Finite Group Databasebeta

← C31×#9988 #21466747

Subgroups — C31×#9988 #21466747

Order 7936.

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

normal subgroups only

This group has 750 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
2C2×C31×#448C212yesmaximal, complemented
3C31×#1784C211yesmaximal, characteristic, complemented
4C31×#1471C212yesmaximal, complemented
5C31×#1298C211yesmaximal, characteristic, complemented
6C23×Q8×C31C2211yescharacteristic
7C22×SD16×C31—22no—
8C2×C31×#121C2212yes—
9C31×#481—22no—
10C31×#469—22no—
11C31×#448—24no—
12C31×#448—24no—
13C31×#447—22no—
14C2×C31×#103—22no—
15C2×C31×#101—22no—
16C31×#407C2211yescharacteristic
17C31×#407C2211yescharacteristic
18C2×C31×C22⋊C8C2212yes—
19C31×#323—24no—
20C22×Q8×C31—21no—
21C22×Q8×C31—22no—
22C22×Q8×C31—22no—
23C22×Q8×C31—44no—
24C22×Q8×C31D412yes—
25C22×D4×C31—22no—
26C22×D4×C31D412yes—
27C23×C4×C31—21no—
28C23×C4×C31C2311yescharacteristic
29C2×SD16×C31—28no—
30C2×SD16×C31—48no—
31C2×D8×C31—22no—
32C2×M16×C31—21no—
33C2×M16×C31D412yes—
34C22×C8×C31—22no—
35C31×#121—22no—
36C31×#121—22no—
37C31×#121—22no—
38C31×#121—22no—
39C2×C4⋊C4×C31D412yes—
40C2×C22⋊C4×C31—22no—
41C31×#106—22no—
42C31×#103—28no—
43C31×#102—24no—
44C31×#101—44no—
45C31×C22⋊C8—22no—
46C31×C22⋊C8—22no—
47C31×C22⋊C8—22no—
48C31×C22⋊C8D414yes—
49C24×C31—22no—
50C24×C31#4111yescharacteristic
51C2×Q8×C31—24no—
52C2×Q8×C31—28no—
53C2×Q8×C31—24no—
54C2×Q8×C31—44no—
55C2×Q8×C31—44no—
56C2×Q8×C31—42no—
57C2×Q8×C31—48no—
58C2×Q8×C31—41no—
59C2×Q8×C31—41no—
60C2×D4×C31—22no—
61C2×D4×C31—22no—
62C2×D4×C31—44no—
63C2×D4×C31—42no—
64C2×D4×C31—42no—
65C2×D4×C31—84no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31—42no—
69C22×C4×C31—44no—
70C22×C4×C31—42no—
71C22×C4×C31C2×D412yes—
72C22×C4×C31C2×D411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D412yes—
75SD16×C31—416no—
76D8×C31—82no—
77M16×C31—42no—
78M16×C31—42no—
79C2×C8×C31—22no—
80C2×C8×C31—24no—
81C2×C8×C31—44no—
82C2×C8×C31—41no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—22no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87#9988C3111yesmaximal, characteristic, direct product factor, Sylow
88C23×C31—24no—
89C23×C31—41no—
90C23×C31—42no—
91C23×C31—44no—
92C23×C31—82no—
93C23×C31—82no—
94C23×C31#12111yescharacteristic
95C23×C31#12112yes—
96Q8×C31—28no—
97Q8×C31—44no—
98Q8×C31—44no—
99Q8×C31—44no—
100D4×C31—44no—
101D4×C31—44no—
102D4×C31—84no—
103D4×C31—84no—
104C2×C4×C31—24no—
105C2×C4×C31—22no—
106C2×C4×C31—24no—
107C2×C4×C31—41no—
108C2×C4×C31—42no—
109C2×C4×C31—42no—
110C2×C4×C31—48no—
111C2×C4×C31—42no—
112C2×C4×C31—42no—
113C2×C4×C31—82no—
114C2×C4×C31#12012yes—
115C2×C4×C31#12011yescharacteristic
116C2×C4×C31#12711yescharacteristic
117C8×C31—44no—
118C8×C31—42no—
119C8×C31—81no—
120#3192C2×C3111yescharacteristic
121C2×#448C2×C3112yescomplemented
122#1784C2×C3111yescharacteristic, complemented
123#1471C2×C3112yescomplemented
124#1298C2×C3111yescharacteristic, complemented
125C22×C31—22no—
126C22×C31—22no—
127C22×C31—22no—
128C22×C31—21no—
129C22×C31—44no—
130C22×C31—82no—
131C22×C31—82no—
132C22×C31—82no—
133C22×C31—164no—
134C22×C31#39211yescharacteristic
135C22×C31#47114yes—
136C4×C31—22no—
137C4×C31—21no—
138C4×C31—44no—
139C4×C31—82no—
140C4×C31B(4,𝔽2)12yes—
141C23×Q8C22×C3111yescharacteristic
142C22×SD16—22no—
143C2×#121C22×C3112yes—
144#481—22no—
145#469—22no—
146#448—24no—
147#448—24no—
148#447—22no—
149C2×#103—22no—
150C2×#101—22no—
151#407C22×C3111yescharacteristic
152#407C22×C3111yescharacteristic
153C2×C22⋊C8C22×C3112yes—
154#323—24no—
155C2×C31—24no—
156C2×C31—41no—
157C2×C31—162no—
158C2×C31#201912yes—
159C2×C31#202111yescharacteristic
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—44no—
164C22×Q8D4×C3112yes—
165C22×D4—22no—
166C22×D4D4×C3112yes—
167C23×C4—21no—
168C23×C4C23×C3111yescharacteristic
169C2×SD16—28no—
170C2×SD16—48no—
171C2×D8—22no—
172C2×M16—21no—
173C2×M16D4×C3112yes—
174C22×C8—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179C2×C4⋊C4D4×C3112yes—
180C2×C22⋊C4—22no—
181#106—22no—
182#103—28no—
183#102—24no—
184#101—44no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8D4×C3114yes—
189C31#998811yescharacteristic, direct product factor, Sylow
190C24—22no—
191C24#41×C3111yescharacteristic
192C2×Q8—24no—
193C2×Q8—28no—
194C2×Q8—24no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—48no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—44no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—84no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4C2×D4×C3112yes—
213C22×C4C2×D4×C3111yescharacteristic
214C22×C4C2×D4×C3111yescharacteristic
215C22×C4C2×D4×C3112yes—
216SD16—416no—
217D8—82no—
218M16—42no—
219M16—42no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—44no—
223C2×C8—41no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C23—24no—
229C23—41no—
230C23—42no—
231C23—44no—
232C23—82no—
233C23—82no—
234C23C31×#12111yescharacteristic
235C23C31×#12112yes—
236Q8—28no—
237Q8—44no—
238Q8—44no—
239Q8—44no—
240D4—44no—
241D4—44no—
242D4—84no—
243D4—84no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—48no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4C31×#12012yes—
255C2×C4C31×#12011yescharacteristic
256C2×C4C31×#12711yescharacteristic
257C8—44no—
258C8—42no—
259C8—81no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—44no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—164no—
269C22C31×#39211yescharacteristic
270C22C31×#47114yes—
271C4—22no—
272C4—21no—
273C4—44no—
274C4—82no—
275C4C31×B(4,𝔽2)12yes—
276C2—24no—
277C2—41no—
278C2—162no—
279C2C31×#201912yes—
280C2C31×#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. C2×C31×#448C2semidirect
3. C31×#1784C2semidirect
4. C31×#1471C2semidirect
5. C31×#1298C2semidirect
87. #9988C31direct (N × H)
121. C2×#448C2×C31semidirect
122. #1784C2×C31semidirect
123. #1471C2×C31semidirect
124. #1298C2×C31semidirect
189. C31#9988direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#31922
C2×C31×#4482
C31×#17842
C31×#14712
C31×#12982
#998831