Finite Group Databasebeta

← C31×#12171 #21468930

Subgroups — C31×#12171 #21468930

Order 7936.

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

normal subgroups only

This group has 332 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×#3261C211yesmaximal, characteristic, complemented
2C31×#2191C212yesmaximal
3C31×#2053C211yesmaximal, characteristic
4C31×#1340C212yesmaximal, complemented
5C31×#1328C211yesmaximal, characteristic, complemented
6C31×#575C2211yescharacteristic
7C31×#541—21no—
8C31×#510—22no—
9C31×#479—22no—
10C31×#470C2212yes—
11C31×#467—21no—
12C31×#462—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×Q16×C31—21no—
24C2×Q16×C31D412yes—
25C2×M16×C31C2311yescharacteristic
26C22×C8×C31D411yescharacteristic
27C31×C4⋊Q8—21no—
28C31×#126—42no—
29C31×#125—21no—
30C31×#125—42no—
31C31×C22⋊Q8—22no—
32C4×Q8×C31—22no—
33C31×#117—21no—
34C31×#117D411yescharacteristic
35C31×Q32—42no—
36C31×SD32—42no—
37C31×M32—22no—
38C31×#107—21no—
39C31×#106D412yes—
40C31×C4⋊C8—22no—
41C31×#103—22no—
42C31×#103—22no—
43C31×#103—41no—
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×Q8×C31—22no—
52C2×D4×C31—21no—
53C2×D4×C31—41no—
54C22×C4×C31—41no—
55C22×C4×C31C2×D411yescharacteristic
56Q16×C31—22no—
57Q16×C31—42no—
58Q16×C31—81no—
59SD16×C31—42no—
60D8×C31—22no—
61M16×C31—41no—
62C2×C8×C31—21no—
63C2×C8×C31—41no—
64C2×C8×C31C2×D412yes—
65C4⋊C4×C31—21no—
66C4⋊C4×C31—22no—
67C4⋊C4×C31—42no—
68C4⋊C4×C31—42no—
69C4⋊C4×C31—42no—
70C4⋊C4×C31—41no—
71C4⋊C4×C31—42no—
72C22⋊C4×C31—42no—
73C22⋊C4×C31—41no—
74C22⋊C4×C31D812yes—
75C42×C31—21no—
76C42×C31—42no—
77C42×C31—41no—
78C16×C31—42no—
79#12171C3111yesmaximal, characteristic, direct product factor, Sylow
80C23×C31—41no—
81C23×C31C2×D811yescharacteristic
82Q8×C31—41no—
83Q8×C31—42no—
84Q8×C31—41no—
85Q8×C31—42no—
86D4×C31—41no—
87D4×C31—41no—
88D4×C31—81no—
89D4×C31—81no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—22no—
93C2×C4×C31—41no—
94C2×C4×C31—42no—
95C2×C4×C31—42no—
96C2×C4×C31—81no—
97C2×C4×C31—81no—
98C2×C4×C31—81no—
99C2×C4×C31#12011yescharacteristic
100C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
101C8×C31—22no—
102C8×C31—41no—
103#3261C2×C3111yescharacteristic, complemented
104#2191C2×C3112yes—
105#2053C2×C3111yescharacteristic
106#1340C2×C3112yescomplemented
107#1328C2×C3111yescharacteristic, complemented
108C22×C31—21no—
109C22×C31—41no—
110C22×C31—81no—
111C22×C31—161no—
112C22×C31#44711yescharacteristic
113C4×C31—21no—
114C4×C31—42no—
115C4×C31—82no—
116C4×C31—81no—
117C4×C31—82no—
118C4×C31—161no—
119C4×C31#45411yescharacteristic
120C4×C31B(4,𝔽2)11yescharacteristic
121#575C22×C3111yescharacteristic
122#541—21no—
123#510—22no—
124#479—22no—
125#470C22×C3112yes—
126#467—21no—
127#462—22no—
128#424C22×C3111yescharacteristic
129#416—21no—
130#410—21no—
131#360—22no—
132#349C22×C3112yes—
133H(3,ℤ/4)C22×C3111yescharacteristic
134C2×C31—21no—
135C2×C31—41no—
136C2×C31—161no—
137C2×C31#159811yescharacteristic
138C2×#41—21no—
139#137—41no—
140AGL(1,ℤ/8)—41no—
141#135—22no—
142C2×Q16—21no—
143C2×Q16D4×C3112yes—
144C2×M16C23×C3111yescharacteristic
145C22×C8D4×C3111yescharacteristic
146C4⋊Q8—21no—
147#126—42no—
148#125—21no—
149#125—42no—
150C22⋊Q8—22no—
151C4×Q8—22no—
152#117—21no—
153#117D4×C3111yescharacteristic
154Q32—42no—
155SD32—42no—
156M32—22no—
157#107—21no—
158#106D4×C3112yes—
159C4⋊C8—22no—
160#103—22no—
161#103—22no—
162#103—41no—
163#102—41no—
164#99—42no—
165C31#1217111yescharacteristic, direct product factor, Sylow
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×D4—21no—
173C2×D4—41no—
174C22×C4—41no—
175C22×C4C2×D4×C3111yescharacteristic
176Q16—22no—
177Q16—42no—
178Q16—81no—
179SD16—42no—
180D8—22no—
181M16—41no—
182C2×C8—21no—
183C2×C8—41no—
184C2×C8C2×D4×C3112yes—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—41no—
194C22⋊C4D8×C3112yes—
195C42—21no—
196C42—42no—
197C42—41no—
198C16—42no—
199C23—41no—
200C23C2×D8×C3111yescharacteristic
201Q8—41no—
202Q8—42no—
203Q8—41no—
204Q8—42no—
205D4—41no—
206D4—41no—
207D4—81no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4C31×#12011yescharacteristic
219C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
220C8—22no—
221C8—41no—
222C22—21no—
223C22—41no—
224C22—81no—
225C22—161no—
226C22C31×#44711yescharacteristic
227C4—21no—
228C4—42no—
229C4—82no—
230C4—81no—
231C4—82no—
232C4—161no—
233C4C31×#45411yescharacteristic
234C4C31×B(4,𝔽2)11yescharacteristic
235C2—21no—
236C2—41no—
237C2—161no—
238C2C31×#159811yescharacteristic
subgroupG/H
center, socleC2×C31#1598
commutator, Frattini, cosocleC2×M16C23×C31
prime radicalC31×#575C22
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×#3261C2semidirect
4. C31×#1340C2semidirect
5. C31×#1328C2semidirect
79. #12171C31direct (N × H)
103. #3261C2×C31semidirect
106. #1340C2×C31semidirect
107. #1328C2×C31semidirect
165. C31#12171direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#32612
C31×#21912
C31×#20532
C31×#13402
C31×#13282
#1217131