Finite Group Databasebeta

← C31×#10327 #21467086

Subgroups — C31×#10327 #21467086

Order 7936.

806 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2436C212yesmaximal, complemented
3C31×#1784C211yesmaximal, characteristic, complemented
4C31×#1507C212yesmaximal, complemented
5C31×#1297C211yesmaximal, characteristic, complemented
6C23×Q8×C31C2211yescharacteristic
7C2×C31×#124—22no—
8C2×C31×#121C2212yes—
9C31×#481—22no—
10C31×#469—22no—
11C31×#447—22no—
12C2×C31×#101—22no—
13C31×#407C2211yescharacteristic
14C31×#407C2211yescharacteristic
15C31×#393—24no—
16C31×#391—22no—
17C31×#386—24no—
18C4×C22⋊C4×C31C2212yes—
19C31×#327—24no—
20C22×Q8×C31—21no—
21C22×Q8×C31—22no—
22C22×Q8×C31—22no—
23C22×Q8×C31—21no—
24C22×Q8×C31—44no—
25C22×D4×C31—22no—
26C22×D4×C31D412yes—
27C23×C4×C31D412yes—
28C23×C4×C31C2311yescharacteristic
29C2×D8×C31—22no—
30C2×M16×C31—21no—
31C2×M16×C31D412yes—
32C31×#124—48no—
33C31×#121—22no—
34C31×#121—22no—
35C31×#121—22no—
36C31×#121—22no—
37C2×C4⋊C4×C31D412yes—
38C2×C22⋊C4×C31—28no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C42×C31—22no—
42C31×#106—22no—
43C31×#102—24no—
44C31×#101—44no—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×#95—28no—
48C31×#95D414yes—
49C24×C31—22no—
50C24×C31C2×D411yescharacteristic
51C2×Q8×C31—24no—
52C2×Q8×C31—28no—
53C2×Q8×C31—44no—
54C2×Q8×C31—42no—
55C2×Q8×C31—48no—
56C2×Q8×C31—44no—
57C2×Q8×C31—41no—
58C2×Q8×C31—42no—
59C2×Q8×C31—41no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—44no—
63C2×D4×C31—48no—
64C2×D4×C31—42no—
65C2×D4×C31—42no—
66C22×C4×C31—24no—
67C22×C4×C31—22no—
68C22×C4×C31—22no—
69C22×C4×C31—24no—
70C22×C4×C31—44no—
71C22×C4×C31—42no—
72C22×C4×C31C2×D412yes—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D412yes—
75C22×C4×C31#4111yescharacteristic
76D8×C31—82no—
77M16×C31—42no—
78M16×C31—42no—
79C2×C8×C31—22no—
80C2×C8×C31—41no—
81C4⋊C4×C31—22no—
82C4⋊C4×C31—22no—
83C22⋊C4×C31—416no—
84C22⋊C4×C31—44no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C42×C31—44no—
88#10327C3111yesmaximal, characteristic, direct product factor, Sylow
89C23×C31—24no—
90C23×C31—41no—
91C23×C31—44no—
92C23×C31—44no—
93C23×C31—42no—
94C23×C31—44no—
95C23×C31#12012yes—
96C23×C31#12111yescharacteristic
97Q8×C31—44no—
98Q8×C31—48no—
99Q8×C31—44no—
100D4×C31—44no—
101D4×C31—82no—
102D4×C31—84no—
103D4×C31—84no—
104C2×C4×C31—24no—
105C2×C4×C31—22no—
106C2×C4×C31—28no—
107C2×C4×C31—24no—
108C2×C4×C31—42no—
109C2×C4×C31—41no—
110C2×C4×C31—44no—
111C2×C4×C31—42no—
112C2×C4×C31—42no—
113C2×C4×C31—44no—
114C2×C4×C31—44no—
115C2×C4×C31—44no—
116C2×C4×C31—42no—
117C2×C4×C31—44no—
118C2×C4×C31#12011yescharacteristic
119C2×C4×C31#12112yes—
120C2×C4×C31#12711yescharacteristic
121C8×C31—42no—
122C8×C31—81no—
123#3192C2×C3111yescharacteristic
124#2436C2×C3112yescomplemented
125#1784C2×C3111yescharacteristic, complemented
126#1507C2×C3112yescomplemented
127#1297C2×C3111yescharacteristic, complemented
128C22×C31—22no—
129C22×C31—22no—
130C22×C31—22no—
131C22×C31—21no—
132C22×C31—44no—
133C22×C31—88no—
134C22×C31—82no—
135C22×C31—82no—
136C22×C31—82no—
137C22×C31#39211yescharacteristic
138C22×C31#45414yes—
139C4×C31—21no—
140C4×C31—22no—
141C4×C31—44no—
142C4×C31—84no—
143C4×C31—82no—
144C4×C31B(4,𝔽2)12yes—
145C23×Q8C22×C3111yescharacteristic
146C2×#124—22no—
147C2×#121C22×C3112yes—
148#481—22no—
149#469—22no—
150#447—22no—
151C2×#101—22no—
152#407C22×C3111yescharacteristic
153#407C22×C3111yescharacteristic
154#393—24no—
155#391—22no—
156#386—24no—
157C4×C22⋊C4C22×C3112yes—
158#327—24no—
159C2×C31—24no—
160C2×C31—41no—
161C2×C31—162no—
162C2×C31#200312yes—
163C2×C31#202111yescharacteristic
164C22×Q8—22no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—21no—
168C22×Q8—44no—
169C22×D4—22no—
170C22×D4D4×C3112yes—
171C23×C4D4×C3112yes—
172C23×C4C23×C3111yescharacteristic
173C2×D8—22no—
174C2×M16—21no—
175C2×M16D4×C3112yes—
176#124—48no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181C2×C4⋊C4D4×C3112yes—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C42—22no—
186#106—22no—
187#102—24no—
188#101—44no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#95—28no—
192#95D4×C3114yes—
193C31#1032711yescharacteristic, direct product factor, Sylow
194C24—22no—
195C24C2×D4×C3111yescharacteristic
196C2×Q8—24no—
197C2×Q8—28no—
198C2×Q8—44no—
199C2×Q8—42no—
200C2×Q8—48no—
201C2×Q8—44no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—42no—
210C2×D4—42no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4C2×D4×C3112yes—
218C22×C4C2×D4×C3111yescharacteristic
219C22×C4C2×D4×C3112yes—
220C22×C4#41×C3111yescharacteristic
221D8—82no—
222M16—42no—
223M16—42no—
224C2×C8—22no—
225C2×C8—41no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C22⋊C4—416no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—44no—
233C23—24no—
234C23—41no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23C31×#12012yes—
240C23C31×#12111yescharacteristic
241Q8—44no—
242Q8—48no—
243Q8—44no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4C31×#12011yescharacteristic
263C2×C4C31×#12112yes—
264C2×C4C31×#12711yescharacteristic
265C8—42no—
266C8—81no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—21no—
271C22—44no—
272C22—88no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22C31×#39211yescharacteristic
277C22C31×#45414yes—
278C4—21no—
279C4—22no—
280C4—44no—
281C4—84no—
282C4—82no—
283C4C31×B(4,𝔽2)12yes—
284C2—24no—
285C2—41no—
286C2—162no—
287C2C31×#200312yes—
288C2C31×#202111yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC23×C4C23×C31
prime radicalC31×#3192C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2436C2semidirect
3. C31×#1784C2semidirect
4. C31×#1507C2semidirect
5. C31×#1297C2semidirect
88. #10327C31direct (N × H)
124. #2436C2×C31semidirect
125. #1784C2×C31semidirect
126. #1507C2×C31semidirect
127. #1297C2×C31semidirect
193. C31#10327direct (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×#24362
C31×#17842
C31×#15072
C31×#12972
#1032731