Finite Group Databasebeta

← C31×#9444 #21466203

Subgroups — C31×#9444 #21466203

Order 7936.

474 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic
2C31×#2499C211yesmaximal, characteristic
3C31×#1820C211yesmaximal, characteristic
4C31×#1511C212yesmaximal
5C31×#1284C212yesmaximal
6C2×C31×C22⋊Q8C2211yescharacteristic
7C2×C31×#117C2211yescharacteristic
8C31×#433—24no—
9C31×#423—21no—
10C31×#407C2212yes—
11C4×M16×C31—21no—
12C31×#397—21no—
13C31×#397—21no—
14C31×#391C411yescharacteristic, complemented
15C31×#391C411yescharacteristic, complemented
16C31×#387—21no—
17C31×#387—21no—
18C31×#384C411yescharacteristic, complemented
19C31×#382C411yescharacteristic, complemented
20C31×#381C2212yes—
21C4×C22⋊C4×C31C2211yescharacteristic
22C31×#342—21no—
23C31×#342—21no—
24C31×#328—24no—
25C22×Q8×C31C2×C411yescharacteristic
26C23×C4×C31C2311yescharacteristic
27C2×M16×C31—22no—
28C31×C22⋊Q8—22no—
29C31×C22⋊Q8—22no—
30C31×#117—22no—
31C31×#117—22no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C4⋊C4×C31C2×C411yescharacteristic
37C2×C4⋊C4×C31C2×C411yescharacteristic
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31D411yescharacteristic
42C2×C22⋊C4×C31D411yescharacteristic
43C2×C42×C31C2×C411yescharacteristic
44C2×C42×C31C2×C411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C2×C42×C31D411yescharacteristic
47C31×C4⋊C8—24no—
48C31×C22⋊C8—22no—
49C31×C22⋊C8D414yes—
50C31×#97—22no—
51C4×C8×C31—22no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—22no—
60C31×#95—22no—
61C24×C31C2×D411yescharacteristic
62C2×Q8×C31—22no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—41no—
70C22×C4×C31—41no—
71C22×C4×C31—42no—
72C22×C4×C31C22⋊C411yescharacteristic
73C22×C4×C31C22⋊C411yescharacteristic
74C22×C4×C31C22⋊C411yescharacteristic
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22×C411yescharacteristic
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D412yes—
79C22×C4×C31#4112yes—
80M16×C31—42no—
81M16×C31—42no—
82C2×C8×C31—24no—
83C4⋊C4×C31—22no—
84C4⋊C4×C31—22no—
85C4⋊C4×C31—22no—
86C4⋊C4×C31—42no—
87C4⋊C4×C31—42no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C22⋊C4×C31—22no—
91C22⋊C4×C31—22no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C42×C31—21no—
97C42×C31—21no—
98C42×C31—21no—
99C42×C31—21no—
100C42×C31—41no—
101C42×C31—41no—
102#9444C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—41no—
106C23×C31—41no—
107C23×C31C2×C22⋊C411yescharacteristic
108C23×C31#12112yes—
109Q8×C31—82no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—22no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—41no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—82no—
137C2×C4×C31—82no—
138C2×C4×C31—82no—
139C2×C4×C31C4×D412yes—
140C2×C4×C31#12011yescharacteristic
141C2×C4×C31#12311yescharacteristic
142C8×C31—44no—
143#2921C2×C3111yescharacteristic
144#2499C2×C3111yescharacteristic
145#1820C2×C3111yescharacteristic
146#1511C2×C3112yes—
147#1284C2×C3112yes—
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—22no—
151C22×C31—22no—
152C22×C31—22no—
153C22×C31—81no—
154C22×C31—81no—
155C22×C31#38611yescharacteristic
156C22×C31#41011yescharacteristic
157C22×C31#41011yescharacteristic
158C4×C31—21no—
159C4×C31—22no—
160C4×C31—21no—
161C4×C31—42no—
162C4×C31—41no—
163C4×C31—41no—
164C4×C31—41no—
165C4×C31—41no—
166C4×C31—81no—
167C4×C31—81no—
168C4×C31—81no—
169C4×C31—162no—
170C2×C22⋊Q8C22×C3111yescharacteristic
171C2×#117C22×C3111yescharacteristic
172#433—24no—
173#423—21no—
174#407C22×C3112yes—
175C4×M16—21no—
176#397—21no—
177#397—21no—
178#391C4×C3111yescharacteristic, complemented
179#391C4×C3111yescharacteristic, complemented
180#387—21no—
181#387—21no—
182#384C4×C3111yescharacteristic, complemented
183#382C4×C3111yescharacteristic, complemented
184#381C22×C3112yes—
185C4×C22⋊C4C22×C3111yescharacteristic
186#342—21no—
187#342—21no—
188#328—24no—
189C2×C31—21no—
190C2×C31—21no—
191C2×C31—42no—
192C2×C31#188411yescharacteristic
193C2×C31#189911yescharacteristic
194C2×C31#190911yescharacteristic
195C22×Q8C2×C4×C3111yescharacteristic
196C23×C4C23×C3111yescharacteristic
197C2×M16—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×C4×C3111yescharacteristic
206C2×C4⋊C4C2×C4×C3111yescharacteristic
207C2×C4⋊C4C2×C4×C3111yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4D4×C3111yescharacteristic
212C2×C22⋊C4D4×C3111yescharacteristic
213C2×C42C2×C4×C3111yescharacteristic
214C2×C42C2×C4×C3111yescharacteristic
215C2×C42D4×C3111yescharacteristic
216C2×C42D4×C3111yescharacteristic
217C4⋊C8—24no—
218C22⋊C8—22no—
219C22⋊C8D4×C3114yes—
220#97—22no—
221C4×C8—22no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231C31#944411yescharacteristic, direct product factor, Sylow
232C24C2×D4×C3111yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4C22⋊C4×C3111yescharacteristic
244C22×C4C22⋊C4×C3111yescharacteristic
245C22×C4C22⋊C4×C3111yescharacteristic
246C22×C4C22⋊C4×C3111yescharacteristic
247C22×C4C22×C4×C3111yescharacteristic
248C22×C4C2×D4×C3111yescharacteristic
249C22×C4C2×D4×C3112yes—
250C22×C4#41×C3112yes—
251M16—42no—
252M16—42no—
253C2×C8—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊C4×C3111yescharacteristic
278C23C31×#12112yes—
279Q8—82no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C4×D4×C3112yes—
310C2×C4C31×#12011yescharacteristic
311C2×C4C31×#12311yescharacteristic
312C8—44no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22C31×#38611yescharacteristic
321C22C31×#41011yescharacteristic
322C22C31×#41011yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—162no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C31×#188411yescharacteristic
339C2C31×#189911yescharacteristic
340C2C31×#190911yescharacteristic
subgroupG/H
center, socleC22×C31#386
commutatorC22×C4C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC24×C31C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
14. C31×#391C4semidirect
15. C31×#391C4semidirect
18. C31×#384C4semidirect
19. C31×#382C4semidirect
102. #9444C31direct (N × H)
178. #391C4×C31semidirect
179. #391C4×C31semidirect
182. #384C4×C31semidirect
183. #382C4×C31semidirect
231. C31#9444direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#29212
C31×#24992
C31×#18202
C31×#15112
C31×#12842
#944431