Finite Group Databasebeta

← C31×#9001 #21465760

Subgroups — C31×#9001 #21465760

Order 7936.

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

normal subgroups only

This group has 662 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×#496C211yesmaximal, characteristic
2C31×#2379C211yesmaximal, characteristic, complemented
3C31×#1961C212yesmaximal, complemented
4C2×C31×#329C211yesmaximal, characteristic, complemented
5C31×#1295C212yesmaximal, complemented
6C22×SD16×C31—21no—
7C22×D8×C31—21no—
8C2×C31×#127C2211yescharacteristic
9C2×C31×#124C2211yescharacteristic
10C31×#496—22no—
11C31×#496C414yescomplemented
12C2×C31×#102—22no—
13C2×C31×#102—22no—
14C2×C31×#97C2212yes—
15C2×C31×#97C2211yescharacteristic
16C31×#391—21no—
17C31×#388—22no—
18C31×#386—21no—
19C31×#385—22no—
20C31×#382C2212yes—
21C31×#329—22no—
22C22×Q8×C31D411yescharacteristic
23C22×D4×C31—22no—
24C22×D4×C31—21no—
25C22×D4×C31D411yescharacteristic
26C23×C4×C31—21no—
27C2×SD16×C31—24no—
28C2×SD16×C31—42no—
29C2×SD16×C31—42no—
30C2×D8×C31—24no—
31C2×D8×C31—42no—
32C2×D8×C31—42no—
33C22×C8×C31—21no—
34C22×C8×C31—22no—
35C31×#127—21no—
36C31×#127—22no—
37C31×#127C2×C412yescomplemented
38C31×#124—22no—
39C31×#124C2×C412yes—
40C31×#124D412yes—
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×#102—44no—
50C31×#102—44no—
51C31×#97—21no—
52C31×#97—22no—
53C31×#97—22no—
54C31×#97C2×C412yescomplemented
55C31×#95—22no—
56C31×#95—22no—
57C31×#95—22no—
58C24×C31—21no—
59C24×C31—41no—
60C2×Q8×C31—21no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×Q8×C31C22⋊C412yes—
64C2×D4×C31—22no—
65C2×D4×C31—24no—
66C2×D4×C31—21no—
67C2×D4×C31—41no—
68C2×D4×C31—41no—
69C2×D4×C31—42no—
70C2×D4×C31—42no—
71C2×D4×C31—41no—
72C2×D4×C31—41no—
73C2×D4×C31—41no—
74C2×D4×C31—41no—
75C2×D4×C31—42no—
76C2×D4×C31—42no—
77C2×D4×C31—42no—
78C2×D4×C31—41no—
79C2×D4×C31C22⋊C412yes—
80C22×C4×C31—21no—
81C22×C4×C31—22no—
82C22×C4×C31—22no—
83C22×C4×C31—82no—
84C22×C4×C31—81no—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D412yes—
87SD16×C31—48no—
88D8×C31—48no—
89C2×C8×C31—22no—
90C2×C8×C31—22no—
91C2×C8×C31—22no—
92C2×C8×C31—24no—
93C2×C8×C31—42no—
94C2×C8×C31—42no—
95C4⋊C4×C31—42no—
96C4⋊C4×C31—42no—
97C4⋊C4×C31—81no—
98C22⋊C4×C31—24no—
99C22⋊C4×C31—24no—
100C22⋊C4×C31—81no—
101C22⋊C4×C31—82no—
102C42×C31—81no—
103C42×C31C22×C411yescharacteristic
104C42×C31C2×D411yescharacteristic
105C42×C31#4112yes—
106#9001C3111yesmaximal, characteristic, direct product factor, Sylow
107C23×C31—22no—
108C23×C31—42no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—42no—
112C23×C31—42no—
113C23×C31—82no—
114C23×C31—81no—
115C23×C31—81no—
116C23×C31—82no—
117C23×C31#12011yescharacteristic
118Q8×C31—42no—
119Q8×C31—81no—
120D4×C31—42no—
121D4×C31—42no—
122D4×C31—42no—
123D4×C31—44no—
124D4×C31—44no—
125D4×C31—81no—
126D4×C31—82no—
127D4×C31—82no—
128D4×C31—81no—
129D4×C31—81no—
130C2×C4×C31—22no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—22no—
134C2×C4×C31—22no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—42no—
138C2×C4×C31—42no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—84no—
144C2×C4×C31C2×C22⋊C411yescharacteristic
145C2×C4×C31C4×D412yes—
146C2×C4×C31#12112yes—
147C2×C4×C31#12311yescharacteristic
148C8×C31—24no—
149C8×C31—44no—
150C2×#496C2×C3111yescharacteristic
151#2379C2×C3111yescharacteristic, complemented
152#1961C2×C3112yescomplemented
153C2×#329C2×C3111yescharacteristic, complemented
154#1295C2×C3112yescomplemented
155C22×C31—44no—
156C22×C31—42no—
157C22×C31—81no—
158C22×C31—82no—
159C22×C31—81no—
160C22×C31—84no—
161C22×C31—82no—
162C22×C31—161no—
163C22×C31—162no—
164C22×C31—161no—
165C22×C31#38611yescharacteristic
166C22×C31#45312yes—
167C22×C31#45311yescharacteristic
168C22×C31#45311yescharacteristic
169C22×C31#45412yes—
170C4×C31—22no—
171C4×C31—22no—
172C4×C31—21no—
173C4×C31—21no—
174C4×C31—42no—
175C4×C31—82no—
176C4×C31—82no—
177C22×SD16—21no—
178C22×D8—21no—
179C2×#127C22×C3111yescharacteristic
180C2×#124C22×C3111yescharacteristic
181#496—22no—
182#496C4×C3114yescomplemented
183C2×#102—22no—
184C2×#102—22no—
185C2×#97C22×C3112yes—
186C2×#97C22×C3111yescharacteristic
187#391—21no—
188#388—22no—
189#386—21no—
190#385—22no—
191#382C22×C3112yes—
192#329—22no—
193C2×C31—82no—
194C2×C31—162no—
195C2×C31#165511yescharacteristic
196C2×C31#165612yes—
197C2×C31#165711yescharacteristic
198C2×C31#188312yes—
199C2×C31#188711yescharacteristic
200C22×Q8D4×C3111yescharacteristic
201C22×D4—21no—
202C22×D4—22no—
203C22×D4D4×C3111yescharacteristic
204C23×C4—21no—
205C2×SD16—24no—
206C2×SD16—42no—
207C2×SD16—42no—
208C2×D8—24no—
209C2×D8—42no—
210C2×D8—42no—
211C22×C8—21no—
212C22×C8—22no—
213#127—21no—
214#127—22no—
215#127C2×C4×C3112yescomplemented
216#124—22no—
217#124C2×C4×C3112yes—
218#124D4×C3112yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4D4×C3112yes—
221C2×C4⋊C4D4×C3112yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42C23×C3111yescharacteristic
227#102—44no—
228#102—44no—
229#97—21no—
230#97—22no—
231#97—22no—
232#97C2×C4×C3112yescomplemented
233#95—22no—
234#95—22no—
235#95—22no—
236C31#900111yescharacteristic, direct product factor, Sylow
237C24—21no—
238C24—41no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8C22⋊C4×C3112yes—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4C22⋊C4×C3112yes—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4C2×D4×C3111yescharacteristic
265C22×C4C2×D4×C3112yes—
266SD16—48no—
267D8—48no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—81no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—81no—
280C22⋊C4—82no—
281C42—81no—
282C42C22×C4×C3111yescharacteristic
283C42C2×D4×C3111yescharacteristic
284C42#41×C3112yes—
285C23—22no—
286C23—42no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—82no—
292C23—81no—
293C23—81no—
294C23—82no—
295C23C31×#12011yescharacteristic
296Q8—42no—
297Q8—81no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—44no—
302D4—44no—
303D4—81no—
304D4—82no—
305D4—82no—
306D4—81no—
307D4—81no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—84no—
322C2×C4C2×C22⋊C4×C3111yescharacteristic
323C2×C4C4×D4×C3112yes—
324C2×C4C31×#12112yes—
325C2×C4C31×#12311yescharacteristic
326C8—24no—
327C8—44no—
328C22—44no—
329C22—42no—
330C22—81no—
331C22—82no—
332C22—81no—
333C22—84no—
334C22—82no—
335C22—161no—
336C22—162no—
337C22—161no—
338C22C31×#38611yescharacteristic
339C22C31×#45312yes—
340C22C31×#45311yescharacteristic
341C22C31×#45311yescharacteristic
342C22C31×#45412yes—
343C4—22no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—82no—
349C4—82no—
350C2—82no—
351C2—162no—
352C2C31×#165511yescharacteristic
353C2C31×#165612yes—
354C2C31×#165711yescharacteristic
355C2C31×#188312yes—
356C2C31×#188711yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutatorC42C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#496C2
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
2. C31×#2379C2semidirect
3. C31×#1961C2semidirect
4. C2×C31×#329C2semidirect
5. C31×#1295C2semidirect
11. C31×#496C4semidirect
37. C31×#127C2×C4semidirect
54. C31×#97C2×C4semidirect
106. #9001C31direct (N × H)
151. #2379C2×C31semidirect
152. #1961C2×C31semidirect
153. C2×#329C2×C31semidirect
154. #1295C2×C31semidirect
182. #496C4×C31semidirect
215. #127C2×C4×C31semidirect
232. #97C2×C4×C31semidirect
236. C31#9001direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#4962
C31×#23792
C31×#19612
C2×C31×#3292
C31×#12952
#900131