Finite Group Databasebeta

← C31×#7502 #21464261

Subgroups — C31×#7502 #21464261

Order 7936.

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

normal subgroups only

This group has 286 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×#1958C211yesmaximal, characteristic
4C2×C31×#124C2211yescharacteristic
5C2×C31×#103—21no—
6C2×C31×#102—21no—
7C2×C31×#101—21no—
8C2×C31×#99—21no—
9C2×C4×C8×C31C411yescharacteristic, complemented
10C31×#382C411yescharacteristic, 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×C22⋊C4×C31—41no—
26C2×C22⋊C4×C31—41no—
27C2×C42×C31C2×C411yescharacteristic
28C31×#103—42no—
29C31×#102—42no—
30C31×#101—22no—
31C31×#101—41no—
32C31×#99—22no—
33C31×#99—41no—
34C4×C8×C31—41no—
35C31×#95—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—
60C22⋊C4×C31—41no—
61C22⋊C4×C31—41no—
62C22⋊C4×C31—41no—
63C22⋊C4×C31—41no—
64C22⋊C4×C31—42no—
65C22⋊C4×C31—81no—
66C42×C31—21no—
67C42×C31C4211yescharacteristic
68C42×C31C4⋊C411yescharacteristic
69#7502C3111yesmaximal, characteristic, direct product factor, Sylow
70C23×C31—21no—
71C23×C31—21no—
72C23×C31—41no—
73C23×C31—41no—
74C23×C31—82no—
75C23×C31#9911yescharacteristic
76Q8×C31—41no—
77Q8×C31—41no—
78Q8×C31—81no—
79D4×C31—41no—
80D4×C31—41no—
81D4×C31—81no—
82C2×C4×C31—21no—
83C2×C4×C31—21no—
84C2×C4×C31—21no—
85C2×C4×C31—21no—
86C2×C4×C31—21no—
87C2×C4×C31—21no—
88C2×C4×C31—42no—
89C2×C4×C31—41no—
90C2×C4×C31—41no—
91C2×C4×C31—82no—
92C2×C4×C31—82no—
93C2×C4×C31—81no—
94C2×C4×C31—162no—
95C2×C4×C31#9511yescharacteristic
96C2×C4×C31#9911yescharacteristic
97C8×C31—42no—
98C8×C31—82no—
99C2×#355C2×C3111yescharacteristic
100C2×#354C2×C3111yescharacteristic
101#1958C2×C3111yescharacteristic
102C22×C31—22no—
103C22×C31—41no—
104C22×C31—41no—
105C22×C31—81no—
106C22×C31—81no—
107C22×C31—82no—
108C22×C31—161no—
109C22×C31#34211yescharacteristic
110C22×C31#35311yescharacteristic
111C22×C31#35411yescharacteristic
112C4×C31—21no—
113C4×C31—21no—
114C4×C31—41no—
115C4×C31—41no—
116C4×C31—41no—
117C4×C31—41no—
118C4×C31—161no—
119C4×C31—162no—
120C2×#124C22×C3111yescharacteristic
121C2×#103—21no—
122C2×#102—21no—
123C2×#101—21no—
124C2×#99—21no—
125C2×C4×C8C4×C3111yescharacteristic, complemented
126#382C4×C3111yescharacteristic, complemented
127#355—21no—
128#355C4×C3112yescomplemented
129#354—21no—
130#354C4×C3112yescomplemented
131C2×C31—21no—
132C2×C31—21no—
133C2×C31—81no—
134C2×C31—81no—
135C2×C31#139311yescharacteristic
136C2×C31#141412yes—
137C22×Q8D4×C3111yescharacteristic
138C22×D4D4×C3111yescharacteristic
139C2×M16—41no—
140C22×C8—21no—
141#124—41no—
142#124C2×C4×C3111yescharacteristic
143#124C2×C4×C3111yescharacteristic
144#124D4×C3111yescharacteristic
145#124Q8×C3111yescharacteristic
146C2×C4⋊C4—21no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42C2×C4×C3111yescharacteristic
150#103—42no—
151#102—42no—
152#101—22no—
153#101—41no—
154#99—22no—
155#99—41no—
156C4×C8—41no—
157#95—41no—
158C31#750211yescharacteristic, direct product factor, Sylow
159C24—21no—
160C2×Q8—21no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8C22⋊C4×C3111yescharacteristic
164C2×Q8C4⋊C4×C3111yescharacteristic
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4C22⋊C4×C3111yescharacteristic
169C2×D4C4⋊C4×C3111yescharacteristic
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—81no—
174C22×C4C22⋊C4×C3111yescharacteristic
175M16—42no—
176M16—81no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—81no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—81no—
189C42—21no—
190C42C42×C3111yescharacteristic
191C42C4⋊C4×C3111yescharacteristic
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—82no—
197C23C31×#9911yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—81no—
201D4—41no—
202D4—41no—
203D4—81no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—162no—
217C2×C4C31×#9511yescharacteristic
218C2×C4C31×#9911yescharacteristic
219C8—42no—
220C8—82no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—82no—
227C22—161no—
228C22C31×#34211yescharacteristic
229C22C31×#35311yescharacteristic
230C22C31×#35411yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—161no—
238C4—162no—
239C2—21no—
240C2—21no—
241C2—81no—
242C2—81no—
243C2C31×#139311yescharacteristic
244C2C31×#141412yes—
subgroupG/H
center, socleC22×C31#354
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×C4×C8×C31C4semidirect
10. C31×#382C4semidirect
12. C31×#355C4semidirect
14. C31×#354C4semidirect
69. #7502C31direct (N × H)
125. C2×C4×C8C4×C31semidirect
126. #382C4×C31semidirect
128. #355C4×C31semidirect
130. #354C4×C31semidirect
158. C31#7502direct (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×#19582
#750231