Finite Group Databasebeta

← C31×#8494 #21465253

Subgroups — C31×#8494 #21465253

Order 7936.

732 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3548C211yesmaximal, characteristic
2C31×#1995C211yesmaximal, characteristic, complemented
3C31×#1557C212yesmaximal, complemented
4C31×#1537C211yesmaximal, characteristic, complemented
5C31×#1505C212yesmaximal, complemented
6C31×#562—22no—
7C31×#554—22no—
8C31×#550—22no—
9C31×#547C414yescomplemented
10C31×#540—22no—
11C31×#537—24no—
12C31×#533C2211yescharacteristic
13C31×#532C2212yes—
14C31×#432C2212yes—
15C31×#423C2211yescharacteristic
16C31×#376C2211yescharacteristic
17C31×#339—22no—
18C31×#332—22no—
19C31×#331—22no—
20C31×#330—22no—
21C31×#327—24no—
22C2×#41×C31—24no—
23C2×#41×C31—22no—
24C2×M16×C31—21no—
25C31×C4⋊Q8C2×C412yes—
26C31×C4⋊Q8D412yes—
27C31×#127—22no—
28C31×#127D412yes—
29C31×#125—22no—
30C31×#125D412yes—
31C31×#124—22no—
32C31×#123—28no—
33C31×C22⋊Q8—28no—
34C31×C22⋊Q8—22no—
35C31×C22⋊Q8—24no—
36C31×#121—28no—
37C31×#121—28no—
38C31×#121—24no—
39C31×#121—22no—
40C31×#121D414yes—
41C4×Q8×C31—22no—
42C4×D4×C31—24no—
43C4×D4×C31—22no—
44C4×D4×C31C2×C414yes—
45C31×#117D412yes—
46C2×C4⋊C4×C31—24no—
47C2×C42×C31—21no—
48C2×C42×C31C2311yescharacteristic
49C31×C4⋊C8—22no—
50C31×C4⋊C8—22no—
51C31×C22⋊C8—22no—
52C31×#97—22no—
53C31×#95—22no—
54#41×C31—84no—
55#41×C31—88no—
56C2×Q8×C31—22no—
57C2×Q8×C31—22no—
58C2×D4×C31—28no—
59C2×D4×C31—24no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31C22⋊C414yes—
63C22×C4×C31—28no—
64C22×C4×C31—24no—
65C22×C4×C31—41no—
66C22×C4×C31C2×D411yescharacteristic
67C22×C4×C31C2×D412yes—
68M16×C31—42no—
69C2×C8×C31—42no—
70C2×C8×C31—41no—
71C4⋊C4×C31—28no—
72C4⋊C4×C31—22no—
73C4⋊C4×C31—22no—
74C4⋊C4×C31—24no—
75C4⋊C4×C31C22⋊C414yes—
76C4⋊C4×C31C22⋊C414yes—
77C22⋊C4×C31—28no—
78C22⋊C4×C31—22no—
79C22⋊C4×C31—24no—
80C22⋊C4×C31—24no—
81C42×C31—22no—
82C42×C31—22no—
83C42×C31—41no—
84C42×C31C22×C411yescharacteristic
85C42×C31C2×D411yescharacteristic
86C42×C31C2×D412yes—
87#8494C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—24no—
89C23×C31#12011yescharacteristic
90Q8×C31—44no—
91Q8×C31—82no—
92D4×C31—44no—
93D4×C31—42no—
94D4×C31—42no—
95D4×C31—82no—
96D4×C31—84no—
97D4×C31—88no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—24no—
101C2×C4×C31—24no—
102C2×C4×C31—42no—
103C2×C4×C31—48no—
104C2×C4×C31—42no—
105C2×C4×C31—88no—
106C2×C4×C31—82no—
107C2×C4×C31C2×C22⋊C411yescharacteristic
108C2×C4×C31C2×C22⋊C412yes—
109C2×C4×C31#12012yes—
110C2×C4×C31#12011yescharacteristic
111C8×C31—82no—
112C8×C31—81no—
113#3548C2×C3111yescharacteristic
114#1995C2×C3111yescharacteristic, complemented
115#1557C2×C3112yescomplemented
116#1537C2×C3111yescharacteristic, complemented
117#1505C2×C3112yescomplemented
118C22×C31—21no—
119C22×C31—22no—
120C22×C31—44no—
121C22×C31—44no—
122C22×C31—44no—
123C22×C31#37911yescharacteristic
124C4×C31—21no—
125C4×C31—22no—
126C4×C31—22no—
127C4×C31—44no—
128C4×C31—44no—
129C4×C31—82no—
130C4×C31#45411yescharacteristic
131C4×C31#45611yescharacteristic
132#562—22no—
133#554—22no—
134#550—22no—
135#547C4×C3114yescomplemented
136#540—22no—
137#537—24no—
138#533C22×C3111yescharacteristic
139#532C22×C3112yes—
140#432C22×C3112yes—
141#423C22×C3111yescharacteristic
142#376C22×C3111yescharacteristic
143#339—22no—
144#332—22no—
145#331—22no—
146#330—22no—
147#327—24no—
148C2×C31—41no—
149C2×C31—84no—
150C2×C31#179311yescharacteristic
151C2×C31#179512yes—
152C2×#41—24no—
153C2×#41—22no—
154C2×M16—21no—
155C4⋊Q8C2×C4×C3112yes—
156C4⋊Q8D4×C3112yes—
157#127—22no—
158#127D4×C3112yes—
159#125—22no—
160#125D4×C3112yes—
161#124—22no—
162#123—28no—
163C22⋊Q8—28no—
164C22⋊Q8—22no—
165C22⋊Q8—24no—
166#121—28no—
167#121—28no—
168#121—24no—
169#121—22no—
170#121D4×C3114yes—
171C4×Q8—22no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4C2×C4×C3114yes—
175#117D4×C3112yes—
176C2×C4⋊C4—24no—
177C2×C42—21no—
178C2×C42C23×C3111yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C22⋊C8—22no—
182#97—22no—
183#95—22no—
184C31#849411yescharacteristic, direct product factor, Sylow
185#41—84no—
186#41—88no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4C22⋊C4×C3114yes—
194C22×C4—28no—
195C22×C4—24no—
196C22×C4—41no—
197C22×C4C2×D4×C3111yescharacteristic
198C22×C4C2×D4×C3112yes—
199M16—42no—
200C2×C8—42no—
201C2×C8—41no—
202C4⋊C4—28no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4C22⋊C4×C3114yes—
207C4⋊C4C22⋊C4×C3114yes—
208C22⋊C4—28no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—24no—
212C42—22no—
213C42—22no—
214C42—41no—
215C42C22×C4×C3111yescharacteristic
216C42C2×D4×C3111yescharacteristic
217C42C2×D4×C3112yes—
218C23—24no—
219C23C31×#12011yescharacteristic
220Q8—44no—
221Q8—82no—
222D4—44no—
223D4—42no—
224D4—42no—
225D4—82no—
226D4—84no—
227D4—88no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—42no—
235C2×C4—88no—
236C2×C4—82no—
237C2×C4C2×C22⋊C4×C3111yescharacteristic
238C2×C4C2×C22⋊C4×C3112yes—
239C2×C4C31×#12012yes—
240C2×C4C31×#12011yescharacteristic
241C8—82no—
242C8—81no—
243C22—22no—
244C22—21no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22C31×#37911yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4C31×#45411yescharacteristic
256C4C31×#45611yescharacteristic
257C2—41no—
258C2—84no—
259C2C31×#179311yescharacteristic
260C2C31×#179512yes—
subgroupG/H
center, socleC22×C31#379
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC31×#3548C2
LCSG ▷ 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×#1995C2semidirect
3. C31×#1557C2semidirect
4. C31×#1537C2semidirect
5. C31×#1505C2semidirect
9. C31×#547C4semidirect
87. #8494C31direct (N × H)
114. #1995C2×C31semidirect
115. #1557C2×C31semidirect
116. #1537C2×C31semidirect
117. #1505C2×C31semidirect
135. #547C4×C31semidirect
184. C31#8494direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#35482
C31×#19952
C31×#15572
C31×#15372
C31×#15052
#849431