Finite Group Databasebeta

← C31×#12970 #21469729

Subgroups — C31×#12970 #21469729

Order 7936.

472 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2379C211yesmaximal, characteristic
2C2×C31×#355C211yesmaximal, characteristic
3C31×#1888C211yesmaximal, characteristic
4C31×#1857C212yesmaximal, complemented
5C31×#1394C212yesmaximal, complemented
6C2×C31×#124C2211yescharacteristic
7C31×#519C2211yescharacteristic
8C31×#428—22no—
9C31×#418—21no—
10C2×C31×#101C2211yescharacteristic
11C2×C31×#101C2212yes—
12C31×#391C2212yes—
13C31×#388—22no—
14C31×#386—21no—
15C31×#385—22no—
16C31×#382—21no—
17C31×#355C414yescomplemented
18C31×#339—22no—
19C31×#339—21no—
20C22×Q8×C31C2311yescharacteristic
21C22×D4×C31D411yescharacteristic
22C23×C4×C31—21no—
23C2×M16×C31—22no—
24C2×M16×C31—21no—
25C31×#124—22no—
26C31×#124C2×C412yes—
27C31×#124Q812yes—
28C4×Q8×C31—41no—
29C4×Q8×C31—41no—
30C31×#117—21no—
31C31×#117—22no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31D412yes—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—21no—
38C2×C42×C31D411yescharacteristic
39C2×C42×C31D412yes—
40C31×#107—42no—
41C31×#106—42no—
42C31×#102—41no—
43C31×#102—41no—
44C31×#101—22no—
45C31×#101—22no—
46C31×#101C2×C414yescomplemented
47C31×#95—22no—
48C31×#95—22no—
49C31×#95—22no—
50C24×C31—21no—
51C2×Q8×C31—41no—
52C2×Q8×C31—41no—
53C2×Q8×C31C22×C411yescharacteristic
54C2×Q8×C31C2×Q811yescharacteristic
55C2×Q8×C31#4112yes—
56C2×D4×C31—21no—
57C2×D4×C31—41no—
58C2×D4×C31—41no—
59C2×D4×C31C4⋊C412yes—
60C22×C4×C31—21no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—82no—
64C22×C4×C31—81no—
65C22×C4×C31C2×D411yescharacteristic
66C22×C4×C31C2×D412yes—
67M16×C31—24no—
68M16×C31—42no—
69M16×C31—41no—
70M16×C31—41no—
71C2×C8×C31—42no—
72C2×C8×C31—41no—
73C4⋊C4×C31—21no—
74C4⋊C4×C31—21no—
75C4⋊C4×C31—22no—
76C4⋊C4×C31—22no—
77C4⋊C4×C31—42no—
78C4⋊C4×C31—81no—
79C4⋊C4×C31—81no—
80C22⋊C4×C31—24no—
81C22⋊C4×C31—24no—
82C22⋊C4×C31—41no—
83C22⋊C4×C31—82no—
84C22⋊C4×C31—81no—
85C42×C31—21no—
86C42×C31—42no—
87C42×C31—41no—
88C42×C31—42no—
89C42×C31C4⋊C412yes—
90#12970C3111yesmaximal, characteristic, direct product factor, Sylow
91C23×C31—22no—
92C23×C31—42no—
93C23×C31—42no—
94C23×C31—41no—
95C23×C31—41no—
96C23×C31#12011yescharacteristic
97Q8×C31—41no—
98Q8×C31—41no—
99Q8×C31—42no—
100D4×C31—42no—
101D4×C31—81no—
102C2×C4×C31—22no—
103C2×C4×C31—21no—
104C2×C4×C31—22no—
105C2×C4×C31—21no—
106C2×C4×C31—22no—
107C2×C4×C31—41no—
108C2×C4×C31—41no—
109C2×C4×C31—42no—
110C2×C4×C31—42no—
111C2×C4×C31—42no—
112C2×C4×C31—42no—
113C2×C4×C31—41no—
114C2×C4×C31—41no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—84no—
118C2×C4×C31C2×C4⋊C411yescharacteristic
119C2×C4×C31C4×D412yes—
120C2×C4×C31C22⋊Q812yes—
121C2×C4×C31#12311yescharacteristic
122C8×C31—44no—
123C8×C31—81no—
124#2379C2×C3111yescharacteristic
125C2×#355C2×C3111yescharacteristic
126#1888C2×C3111yescharacteristic
127#1857C2×C3112yescomplemented
128#1394C2×C3112yescomplemented
129C22×C31—21no—
130C22×C31—21no—
131C22×C31—44no—
132C22×C31—42no—
133C22×C31—82no—
134C22×C31—81no—
135C22×C31—81no—
136C22×C31#38511yescharacteristic
137C22×C31B(4,𝔽2)11yescharacteristic
138C22×C31#45811yescharacteristic
139C4×C31—22no—
140C4×C31—21no—
141C4×C31—21no—
142C4×C31—22no—
143C4×C31—42no—
144C4×C31—82no—
145C4×C31—81no—
146C4×C31—82no—
147C4×C31—81no—
148C2×#124C22×C3111yescharacteristic
149#519C22×C3111yescharacteristic
150#428—22no—
151#418—21no—
152C2×#101C22×C3111yescharacteristic
153C2×#101C22×C3112yes—
154#391C22×C3112yes—
155#388—22no—
156#386—21no—
157#385—22no—
158#382—21no—
159#355C4×C3114yescomplemented
160#339—22no—
161#339—21no—
162C2×C31—21no—
163C2×C31—21no—
164C2×C31—82no—
165C2×C31#186711yescharacteristic
166C2×C31#220211yescharacteristic
167C2×C31#220311yescharacteristic
168C22×Q8C23×C3111yescharacteristic
169C22×D4D4×C3111yescharacteristic
170C23×C4—21no—
171C2×M16—22no—
172C2×M16—21no—
173#124—22no—
174#124C2×C4×C3112yes—
175#124Q8×C3112yes—
176C4×Q8—41no—
177C4×Q8—41no—
178#117—21no—
179#117—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4D4×C3112yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C42D4×C3111yescharacteristic
187C2×C42D4×C3112yes—
188#107—42no—
189#106—42no—
190#102—41no—
191#102—41no—
192#101—22no—
193#101—22no—
194#101C2×C4×C3114yescomplemented
195#95—22no—
196#95—22no—
197#95—22no—
198C31#1297011yescharacteristic, direct product factor, Sylow
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C22×C4×C3111yescharacteristic
203C2×Q8C2×Q8×C3111yescharacteristic
204C2×Q8#41×C3112yes—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊C4×C3112yes—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—82no—
213C22×C4—81no—
214C22×C4C2×D4×C3111yescharacteristic
215C22×C4C2×D4×C3112yes—
216M16—24no—
217M16—42no—
218M16—41no—
219M16—41no—
220C2×C8—42no—
221C2×C8—41no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—41no—
232C22⋊C4—82no—
233C22⋊C4—81no—
234C42—21no—
235C42—42no—
236C42—42no—
237C42—41no—
238C42C4⋊C4×C3112yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C31×#12011yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—42no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—84no—
266C2×C4C2×C4⋊C4×C3111yescharacteristic
267C2×C4C4×D4×C3112yes—
268C2×C4C31×C22⋊Q812yes—
269C2×C4C31×#12311yescharacteristic
270C8—44no—
271C8—81no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22C31×#38511yescharacteristic
280C22C31×B(4,𝔽2)11yescharacteristic
281C22C31×#45811yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—82no—
288C4—82no—
289C4—81no—
290C4—81no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2C31×#186711yescharacteristic
295C2C31×#220211yescharacteristic
296C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×Q8C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1857C2semidirect
5. C31×#1394C2semidirect
17. C31×#355C4semidirect
46. C31×#101C2×C4semidirect
90. #12970C31direct (N × H)
127. #1857C2×C31semidirect
128. #1394C2×C31semidirect
159. #355C4×C31semidirect
194. #101C2×C4×C31semidirect
198. C31#12970direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23792
C2×C31×#3552
C31×#18882
C31×#18572
C31×#13942
#1297031