Finite Group Databasebeta

← C31×#7322 #21464081

Subgroups — C31×#7322 #21464081

Order 7936.

338 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1955C211yesmaximal, characteristic
2C31×#1799C211yesmaximal, characteristic
3C31×#1465C211yesmaximal, characteristic
4C2×C4×D4×C31C2211yescharacteristic
5C2×C31×C4⋊C8C411yescharacteristic, complemented
6C2×C31×#99C412yes—
7C31×#407—21no—
8C2×C31×C22⋊C8—21no—
9C2×C31×C22⋊C8—21no—
10C2×C31×#97C411yescharacteristic, complemented
11C31×#380—21no—
12C31×#342—21no—
13C31×#324C412yescomplemented
14C31×#323—21no—
15C22×D4×C31C2×C411yescharacteristic
16C23×C4×C31D411yescharacteristic
17C23×C4×C31D411yescharacteristic
18C2×M16×C31—41no—
19C22×C8×C31—21no—
20C22×C8×C31—21no—
21C4×D4×C31—41no—
22C4×D4×C31—41no—
23C2×C4⋊C4×C31—41no—
24C2×C4⋊C4×C31C2×C411yescharacteristic
25C2×C22⋊C4×C31—22no—
26C2×C22⋊C4×C31D411yescharacteristic
27C2×C22⋊C4×C31Q811yescharacteristic
28C2×C42×C31C2×C411yescharacteristic
29C31×C4⋊C8—41no—
30C31×#99—42no—
31C31×C22⋊C8—22no—
32C31×C22⋊C8—42no—
33C31×C22⋊C8—42no—
34C31×C22⋊C8—41no—
35C31×#97—41no—
36C31×#95—41no—
37C24×C31C22⋊C411yescharacteristic
38C24×C31C4⋊C411yescharacteristic
39C2×D4×C31—21no—
40C2×D4×C31—21no—
41C2×D4×C31—41no—
42C2×D4×C31—41no—
43C22×C4×C31—21no—
44C22×C4×C31—21no—
45C22×C4×C31—21no—
46C22×C4×C31—21no—
47C22×C4×C31—41no—
48C22×C4×C31—41no—
49C22×C4×C31—41no—
50C22×C4×C31—41no—
51C22×C4×C31—42no—
52C22×C4×C31C4211yescharacteristic
53C22×C4×C31C22⋊C411yescharacteristic
54C22×C4×C31C22⋊C411yescharacteristic
55C22×C4×C31C4⋊C411yescharacteristic
56C22×C4×C31C4⋊C411yescharacteristic
57M16×C31—81no—
58M16×C31—81no—
59C2×C8×C31—22no—
60C2×C8×C31—41no—
61C2×C8×C31—41no—
62C2×C8×C31—42no—
63C2×C8×C31—41no—
64C2×C8×C31—41no—
65C2×C8×C31—41no—
66C4⋊C4×C31—41no—
67C4⋊C4×C31—81no—
68C4⋊C4×C31—81no—
69C22⋊C4×C31—41no—
70C22⋊C4×C31—41no—
71C22⋊C4×C31—42no—
72C22⋊C4×C31—42no—
73C42×C31—21no—
74C42×C31—21no—
75#7322C3111yesmaximal, characteristic, direct product factor, Sylow
76C23×C31—21no—
77C23×C31—21no—
78C23×C31—21no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—41no—
82C23×C31—41no—
83C23×C31—41no—
84C23×C31—41no—
85C23×C31#9511yescharacteristic
86C23×C31#10011yescharacteristic
87C23×C31#10111yescharacteristic
88D4×C31—42no—
89D4×C31—42no—
90C2×C4×C31—21no—
91C2×C4×C31—21no—
92C2×C4×C31—21no—
93C2×C4×C31—21no—
94C2×C4×C31—21no—
95C2×C4×C31—21no—
96C2×C4×C31—21no—
97C2×C4×C31—21no—
98C2×C4×C31—41no—
99C2×C4×C31—41no—
100C2×C4×C31—41no—
101C2×C4×C31—41no—
102C2×C4×C31—41no—
103C2×C4×C31—41no—
104C2×C4×C31—41no—
105C2×C4×C31—81no—
106C2×C4×C31—81no—
107C2×C4×C31—82no—
108C2×C4×C31—82no—
109C2×C4×C31—82no—
110C2×C4×C31#9911yescharacteristic
111C2×C4×C31#9911yescharacteristic
112C8×C31—42no—
113C8×C31—82no—
114C8×C31—81no—
115#1955C2×C3111yescharacteristic
116#1799C2×C3111yescharacteristic
117#1465C2×C3111yescharacteristic
118C22×C31—21no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—41no—
123C22×C31—41no—
124C22×C31—42no—
125C22×C31—41no—
126C22×C31—41no—
127C22×C31—41no—
128C22×C31—41no—
129C22×C31—41no—
130C22×C31—81no—
131C22×C31—81no—
132C22×C31#34211yescharacteristic
133C22×C31#34311yescharacteristic
134C22×C31#34411yescharacteristic
135C4×C31—21no—
136C4×C31—41no—
137C4×C31—41no—
138C4×C31—41no—
139C4×C31—41no—
140C4×C31—162no—
141C4×C31#35311yescharacteristic
142C4×C31#35411yescharacteristic
143C2×C4×D4C22×C3111yescharacteristic
144C2×C4⋊C8C4×C3111yescharacteristic, complemented
145C2×#99C4×C3112yes—
146#407—21no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8—21no—
149C2×#97C4×C3111yescharacteristic, complemented
150#380—21no—
151#342—21no—
152#324C4×C3112yescomplemented
153#323—21no—
154C2×C31—21no—
155C2×C31—21no—
156C2×C31—41no—
157C2×C31—41no—
158C2×C31—81no—
159C2×C31#130511yescharacteristic
160C2×C31#139311yescharacteristic
161C2×C31#139511yescharacteristic
162C22×D4C2×C4×C3111yescharacteristic
163C23×C4D4×C3111yescharacteristic
164C23×C4D4×C3111yescharacteristic
165C2×M16—41no—
166C22×C8—21no—
167C22×C8—21no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4C2×C4×C3111yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4D4×C3111yescharacteristic
174C2×C22⋊C4Q8×C3111yescharacteristic
175C2×C42C2×C4×C3111yescharacteristic
176C4⋊C8—41no—
177#99—42no—
178C22⋊C8—22no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—41no—
182#97—41no—
183#95—41no—
184C31#732211yescharacteristic, direct product factor, Sylow
185C24C22⋊C4×C3111yescharacteristic
186C24C4⋊C4×C3111yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4C42×C3111yescharacteristic
201C22×C4C22⋊C4×C3111yescharacteristic
202C22×C4C22⋊C4×C3111yescharacteristic
203C22×C4C4⋊C4×C3111yescharacteristic
204C22×C4C4⋊C4×C3111yescharacteristic
205M16—81no—
206M16—81no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C31×#9511yescharacteristic
233C23C31×#10011yescharacteristic
234C23C31×#10111yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4C31×#9911yescharacteristic
258C2×C4C31×#9911yescharacteristic
259C8—42no—
260C8—82no—
261C8—81no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22C31×#34211yescharacteristic
277C22C31×#34311yescharacteristic
278C22C31×#34411yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—162no—
285C4C31×#35311yescharacteristic
286C4C31×#35411yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2C31×#130511yescharacteristic
293C2C31×#139311yescharacteristic
294C2C31×#139511yescharacteristic
subgroupG/H
center, socleC22×C31#342
commutatorC22×C4C42×C31
Frattini, cosocleC2×C4×D4C22×C31
prime radicalC22×D4×C31C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C31×C4⋊C8C4semidirect
10. C2×C31×#97C4semidirect
13. C31×#324C4semidirect
75. #7322C31direct (N × H)
144. C2×C4⋊C8C4×C31semidirect
149. C2×#97C4×C31semidirect
152. #324C4×C31semidirect
184. C31#7322direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#19552
C31×#17992
C31×#14652
#732231