Finite Group Databasebeta

← C31×#7398 #21464157

Subgroups — C31×#7398 #21464157

Order 7936.

258 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2018C211yesmaximal, characteristic
2C31×#1391C211yesmaximal, characteristic, complemented
3C31×#1321C211yesmaximal, characteristic, complemented
4C2×C31×AGL(1,ℤ/8)—21no—
5C22×D8×C31C411yescharacteristic, complemented
6C2×C31×#100—21no—
7C31×#410—21no—
8C31×#408C2211yescharacteristic
9C31×#349—21no—
10C31×#341C411yescharacteristic, complemented
11C2×#41×C31D411yescharacteristic
12C22×D4×C31—21no—
13C31×AGL(1,ℤ/8)—42no—
14C31×AGL(1,ℤ/8)—42no—
15C2×SD16×C31—41no—
16C2×D8×C31—41no—
17C2×D8×C31—41no—
18C2×D8×C31—42no—
19C2×M16×C31—21no—
20C2×M16×C31D411yescharacteristic
21C22×C8×C31C2×C411yescharacteristic
22C31×#117—41no—
23C31×M32—41no—
24C31×#100—42no—
25C31×#99—22no—
26C31×#99—41no—
27C31×C22⋊C8—41no—
28C24×C31—41no—
29#41×C31—41no—
30#41×C31—41no—
31C2×Q8×C31SD1611yescharacteristic
32C2×D4×C31—22no—
33C2×D4×C31—21no—
34C2×D4×C31—41no—
35C2×D4×C31—41no—
36C2×D4×C31—41no—
37C2×D4×C31—41no—
38C2×D4×C31—41no—
39C2×D4×C31D811yescharacteristic
40C22×C4×C31—21no—
41C22×C4×C31C22⋊C411yescharacteristic
42SD16×C31—44no—
43D8×C31—42no—
44D8×C31—44no—
45D8×C31—42no—
46M16×C31—22no—
47M16×C31—41no—
48M16×C31—41no—
49C2×C8×C31—21no—
50C2×C8×C31—21no—
51C2×C8×C31—41no—
52C2×C8×C31—41no—
53C4⋊C4×C31—81no—
54C22⋊C4×C31—42no—
55C42×C31—81no—
56C16×C31—81no—
57#7398C3111yesmaximal, characteristic, direct product factor, Sylow
58C23×C31—21no—
59C23×C31—42no—
60C23×C31—81no—
61C23×C31—81no—
62C23×C31—162no—
63C23×C31#10411yescharacteristic
64Q8×C31—41no—
65D4×C31—41no—
66D4×C31—42no—
67D4×C31—42no—
68D4×C31—81no—
69D4×C31—81no—
70D4×C31—81no—
71C2×C4×C31—21no—
72C2×C4×C31—21no—
73C2×C4×C31—21no—
74C2×C4×C31—41no—
75C2×C4×C31—41no—
76C2×C4×C31—82no—
77C2×C4×C31#9911yescharacteristic
78C2×C4×C31#10211yescharacteristic
79C8×C31—42no—
80C8×C31—41no—
81C8×C31—41no—
82#2018C2×C3111yescharacteristic
83#1391C2×C3111yescharacteristic, complemented
84#1321C2×C3111yescharacteristic, complemented
85C22×C31—21no—
86C22×C31—41no—
87C22×C31—41no—
88C22×C31—81no—
89C22×C31—82no—
90C22×C31—162no—
91C22×C31—162no—
92C22×C31—162no—
93C22×C31#32711yescharacteristic
94C4×C31—21no—
95C4×C31—41no—
96C4×C31—162no—
97C4×C31AΣL(1,𝔽16)11yescharacteristic
98C4×C31#35311yescharacteristic
99C2×AGL(1,ℤ/8)—21no—
100C22×D8C4×C3111yescharacteristic, complemented
101C2×#100—21no—
102#410—21no—
103#408C22×C3111yescharacteristic
104#349—21no—
105#341C4×C3111yescharacteristic, complemented
106C2×C31—21no—
107C2×C31—41no—
108C2×C31—81no—
109C2×C31—162no—
110C2×C31#134311yescharacteristic
111C2×#41D4×C3111yescharacteristic
112C22×D4—21no—
113AGL(1,ℤ/8)—42no—
114AGL(1,ℤ/8)—42no—
115C2×SD16—41no—
116C2×D8—41no—
117C2×D8—41no—
118C2×D8—42no—
119C2×M16—21no—
120C2×M16D4×C3111yescharacteristic
121C22×C8C2×C4×C3111yescharacteristic
122#117—41no—
123M32—41no—
124#100—42no—
125#99—22no—
126#99—41no—
127C22⋊C8—41no—
128C31#739811yescharacteristic, direct product factor, Sylow
129C24—41no—
130#41—41no—
131#41—41no—
132C2×Q8SD16×C3111yescharacteristic
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4D8×C3111yescharacteristic
141C22×C4—21no—
142C22×C4C22⋊C4×C3111yescharacteristic
143SD16—44no—
144D8—42no—
145D8—42no—
146D8—44no—
147M16—22no—
148M16—41no—
149M16—41no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—41no—
153C2×C8—41no—
154C4⋊C4—81no—
155C22⋊C4—42no—
156C42—81no—
157C16—81no—
158C23—21no—
159C23—42no—
160C23—81no—
161C23—81no—
162C23—162no—
163C23C31×#10411yescharacteristic
164Q8—41no—
165D4—41no—
166D4—42no—
167D4—42no—
168D4—81no—
169D4—81no—
170D4—81no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—82no—
177C2×C4C31×#9911yescharacteristic
178C2×C4C31×#10211yescharacteristic
179C8—42no—
180C8—41no—
181C8—41no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—81no—
186C22—82no—
187C22—162no—
188C22—162no—
189C22—162no—
190C22C31×#32711yescharacteristic
191C4—21no—
192C4—41no—
193C4—162no—
194C4C31×AΣL(1,𝔽16)11yescharacteristic
195C4C31×#35311yescharacteristic
196C2—21no—
197C2—41no—
198C2—81no—
199C2—162no—
200C2C31×#134311yescharacteristic
subgroupG/H
center, socleC2×C31#1343
commutatorC22×C8C2×C4×C31
Frattini, cosocle#408C22×C31
prime radicalC31×#2018C2
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C22×C4×C31 ◁ C31×#408 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1391C2semidirect
3. C31×#1321C2semidirect
5. C22×D8×C31C4semidirect
10. C31×#341C4semidirect
57. #7398C31direct (N × H)
83. #1391C2×C31semidirect
84. #1321C2×C31semidirect
100. C22×D8C4×C31semidirect
105. #341C4×C31semidirect
128. C31#7398direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#20182
C31×#13912
C31×#13212
#739831