Finite Group Databasebeta

← C31×#9003 #21465762

Subgroups — C31×#9003 #21465762

Order 7936.

578 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 134 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×#497C211yesmaximal, characteristic
2C31×#2379C211yesmaximal, characteristic
3C31×#1993C212yesmaximal, complemented
4C2×C31×#329C211yesmaximal, characteristic
5C31×#1295C212yesmaximal, complemented
6C22×Q16×C31—21no—
7C22×SD16×C31—21no—
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C31×#497—22no—
11C31×#497C414yescomplemented
12C2×C31×#97C2212yes—
13C2×C31×#97C2211yescharacteristic
14C31×#391—21no—
15C31×#388—22no—
16C31×#386—21no—
17C31×#385—22no—
18C31×#382C2212yes—
19C31×#340—22no—
20C31×#340—22no—
21C31×#329—22no—
22C22×Q8×C31—21no—
23C22×Q8×C31D411yescharacteristic
24C22×D4×C31D411yescharacteristic
25C23×C4×C31—21no—
26C2×Q16×C31—24no—
27C2×Q16×C31—42no—
28C2×Q16×C31—42no—
29C2×SD16×C31—24no—
30C2×SD16×C31—42no—
31C2×SD16×C31—42no—
32C22×C8×C31—21no—
33C22×C8×C31—22no—
34C31×C4⋊Q8—21no—
35C31×C4⋊Q8—22no—
36C31×C4⋊Q8C2×C412yescomplemented
37C31×#124—22no—
38C31×#124C2×C412yes—
39C31×#124D412yes—
40C2×C4⋊C4×C31—22no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31D412yes—
43C2×C4⋊C4×C31D412yes—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—21no—
47C2×C42×C31—21no—
48C2×C42×C31C2311yescharacteristic
49C31×#97—21no—
50C31×#97—22no—
51C31×#97—22no—
52C31×#97C2×C412yescomplemented
53C31×#95—22no—
54C31×#95—22no—
55C31×#95—22no—
56C24×C31—21no—
57C2×Q8×C31—21no—
58C2×Q8×C31—22no—
59C2×Q8×C31—41no—
60C2×Q8×C31—41no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C2×Q8×C31C22⋊C412yes—
67C2×D4×C31—21no—
68C2×D4×C31—41no—
69C2×D4×C31—41no—
70C2×D4×C31C22⋊C412yes—
71C22×C4×C31—21no—
72C22×C4×C31—22no—
73C22×C4×C31—22no—
74C22×C4×C31—41no—
75C22×C4×C31—81no—
76C22×C4×C31—82no—
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D412yes—
79Q16×C31—48no—
80SD16×C31—48no—
81C2×C8×C31—22no—
82C2×C8×C31—22no—
83C2×C8×C31—22no—
84C2×C8×C31—24no—
85C2×C8×C31—42no—
86C2×C8×C31—42no—
87C4⋊C4×C31—24no—
88C4⋊C4×C31—42no—
89C4⋊C4×C31—42no—
90C4⋊C4×C31—42no—
91C4⋊C4×C31—81no—
92C22⋊C4×C31—24no—
93C22⋊C4×C31—24no—
94C22⋊C4×C31—81no—
95C22⋊C4×C31—82no—
96C42×C31—81no—
97C42×C31C22×C411yescharacteristic
98C42×C31C2×D411yescharacteristic
99C42×C31#4112yes—
100#9003C3111yesmaximal, characteristic, direct product factor, Sylow
101C23×C31—22no—
102C23×C31—41no—
103C23×C31—41no—
104C23×C31—42no—
105C23×C31—42no—
106C23×C31#12011yescharacteristic
107Q8×C31—42no—
108Q8×C31—42no—
109Q8×C31—42no—
110Q8×C31—81no—
111Q8×C31—81no—
112Q8×C31—81no—
113D4×C31—42no—
114D4×C31—81no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—22no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—42no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—84no—
130C2×C4×C31—81no—
131C2×C4×C31—81no—
132C2×C4×C31C2×C22⋊C411yescharacteristic
133C2×C4×C31C4×D412yes—
134C2×C4×C31#12112yes—
135C2×C4×C31#12311yescharacteristic
136C8×C31—24no—
137C8×C31—44no—
138C2×#497C2×C3111yescharacteristic
139#2379C2×C3111yescharacteristic
140#1993C2×C3112yescomplemented
141C2×#329C2×C3111yescharacteristic
142#1295C2×C3112yescomplemented
143C22×C31—44no—
144C22×C31—42no—
145C22×C31—81no—
146C22×C31—81no—
147C22×C31—82no—
148C22×C31#38611yescharacteristic
149C22×C31#45311yescharacteristic
150C22×C31#45311yescharacteristic
151C22×C31#45512yes—
152C22×C31#45612yes—
153C4×C31—22no—
154C4×C31—22no—
155C4×C31—21no—
156C4×C31—21no—
157C4×C31—42no—
158C4×C31—82no—
159C4×C31—82no—
160C4×C31—82no—
161C22×Q16—21no—
162C22×SD16—21no—
163C2×C4⋊Q8C22×C3111yescharacteristic
164C2×#124C22×C3111yescharacteristic
165#497—22no—
166#497C4×C3114yescomplemented
167C2×#97C22×C3112yes—
168C2×#97C22×C3111yescharacteristic
169#391—21no—
170#388—22no—
171#386—21no—
172#385—22no—
173#382C22×C3112yes—
174#340—22no—
175#340—22no—
176#329—22no—
177C2×C31—82no—
178C2×C31#165911yescharacteristic
179C2×C31#166011yescharacteristic
180C2×C31#166112yes—
181C2×C31#188212yes—
182C2×C31#188711yescharacteristic
183C22×Q8—21no—
184C22×Q8D4×C3111yescharacteristic
185C22×D4D4×C3111yescharacteristic
186C23×C4—21no—
187C2×Q16—24no—
188C2×Q16—42no—
189C2×Q16—42no—
190C2×SD16—24no—
191C2×SD16—42no—
192C2×SD16—42no—
193C22×C8—21no—
194C22×C8—22no—
195C4⋊Q8—21no—
196C4⋊Q8—22no—
197C4⋊Q8C2×C4×C3112yescomplemented
198#124—22no—
199#124C2×C4×C3112yes—
200#124D4×C3112yes—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4D4×C3112yes—
204C2×C4⋊C4D4×C3112yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42C23×C3111yescharacteristic
210#97—21no—
211#97—22no—
212#97—22no—
213#97C2×C4×C3112yescomplemented
214#95—22no—
215#95—22no—
216#95—22no—
217C31#900311yescharacteristic, direct product factor, Sylow
218C24—21no—
219C2×Q8—21no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8C22⋊C4×C3112yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4C22⋊C4×C3112yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—81no—
238C22×C4—82no—
239C22×C4C2×D4×C3111yescharacteristic
240C22×C4C2×D4×C3112yes—
241Q16—48no—
242SD16—48no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—24no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—81no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—81no—
259C42C22×C4×C3111yescharacteristic
260C42C2×D4×C3111yescharacteristic
261C42#41×C3112yes—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23C31×#12011yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—42no—
271Q8—81no—
272Q8—81no—
273Q8—81no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4C2×C22⋊C4×C3111yescharacteristic
294C2×C4C4×D4×C3112yes—
295C2×C4C31×#12112yes—
296C2×C4C31×#12311yescharacteristic
297C8—24no—
298C8—44no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22C31×#38611yescharacteristic
305C22C31×#45311yescharacteristic
306C22C31×#45311yescharacteristic
307C22C31×#45512yes—
308C22C31×#45612yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—82no—
317C2—82no—
318C2C31×#165911yescharacteristic
319C2C31×#166011yescharacteristic
320C2C31×#166112yes—
321C2C31×#188212yes—
322C2C31×#188711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#1993C2semidirect
5. C31×#1295C2semidirect
11. C31×#497C4semidirect
36. C31×C4⋊Q8C2×C4semidirect
52. C31×#97C2×C4semidirect
100. #9003C31direct (N × H)
140. #1993C2×C31semidirect
142. #1295C2×C31semidirect
166. #497C4×C31semidirect
197. C4⋊Q8C2×C4×C31semidirect
213. #97C2×C4×C31semidirect
217. C31#9003direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4972
C31×#23792
C31×#19932
C2×C31×#3292
C31×#12952
#900331