Finite Group Databasebeta

← C31×#7508 #21464267

Subgroups — C31×#7508 #21464267

Order 7936.

290 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×C31×#355C211yesmaximal, characteristic
2C2×C31×#354C211yesmaximal, characteristic
3C31×#1959C211yesmaximal, characteristic
4C2×C31×#124C2211yescharacteristic
5C2×C31×#103—21no—
6C2×C31×#102—21no—
7C2×C31×#101—21no—
8C2×C31×#99—21no—
9C2×C31×#97C411yescharacteristic, complemented
10C31×#384C411yescharacteristic, complemented
11C31×#355—21no—
12C31×#355C412yescomplemented
13C31×#354—21no—
14C31×#354C412yescomplemented
15C22×Q8×C31D411yescharacteristic
16C22×D4×C31D411yescharacteristic
17C2×M16×C31—41no—
18C22×C8×C31—21no—
19C31×#124—41no—
20C31×#124C2×C411yescharacteristic
21C31×#124C2×C411yescharacteristic
22C31×#124D411yescharacteristic
23C31×#124Q811yescharacteristic
24C2×C4⋊C4×C31—21no—
25C2×C4⋊C4×C31—41no—
26C2×C22⋊C4×C31—41no—
27C2×C22⋊C4×C31—41no—
28C2×C42×C31C2×C411yescharacteristic
29C31×#103—42no—
30C31×#102—42no—
31C31×#101—22no—
32C31×#101—41no—
33C31×#99—22no—
34C31×#99—41no—
35C31×#97—41no—
36C24×C31—21no—
37C2×Q8×C31—21no—
38C2×Q8×C31—41no—
39C2×Q8×C31—41no—
40C2×Q8×C31C22⋊C411yescharacteristic
41C2×Q8×C31C4⋊C411yescharacteristic
42C2×D4×C31—21no—
43C2×D4×C31—41no—
44C2×D4×C31—41no—
45C2×D4×C31C22⋊C411yescharacteristic
46C2×D4×C31C4⋊C411yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31—21no—
49C22×C4×C31—41no—
50C22×C4×C31—81no—
51C22×C4×C31C22⋊C411yescharacteristic
52M16×C31—42no—
53M16×C31—81no—
54C2×C8×C31—22no—
55C2×C8×C31—41no—
56C2×C8×C31—41no—
57C2×C8×C31—81no—
58C4⋊C4×C31—41no—
59C4⋊C4×C31—41no—
60C4⋊C4×C31—82no—
61C22⋊C4×C31—41no—
62C22⋊C4×C31—41no—
63C22⋊C4×C31—41no—
64C22⋊C4×C31—41no—
65C22⋊C4×C31—42no—
66C22⋊C4×C31—81no—
67C42×C31—21no—
68C42×C31C4211yescharacteristic
69C42×C31C4⋊C411yescharacteristic
70#7508C3111yesmaximal, characteristic, direct product factor, Sylow
71C23×C31—21no—
72C23×C31—21no—
73C23×C31—41no—
74C23×C31—41no—
75C23×C31—81no—
76C23×C31—81no—
77C23×C31#9911yescharacteristic
78Q8×C31—41no—
79Q8×C31—41no—
80Q8×C31—81no—
81D4×C31—41no—
82D4×C31—41no—
83D4×C31—81no—
84C2×C4×C31—21no—
85C2×C4×C31—21no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—21no—
89C2×C4×C31—21no—
90C2×C4×C31—41no—
91C2×C4×C31—41no—
92C2×C4×C31—41no—
93C2×C4×C31—41no—
94C2×C4×C31—82no—
95C2×C4×C31—82no—
96C2×C4×C31—81no—
97C2×C4×C31—161no—
98C2×C4×C31—161no—
99C2×C4×C31#9511yescharacteristic
100C2×C4×C31#9911yescharacteristic
101C8×C31—42no—
102C8×C31—82no—
103C2×#355C2×C3111yescharacteristic
104C2×#354C2×C3111yescharacteristic
105#1959C2×C3111yescharacteristic
106C22×C31—21no—
107C22×C31—21no—
108C22×C31—41no—
109C22×C31—41no—
110C22×C31—81no—
111C22×C31—81no—
112C22×C31—81no—
113C22×C31—81no—
114C22×C31—161no—
115C22×C31#34211yescharacteristic
116C22×C31#35311yescharacteristic
117C22×C31#35411yescharacteristic
118C4×C31—21no—
119C4×C31—21no—
120C4×C31—41no—
121C4×C31—41no—
122C4×C31—41no—
123C4×C31—41no—
124C4×C31—162no—
125C4×C31—161no—
126C2×#124C22×C3111yescharacteristic
127C2×#103—21no—
128C2×#102—21no—
129C2×#101—21no—
130C2×#99—21no—
131C2×#97C4×C3111yescharacteristic, complemented
132#384C4×C3111yescharacteristic, complemented
133#355—21no—
134#355C4×C3112yescomplemented
135#354—21no—
136#354C4×C3112yescomplemented
137C2×C31—21no—
138C2×C31—21no—
139C2×C31—81no—
140C2×C31—81no—
141C2×C31#139311yescharacteristic
142C2×C31#140611yescharacteristic
143C2×C31#140711yescharacteristic
144C22×Q8D4×C3111yescharacteristic
145C22×D4D4×C3111yescharacteristic
146C2×M16—41no—
147C22×C8—21no—
148#124—41no—
149#124C2×C4×C3111yescharacteristic
150#124C2×C4×C3111yescharacteristic
151#124D4×C3111yescharacteristic
152#124Q8×C3111yescharacteristic
153C2×C4⋊C4—21no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42C2×C4×C3111yescharacteristic
158#103—42no—
159#102—42no—
160#101—22no—
161#101—41no—
162#99—22no—
163#99—41no—
164#97—41no—
165C31#750811yescharacteristic, direct product factor, Sylow
166C24—21no—
167C2×Q8—21no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8C22⋊C4×C3111yescharacteristic
171C2×Q8C4⋊C4×C3111yescharacteristic
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4C22⋊C4×C3111yescharacteristic
176C2×D4C4⋊C4×C3111yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—41no—
180C22×C4—81no—
181C22×C4C22⋊C4×C3111yescharacteristic
182M16—42no—
183M16—81no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—81no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—82no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C42—21no—
198C42C42×C3111yescharacteristic
199C42C4⋊C4×C3111yescharacteristic
200C23—21no—
201C23—21no—
202C23—41no—
203C23—41no—
204C23—81no—
205C23—81no—
206C23C31×#9911yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—81no—
210D4—41no—
211D4—41no—
212D4—81no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—161no—
227C2×C4—161no—
228C2×C4C31×#9511yescharacteristic
229C2×C4C31×#9911yescharacteristic
230C8—42no—
231C8—82no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—161no—
241C22C31×#34211yescharacteristic
242C22C31×#35311yescharacteristic
243C22C31×#35411yescharacteristic
244C4—21no—
245C4—21no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—162no—
251C4—161no—
252C2—21no—
253C2—21no—
254C2—81no—
255C2—81no—
256C2C31×#139311yescharacteristic
257C2C31×#140611yescharacteristic
258C2C31×#140711yescharacteristic
subgroupG/H
center, socleC22×C31#353
commutatorC42C42×C31
Frattini, cosocleC2×#124C22×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×C4×C31 ◁ C2×C31×#124 ◁ G

Complements

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

NH ≅ G/Ntype
9. C2×C31×#97C4semidirect
10. C31×#384C4semidirect
12. C31×#355C4semidirect
14. C31×#354C4semidirect
70. #7508C31direct (N × H)
131. C2×#97C4×C31semidirect
132. #384C4×C31semidirect
134. #355C4×C31semidirect
136. #354C4×C31semidirect
165. C31#7508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3552
C2×C31×#3542
C31×#19592
#750831