Finite Group Databasebeta

← C31×#12197 #21468956

Subgroups — C31×#12197 #21468956

Order 7936.

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

normal subgroups only

This group has 426 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×#2586C211yesmaximal, characteristic, complemented
2C31×#2007C211yesmaximal, characteristic
3C31×#1610C211yesmaximal, characteristic
4C31×#1604C211yesmaximal, characteristic
5C31×#1354C211yesmaximal, characteristic, complemented
6C31×#1346C211yesmaximal, characteristic, complemented
7C31×#1319C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#537C2211yescharacteristic
10C31×#480—21no—
11C31×#448—21no—
12C2×C31×#104—21no—
13C31×#398C2211yescharacteristic
14C31×#395—21no—
15C31×#395C2211yescharacteristic
16C31×#391—21no—
17C31×#391C2211yescharacteristic
18C31×#389—21no—
19C31×#387—21no—
20C31×#387—21no—
21C4×C4⋊C4×C31—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×Q8×C31D411yescharacteristic
30C31×#137—42no—
31C31×#137—42no—
32C2×Q16×C31—41no—
33C2×SD16×C31—41no—
34C2×SD16×C31—41no—
35C2×M16×C31—21no—
36C31×C22⋊Q8—41no—
37C31×#121—21no—
38C31×#121—21no—
39C31×#121—41no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31—21no—
43C2×C4⋊C4×C31—21no—
44C2×C4⋊C4×C31—21no—
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C4⋊C4×C31C2311yescharacteristic
47C2×C42×C31—21no—
48C2×C42×C31—21no—
49C2×C42×C31D411yescharacteristic
50C31×#107—41no—
51C31×#104—42no—
52C31×#103—41no—
53C31×#102—41no—
54C31×C22⋊C8—21no—
55C31×C22⋊C8—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—42no—
73C2×Q8×C31—41no—
74C2×Q8×C31—41no—
75C2×Q8×C31—41no—
76C2×D4×C31—21no—
77C2×D4×C31—21no—
78C2×D4×C31—21no—
79C2×D4×C31—41no—
80C22×C4×C31—21no—
81C22×C4×C31—21no—
82C22×C4×C31—21no—
83C22×C4×C31—21no—
84C22×C4×C31—21no—
85C22×C4×C31—21no—
86C22×C4×C31—41no—
87C22×C4×C31C2×D411yescharacteristic
88C22×C4×C31C2×D411yescharacteristic
89C22×C4×C31C2×D411yescharacteristic
90Q16×C31—44no—
91SD16×C31—44no—
92SD16×C31—82no—
93M16×C31—22no—
94M16×C31—41no—
95C2×C8×C31—41no—
96C2×C8×C31—41no—
97C2×C8×C31—41no—
98C4⋊C4×C31—24no—
99C4⋊C4×C31—21no—
100C4⋊C4×C31—21no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—42no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—41no—
106C4⋊C4×C31—81no—
107C4⋊C4×C31—81no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C42×C31—42no—
111C42×C31—41no—
112C42×C31—81no—
113#12197C3111yesmaximal, characteristic, direct product factor, Sylow
114C23×C31—41no—
115C23×C31#12011yescharacteristic
116Q8×C31—42no—
117Q8×C31—41no—
118Q8×C31—41no—
119Q8×C31—42no—
120Q8×C31—81no—
121Q8×C31—81no—
122D4×C31—41no—
123D4×C31—41no—
124D4×C31—81no—
125D4×C31—81no—
126D4×C31—81no—
127D4×C31—81no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—22no—
131C2×C4×C31—22no—
132C2×C4×C31—22no—
133C2×C4×C31—22no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
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—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—81no—
154C2×C4×C31—81no—
155C2×C4×C31#13511yescharacteristic
156C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
157C8×C31—42no—
158C8×C31—81no—
159C8×C31—81no—
160#2586C2×C3111yescharacteristic, complemented
161#2007C2×C3111yescharacteristic
162#1610C2×C3111yescharacteristic
163#1604C2×C3111yescharacteristic
164#1354C2×C3111yescharacteristic, complemented
165#1346C2×C3111yescharacteristic, complemented
166#1319C2×C3111yescharacteristic, complemented
167C22×C31—21no—
168C22×C31—21no—
169C22×C31—81no—
170C22×C31—81no—
171C22×C31—81no—
172C22×C31#44911yescharacteristic
173C22×C31#45311yescharacteristic
174C22×C31B(4,𝔽2)11yescharacteristic
175C4×C31—21no—
176C4×C31—21no—
177C4×C31—42no—
178C4×C31—42no—
179C4×C31—42no—
180C4×C31—41no—
181C4×C31—41no—
182C4×C31—42no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31—81no—
186C2×#137—21no—
187#537C22×C3111yescharacteristic
188#480—21no—
189#448—21no—
190C2×#104—21no—
191#398C22×C3111yescharacteristic
192#395—21no—
193#395C22×C3111yescharacteristic
194#391—21no—
195#391C22×C3111yescharacteristic
196#389—21no—
197#387—21no—
198#387—21no—
199C4×C4⋊C4—21no—
200#343C22×C3111yescharacteristic
201#327—21no—
202#327—21no—
203#324C22×C3111yescharacteristic
204#324C22×C3111yescharacteristic
205C2×C31—21no—
206C2×C31—21no—
207C2×C31—161no—
208C2×C31#160011yescharacteristic
209C2×C31#219711yescharacteristic
210C2×C31#220511yescharacteristic
211C2×#41—21no—
212C22×Q8D4×C3111yescharacteristic
213C22×Q8D4×C3111yescharacteristic
214#137—42no—
215#137—42no—
216C2×Q16—41no—
217C2×SD16—41no—
218C2×SD16—41no—
219C2×M16—21no—
220C22⋊Q8—41no—
221#121—21no—
222#121—21no—
223#121—41no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C3111yescharacteristic
230C2×C4⋊C4C23×C3111yescharacteristic
231C2×C42—21no—
232C2×C42—21no—
233C2×C42D4×C3111yescharacteristic
234#107—41no—
235#104—42no—
236#103—41no—
237#102—41no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95D4×C3111yescharacteristic
248#95D4×C3111yescharacteristic
249C31#1219711yescharacteristic, direct product factor, Sylow
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4C2×D4×C3111yescharacteristic
273C22×C4C2×D4×C3111yescharacteristic
274C22×C4C2×D4×C3111yescharacteristic
275Q16—44no—
276SD16—44no—
277SD16—82no—
278M16—22no—
279M16—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—24no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C42—41no—
296C42—42no—
297C42—81no—
298C23—41no—
299C23C31×#12011yescharacteristic
300Q8—42no—
301Q8—41no—
302Q8—41no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—81no—
310D4—81no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C31×#13511yescharacteristic
340C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
341C8—42no—
342C8—81no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C31×#44911yescharacteristic
350C22C31×#45311yescharacteristic
351C22C31×B(4,𝔽2)11yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—161no—
366C2C31×#160011yescharacteristic
367C2C31×#219711yescharacteristic
368C2C31×#220511yescharacteristic
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×#2586C2semidirect
5. C31×#1354C2semidirect
6. C31×#1346C2semidirect
7. C31×#1319C2semidirect
113. #12197C31direct (N × H)
160. #2586C2×C31semidirect
164. #1354C2×C31semidirect
165. #1346C2×C31semidirect
166. #1319C2×C31semidirect
249. C31#12197direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25862
C31×#20072
C31×#16102
C31×#16042
C31×#13542
C31×#13462
C31×#13192
#1219731