Finite Group Databasebeta

← C31×#7249 #21464008

Subgroups — C31×#7249 #21464008

Order 7936.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1954C211yesmaximal, characteristic
2C31×#1465C212yesmaximal
3C2×C4×D4×C31C2211yescharacteristic
4C2×C31×C4⋊C8C411yescharacteristic
5C31×#407—22no—
6C2×C31×C22⋊C8—22no—
7C2×C4×C8×C31C411yescharacteristic
8C31×#324—22no—
9C31×#323C414yes—
10C22×D4×C31Q811yescharacteristic
11C23×C4×C31D412yes—
12C2×M16×C31—42no—
13C22×C8×C31—21no—
14C22×C8×C31—21no—
15C4×D4×C31—41no—
16C4×D4×C31—41no—
17C2×C4⋊C4×C31D411yescharacteristic
18C2×C22⋊C4×C31C2×C412yes—
19C2×C42×C31C2×C411yescharacteristic
20C31×C4⋊C8—41no—
21C31×C22⋊C8—24no—
22C31×C22⋊C8—44no—
23C31×C22⋊C8—42no—
24C4×C8×C31—41no—
25C24×C31C4⋊C412yes—
26C2×D4×C31—22no—
27C2×D4×C31—41no—
28C2×D4×C31—41no—
29C22×C4×C31—22no—
30C22×C4×C31—22no—
31C22×C4×C31—42no—
32C22×C4×C31—42no—
33C22×C4×C31C4211yescharacteristic
34C22×C4×C31C22⋊C412yes—
35C22×C4×C31C22⋊C411yescharacteristic
36C22×C4×C31C4⋊C411yescharacteristic
37M16×C31—82no—
38M16×C31—82no—
39C2×C8×C31—22no—
40C2×C8×C31—41no—
41C2×C8×C31—41no—
42C2×C8×C31—44no—
43C2×C8×C31—42no—
44C2×C8×C31—41no—
45C4⋊C4×C31—41no—
46C22⋊C4×C31—42no—
47C42×C31—21no—
48C42×C31—21no—
49#7249C3111yesmaximal, characteristic, direct product factor, Sylow
50C23×C31—22no—
51C23×C31—22no—
52C23×C31—22no—
53C23×C31—42no—
54C23×C31—42no—
55C23×C31#9511yescharacteristic
56D4×C31—44no—
57C2×C4×C31—21no—
58C2×C4×C31—21no—
59C2×C4×C31—21no—
60C2×C4×C31—21no—
61C2×C4×C31—21no—
62C2×C4×C31—22no—
63C2×C4×C31—42no—
64C2×C4×C31—42no—
65C2×C4×C31—42no—
66C2×C4×C31—41no—
67C2×C4×C31—82no—
68C2×C4×C31#9911yescharacteristic
69C2×C4×C31#9911yescharacteristic
70C2×C4×C31#10812yes—
71C8×C31—42no—
72C8×C31—84no—
73C8×C31—81no—
74#1954C2×C3111yescharacteristic
75#1465C2×C3112yes—
76C22×C31—21no—
77C22×C31—21no—
78C22×C31—44no—
79C22×C31—42no—
80C22×C31—42no—
81C22×C31—42no—
82C22×C31—82no—
83C22×C31#33811yescharacteristic
84C22×C31#34111yescharacteristic
85C22×C31#34211yescharacteristic
86C4×C31—22no—
87C4×C31—21no—
88C4×C31—41no—
89C4×C31—42no—
90C4×C31#35311yescharacteristic
91C4×C31#35411yescharacteristic
92C2×C4×D4C22×C3111yescharacteristic
93C2×C4⋊C8C4×C3111yescharacteristic
94#407—22no—
95C2×C22⋊C8—22no—
96C2×C4×C8C4×C3111yescharacteristic
97#324—22no—
98#323C4×C3114yes—
99C2×C31—21no—
100C2×C31—21no—
101C2×C31—82no—
102C2×C31#129311yescharacteristic
103C2×C31#139211yescharacteristic
104C2×C31#139311yescharacteristic
105C22×D4Q8×C3111yescharacteristic
106C23×C4D4×C3112yes—
107C2×M16—42no—
108C22×C8—21no—
109C22×C8—21no—
110C4×D4—41no—
111C4×D4—41no—
112C2×C4⋊C4D4×C3111yescharacteristic
113C2×C22⋊C4C2×C4×C3112yes—
114C2×C42C2×C4×C3111yescharacteristic
115C4⋊C8—41no—
116C22⋊C8—24no—
117C22⋊C8—44no—
118C22⋊C8—42no—
119C4×C8—41no—
120C31#724911yescharacteristic, direct product factor, Sylow
121C24C4⋊C4×C3112yes—
122C2×D4—22no—
123C2×D4—41no—
124C2×D4—41no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C42×C3111yescharacteristic
130C22×C4C22⋊C4×C3111yescharacteristic
131C22×C4C22⋊C4×C3112yes—
132C22×C4C4⋊C4×C3111yescharacteristic
133M16—82no—
134M16—82no—
135C2×C8—22no—
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—44no—
139C2×C8—42no—
140C2×C8—41no—
141C4⋊C4—41no—
142C22⋊C4—42no—
143C42—21no—
144C42—21no—
145C23—22no—
146C23—22no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23C31×#9511yescharacteristic
151D4—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—82no—
163C2×C4C31×#9911yescharacteristic
164C2×C4C31×#9911yescharacteristic
165C2×C4C31×#10812yes—
166C8—42no—
167C8—84no—
168C8—81no—
169C22—21no—
170C22—21no—
171C22—44no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—82no—
176C22C31×#33811yescharacteristic
177C22C31×#34111yescharacteristic
178C22C31×#34211yescharacteristic
179C4—22no—
180C4—21no—
181C4—41no—
182C4—42no—
183C4C31×#35311yescharacteristic
184C4C31×#35411yescharacteristic
185C2—21no—
186C2—21no—
187C2—82no—
188C2C31×#129311yescharacteristic
189C2C31×#139211yescharacteristic
190C2C31×#139311yescharacteristic
subgroupG/H
center, socleC22×C31#342
commutatorC22×C4C42×C31
Frattini, cosocleC2×C4×D4C22×C31
prime radicalC22×D4×C31Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
49. #7249C31direct (N × H)
120. C31#7249direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#19542
C31×#14652
#724931