Finite Group Databasebeta

← C31×#9005 #21465764

Subgroups — C31×#9005 #21465764

Order 7936.

416 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3135C211yesmaximal, characteristic
2C31×#2980C211yesmaximal, characteristic
3C31×#1543C211yesmaximal, characteristic
4C31×#1497C212yesmaximal, complemented
5C31×#1302C212yesmaximal, complemented
6C31×#533C2211yescharacteristic
7C31×#532C2211yescharacteristic
8C31×#491—21no—
9C31×#490—21no—
10C31×#489C414yes—
11C31×#435—24no—
12C31×#432C2212yes—
13C31×#431C2212yes—
14C31×#416—22no—
15C31×#408—22no—
16C2×C31×#97C2211yescharacteristic
17C31×#333—21no—
18C31×#331—21no—
19C31×Q8⋊C8—24no—
20C2×#41×C31D412yes—
21C2×M16×C31—22no—
22C22×C8×C31—21no—
23C22×C8×C31—22no—
24C31×C4⋊Q8C2×C412yes—
25C31×C4⋊Q8D411yescharacteristic
26C31×#127D411yescharacteristic
27C31×#125—21no—
28C31×#124C2×C412yes—
29C31×C22⋊Q8—22no—
30C31×#121—22no—
31C4×Q8×C31—22no—
32C4×D4×C31—22no—
33C31×#117—21no—
34C2×C42×C31C2311yescharacteristic
35C31×C4⋊C8—22no—
36C31×C4⋊C8D414yes—
37C31×#103—24no—
38C31×#102—24no—
39C31×C22⋊C8—24no—
40C31×C22⋊C8—22no—
41C31×C22⋊C8—22no—
42C31×#97—21no—
43C31×#97—22no—
44C31×#97C2×C412yes—
45C4×C8×C31—22no—
46#41×C31—42no—
47#41×C31—42no—
48C2×Q8×C31—22no—
49C2×Q8×C31C22⋊C412yes—
50C2×D4×C31—22no—
51C2×D4×C31—22no—
52C2×D4×C31C22⋊C412yes—
53C22×C4×C31—22no—
54C22×C4×C31C2×D411yescharacteristic
55C22×C4×C31#4112yes—
56M16×C31—82no—
57C2×C8×C31—22no—
58C2×C8×C31—24no—
59C2×C8×C31—42no—
60C2×C8×C31—42no—
61C2×C8×C31—82no—
62C4⋊C4×C31—22no—
63C4⋊C4×C31—22no—
64C4⋊C4×C31—22no—
65C4⋊C4×C31—42no—
66C22⋊C4×C31—24no—
67C22⋊C4×C31—41no—
68C42×C31—41no—
69C42×C31C22×C411yescharacteristic
70C42×C31C2×D411yescharacteristic
71C42×C31C2×D412yes—
72#9005C3111yesmaximal, characteristic, direct product factor, Sylow
73C23×C31—22no—
74C23×C31#12311yescharacteristic
75Q8×C31—42no—
76Q8×C31—44no—
77D4×C31—42no—
78D4×C31—82no—
79D4×C31—82no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—42no—
83C2×C4×C31—42no—
84C2×C4×C31—42no—
85C2×C4×C31—42no—
86C2×C4×C31C2×C22⋊C411yescharacteristic
87C2×C4×C31C4×D412yes—
88C2×C4×C31#12011yescharacteristic
89C2×C4×C31#12112yes—
90C8×C31—42no—
91C8×C31—44no—
92C8×C31—82no—
93#3135C2×C3111yescharacteristic
94#2980C2×C3111yescharacteristic
95#1543C2×C3111yescharacteristic
96#1497C2×C3112yescomplemented
97#1302C2×C3112yescomplemented
98C22×C31—22no—
99C22×C31—21no—
100C22×C31—44no—
101C22×C31—42no—
102C22×C31#38611yescharacteristic
103C4×C31—22no—
104C4×C31—22no—
105C4×C31—21no—
106C4×C31—42no—
107C4×C31—82no—
108C4×C31#45512yes—
109#533C22×C3111yescharacteristic
110#532C22×C3111yescharacteristic
111#491—21no—
112#490—21no—
113#489C4×C3114yes—
114#435—24no—
115#432C22×C3112yes—
116#431C22×C3112yes—
117#416—22no—
118#408—22no—
119C2×#97C22×C3111yescharacteristic
120#333—21no—
121#331—21no—
122Q8⋊C8—24no—
123C2×C31—41no—
124C2×C31—82no—
125C2×C31#188112yes—
126C2×C31#188511yescharacteristic
127C2×#41D4×C3112yes—
128C2×M16—22no—
129C22×C8—21no—
130C22×C8—22no—
131C4⋊Q8C2×C4×C3112yes—
132C4⋊Q8D4×C3111yescharacteristic
133#127D4×C3111yescharacteristic
134#125—21no—
135#124C2×C4×C3112yes—
136C22⋊Q8—22no—
137#121—22no—
138C4×Q8—22no—
139C4×D4—22no—
140#117—21no—
141C2×C42C23×C3111yescharacteristic
142C4⋊C8—22no—
143C4⋊C8D4×C3114yes—
144#103—24no—
145#102—24no—
146C22⋊C8—24no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149#97—21no—
150#97—22no—
151#97C2×C4×C3112yes—
152C4×C8—22no—
153C31#900511yescharacteristic, direct product factor, Sylow
154#41—42no—
155#41—42no—
156C2×Q8—22no—
157C2×Q8C22⋊C4×C3112yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4C22⋊C4×C3112yes—
161C22×C4—22no—
162C22×C4C2×D4×C3111yescharacteristic
163C22×C4#41×C3112yes—
164M16—82no—
165C2×C8—22no—
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—82no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C22⋊C4—24no—
175C22⋊C4—41no—
176C42—41no—
177C42C22×C4×C3111yescharacteristic
178C42C2×D4×C3111yescharacteristic
179C42C2×D4×C3112yes—
180C23—22no—
181C23C31×#12311yescharacteristic
182Q8—42no—
183Q8—44no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4C2×C22⋊C4×C3111yescharacteristic
194C2×C4C4×D4×C3112yes—
195C2×C4C31×#12011yescharacteristic
196C2×C4C31×#12112yes—
197C8—42no—
198C8—44no—
199C8—82no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22C31×#38611yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—82no—
210C4C31×#45512yes—
211C2—41no—
212C2—82no—
213C2C31×#188112yes—
214C2C31×#188511yescharacteristic
subgroupG/H
centerC2×C4×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
socleC22×C31#386
prime radicalC31×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#1497C2semidirect
5. C31×#1302C2semidirect
72. #9005C31direct (N × H)
96. #1497C2×C31semidirect
97. #1302C2×C31semidirect
153. C31#9005direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31352
C31×#29802
C31×#15432
C31×#14972
C31×#13022
#900531