Finite Group Databasebeta

← C31×#13309 #21470068

Subgroups — C31×#13309 #21470068

Order 7936.

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3414C211yesmaximal, characteristic, complemented
2C31×#3050C212yesmaximal, complemented
3C31×#2145C212yesmaximal, complemented
4C31×#2112C211yesmaximal, characteristic, complemented
5C31×#1384C211yesmaximal, characteristic, complemented
6C2×C31×#135C2212yes—
7C23×C8×C31C2211yescharacteristic
8C31×#525—22no—
9C31×#510—22no—
10C31×#509—22no—
11C31×#508—22no—
12C2×C31×M32C2211yescharacteristic
13C2×C31×M32C2211yescharacteristic
14C31×#467—22no—
15C31×#466—22no—
16C31×#465—24no—
17C31×#427C2212yes—
18C31×#416—22no—
19C31×#360—24no—
20C31×C22⋊C16—22no—
21C2×#41×C31—22no—
22C23×C4×C31D411yescharacteristic
23C31×#135—22no—
24C31×#135—22no—
25C31×#135—22no—
26C31×#135—22no—
27C2×Q16×C31D412yescomplemented
28C2×SD16×C31—22no—
29C2×D8×C31D412yescomplemented
30C22×C8×C31—22no—
31C22×C8×C31—22no—
32C22×C8×C31—21no—
33C22×C8×C31—21no—
34C22×C8×C31D411yescharacteristic
35C22×C8×C31C2311yescharacteristic
36C31×#123—42no—
37C31×C22⋊Q8—24no—
38C31×#121—24no—
39C31×#120—42no—
40C4×D4×C31—44no—
41C31×#117—22no—
42C31×Q32—22no—
43C31×SD32—24no—
44C31×D16—22no—
45C31×M32—22no—
46C31×M32—22no—
47C2×C16×C31—21no—
48C2×C16×C31D412yescomplemented
49C31×#107—22no—
50C31×#106D414yescomplemented
51C31×#103—24no—
52C31×#102—24no—
53C24×C31D811yescharacteristic
54#41×C31—42no—
55#41×C31—42no—
56#41×C31—42no—
57#41×C31—42no—
58C2×Q8×C31—22no—
59C2×D4×C31—22no—
60C2×D4×C31—44no—
61C2×D4×C31—42no—
62C22×C4×C31—22no—
63C22×C4×C31—21no—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—42no—
67C22×C4×C31D811yescharacteristic
68C22×C4×C31C2×D411yescharacteristic
69Q16×C31—22no—
70Q16×C31—22no—
71SD16×C31—42no—
72SD16×C31—42no—
73D8×C31—22no—
74D8×C31—22no—
75C2×C8×C31—22no—
76C2×C8×C31—21no—
77C2×C8×C31—41no—
78C2×C8×C31—42no—
79C2×C8×C31—41no—
80C2×C8×C31C2×D412yes—
81C2×C8×C31C2×D411yescharacteristic
82C2×C8×C31C2×D412yes—
83C2×C8×C31#4111yescharacteristic
84C4⋊C4×C31—24no—
85C4⋊C4×C31—44no—
86C22⋊C4×C31—44no—
87C22⋊C4×C31—44no—
88C22⋊C4×C31—42no—
89C42×C31—42no—
90C16×C31—22no—
91C16×C31—41no—
92#13309C3111yesmaximal, characteristic, direct product factor, Sylow
93C23×C31—22no—
94C23×C31—21no—
95C23×C31—42no—
96C23×C31—42no—
97C23×C31C2×D811yescharacteristic
98Q8×C31—42no—
99Q8×C31—42no—
100D4×C31—42no—
101D4×C31—42no—
102D4×C31—88no—
103D4×C31—82no—
104D4×C31—82no—
105C2×C4×C31—22no—
106C2×C4×C31—21no—
107C2×C4×C31—41no—
108C2×C4×C31—42no—
109C2×C4×C31—41no—
110C2×C4×C31—44no—
111C2×C4×C31—42no—
112C2×C4×C31—82no—
113C2×C4×C31—82no—
114C2×C4×C31#12012yes—
115C2×C4×C31#12111yescharacteristic
116C2×C4×C31#13511yescharacteristic
117C2×C4×C31AGL(1,ℤ/8)12yes—
118C8×C31—41no—
119C8×C31#12011yescharacteristic
120C8×C31#12112yes—
121C8×C31#12711yescharacteristic
122#3414C2×C3111yescharacteristic, complemented
123#3050C2×C3112yescomplemented
124#2145C2×C3112yescomplemented
125#2112C2×C3111yescharacteristic, complemented
126#1384C2×C3111yescharacteristic, complemented
127C22×C31—24no—
128C22×C31—22no—
129C22×C31—41no—
130C22×C31—42no—
131C22×C31—41no—
132C22×C31—41no—
133C22×C31—84no—
134C22×C31—82no—
135C22×C31#44712yes—
136C22×C31#46611yescharacteristic
137C4×C31—41no—
138C4×C31—84no—
139C4×C31—82no—
140C4×C31#39211yescharacteristic
141C4×C31#44912yes—
142C4×C31#46911yescharacteristic
143C2×#135C22×C3112yes—
144C23×C8C22×C3111yescharacteristic
145#525—22no—
146#510—22no—
147#509—22no—
148#508—22no—
149C2×M32C22×C3111yescharacteristic
150C2×M32C22×C3111yescharacteristic
151#467—22no—
152#466—22no—
153#465—24no—
154#427C22×C3112yes—
155#416—22no—
156#360—24no—
157C22⋊C16—22no—
158C2×C31—22no—
159C2×C31—21no—
160C2×C31—42no—
161C2×C31—162no—
162C2×C31#199911yescharacteristic
163C2×#41—22no—
164C23×C4D4×C3111yescharacteristic
165#135—22no—
166#135—22no—
167#135—22no—
168#135—22no—
169C2×Q16D4×C3112yescomplemented
170C2×SD16—22no—
171C2×D8D4×C3112yescomplemented
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8D4×C3111yescharacteristic
177C22×C8C23×C3111yescharacteristic
178#123—42no—
179C22⋊Q8—24no—
180#121—24no—
181#120—42no—
182C4×D4—44no—
183#117—22no—
184Q32—22no—
185SD32—24no—
186D16—22no—
187M32—22no—
188M32—22no—
189C2×C16—21no—
190C2×C16D4×C3112yescomplemented
191#107—22no—
192#106D4×C3114yescomplemented
193#103—24no—
194#102—24no—
195C31#1330911yescharacteristic, direct product factor, Sylow
196C24D8×C3111yescharacteristic
197#41—42no—
198#41—42no—
199#41—42no—
200#41—42no—
201C2×Q8—22no—
202C2×D4—22no—
203C2×D4—44no—
204C2×D4—42no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4D8×C3111yescharacteristic
211C22×C4C2×D4×C3111yescharacteristic
212Q16—22no—
213Q16—22no—
214SD16—42no—
215SD16—42no—
216D8—22no—
217D8—22no—
218C2×C8—22no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8C2×D4×C3112yes—
224C2×C8C2×D4×C3111yescharacteristic
225C2×C8C2×D4×C3112yes—
226C2×C8#41×C3111yescharacteristic
227C4⋊C4—24no—
228C4⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C42—42no—
233C16—22no—
234C16—41no—
235C23—22no—
236C23—21no—
237C23—42no—
238C23—42no—
239C23C2×D8×C3111yescharacteristic
240Q8—42no—
241Q8—42no—
242D4—42no—
243D4—42no—
244D4—88no—
245D4—82no—
246D4—82no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C31×#12012yes—
257C2×C4C31×#12111yescharacteristic
258C2×C4C31×#13511yescharacteristic
259C2×C4C31×AGL(1,ℤ/8)12yes—
260C8—41no—
261C8C31×#12011yescharacteristic
262C8C31×#12112yes—
263C8C31×#12711yescharacteristic
264C22—24no—
265C22—22no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—84no—
271C22—82no—
272C22C31×#44712yes—
273C22C31×#46611yescharacteristic
274C4—41no—
275C4—84no—
276C4—82no—
277C4C31×#39211yescharacteristic
278C4C31×#44912yes—
279C4C31×#46911yescharacteristic
280C2—22no—
281C2—21no—
282C2—42no—
283C2—162no—
284C2C31×#199911yescharacteristic
subgroupG/H
centerC4×C31#469
commutator, Frattini, cosocleC22×C8C23×C31
socleC2×C31#1999
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C31 ◁ C22×C4×C31 ◁ C23×C8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3414C2semidirect
2. C31×#3050C2semidirect
3. C31×#2145C2semidirect
4. C31×#2112C2semidirect
5. C31×#1384C2semidirect
27. C2×Q16×C31D4semidirect
29. C2×D8×C31D4semidirect
48. C2×C16×C31D4semidirect
50. C31×#106D4semidirect
92. #13309C31direct (N × H)
122. #3414C2×C31semidirect
123. #3050C2×C31semidirect
124. #2145C2×C31semidirect
125. #2112C2×C31semidirect
126. #1384C2×C31semidirect
169. C2×Q16D4×C31semidirect
171. C2×D8D4×C31semidirect
190. C2×C16D4×C31semidirect
192. #106D4×C31semidirect
195. C31#13309direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#34142
C31×#30502
C31×#21452
C31×#21122
C31×#13842
#1330931