C31D4semidirect
45. C31×#95D4semidirect
86. #12124C31direct (N × H)
131. #2842C2×C31semidirect
133. #1613C2×C31semidirect
134. #1352C2×C31semidirect
135. #1316C2×C31semidirect
196. C2×C4⋊C4D4×C31semidirect
211. #95D4×C31semidirect
212. C31#12124direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28422
C31×#20212
C31×#16132
C31×#13522
C31×#13162
#1212431