Finite Group Databasebeta

← C31×#12215 #21468974

Subgroups — C31×#12215 #21468974

Order 7936.

400 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2607C211yesmaximal, characteristic
2C31×#2030C211yesmaximal, characteristic
3C31×#1602C212yesmaximal
4C31×#1351C212yesmaximal
5C31×#1317C211yesmaximal, characteristic
6C31×#543C2211yescharacteristic
7C31×#451—22no—
8C31×#399—21no—
9C31×#393C2212yes—
10C31×#391—22no—
11C31×#387—22no—
12C31×#381—21no—
13C31×#381C2211yescharacteristic
14C4×C22⋊C4×C31—21no—
15C31×#342—21no—
16C31×#342C2211yescharacteristic
17C31×#328—22no—
18C31×#323C2212yes—
19C22×Q8×C31D412yes—
20C23×C4×C31D411yescharacteristic
21C2×Q16×C31—42no—
22C31×C22⋊Q8—22no—
23C31×C22⋊Q8—42no—
24C2×C4⋊C4×C31—22no—
25C2×C4⋊C4×C31—21no—
26C2×C22⋊C4×C31—22no—
27C2×C22⋊C4×C31—21no—
28C2×C22⋊C4×C31—21no—
29C2×C22⋊C4×C31D411yescharacteristic
30C2×C22⋊C4×C31C2311yescharacteristic
31C2×C42×C31—21no—
32C2×C42×C31—21no—
33C31×#103—42no—
34C31×C22⋊C8—22no—
35C31×#95—21no—
36C31×#95—22no—
37C31×#95—22no—
38C31×#95—22no—
39C31×#95—42no—
40C31×#95D412yes—
41C24×C31C2×D411yescharacteristic
42C2×Q8×C31—22no—
43C2×Q8×C31—44no—
44C2×Q8×C31—42no—
45C22×C4×C31—21no—
46C22×C4×C31—22no—
47C22×C4×C31—21no—
48C22×C4×C31—21no—
49C22×C4×C31—22no—
50C22×C4×C31—22no—
51C22×C4×C31—41no—
52C22×C4×C31—41no—
53C22×C4×C31—41no—
54C22×C4×C31Q1612yescomplemented
55C22×C4×C31C2×D412yes—
56Q16×C31—84no—
57C2×C8×C31—42no—
58C4⋊C4×C31—22no—
59C4⋊C4×C31—44no—
60C4⋊C4×C31—42no—
61C4⋊C4×C31—81no—
62C22⋊C4×C31—41no—
63C22⋊C4×C31—44no—
64C22⋊C4×C31—41no—
65C22⋊C4×C31—41no—
66C22⋊C4×C31—41no—
67C22⋊C4×C31—41no—
68C22⋊C4×C31—41no—
69C42×C31—81no—
70C42×C31—81no—
71#12215C3111yesmaximal, characteristic, direct product factor, Sylow
72C23×C31—21no—
73C23×C31—21no—
74C23×C31—42no—
75C23×C31#12011yescharacteristic
76C23×C31C2×Q1611yescharacteristic
77C23×C31#13711yescharacteristic
78Q8×C31—44no—
79Q8×C31—82no—
80Q8×C31—82no—
81C2×C4×C31—24no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—42no—
85C2×C4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31—42no—
88C2×C4×C31—42no—
89C2×C4×C31—42no—
90C2×C4×C31—42no—
91C2×C4×C31—42no—
92C2×C4×C31—42no—
93C2×C4×C31—42no—
94C2×C4×C31—41no—
95C2×C4×C31—42no—
96C2×C4×C31—81no—
97C2×C4×C31—82no—
98C2×C4×C31—81no—
99C2×C4×C31—82no—
100C8×C31—82no—
101#2607C2×C3111yescharacteristic
102#2030C2×C3111yescharacteristic
103#1602C2×C3112yes—
104#1351C2×C3112yes—
105#1317C2×C3111yescharacteristic
106C22×C31—22no—
107C22×C31—21no—
108C22×C31—21no—
109C22×C31—41no—
110C22×C31—41no—
111C22×C31—42no—
112C22×C31—81no—
113C22×C31#45111yescharacteristic
114C22×C31#45511yescharacteristic
115C22×C31B(4,𝔽2)11yescharacteristic
116C4×C31—42no—
117C4×C31—42no—
118C4×C31—44no—
119C4×C31—82no—
120C4×C31—81no—
121C4×C31—82no—
122C4×C31—161no—
123#543C22×C3111yescharacteristic
124#451—22no—
125#399—21no—
126#393C22×C3112yes—
127#391—22no—
128#387—22no—
129#381—21no—
130#381C22×C3111yescharacteristic
131C4×C22⋊C4—21no—
132#342—21no—
133#342C22×C3111yescharacteristic
134#328—22no—
135#323C22×C3112yes—
136C2×C31—21no—
137C2×C31—21no—
138C2×C31—41no—
139C2×C31—41no—
140C2×C31#160211yescharacteristic
141C2×C31#219712yes—
142C22×Q8D4×C3112yes—
143C23×C4D4×C3111yescharacteristic
144C2×Q16—42no—
145C22⋊Q8—22no—
146C22⋊Q8—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4D4×C3111yescharacteristic
153C2×C22⋊C4C23×C3111yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156#103—42no—
157C22⋊C8—22no—
158#95—22no—
159#95—21no—
160#95—22no—
161#95—22no—
162#95—42no—
163#95D4×C3112yes—
164C31#1221511yescharacteristic, direct product factor, Sylow
165C24C2×D4×C3111yescharacteristic
166C2×Q8—22no—
167C2×Q8—44no—
168C2×Q8—42no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4Q16×C3112yescomplemented
179C22×C4C2×D4×C3112yes—
180Q16—84no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C22⋊C4—44no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C42—81no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23C31×#12011yescharacteristic
199C23C2×Q16×C3111yescharacteristic
200C23C31×#13711yescharacteristic
201Q8—44no—
202Q8—82no—
203Q8—82no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—82no—
223C8—82no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22C31×#45111yescharacteristic
232C22C31×#45511yescharacteristic
233C22C31×B(4,𝔽2)11yescharacteristic
234C4—42no—
235C4—42no—
236C4—44no—
237C4—82no—
238C4—81no—
239C4—82no—
240C4—161no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2C31×#160211yescharacteristic
246C2C31×#219712yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
54. C22×C4×C31Q16semidirect
71. #12215C31direct (N × H)
164. C31#12215direct (N × H)
178. C22×C4Q16×C31semidirect

Maximal subgroups

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

subgroupindex
C31×#26072
C31×#20302
C31×#16022
C31×#13512
C31×#13172
#1221531