Finite Group Databasebeta

← C31×#12103 #21468862

Subgroups — C31×#12103 #21468862

Order 7936.

476 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3189C211yesmaximal, characteristic, complemented
2C31×#2187C212yesmaximal, complemented
3C31×#1999C211yesmaximal, characteristic
4C31×#1348C212yesmaximal, complemented
5C31×#1314C211yesmaximal, characteristic, complemented
6C22×D8×C31C2211yescharacteristic
7C2×C31×C22⋊Q8—21no—
8C2×C31×#121—21no—
9C2×C31×SD32—22no—
10C31×#484—22no—
11C31×#467—21no—
12C31×#467C2212yes—
13C31×#451—22no—
14C2×C31×#102—21no—
15C2×C31×C22⋊C8C2211yescharacteristic
16C31×#357—22no—
17C31×C22⋊C16C2212yes—
18C31×#340C2211yescharacteristic
19C22×Q8×C31—21no—
20C22×D4×C31—21no—
21C22×D4×C31—41no—
22C23×C4×C31D411yescharacteristic
23C2×Q16×C31—21no—
24C2×Q16×C31D412yescomplemented
25C2×D8×C31—22no—
26C2×D8×C31—42no—
27C22×C8×C31D411yescharacteristic
28C22×C8×C31C2311yescharacteristic
29C31×C22⋊Q8—22no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8—41no—
32C31×C22⋊Q8—41no—
33C31×#121—42no—
34C31×#121—42no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—41no—
38C2×C22⋊C4×C31—41no—
39C2×C22⋊C4×C31—41no—
40C31×SD32—44no—
41C2×C16×C31—22no—
42C31×#107—21no—
43C31×#107D412yescomplemented
44C31×#103—22no—
45C31×#103—22no—
46C31×#102—42no—
47C31×C22⋊C8—22no—
48C24×C31—41no—
49C24×C31D811yescharacteristic
50C2×Q8×C31—22no—
51C2×Q8×C31—41no—
52C2×Q8×C31—81no—
53C2×Q8×C31—81no—
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—82no—
62C2×D4×C31—82no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—41no—
66C22×C4×C31—41no—
67C22×C4×C31—41no—
68C22×C4×C31—41no—
69C22×C4×C31—41no—
70C22×C4×C31D811yescharacteristic
71C22×C4×C31C2×D411yescharacteristic
72Q16×C31—42no—
73Q16×C31—81no—
74D8×C31—24no—
75D8×C31—42no—
76C2×C8×C31—21no—
77C2×C8×C31—41no—
78C2×C8×C31—41no—
79C2×C8×C31C2×D412yes—
80C4⋊C4×C31—22no—
81C4⋊C4×C31—22no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—42no—
85C4⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—44no—
88C22⋊C4×C31—42no—
89C16×C31—42no—
90#12103C3111yesmaximal, characteristic, direct product factor, Sylow
91C23×C31—21no—
92C23×C31—21no—
93C23×C31—41no—
94C23×C31—41no—
95C23×C31—42no—
96C23×C31—81no—
97C23×C31—81no—
98C23×C31—81no—
99C23×C31—81no—
100C23×C31—81no—
101C23×C31—81no—
102C23×C31SD3212yescomplemented
103C23×C31C2×D811yescharacteristic
104Q8×C31—42no—
105Q8×C31—42no—
106Q8×C31—82no—
107D4×C31—42no—
108D4×C31—42no—
109D4×C31—81no—
110D4×C31—88no—
111D4×C31—81no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—41no—
116C2×C4×C31—41no—
117C2×C4×C31—41no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—81no—
122C2×C4×C31—81no—
123C2×C4×C31—82no—
124C2×C4×C31—82no—
125C2×C4×C31—81no—
126C2×C4×C31#12011yescharacteristic
127C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
128C8×C31—22no—
129C8×C31—41no—
130#3189C2×C3111yescharacteristic, complemented
131#2187C2×C3112yescomplemented
132#1999C2×C3111yescharacteristic
133#1348C2×C3112yescomplemented
134#1314C2×C3111yescharacteristic, complemented
135C22×C31—21no—
136C22×C31—21no—
137C22×C31—22no—
138C22×C31—42no—
139C22×C31—41no—
140C22×C31—41no—
141C22×C31—81no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—161no—
146C22×C31—161no—
147C22×C31—161no—
148C22×C31—161no—
149C22×C31#44711yescharacteristic
150C22×C31C2×SD3211yescharacteristic
151C22×C31#51011yescharacteristic
152C4×C31—21no—
153C4×C31—41no—
154C4×C31—82no—
155C4×C31—82no—
156C4×C31—82no—
157C4×C31#45311yescharacteristic
158C4×C31B(4,𝔽2)11yescharacteristic
159C22×D8C22×C3111yescharacteristic
160C2×C22⋊Q8—21no—
161C2×#121—21no—
162C2×SD32—22no—
163#484—22no—
164#467—21no—
165#467C22×C3112yes—
166#451—22no—
167C2×#102—21no—
168C2×C22⋊C8C22×C3111yescharacteristic
169#357—22no—
170C22⋊C16C22×C3112yes—
171#340C22×C3111yescharacteristic
172C2×C31—21no—
173C2×C31—21no—
174C2×C31—42no—
175C2×C31—162no—
176C2×C31#159511yescharacteristic
177C2×C31#218511yescharacteristic
178C2×C31#219011yescharacteristic
179C22×Q8—21no—
180C22×D4—21no—
181C22×D4—41no—
182C23×C4D4×C3111yescharacteristic
183C2×Q16—21no—
184C2×Q16D4×C3112yescomplemented
185C2×D8—22no—
186C2×D8—42no—
187C22×C8D4×C3111yescharacteristic
188C22×C8C23×C3111yescharacteristic
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193#121—42no—
194#121—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200SD32—44no—
201C2×C16—22no—
202#107—21no—
203#107D4×C3112yescomplemented
204#103—22no—
205#103—22no—
206#102—42no—
207C22⋊C8—22no—
208C31#1210311yescharacteristic, direct product factor, Sylow
209C24—41no—
210C24D8×C3111yescharacteristic
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4D8×C3111yescharacteristic
232C22×C4C2×D4×C3111yescharacteristic
233Q16—42no—
234Q16—81no—
235D8—24no—
236D8—42no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×D4×C3112yes—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C16—42no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23C31×SD3212yescomplemented
263C23C2×D8×C3111yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—82no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—88no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4C31×#12011yescharacteristic
287C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22C31×#44711yescharacteristic
305C22C2×C31×SD3211yescharacteristic
306C22C31×#51011yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4—82no—
312C4C31×#45311yescharacteristic
313C4C31×B(4,𝔽2)11yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—162no—
318C2C31×#159511yescharacteristic
319C2C31×#218511yescharacteristic
320C2C31×#219011yescharacteristic
subgroupG/H
center, socleC22×C31#447
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC31×#1999C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×C22⋊C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3189C2semidirect
2. C31×#2187C2semidirect
4. C31×#1348C2semidirect
5. C31×#1314C2semidirect
24. C2×Q16×C31D4semidirect
43. C31×#107D4semidirect
90. #12103C31direct (N × H)
102. C23×C31SD32semidirect
130. #3189C2×C31semidirect
131. #2187C2×C31semidirect
133. #1348C2×C31semidirect
134. #1314C2×C31semidirect
184. C2×Q16D4×C31semidirect
203. #107D4×C31semidirect
208. C31#12103direct (N × H)
262. C23C31×SD32semidirect

Maximal subgroups

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

subgroupindex
C31×#31892
C31×#21872
C31×#19992
C31×#13482
C31×#13142
#1210331