Finite Group Databasebeta

← C31×#12170 #21468929

Subgroups — C31×#12170 #21468929

Order 7936.

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

normal subgroups only

This group has 356 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
1C31×#3259C211yesmaximal, characteristic, complemented
2C31×#2190C212yesmaximal, complemented
3C31×#2053C211yesmaximal, characteristic, complemented
4C31×#1339C212yesmaximal, complemented
5C31×#1328C211yesmaximal, characteristic, complemented
6C31×#575C2211yescharacteristic
7C31×#540—21no—
8C31×#509—22no—
9C31×#478—22no—
10C31×#469C2212yes—
11C31×#466—21no—
12C31×#459—22no—
13C31×#424C2211yescharacteristic
14C31×#416—21no—
15C31×#410—21no—
16C31×#360—22no—
17C31×#349C2212yes—
18C31×H(3,ℤ/4)C2211yescharacteristic
19C2×#41×C31—21no—
20C31×#137—41no—
21C31×AGL(1,ℤ/8)—41no—
22C31×#135—22no—
23C2×D8×C31—21no—
24C2×D8×C31D412yes—
25C2×M16×C31C2311yescharacteristic
26C22×C8×C31D411yescharacteristic
27C31×#127—21no—
28C31×#125—21no—
29C31×#123—42no—
30C31×#121—22no—
31C31×#121—42no—
32C4×D4×C31—22no—
33C31×#117—21no—
34C31×#117D411yescharacteristic
35C31×SD32—42no—
36C31×D16—42no—
37C31×M32—22no—
38C31×#107—21no—
39C31×#106D412yes—
40C31×C4⋊C8—22no—
41C31×#103—41no—
42C31×#102—22no—
43C31×#102—22no—
44C31×#102—41no—
45C31×#99—42no—
46#41×C31—41no—
47#41×C31—41no—
48#41×C31—41no—
49#41×C31—41no—
50C2×Q8×C31—21no—
51C2×D4×C31—21no—
52C2×D4×C31—22no—
53C2×D4×C31—42no—
54C2×D4×C31—41no—
55C2×D4×C31—42no—
56C22×C4×C31—41no—
57C22×C4×C31—42no—
58C22×C4×C31C2×D411yescharacteristic
59Q16×C31—22no—
60SD16×C31—42no—
61D8×C31—22no—
62D8×C31—42no—
63D8×C31—81no—
64M16×C31—41no—
65C2×C8×C31—21no—
66C2×C8×C31—41no—
67C2×C8×C31C2×D412yes—
68C4⋊C4×C31—21no—
69C4⋊C4×C31—22no—
70C4⋊C4×C31—42no—
71C4⋊C4×C31—41no—
72C22⋊C4×C31—42no—
73C22⋊C4×C31—42no—
74C22⋊C4×C31—41no—
75C22⋊C4×C31D812yes—
76C42×C31—21no—
77C42×C31—41no—
78C16×C31—42no—
79#12170C3111yesmaximal, characteristic, direct product factor, Sylow
80C23×C31—41no—
81C23×C31—42no—
82C23×C31C2×D811yescharacteristic
83Q8×C31—41no—
84Q8×C31—41no—
85D4×C31—41no—
86D4×C31—42no—
87D4×C31—41no—
88D4×C31—42no—
89D4×C31—81no—
90D4×C31—81no—
91D4×C31—162no—
92D4×C31—162no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—22no—
96C2×C4×C31—41no—
97C2×C4×C31—42no—
98C2×C4×C31—81no—
99C2×C4×C31—81no—
100C2×C4×C31—81no—
101C2×C4×C31—162no—
102C2×C4×C31#12011yescharacteristic
103C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
104C8×C31—22no—
105C8×C31—41no—
106#3259C2×C3111yescharacteristic, complemented
107#2190C2×C3112yescomplemented
108#2053C2×C3111yescharacteristic, complemented
109#1339C2×C3112yescomplemented
110#1328C2×C3111yescharacteristic, complemented
111C22×C31—21no—
112C22×C31—41no—
113C22×C31—82no—
114C22×C31—82no—
115C22×C31—81no—
116C22×C31—82no—
117C22×C31—161no—
118C22×C31#44711yescharacteristic
119C4×C31—21no—
120C4×C31—42no—
121C4×C31—82no—
122C4×C31—81no—
123C4×C31—161no—
124C4×C31#45411yescharacteristic
125C4×C31B(4,𝔽2)11yescharacteristic
126#575C22×C3111yescharacteristic
127#540—21no—
128#509—22no—
129#478—22no—
130#469C22×C3112yes—
131#466—21no—
132#459—22no—
133#424C22×C3111yescharacteristic
134#416—21no—
135#410—21no—
136#360—22no—
137#349C22×C3112yes—
138H(3,ℤ/4)C22×C3111yescharacteristic
139C2×C31—21no—
140C2×C31—41no—
141C2×C31—162no—
142C2×C31—161no—
143C2×C31#159811yescharacteristic
144C2×#41—21no—
145#137—41no—
146AGL(1,ℤ/8)—41no—
147#135—22no—
148C2×D8—21no—
149C2×D8D4×C3112yes—
150C2×M16C23×C3111yescharacteristic
151C22×C8D4×C3111yescharacteristic
152#127—21no—
153#125—21no—
154#123—42no—
155#121—22no—
156#121—42no—
157C4×D4—22no—
158#117—21no—
159#117D4×C3111yescharacteristic
160SD32—42no—
161D16—42no—
162M32—22no—
163#107—21no—
164#106D4×C3112yes—
165C4⋊C8—22no—
166#103—41no—
167#102—22no—
168#102—22no—
169#102—41no—
170#99—42no—
171C31#1217011yescharacteristic, direct product factor, Sylow
172#41—41no—
173#41—41no—
174#41—41no—
175#41—41no—
176C2×Q8—21no—
177C2×D4—21no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4C2×D4×C3111yescharacteristic
185Q16—22no—
186SD16—42no—
187D8—22no—
188D8—42no—
189D8—81no—
190M16—41no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8C2×D4×C3112yes—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4D8×C3112yes—
202C42—21no—
203C42—41no—
204C16—42no—
205C23—41no—
206C23—42no—
207C23C2×D8×C3111yescharacteristic
208Q8—41no—
209Q8—41no—
210D4—41no—
211D4—42no—
212D4—41no—
213D4—42no—
214D4—81no—
215D4—81no—
216D4—162no—
217D4—162no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—162no—
227C2×C4C31×#12011yescharacteristic
228C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
229C8—22no—
230C8—41no—
231C22—21no—
232C22—41no—
233C22—82no—
234C22—82no—
235C22—81no—
236C22—82no—
237C22—161no—
238C22C31×#44711yescharacteristic
239C4—21no—
240C4—42no—
241C4—82no—
242C4—81no—
243C4—161no—
244C4C31×#45411yescharacteristic
245C4C31×B(4,𝔽2)11yescharacteristic
246C2—21no—
247C2—41no—
248C2—162no—
249C2—161no—
250C2C31×#159811yescharacteristic
subgroupG/H
center, socleC2×C31#1598
commutator, Frattini, cosocleC2×M16C23×C31
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C31 ◁ C2×C4×C31 ◁ C2×M16×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#3259C2semidirect
2. C31×#2190C2semidirect
3. C31×#2053C2semidirect
4. C31×#1339C2semidirect
5. C31×#1328C2semidirect
79. #12170C31direct (N × H)
106. #3259C2×C31semidirect
107. #2190C2×C31semidirect
108. #2053C2×C31semidirect
109. #1339C2×C31semidirect
110. #1328C2×C31semidirect
171. C31#12170direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32592
C31×#21902
C31×#20532
C31×#13392
C31×#13282
#1217031