Finite Group Databasebeta

← C31×#9028 #21465787

Subgroups — C31×#9028 #21465787

Order 7936.

696 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C31×#397C211yesmaximal, characteristic
2C31×#2312C211yesmaximal, characteristic
3C31×#1823C212yesmaximal, complemented
4C2×C31×#323C211yesmaximal, characteristic
5C31×#1294C212yesmaximal, complemented
6C22×C4⋊C4×C31C2211yescharacteristic
7C22×C22⋊C4×C31—21no—
8C22×C22⋊C4×C31C2211yescharacteristic
9C2×C31×C22⋊C8C2212yes—
10C31×#397—22no—
11C31×#397C414yescomplemented
12C31×#388—24no—
13C31×#380C2212yes—
14C31×#379—21no—
15C4×C22⋊C4×C31—21no—
16C2×C31×#95—21no—
17C2×C31×#95C2211yescharacteristic
18C31×#340—24no—
19C31×#323—22no—
20C25×C31D411yescharacteristic
21C23×C4×C31—21no—
22C23×C4×C31—21no—
23C23×C4×C31D411yescharacteristic
24C23×C4×C31C2311yescharacteristic
25C22×C8×C31—22no—
26C2×C4⋊C4×C31—22no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—21no—
29C2×C4⋊C4×C31C2×C412yes—
30C2×C4⋊C4×C31D414yes—
31C2×C22⋊C4×C31—22no—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—24no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—44no—
38C2×C22⋊C4×C31C2×C412yes—
39C2×C22⋊C4×C31D412yes—
40C2×C42×C31—22no—
41C31×C22⋊C8—22no—
42C31×C22⋊C8—22no—
43C31×#95—22no—
44C31×#95—24no—
45C31×#95—21no—
46C31×#95—22no—
47C31×#95C2×C412yes—
48C24×C31—21no—
49C24×C31—22no—
50C24×C31—21no—
51C24×C31—22no—
52C24×C31—42no—
53C24×C31—42no—
54C24×C31C22⋊C412yes—
55C24×C31C2×D411yescharacteristic
56C22×C4×C31—21no—
57C22×C4×C31—21no—
58C22×C4×C31—22no—
59C22×C4×C31—22no—
60C22×C4×C31—24no—
61C22×C4×C31—42no—
62C22×C4×C31—42no—
63C22×C4×C31—42no—
64C22×C4×C31—42no—
65C22×C4×C31—41no—
66C22×C4×C31—41no—
67C22×C4×C31—42no—
68C22×C4×C31—42no—
69C22×C4×C31—42no—
70C22×C4×C31C22⋊C412yes—
71C22×C4×C31C22×C411yescharacteristic
72C22×C4×C31C2×D412yes—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31#4112yes—
75C2×C8×C31—44no—
76C2×C8×C31—42no—
77C4⋊C4×C31—24no—
78C4⋊C4×C31—44no—
79C4⋊C4×C31—42no—
80C22⋊C4×C31—24no—
81C22⋊C4×C31—24no—
82C22⋊C4×C31—48no—
83C22⋊C4×C31—42no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—82no—
86C22⋊C4×C31—82no—
87C42×C31—82no—
88#9028C3111yesmaximal, characteristic, direct product factor, Sylow
89C23×C31—22no—
90C23×C31—22no—
91C23×C31—22no—
92C23×C31—22no—
93C23×C31—22no—
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—48no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—81no—
102C23×C31—81no—
103C23×C31—84no—
104C23×C31C2×C22⋊C411yescharacteristic
105C23×C31#12011yescharacteristic
106C23×C31#12311yescharacteristic
107C23×C31#13512yes—
108C23×C31AGL(1,ℤ/8)12yes—
109C2×C4×C31—24no—
110C2×C4×C31—22no—
111C2×C4×C31—41no—
112C2×C4×C31—41no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—42no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—44no—
123C2×C4×C31—44no—
124C2×C4×C31—44no—
125C2×C4×C31—81no—
126C2×C4×C31—84no—
127C2×C4×C31—82no—
128C2×C4×C31—82no—
129C2×C4×C31C4×D412yes—
130C2×C4×C31#12112yes—
131C8×C31—82no—
132C2×#397C2×C3111yescharacteristic
133#2312C2×C3111yescharacteristic
134#1823C2×C3112yescomplemented
135C2×#323C2×C3111yescharacteristic
136#1294C2×C3112yescomplemented
137C22×C31—24no—
138C22×C31—24no—
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—44no—
144C22×C31—42no—
145C22×C31—42no—
146C22×C31—42no—
147C22×C31—82no—
148C22×C31—84no—
149C22×C31—84no—
150C22×C31#38611yescharacteristic
151C22×C31#41611yescharacteristic
152C22×C31#41811yescharacteristic
153C22×C31#44912yes—
154C22×C31#45512yes—
155C22×C31#45812yes—
156C22×C31#48212yes—
157C4×C31—22no—
158C4×C31—22no—
159C4×C31—42no—
160C4×C31—82no—
161C4×C31—84no—
162C4×C31—82no—
163C22×C4⋊C4C22×C3111yescharacteristic
164C22×C22⋊C4—21no—
165C22×C22⋊C4C22×C3111yescharacteristic
166C2×C22⋊C8C22×C3112yes—
167#397—22no—
168#397C4×C3114yescomplemented
169#388—24no—
170#380C22×C3112yes—
171#379—21no—
172C4×C22⋊C4—21no—
173C2×#95—21no—
174C2×#95C22×C3111yescharacteristic
175#340—24no—
176#323—22no—
177C2×C31—21no—
178C2×C31—21no—
179C2×C31—22no—
180C2×C31—82no—
181C2×C31#161214yes—
182C2×C31#187511yescharacteristic
183C2×C31#188511yescharacteristic
184C2×C31#189011yescharacteristic
185C25D4×C3111yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4D4×C3111yescharacteristic
189C23×C4C23×C3111yescharacteristic
190C22×C8—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×C4×C3112yes—
195C2×C4⋊C4D4×C3114yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4C2×C4×C3112yes—
204C2×C22⋊C4D4×C3112yes—
205C2×C42—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#95—22no—
209#95—24no—
210#95—21no—
211#95—22no—
212#95C2×C4×C3112yes—
213C31#902811yescharacteristic, direct product factor, Sylow
214C24—21no—
215C24—22no—
216C24—21no—
217C24—22no—
218C24—42no—
219C24—42no—
220C24C22⋊C4×C3112yes—
221C24C2×D4×C3111yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊C4×C3112yes—
237C22×C4C22×C4×C3111yescharacteristic
238C22×C4C2×D4×C3112yes—
239C22×C4C2×D4×C3111yescharacteristic
240C22×C4#41×C3112yes—
241C2×C8—44no—
242C2×C8—42no—
243C4⋊C4—24no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—48no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—82no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—48no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23C2×C22⋊C4×C3111yescharacteristic
270C23C31×#12011yescharacteristic
271C23C31×#12311yescharacteristic
272C23C31×#13512yes—
273C23C31×AGL(1,ℤ/8)12yes—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C4×D4×C3112yes—
295C2×C4C31×#12112yes—
296C8—82no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22C31×#38611yescharacteristic
311C22C31×#41611yescharacteristic
312C22C31×#41811yescharacteristic
313C22C31×#44912yes—
314C22C31×#45512yes—
315C22C31×#45812yes—
316C22C31×#48212yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—84no—
322C4—82no—
323C2—21no—
324C2—21no—
325C2—22no—
326C2—82no—
327C2C31×#161214yes—
328C2C31×#187511yescharacteristic
329C2C31×#188511yescharacteristic
330C2C31×#189011yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC25×C31D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1823C2semidirect
5. C31×#1294C2semidirect
11. C31×#397C4semidirect
88. #9028C31direct (N × H)
134. #1823C2×C31semidirect
136. #1294C2×C31semidirect
168. #397C4×C31semidirect
213. C31#9028direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3972
C31×#23122
C31×#18232
C2×C31×#3232
C31×#12942
#902831