Finite Group Databasebeta

← C31×#13306 #21470065

Subgroups — C31×#13306 #21470065

Order 7936.

566 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C22×C31×Q32C211yesmaximal, characteristic, complemented
2C2×C31×#467C212yesmaximal
3C2×C31×#358C212yesmaximal, complemented
4C2×C31×C22⋊C16C211yesmaximal, characteristic
5C31×#1380C211yesmaximal, characteristic, complemented
6C22×Q16×C31C2212yes—
7C23×C8×C31C2211yescharacteristic
8C2×C31×C22⋊Q8—22no—
9C2×C31×Q32—24no—
10C2×C31×Q32—22no—
11C22×C16×C31C2211yescharacteristic
12C22×C16×C31C2211yescharacteristic
13C31×#467—24no—
14C31×#467—24no—
15C2×C31×#107C2212yes—
16C2×C31×#103—22no—
17C31×#358—24no—
18C31×C22⋊C16—22no—
19C22×Q8×C31—22no—
20C23×C4×C31D411yescharacteristic
21C2×Q16×C31—22no—
22C2×Q16×C31—22no—
23C2×Q16×C31—22no—
24C2×Q16×C31—22no—
25C2×Q16×C31—22no—
26C2×Q16×C31D414yes—
27C22×C8×C31—22no—
28C22×C8×C31—22no—
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C22×C8×C31D411yescharacteristic
32C22×C8×C31C2311yescharacteristic
33C31×C22⋊Q8—28no—
34C31×C22⋊Q8—44no—
35C2×C4⋊C4×C31—22no—
36C2×C4⋊C4×C31—42no—
37C2×C22⋊C4×C31—42no—
38C31×Q32—28no—
39C2×C16×C31—22no—
40C2×C16×C31—21no—
41C2×C16×C31—22no—
42C2×C16×C31D412yes—
43C31×#107—22no—
44C31×#107D414yes—
45C31×#103—28no—
46C24×C31D811yescharacteristic
47C2×Q8×C31—24no—
48C2×Q8×C31—42no—
49C2×Q8×C31—42no—
50C2×Q8×C31—42no—
51C2×Q8×C31—42no—
52C2×Q8×C31—42no—
53C22×C4×C31—22no—
54C22×C4×C31—21no—
55C22×C4×C31—21no—
56C22×C4×C31—22no—
57C22×C4×C31—42no—
58C22×C4×C31—42no—
59C22×C4×C31D811yescharacteristic
60C22×C4×C31C2×D411yescharacteristic
61Q16×C31—24no—
62Q16×C31—24no—
63Q16×C31—42no—
64Q16×C31—42no—
65C2×C8×C31—22no—
66C2×C8×C31—21no—
67C2×C8×C31—41no—
68C2×C8×C31—42no—
69C2×C8×C31—41no—
70C2×C8×C31C2×D412yes—
71C2×C8×C31C2×D411yescharacteristic
72C2×C8×C31C2×D412yes—
73C2×C8×C31#4111yescharacteristic
74C4⋊C4×C31—24no—
75C4⋊C4×C31—48no—
76C4⋊C4×C31—42no—
77C22⋊C4×C31—48no—
78C16×C31—22no—
79C16×C31—41no—
80#13306C3111yesmaximal, characteristic, direct product factor, Sylow
81C23×C31—22no—
82C23×C31—21no—
83C23×C31—21no—
84C23×C31—22no—
85C23×C31Q3212yescomplemented
86C23×C31C2×D811yescharacteristic
87Q8×C31—44no—
88Q8×C31—44no—
89Q8×C31—82no—
90Q8×C31—82no—
91C2×C4×C31—22no—
92C2×C4×C31—21no—
93C2×C4×C31—41no—
94C2×C4×C31—42no—
95C2×C4×C31—41no—
96C2×C4×C31—44no—
97C2×C4×C31—44no—
98C2×C4×C31—82no—
99C2×C4×C31—82no—
100C2×C4×C31—82no—
101C2×C4×C31—82no—
102C2×C4×C31#12012yes—
103C2×C4×C31#12111yescharacteristic
104C2×C4×C31#13511yescharacteristic
105C2×C4×C31AGL(1,ℤ/8)12yes—
106C8×C31—41no—
107C8×C31#12011yescharacteristic
108C8×C31#12112yes—
109C8×C31#12711yescharacteristic
110C22×Q32C2×C3111yescharacteristic, complemented
111C2×#467C2×C3112yes—
112C2×#358C2×C3112yescomplemented
113C2×C22⋊C16C2×C3111yescharacteristic
114#1380C2×C3111yescharacteristic, complemented
115C22×C31—24no—
116C22×C31—22no—
117C22×C31—42no—
118C22×C31—41no—
119C22×C31—41no—
120C22×C31#44712yes—
121C22×C31#46611yescharacteristic
122C22×C31C2×Q3211yescharacteristic
123C22×C31#50811yescharacteristic
124C22×C31#51012yes—
125C4×C31—41no—
126C4×C31—84no—
127C4×C31—84no—
128C4×C31#39211yescharacteristic
129C4×C31#44912yes—
130C4×C31#46911yescharacteristic
131C22×Q16C22×C3112yes—
132C23×C8C22×C3111yescharacteristic
133C2×C22⋊Q8—22no—
134C2×Q32—24no—
135C2×Q32—22no—
136C22×C16C22×C3111yescharacteristic
137C22×C16C22×C3111yescharacteristic
138#467—24no—
139#467—24no—
140C2×#107C22×C3112yes—
141C2×#103—22no—
142#358—24no—
143C22⋊C16—22no—
144C2×C31—42no—
145C2×C31#199911yescharacteristic
146C2×C31#218812yes—
147C2×C31#218912yes—
148C2×C31#221611yescharacteristic
149C2×C31#221911yescharacteristic
150C22×Q8—22no—
151C23×C4D4×C3111yescharacteristic
152C2×Q16—22no—
153C2×Q16—22no—
154C2×Q16—22no—
155C2×Q16—22no—
156C2×Q16—22no—
157C2×Q16D4×C3114yes—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8D4×C3111yescharacteristic
163C22×C8C23×C3111yescharacteristic
164C22⋊Q8—28no—
165C22⋊Q8—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C22⋊C4—42no—
169Q32—28no—
170C2×C16—22no—
171C2×C16—21no—
172C2×C16—22no—
173C2×C16D4×C3112yes—
174#107—22no—
175#107D4×C3114yes—
176#103—28no—
177C31#1330611yescharacteristic, direct product factor, Sylow
178C24D8×C3111yescharacteristic
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4D8×C3111yescharacteristic
192C22×C4C2×D4×C3111yescharacteristic
193Q16—24no—
194Q16—24no—
195Q16—42no—
196Q16—42no—
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8C2×D4×C3112yes—
203C2×C8C2×D4×C3111yescharacteristic
204C2×C8C2×D4×C3112yes—
205C2×C8#41×C3111yescharacteristic
206C4⋊C4—24no—
207C4⋊C4—48no—
208C4⋊C4—42no—
209C22⋊C4—48no—
210C16—22no—
211C16—41no—
212C23—22no—
213C23—21no—
214C23—21no—
215C23—22no—
216C23C31×Q3212yescomplemented
217C23C2×D8×C3111yescharacteristic
218Q8—44no—
219Q8—44no—
220Q8—82no—
221Q8—82no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C31×#12012yes—
234C2×C4C31×#12111yescharacteristic
235C2×C4C31×#13511yescharacteristic
236C2×C4C31×AGL(1,ℤ/8)12yes—
237C8—41no—
238C8C31×#12011yescharacteristic
239C8C31×#12112yes—
240C8C31×#12711yescharacteristic
241C22—24no—
242C22—22no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22C31×#44712yes—
247C22C31×#46611yescharacteristic
248C22C2×C31×Q3211yescharacteristic
249C22C31×#50811yescharacteristic
250C22C31×#51012yes—
251C4—41no—
252C4—84no—
253C4—84no—
254C4C31×#39211yescharacteristic
255C4C31×#44912yes—
256C4C31×#46911yescharacteristic
257C2—42no—
258C2C31×#199911yescharacteristic
259C2C31×#218812yes—
260C2C31×#218912yes—
261C2C31×#221611yescharacteristic
262C2C31×#221911yescharacteristic
subgroupG/H
center, socleC23×C31C2×D8
commutator, Frattini, cosocleC22×C8C23×C31
prime radicalC24×C31D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ C23×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C31×Q32C2semidirect
3. C2×C31×#358C2semidirect
5. C31×#1380C2semidirect
80. #13306C31direct (N × H)
85. C23×C31Q32semidirect
110. C22×Q32C2×C31semidirect
112. C2×#358C2×C31semidirect
114. #1380C2×C31semidirect
177. C31#13306direct (N × H)
216. C23C31×Q32semidirect

Maximal subgroups

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

subgroupindex
C22×C31×Q322
C2×C31×#4672
C2×C31×#3582
C2×C31×C22⋊C162
C31×#13802
#1330631