Finite Group Databasebeta

← C31×#11804 #21468563

Subgroups — C31×#11804 #21468563

Order 7936.

472 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2392C211yesmaximal, characteristic
2C31×#1822C211yesmaximal, characteristic, complemented
3C31×#1464C211yesmaximal, characteristic, complemented
4C31×#1346C212yesmaximal, complemented
5C31×#1345C212yesmaximal, complemented
6C2×C4×D4×C31—21no—
7C2×C4×D4×C31C411yescharacteristic
8C2×C31×#117C2211yescharacteristic
9C31×#432—21no—
10C31×#407—21no—
11C31×#398C2211yescharacteristic
12C31×#397C411yescharacteristic
13C31×#396—21no—
14C31×#394—21no—
15C31×#394C2211yescharacteristic
16C31×#392C411yescharacteristic
17C4×C4⋊C4×C31C411yescharacteristic
18C4×C22⋊C4×C31—21no—
19C31×#343—21no—
20C31×#343C2212yes—
21C31×#324C2212yes—
22C31×#323—21no—
23C22×D4×C31—21no—
24C22×D4×C31C2×C411yescharacteristic
25C23×C4×C31—21no—
26C23×C4×C31D411yescharacteristic
27C2×M16×C31—22no—
28C4×D4×C31—41no—
29C4×D4×C31—41no—
30C4×D4×C31—44no—
31C31×#117—41no—
32C31×#117—41no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31C2×C411yescharacteristic
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C4⋊C4×C31C2311yescharacteristic
38C2×C22⋊C4×C31—21no—
39C2×C22⋊C4×C31—21no—
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31C2×C411yescharacteristic
44C2×C22⋊C4×C31D411yescharacteristic
45C2×C42×C31—21no—
46C2×C42×C31C2×C411yescharacteristic
47C2×C42×C31D411yescharacteristic
48C31×C4⋊C8—41no—
49C31×C22⋊C8—22no—
50C31×C22⋊C8—41no—
51C31×C22⋊C8—41no—
52C31×#97—41no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95C2×C411yescharacteristic
56C31×#95C2×C411yescharacteristic
57C24×C31—21no—
58C24×C31C22⋊C411yescharacteristic
59C2×D4×C31—42no—
60C2×D4×C31—41no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C2×D4×C31—42no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—21no—
67C22×C4×C31—22no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—41no—
71C22×C4×C31—42no—
72C22×C4×C31—42no—
73C22×C4×C31—41no—
74C22×C4×C31—41no—
75C22×C4×C31C22⋊C411yescharacteristic
76C22×C4×C31C22⋊C411yescharacteristic
77C22×C4×C31C22⋊C411yescharacteristic
78C22×C4×C31C22×C411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81M16×C31—42no—
82M16×C31—42no—
83C2×C8×C31—42no—
84C2×C8×C31—42no—
85C4⋊C4×C31—41no—
86C4⋊C4×C31—41no—
87C4⋊C4×C31—42no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—42no—
90C22⋊C4×C31—41no—
91C22⋊C4×C31—41no—
92C22⋊C4×C31—42no—
93C22⋊C4×C31—42no—
94C22⋊C4×C31—42no—
95C22⋊C4×C31—42no—
96C22⋊C4×C31—42no—
97C42×C31—21no—
98C42×C31—21no—
99C42×C31—21no—
100C42×C31—21no—
101C42×C31—42no—
102#11804C3111yesmaximal, characteristic, direct product factor, Sylow
103C23×C31—22no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—21no—
107C23×C31—41no—
108C23×C31—41no—
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31C2×C22⋊C411yescharacteristic
116D4×C31—42no—
117D4×C31—48no—
118D4×C31—42no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—22no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—42no—
129C2×C4×C31—42no—
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—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—82no—
143C2×C4×C31—82no—
144C2×C4×C31#9911yescharacteristic
145C2×C4×C31#9911yescharacteristic
146C2×C4×C31AGL(1,ℤ/8)12yes—
147C8×C31—82no—
148C8×C31—82no—
149#2392C2×C3111yescharacteristic
150#1822C2×C3111yescharacteristic, complemented
151#1464C2×C3111yescharacteristic, complemented
152#1346C2×C3112yescomplemented
153#1345C2×C3112yescomplemented
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—42no—
157C22×C31—42no—
158C22×C31—42no—
159C22×C31—42no—
160C22×C31—41no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31—81no—
167C22×C31—81no—
168C22×C31C2×#9911yescharacteristic
169C22×C31#41811yescharacteristic
170C22×C31#42111yescharacteristic
171C4×C31—22no—
172C4×C31—21no—
173C4×C31—21no—
174C4×C31—42no—
175C4×C31—41no—
176C4×C31—41no—
177C4×C31—41no—
178C4×C31—41no—
179C4×C31—81no—
180C4×C31—81no—
181C2×C4×D4—21no—
182C2×C4×D4C4×C3111yescharacteristic
183C2×#117C22×C3111yescharacteristic
184#432—21no—
185#407—21no—
186#398C22×C3111yescharacteristic
187#397C4×C3111yescharacteristic
188#396—21no—
189#394—21no—
190#394C22×C3111yescharacteristic
191#392C4×C3111yescharacteristic
192C4×C4⋊C4C4×C3111yescharacteristic
193C4×C22⋊C4—21no—
194#343—21no—
195#343C22×C3112yes—
196#324C22×C3112yes—
197#323—21no—
198C2×C31—21no—
199C2×C31—21no—
200C2×C31—81no—
201C2×C31—82no—
202C2×C31#151011yescharacteristic
203C2×C31#212011yescharacteristic
204C2×C31#213211yescharacteristic
205C22×D4—21no—
206C22×D4C2×C4×C3111yescharacteristic
207C23×C4—21no—
208C23×C4D4×C3111yescharacteristic
209C2×M16—22no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—44no—
213#117—41no—
214#117—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×C4×C3111yescharacteristic
218C2×C4⋊C4D4×C3111yescharacteristic
219C2×C4⋊C4C23×C3111yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C2×C4×C3111yescharacteristic
226C2×C22⋊C4D4×C3111yescharacteristic
227C2×C42—21no—
228C2×C42C2×C4×C3111yescharacteristic
229C2×C42D4×C3111yescharacteristic
230C4⋊C8—41no—
231C22⋊C8—22no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—41no—
235#95—21no—
236#95—21no—
237#95C2×C4×C3111yescharacteristic
238#95C2×C4×C3111yescharacteristic
239C31#1180411yescharacteristic, direct product factor, Sylow
240C24—21no—
241C24C22⋊C4×C3111yescharacteristic
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊C4×C3111yescharacteristic
259C22×C4C22⋊C4×C3111yescharacteristic
260C22×C4C22⋊C4×C3111yescharacteristic
261C22×C4C22×C4×C3111yescharacteristic
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C2×C8—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊C4×C3111yescharacteristic
298D4—42no—
299D4—48no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C31×#9911yescharacteristic
327C2×C4C31×#9911yescharacteristic
328C2×C4C31×AGL(1,ℤ/8)12yes—
329C8—82no—
330C8—82no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×C31×#9911yescharacteristic
346C22C31×#41811yescharacteristic
347C22C31×#42111yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—82no—
362C2C31×#151011yescharacteristic
363C2C31×#212011yescharacteristic
364C2C31×#213211yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#392C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1822C2semidirect
3. C31×#1464C2semidirect
4. C31×#1346C2semidirect
5. C31×#1345C2semidirect
102. #11804C31direct (N × H)
150. #1822C2×C31semidirect
151. #1464C2×C31semidirect
152. #1346C2×C31semidirect
153. #1345C2×C31semidirect
239. C31#11804direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#23922
C31×#18222
C31×#14642
C31×#13462
C31×#13452
#1180431