Finite Group Databasebeta

← C31×#10220 #21466979

Subgroups — C31×#10220 #21466979

Order 7936.

542 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3208C212yesmaximal
2C31×#3162C211yesmaximal, characteristic, complemented
3C31×#1961C211yesmaximal, characteristic
4C31×#1518C212yesmaximal, complemented
5C31×#1281C211yesmaximal, characteristic, complemented
6C31×#533C2212yes—
7C2×C31×#127C2211yescharacteristic
8C31×#501—22no—
9C31×#501—22no—
10C31×#492—22no—
11C31×#486—22no—
12C31×#460—24no—
13C31×#450—24no—
14C31×#431C2212yes—
15C31×#427—22no—
16C2×C31×#102—22no—
17C2×C31×#97C2211yescharacteristic
18C31×#382C2211yescharacteristic
19C31×#328—24no—
20C22×D4×C31—21no—
21C22×D4×C31—22no—
22C2×SD16×C31—28no—
23C22×C8×C31D412yes—
24C31×C4⋊Q8D412yes—
25C31×C4⋊Q8D412yes—
26C31×#127—22no—
27C31×#127—21no—
28C31×#127D412yes—
29C31×#125—22no—
30C31×C22⋊Q8—22no—
31C31×C22⋊Q8—22no—
32C4×Q8×C31—22no—
33C4×Q8×C31—22no—
34C31×#117—22no—
35C2×C4⋊C4×C31—21no—
36C2×C42×C31C2311yescharacteristic
37C31×#107—24no—
38C31×#106—22no—
39C31×#106—22no—
40C31×C4⋊C8—22no—
41C31×#102—28no—
42C31×#102—44no—
43C31×C22⋊C8D414yes—
44C31×#97—22no—
45C4×C8×C31—22no—
46C31×#95—22no—
47C24×C31—41no—
48C2×Q8×C31—22no—
49C2×Q8×C31—22no—
50C2×D4×C31—24no—
51C2×D4×C31—22no—
52C2×D4×C31—41no—
53C2×D4×C31—42no—
54C2×D4×C31—42no—
55C2×D4×C31—42no—
56C2×D4×C31—84no—
57C22×C4×C31—41no—
58C22×C4×C31C2×D412yes—
59C22×C4×C31C2×D411yescharacteristic
60SD16×C31—88no—
61C2×C8×C31—22no—
62C2×C8×C31—22no—
63C2×C8×C31—24no—
64C2×C8×C31SD1614yescomplemented
65C4⋊C4×C31—22no—
66C4⋊C4×C31—22no—
67C4⋊C4×C31—22no—
68C4⋊C4×C31—22no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31—42no—
72C22⋊C4×C31—42no—
73C42×C31—42no—
74C42×C31C2×D412yes—
75C42×C31C2×D411yescharacteristic
76C42×C31#4111yescharacteristic
77#10220C3111yesmaximal, characteristic, direct product factor, Sylow
78C23×C31—42no—
79C23×C31—84no—
80C23×C31—81no—
81C23×C31—81no—
82C23×C31#12711yescharacteristic
83Q8×C31—44no—
84Q8×C31—82no—
85D4×C31—48no—
86D4×C31—44no—
87D4×C31—84no—
88D4×C31—82no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—82no—
93C2×C4×C31—81no—
94C2×C4×C31#12012yes—
95C2×C4×C31#12011yescharacteristic
96C2×C4×C31#12112yes—
97C2×C4×C31#12111yescharacteristic
98C2×C4×C31C2×SD1612yes—
99C2×C4×C31#13712yes—
100C8×C31—24no—
101C8×C31—44no—
102#3208C2×C3112yes—
103#3162C2×C3111yescharacteristic, complemented
104#1961C2×C3111yescharacteristic
105#1518C2×C3112yescomplemented
106#1281C2×C3111yescharacteristic, complemented
107C22×C31—22no—
108C22×C31—84no—
109C22×C31—82no—
110C22×C31—162no—
111C22×C31—162no—
112C22×C31#39211yescharacteristic
113C22×C31#49211yescharacteristic
114C22×C31#49711yescharacteristic
115C4×C31—22no—
116C4×C31—22no—
117C4×C31—82no—
118C4×C31—82no—
119C4×C31—161no—
120C4×C31#44812yes—
121C4×C31#46112yes—
122#533C22×C3112yes—
123C2×#127C22×C3111yescharacteristic
124#501—22no—
125#501—22no—
126#492—22no—
127#486—22no—
128#460—24no—
129#450—24no—
130#431C22×C3112yes—
131#427—22no—
132C2×#102—22no—
133C2×#97C22×C3111yescharacteristic
134#382C22×C3111yescharacteristic
135#328—24no—
136C2×C31—21no—
137C2×C31—21no—
138C2×C31—162no—
139C2×C31#200812yes—
140C2×C31#201311yescharacteristic
141C22×D4—21no—
142C22×D4—22no—
143C2×SD16—28no—
144C22×C8D4×C3112yes—
145C4⋊Q8D4×C3112yes—
146C4⋊Q8D4×C3112yes—
147#127—22no—
148#127—21no—
149#127D4×C3112yes—
150#125—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155#117—22no—
156C2×C4⋊C4—21no—
157C2×C42C23×C3111yescharacteristic
158#107—24no—
159#106—22no—
160#106—22no—
161C4⋊C8—22no—
162#102—28no—
163#102—44no—
164C22⋊C8D4×C3114yes—
165#97—22no—
166C4×C8—22no—
167#95—22no—
168C31#1022011yescharacteristic, direct product factor, Sylow
169C24—41no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—84no—
179C22×C4—41no—
180C22×C4C2×D4×C3112yes—
181C22×C4C2×D4×C3111yescharacteristic
182SD16—88no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8SD16×C3114yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C22⋊C4—42no—
195C42—42no—
196C42C2×D4×C3112yes—
197C42C2×D4×C3111yescharacteristic
198C42#41×C3111yescharacteristic
199C23—42no—
200C23—84no—
201C23—81no—
202C23—81no—
203C23C31×#12711yescharacteristic
204Q8—44no—
205Q8—82no—
206D4—48no—
207D4—44no—
208D4—84no—
209D4—82no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4C31×#12012yes—
216C2×C4C31×#12011yescharacteristic
217C2×C4C31×#12112yes—
218C2×C4C31×#12111yescharacteristic
219C2×C4C2×SD16×C3112yes—
220C2×C4C31×#13712yes—
221C8—24no—
222C8—44no—
223C22—22no—
224C22—84no—
225C22—82no—
226C22—162no—
227C22—162no—
228C22C31×#39211yescharacteristic
229C22C31×#49211yescharacteristic
230C22C31×#49711yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—161no—
236C4C31×#44812yes—
237C4C31×#46112yes—
238C2—21no—
239C2—21no—
240C2—162no—
241C2C31×#200812yes—
242C2C31×#201311yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#3162C2semidirect
4. C31×#1518C2semidirect
5. C31×#1281C2semidirect
64. C2×C8×C31SD16semidirect
77. #10220C31direct (N × H)
103. #3162C2×C31semidirect
105. #1518C2×C31semidirect
106. #1281C2×C31semidirect
168. C31#10220direct (N × H)
186. C2×C8SD16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#32082
C31×#31622
C31×#19612
C31×#15182
C31×#12812
#1022031