Finite Group Databasebeta

← C31×#12972 #21469731

Subgroups — C31×#12972 #21469731

Order 7936.

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

normal subgroups only

This group has 492 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×#2372C211yesmaximal, 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—
15C4×C22⋊C4×C31—22no—
16C31×#376—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—42no—
29C31×#117—21no—
30C31×#117—22no—
31C2×C4⋊C4×C31D412yes—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—21no—
35C2×C42×C31—22no—
36C2×C42×C31D411yescharacteristic
37C2×C42×C31D412yes—
38C31×#107—42no—
39C31×#106—42no—
40C31×#102—42no—
41C31×#101—22no—
42C31×#101—22no—
43C31×#101C2×C414yescomplemented
44C31×#95—21no—
45C31×#95—21no—
46C31×#95—22no—
47C31×#95—22no—
48C24×C31—21no—
49C2×Q8×C31—41no—
50C2×Q8×C31—41no—
51C2×Q8×C31C22×C411yescharacteristic
52C2×Q8×C31C2×Q811yescharacteristic
53C2×Q8×C31#4112yes—
54C2×D4×C31—21no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31C4⋊C412yes—
58C22×C4×C31—22no—
59C22×C4×C31—21no—
60C22×C4×C31—22no—
61C22×C4×C31—44no—
62C22×C4×C31—81no—
63C22×C4×C31C2×D411yescharacteristic
64C22×C4×C31C2×D412yes—
65M16×C31—24no—
66M16×C31—42no—
67M16×C31—41no—
68M16×C31—41no—
69C2×C8×C31—42no—
70C2×C8×C31—41no—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—22no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—42no—
75C22⋊C4×C31—24no—
76C22⋊C4×C31—24no—
77C22⋊C4×C31—44no—
78C22⋊C4×C31—41no—
79C22⋊C4×C31—81no—
80C42×C31—21no—
81C42×C31—44no—
82C42×C31—42no—
83C42×C31—41no—
84C42×C31—42no—
85C42×C31C4⋊C412yes—
86#12972C3111yesmaximal, characteristic, direct product factor, Sylow
87C23×C31—22no—
88C23×C31—42no—
89C23×C31—42no—
90C23×C31—41no—
91C23×C31—41no—
92C23×C31#12011yescharacteristic
93Q8×C31—41no—
94Q8×C31—41no—
95Q8×C31—42no—
96D4×C31—42no—
97D4×C31—81no—
98C2×C4×C31—22no—
99C2×C4×C31—21no—
100C2×C4×C31—22no—
101C2×C4×C31—21no—
102C2×C4×C31—24no—
103C2×C4×C31—22no—
104C2×C4×C31—42no—
105C2×C4×C31—42no—
106C2×C4×C31—42no—
107C2×C4×C31—42no—
108C2×C4×C31—42no—
109C2×C4×C31—42no—
110C2×C4×C31—42no—
111C2×C4×C31—84no—
112C2×C4×C31C2×C4⋊C411yescharacteristic
113C2×C4×C31C4×D412yes—
114C2×C4×C31C22⋊Q812yes—
115C2×C4×C31#12311yescharacteristic
116C8×C31—44no—
117C8×C31—81no—
118#2372C2×C3111yescharacteristic
119C2×#355C2×C3111yescharacteristic
120#1888C2×C3111yescharacteristic
121#1857C2×C3112yescomplemented
122#1394C2×C3112yescomplemented
123C22×C31—22no—
124C22×C31—44no—
125C22×C31—42no—
126C22×C31—81no—
127C22×C31—81no—
128C22×C31—82no—
129C22×C31#38511yescharacteristic
130C22×C31B(4,𝔽2)11yescharacteristic
131C22×C31#45811yescharacteristic
132C4×C31—22no—
133C4×C31—21no—
134C4×C31—21no—
135C4×C31—22no—
136C4×C31—42no—
137C4×C31—44no—
138C4×C31—82no—
139C4×C31—82no—
140C2×#124C22×C3111yescharacteristic
141#519C22×C3111yescharacteristic
142#428—22no—
143#418—21no—
144C2×#101C22×C3111yescharacteristic
145C2×#101C22×C3112yes—
146#391C22×C3112yes—
147#388—22no—
148#386—21no—
149C4×C22⋊C4—22no—
150#376—21no—
151#355C4×C3114yescomplemented
152#339—22no—
153#339—21no—
154C2×C31—21no—
155C2×C31—21no—
156C2×C31—82no—
157C2×C31#186711yescharacteristic
158C2×C31#220412yes—
159C22×Q8C23×C3111yescharacteristic
160C22×D4D4×C3111yescharacteristic
161C23×C4—21no—
162C2×M16—22no—
163C2×M16—21no—
164#124—22no—
165#124C2×C4×C3112yes—
166#124Q8×C3112yes—
167C4×Q8—42no—
168#117—21no—
169#117—22no—
170C2×C4⋊C4D4×C3112yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C42—22no—
175C2×C42D4×C3111yescharacteristic
176C2×C42D4×C3112yes—
177#107—42no—
178#106—42no—
179#102—42no—
180#101—22no—
181#101—22no—
182#101C2×C4×C3114yescomplemented
183#95—21no—
184#95—21no—
185#95—22no—
186#95—22no—
187C31#1297211yescharacteristic, direct product factor, Sylow
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C22×C4×C3111yescharacteristic
192C2×Q8C2×Q8×C3111yescharacteristic
193C2×Q8#41×C3112yes—
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4C4⋊C4×C3112yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4C2×D4×C3111yescharacteristic
204C22×C4C2×D4×C3112yes—
205M16—24no—
206M16—42no—
207M16—41no—
208M16—41no—
209C2×C8—42no—
210C2×C8—41no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—44no—
218C22⋊C4—41no—
219C22⋊C4—81no—
220C42—21no—
221C42—42no—
222C42—44no—
223C42—42no—
224C42—41no—
225C42C4⋊C4×C3112yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23C31×#12011yescharacteristic
232Q8—41no—
233Q8—41no—
234Q8—42no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—84no—
251C2×C4C2×C4⋊C4×C3111yescharacteristic
252C2×C4C4×D4×C3112yes—
253C2×C4C31×C22⋊Q812yes—
254C2×C4C31×#12311yescharacteristic
255C8—44no—
256C8—81no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22C31×#38511yescharacteristic
264C22C31×B(4,𝔽2)11yescharacteristic
265C22C31×#45811yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—44no—
271C4—42no—
272C4—82no—
273C4—82no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2C31×#186711yescharacteristic
278C2C31×#220412yes—
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
43. C31×#101C2×C4semidirect
86. #12972C31direct (N × H)
121. #1857C2×C31semidirect
122. #1394C2×C31semidirect
151. #355C4×C31semidirect
182. #101C2×C4×C31semidirect
187. C31#12972direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23722
C2×C31×#3552
C31×#18882
C31×#18572
C31×#13942
#1297231