Finite Group Databasebeta

← C31×#12201 #21468960

Subgroups — C31×#12201 #21468960

Order 7936.

380 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2833C211yesmaximal, characteristic, complemented
2C31×#2007C211yesmaximal, characteristic
3C31×#1616C211yesmaximal, characteristic
4C31×#1610C211yesmaximal, characteristic
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1346C211yesmaximal, characteristic, complemented
7C31×#1319C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#537C2211yescharacteristic
10C31×#481—21no—
11C31×#480—21no—
12C2×C31×#104—21no—
13C31×#401—21no—
14C31×#400—21no—
15C31×#400—21no—
16C31×#400C2211yescharacteristic
17C31×#398—21no—
18C31×#398C2211yescharacteristic
19C31×#395—21no—
20C31×#391C2211yescharacteristic
21C31×#387—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C31×#137—42no—
30C31×#137—42no—
31C2×Q16×C31—41no—
32C2×SD16×C31—41no—
33C2×M16×C31—21no—
34C31×C22⋊Q8—41no—
35C31×#121—21no—
36C31×#121—21no—
37C31×#121—41no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31C2311yescharacteristic
45C2×C42×C31D411yescharacteristic
46C31×#107—41no—
47C31×#106—41no—
48C31×#104—42no—
49C31×#102—41no—
50C31×#102—41no—
51C31×C22⋊C8—21no—
52C31×C22⋊C8—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65#41×C31—41no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69C2×Q8×C31—21no—
70C2×Q8×C31—21no—
71C2×Q8×C31—21no—
72C2×Q8×C31—41no—
73C2×Q8×C31—41no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—21no—
77C2×D4×C31—41no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—41no—
85C22×C4×C31C2×D411yescharacteristic
86C22×C4×C31C2×D411yescharacteristic
87C22×C4×C31C2×D411yescharacteristic
88Q16×C31—44no—
89SD16×C31—44no—
90M16×C31—22no—
91M16×C31—41no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C4⋊C4×C31—21no—
96C4⋊C4×C31—21no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—81no—
103C4⋊C4×C31—81no—
104C4⋊C4×C31—81no—
105C22⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C42×C31—41no—
108#12201C3111yesmaximal, characteristic, direct product factor, Sylow
109C23×C31—41no—
110C23×C31#12011yescharacteristic
111Q8×C31—42no—
112Q8×C31—41no—
113Q8×C31—41no—
114Q8×C31—81no—
115D4×C31—41no—
116D4×C31—41no—
117D4×C31—81no—
118D4×C31—81no—
119D4×C31—81no—
120D4×C31—81no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
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—42no—
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—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—81no—
146C2×C4×C31—81no—
147C2×C4×C31#13511yescharacteristic
148C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
149C8×C31—42no—
150C8×C31—81no—
151C8×C31—81no—
152#2833C2×C3111yescharacteristic, complemented
153#2007C2×C3111yescharacteristic
154#1616C2×C3111yescharacteristic
155#1610C2×C3111yescharacteristic
156#1353C2×C3111yescharacteristic, complemented
157#1346C2×C3111yescharacteristic, complemented
158#1319C2×C3111yescharacteristic, complemented
159C22×C31—21no—
160C22×C31—21no—
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31#44911yescharacteristic
165C22×C31#45311yescharacteristic
166C22×C31B(4,𝔽2)11yescharacteristic
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—41no—
170C4×C31—41no—
171C4×C31—81no—
172C4×C31—81no—
173C4×C31—81no—
174C4×C31—81no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C2×#137—21no—
179#537C22×C3111yescharacteristic
180#481—21no—
181#480—21no—
182C2×#104—21no—
183#401—21no—
184#400—21no—
185#400—21no—
186#400C22×C3111yescharacteristic
187#398—21no—
188#398C22×C3111yescharacteristic
189#395—21no—
190#391C22×C3111yescharacteristic
191#387—21no—
192#343C22×C3111yescharacteristic
193#327—21no—
194#327—21no—
195#324C22×C3111yescharacteristic
196#324C22×C3111yescharacteristic
197C2×C31—21no—
198C2×C31—21no—
199C2×C31—161no—
200C2×C31#160011yescharacteristic
201C2×C31#220111yescharacteristic
202C2×C31#220311yescharacteristic
203C2×#41—21no—
204C22×Q8D4×C3111yescharacteristic
205#137—42no—
206#137—42no—
207C2×Q16—41no—
208C2×SD16—41no—
209C2×M16—21no—
210C22⋊Q8—41no—
211#121—21no—
212#121—21no—
213#121—41no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4D4×C3111yescharacteristic
219C2×C4⋊C4D4×C3111yescharacteristic
220C2×C4⋊C4C23×C3111yescharacteristic
221C2×C42D4×C3111yescharacteristic
222#107—41no—
223#106—41no—
224#104—42no—
225#102—41no—
226#102—41no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95D4×C3111yescharacteristic
240#95D4×C3111yescharacteristic
241C31#1220111yescharacteristic, direct product factor, Sylow
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264C22×C4C2×D4×C3111yescharacteristic
265Q16—44no—
266SD16—44no—
267M16—22no—
268M16—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—41no—
285C23—41no—
286C23C31×#12011yescharacteristic
287Q8—42no—
288Q8—41no—
289Q8—41no—
290Q8—81no—
291D4—41no—
292D4—41no—
293D4—81no—
294D4—81no—
295D4—81no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
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—81no—
322C2×C4—81no—
323C2×C4C31×#13511yescharacteristic
324C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
325C8—42no—
326C8—81no—
327C8—81no—
328C22—21no—
329C22—21no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C31×#44911yescharacteristic
334C22C31×#45311yescharacteristic
335C22C31×B(4,𝔽2)11yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2C31×#160011yescharacteristic
351C2C31×#220111yescharacteristic
352C2C31×#220311yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C2×C4⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2833C2semidirect
5. C31×#1353C2semidirect
6. C31×#1346C2semidirect
7. C31×#1319C2semidirect
108. #12201C31direct (N × H)
152. #2833C2×C31semidirect
156. #1353C2×C31semidirect
157. #1346C2×C31semidirect
158. #1319C2×C31semidirect
241. C31#12201direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28332
C31×#20072
C31×#16162
C31×#16102
C31×#13532
C31×#13462
C31×#13192
#1220131