Finite Group Databasebeta

← C31×#12298 #21469057

Subgroups — C31×#12298 #21469057

Order 7936.

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

normal subgroups only

This group has 376 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×#3192C211yesmaximal, characteristic
2C31×#2238C212yesmaximal, complemented
3C31×#2091C211yesmaximal, characteristic, complemented
4C31×#1341C212yesmaximal, complemented
5C31×#1320C211yesmaximal, characteristic, complemented
6C2×C31×#121—21no—
7C31×#481—22no—
8C31×#469—21no—
9C31×#469C2212yes—
10C31×#447—22no—
11C31×#430—21no—
12C31×#430C2211yescharacteristic, complemented
13C31×#407—21no—
14C31×#407C2211yescharacteristic
15C31×#365—22no—
16C31×#361—22no—
17C31×#349C2212yescomplemented
18C31×#338C2211yescharacteristic, complemented
19C22×D4×C31—21no—
20C22×D4×C31—41no—
21C23×C4×C31D411yescharacteristic
22C2×D8×C31—21no—
23C2×D8×C31D412yes—
24C2×M16×C31—21no—
25C2×M16×C31—21no—
26C2×M16×C31D411yescharacteristic
27C2×M16×C31C2311yescharacteristic
28C31×#121—22no—
29C31×#121—22no—
30C31×#121—41no—
31C31×#121—41no—
32C2×C4⋊C4×C31—21no—
33C2×C22⋊C4×C31—41no—
34C31×M32—22no—
35C31×#108—22no—
36C31×#108—42no—
37C31×#106—21no—
38C31×#106D412yes—
39C31×#102—22no—
40C31×#102—22no—
41C31×C22⋊C8—22no—
42C31×C22⋊C8—42no—
43C24×C31—41no—
44C24×C31SD1611yescharacteristic
45C2×D4×C31—22no—
46C2×D4×C31—41no—
47C2×D4×C31—42no—
48C2×D4×C31—42no—
49C2×D4×C31—81no—
50C2×D4×C31—81no—
51C2×D4×C31—84no—
52C22×C4×C31—21no—
53C22×C4×C31—21no—
54C22×C4×C31—41no—
55C22×C4×C31—41no—
56C22×C4×C31—41no—
57C22×C4×C31SD1611yescharacteristic
58C22×C4×C31C2×D411yescharacteristic
59D8×C31—42no—
60D8×C31—81no—
61M16×C31—41no—
62M16×C31—41no—
63M16×C31—41no—
64M16×C31—41no—
65M16×C31—41no—
66M16×C31—41no—
67C2×C8×C31—22no—
68C2×C8×C31—21no—
69C2×C8×C31—41no—
70C2×C8×C31C2×D412yes—
71C4⋊C4×C31—22no—
72C4⋊C4×C31—41no—
73C22⋊C4×C31—42no—
74C22⋊C4×C31—42no—
75C16×C31—42no—
76#12298C3111yesmaximal, characteristic, direct product factor, Sylow
77C23×C31—21no—
78C23×C31—41no—
79C23×C31—42no—
80C23×C31—42no—
81C23×C31—82no—
82C23×C31—82no—
83C23×C31—82no—
84C23×C31C2×SD1611yescharacteristic
85D4×C31—42no—
86D4×C31—42no—
87D4×C31—82no—
88D4×C31—84no—
89D4×C31—84no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—41no—
94C2×C4×C31—41no—
95C2×C4×C31—41no—
96C2×C4×C31—42no—
97C2×C4×C31—81no—
98C2×C4×C31—82no—
99C2×C4×C31#12011yescharacteristic
100C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
101C8×C31—22no—
102C8×C31—42no—
103C8×C31—41no—
104C8×C31—81no—
105#3192C2×C3111yescharacteristic
106#2238C2×C3112yescomplemented
107#2091C2×C3111yescharacteristic, complemented
108#1341C2×C3112yescomplemented
109#1320C2×C3111yescharacteristic, complemented
110C22×C31—22no—
111C22×C31—21no—
112C22×C31—41no—
113C22×C31—41no—
114C22×C31—42no—
115C22×C31—41no—
116C22×C31—82no—
117C22×C31—82no—
118C22×C31—81no—
119C22×C31—82no—
120C22×C31—164no—
121C22×C31#45011yescharacteristic
122C4×C31—21no—
123C4×C31—41no—
124C4×C31—82no—
125C4×C31#45311yescharacteristic
126C4×C31#45811yescharacteristic
127C2×#121—21no—
128#481—22no—
129#469—21no—
130#469C22×C3112yes—
131#447—22no—
132#430—21no—
133#430C22×C3111yescharacteristic, complemented
134#407—21no—
135#407C22×C3111yescharacteristic
136#365—22no—
137#361—22no—
138#349C22×C3112yescomplemented
139#338C22×C3111yescharacteristic, complemented
140C2×C31—21no—
141C2×C31—41no—
142C2×C31—42no—
143C2×C31—162no—
144C2×C31#161311yescharacteristic
145C22×D4—21no—
146C22×D4—41no—
147C23×C4D4×C3111yescharacteristic
148C2×D8—21no—
149C2×D8D4×C3112yes—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16D4×C3111yescharacteristic
153C2×M16C23×C3111yescharacteristic
154#121—22no—
155#121—22no—
156#121—41no—
157#121—41no—
158C2×C4⋊C4—21no—
159C2×C22⋊C4—41no—
160M32—22no—
161#108—22no—
162#108—42no—
163#106—21no—
164#106D4×C3112yes—
165#102—22no—
166#102—22no—
167C22⋊C8—22no—
168C22⋊C8—42no—
169C31#1229811yescharacteristic, direct product factor, Sylow
170C24—41no—
171C24SD16×C3111yescharacteristic
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—84no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4SD16×C3111yescharacteristic
185C22×C4C2×D4×C3111yescharacteristic
186D8—42no—
187D8—81no—
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193M16—41no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8C2×D4×C3112yes—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C16—42no—
203C23—21no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—82no—
208C23—82no—
209C23—82no—
210C23C2×SD16×C3111yescharacteristic
211D4—42no—
212D4—42no—
213D4—82no—
214D4—84no—
215D4—84no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4C31×#12011yescharacteristic
226C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
227C8—22no—
228C8—42no—
229C8—41no—
230C8—81no—
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—82no—
238C22—82no—
239C22—81no—
240C22—82no—
241C22—164no—
242C22C31×#45011yescharacteristic
243C4—21no—
244C4—41no—
245C4—82no—
246C4C31×#45311yescharacteristic
247C4C31×#45811yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—162no—
252C2C31×#161311yescharacteristic
subgroupG/H
center, socleC2×C31#1613
commutator, Frattini, cosocleC2×M16C23×C31
prime radicalC31×#3192C2
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
2. C31×#2238C2semidirect
3. C31×#2091C2semidirect
4. C31×#1341C2semidirect
5. C31×#1320C2semidirect
12. C31×#430C22semidirect
17. C31×#349C22semidirect
18. C31×#338C22semidirect
76. #12298C31direct (N × H)
106. #2238C2×C31semidirect
107. #2091C2×C31semidirect
108. #1341C2×C31semidirect
109. #1320C2×C31semidirect
133. #430C22×C31semidirect
138. #349C22×C31semidirect
139. #338C22×C31semidirect
169. C31#12298direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31922
C31×#22382
C31×#20912
C31×#13412
C31×#13202
#1229831