Finite Group Databasebeta

← C31×#7252 #21464011

Subgroups — C31×#7252 #21464011

Order 7936.

344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1798C211yesmaximal, characteristic
2C31×#1740C211yesmaximal, characteristic
3C31×#1458C211yesmaximal, characteristic
4C2×C31×#117—21no—
5C2×C31×#117C2211yescharacteristic
6C31×#431—21no—
7C2×C31×C4⋊C8C411yescharacteristic, complemented
8C2×C31×C4⋊C8C411yescharacteristic, complemented
9C2×C31×C22⋊C8—21no—
10C31×#342—21no—
11C31×#342C412yes—
12C31×#336—21no—
13C31×#324C412yescomplemented
14C31×#323—21no—
15C23×C4×C31D411yescharacteristic
16C22×C8×C31—21no—
17C22×C8×C31—21no—
18C31×#117—22no—
19C31×#117—22no—
20C31×#117—41no—
21C31×#117—41no—
22C31×#117—41no—
23C31×#117—41no—
24C2×C4⋊C4×C31—41no—
25C2×C4⋊C4×C31C2×C411yescharacteristic
26C2×C4⋊C4×C31Q811yescharacteristic
27C2×C22⋊C4×C31—22no—
28C2×C22⋊C4×C31C2×C411yescharacteristic
29C2×C22⋊C4×C31D411yescharacteristic
30C2×C42×C31—41no—
31C2×C42×C31C2×C411yescharacteristic
32C2×C42×C31D411yescharacteristic
33C31×C4⋊C8—41no—
34C31×C4⋊C8—41no—
35C31×C4⋊C8—41no—
36C31×C22⋊C8—22no—
37C31×C22⋊C8—41no—
38C31×C22⋊C8—41no—
39C4×C8×C31—41no—
40C24×C31C22⋊C411yescharacteristic
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31—21no—
44C22×C4×C31—21no—
45C22×C4×C31—42no—
46C22×C4×C31C4211yescharacteristic
47C22×C4×C31C22⋊C411yescharacteristic
48C22×C4×C31C22⋊C411yescharacteristic
49C22×C4×C31C4⋊C411yescharacteristic
50C22×C4×C31C4⋊C411yescharacteristic
51C22×C4×C31C4⋊C411yescharacteristic
52C22×C4×C31D811yescharacteristic
53C22×C4×C31SD1611yescharacteristic
54C22×C4×C31SD1611yescharacteristic
55C22×C4×C31Q1611yescharacteristic
56C2×C8×C31—42no—
57C2×C8×C31—41no—
58C2×C8×C31—41no—
59C2×C8×C31—41no—
60C2×C8×C31—41no—
61C2×C8×C31—41no—
62C2×C8×C31—41no—
63C4⋊C4×C31—41no—
64C4⋊C4×C31—41no—
65C4⋊C4×C31—42no—
66C4⋊C4×C31—42no—
67C22⋊C4×C31—24no—
68C22⋊C4×C31—41no—
69C22⋊C4×C31—41no—
70C22⋊C4×C31—42no—
71C42×C31—21no—
72C42×C31—21no—
73C42×C31—21no—
74C42×C31—21no—
75C42×C31—42no—
76C42×C31—42no—
77#7252C3111yesmaximal, characteristic, direct product factor, Sylow
78C23×C31—21no—
79C23×C31—21no—
80C23×C31—41no—
81C23×C31—41no—
82C23×C31#9511yescharacteristic
83C23×C31#10211yescharacteristic
84C23×C31#10311yescharacteristic
85C2×C4×C31—21no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—21no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31—41no—
102C2×C4×C31—41no—
103C2×C4×C31—44no—
104C2×C4×C31—82no—
105C2×C4×C31—82no—
106C2×C4×C31#9911yescharacteristic
107C2×C4×C31#9911yescharacteristic
108C2×C4×C31#10211yescharacteristic
109C2×C4×C31#10311yescharacteristic
110C2×C4×C31#10411yescharacteristic
111C2×C4×C31#10411yescharacteristic
112C2×C4×C31#10611yescharacteristic
113C2×C4×C31#10711yescharacteristic
114C8×C31—82no—
115C8×C31—81no—
116C8×C31—81no—
117#1798C2×C3111yescharacteristic
118#1740C2×C3111yescharacteristic
119#1458C2×C3111yescharacteristic
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—21no—
124C22×C31—41no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31—81no—
129C22×C31#33911yescharacteristic
130C22×C31#34011yescharacteristic
131C22×C31#34211yescharacteristic
132C4×C31—21no—
133C4×C31—21no—
134C4×C31—41no—
135C4×C31—41no—
136C4×C31—41no—
137C4×C31—41no—
138C4×C31—84no—
139C4×C31#32711yescharacteristic
140C4×C31#32711yescharacteristic
141C4×C31#32811yescharacteristic
142C4×C31#32811yescharacteristic
143C2×#117—21no—
144C2×#117C22×C3111yescharacteristic
145#431—21no—
146C2×C4⋊C8C4×C3111yescharacteristic, complemented
147C2×C4⋊C8C4×C3111yescharacteristic, complemented
148C2×C22⋊C8—21no—
149#342—21no—
150#342C4×C3112yes—
151#336—21no—
152#324C4×C3112yescomplemented
153#323—21no—
154C2×C31—21no—
155C2×C31—21no—
156C2×C31—41no—
157C2×C31—41no—
158C2×C31#129411yescharacteristic
159C2×C31#139111yescharacteristic
160C2×C31#139511yescharacteristic
161C23×C4D4×C3111yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164#117—22no—
165#117—22no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C2×C4×C3111yescharacteristic
172C2×C4⋊C4Q8×C3111yescharacteristic
173C2×C22⋊C4—22no—
174C2×C22⋊C4C2×C4×C3111yescharacteristic
175C2×C22⋊C4D4×C3111yescharacteristic
176C2×C42—41no—
177C2×C42C2×C4×C3111yescharacteristic
178C2×C42D4×C3111yescharacteristic
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C4×C8—41no—
186C31#725211yescharacteristic, direct product factor, Sylow
187C24C22⋊C4×C3111yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4C42×C3111yescharacteristic
194C22×C4C22⋊C4×C3111yescharacteristic
195C22×C4C22⋊C4×C3111yescharacteristic
196C22×C4C4⋊C4×C3111yescharacteristic
197C22×C4C4⋊C4×C3111yescharacteristic
198C22×C4C4⋊C4×C3111yescharacteristic
199C22×C4D8×C3111yescharacteristic
200C22×C4SD16×C3111yescharacteristic
201C22×C4SD16×C3111yescharacteristic
202C22×C4Q16×C3111yescharacteristic
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C22⋊C4—24no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C31×#9511yescharacteristic
229C23C31×#10211yescharacteristic
230C23C31×#10311yescharacteristic
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4C31×#9911yescharacteristic
253C2×C4C31×#9911yescharacteristic
254C2×C4C31×#10211yescharacteristic
255C2×C4C31×#10311yescharacteristic
256C2×C4C31×#10411yescharacteristic
257C2×C4C31×#10411yescharacteristic
258C2×C4C31×#10611yescharacteristic
259C2×C4C31×#10711yescharacteristic
260C8—82no—
261C8—81no—
262C8—81no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22C31×#33911yescharacteristic
273C22C31×#34011yescharacteristic
274C22C31×#34211yescharacteristic
275C4—21no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—84no—
282C4C31×#32711yescharacteristic
283C4C31×#32711yescharacteristic
284C4C31×#32811yescharacteristic
285C4C31×#32811yescharacteristic
286C2—21no—
287C2—21no—
288C2—41no—
289C2—41no—
290C2C31×#129411yescharacteristic
291C2C31×#139111yescharacteristic
292C2C31×#139511yescharacteristic
subgroupG/H
centerC2×C4×C31#99
commutatorC22×C4C42×C31
Frattini, cosocleC2×#117C22×C31
socleC22×C31#342
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C31×C4⋊C8C4semidirect
8. C2×C31×C4⋊C8C4semidirect
13. C31×#324C4semidirect
77. #7252C31direct (N × H)
146. C2×C4⋊C8C4×C31semidirect
147. C2×C4⋊C8C4×C31semidirect
152. #324C4×C31semidirect
186. C31#7252direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#17982
C31×#17402
C31×#14582
#725231