Finite Group Databasebeta

← C31×#12196 #21468955

Subgroups — C31×#12196 #21468955

Order 7936.

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

normal subgroups only

This group has 434 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×#2593C211yesmaximal, characteristic, complemented
2C31×#2006C211yesmaximal, characteristic
3C31×#1616C211yesmaximal, characteristic, complemented
4C31×#1610C211yesmaximal, characteristic, complemented
5C31×#1353C211yesmaximal, characteristic, complemented
6C31×#1346C211yesmaximal, characteristic, complemented
7C31×#1318C211yesmaximal, characteristic, complemented
8C2×C31×AGL(1,ℤ/8)—21no—
9C31×#537C2211yescharacteristic
10C31×#481—21no—
11C31×#480—21no—
12C2×C31×#104—21no—
13C31×#400C2211yescharacteristic
14C31×#398C2211yescharacteristic
15C31×#394—21no—
16C31×#390C2211yescharacteristic
17C31×#388—21no—
18C31×#388—21no—
19C31×#388—21no—
20C31×#388—21no—
21C4×C4⋊C4×C31—21no—
22C31×#343C2211yescharacteristic
23C31×#327—21no—
24C31×#327—21no—
25C31×#324C2211yescharacteristic
26C31×#324C2211yescharacteristic
27C2×#41×C31—21no—
28C22×D4×C31D411yescharacteristic
29C31×AGL(1,ℤ/8)—42no—
30C31×AGL(1,ℤ/8)—42no—
31C2×SD16×C31—41no—
32C2×D8×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×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31C2311yescharacteristic
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C42×C31—21no—
50C2×C42×C31—21no—
51C2×C42×C31D411yescharacteristic
52C31×#107—41no—
53C31×#106—41no—
54C31×#104—42no—
55C31×#102—41no—
56C31×#102—41no—
57C31×C22⋊C8—21no—
58C31×C22⋊C8—21no—
59C31×#95—21no—
60C31×#95—21no—
61C31×#95—21no—
62C31×#95—21no—
63C31×#95D411yescharacteristic
64C31×#95D411yescharacteristic
65C24×C31—21no—
66#41×C31—41no—
67#41×C31—41no—
68#41×C31—41no—
69#41×C31—41no—
70C2×Q8×C31—21no—
71C2×D4×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—21no—
75C2×D4×C31—21no—
76C2×D4×C31—41no—
77C2×D4×C31—41no—
78C2×D4×C31—41no—
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
88SD16×C31—44no—
89D8×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—42no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C22⋊C4×C31—42no—
104C22⋊C4×C31—42no—
105C22⋊C4×C31—42no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—81no—
110C22⋊C4×C31—81no—
111C42×C31—42no—
112C42×C31—41no—
113C42×C31—81no—
114#12196C3111yesmaximal, characteristic, direct product factor, Sylow
115C23×C31—41no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31#12011yescharacteristic
124Q8×C31—41no—
125Q8×C31—41no—
126D4×C31—41no—
127D4×C31—42no—
128D4×C31—41no—
129D4×C31—81no—
130D4×C31—81no—
131D4×C31—81no—
132D4×C31—81no—
133D4×C31—81no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—22no—
137C2×C4×C31—22no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
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—41no—
154C2×C4×C31—41no—
155C2×C4×C31—41no—
156C2×C4×C31—81no—
157C2×C4×C31—81no—
158C2×C4×C31#13511yescharacteristic
159C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
160C8×C31—42no—
161C8×C31—81no—
162C8×C31—81no—
163#2593C2×C3111yescharacteristic, complemented
164#2006C2×C3111yescharacteristic
165#1616C2×C3111yescharacteristic, complemented
166#1610C2×C3111yescharacteristic, complemented
167#1353C2×C3111yescharacteristic, complemented
168#1346C2×C3111yescharacteristic, complemented
169#1318C2×C3111yescharacteristic, complemented
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31—81no—
181C22×C31—81no—
182C22×C31#44911yescharacteristic
183C22×C31#45311yescharacteristic
184C22×C31B(4,𝔽2)11yescharacteristic
185C4×C31—21no—
186C4×C31—21no—
187C4×C31—42no—
188C4×C31—42no—
189C4×C31—41no—
190C4×C31—41no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—81no—
194C4×C31—81no—
195C2×AGL(1,ℤ/8)—21no—
196#537C22×C3111yescharacteristic
197#481—21no—
198#480—21no—
199C2×#104—21no—
200#400C22×C3111yescharacteristic
201#398C22×C3111yescharacteristic
202#394—21no—
203#390C22×C3111yescharacteristic
204#388—21no—
205#388—21no—
206#388—21no—
207#388—21no—
208C4×C4⋊C4—21no—
209#343C22×C3111yescharacteristic
210#327—21no—
211#327—21no—
212#324C22×C3111yescharacteristic
213#324C22×C3111yescharacteristic
214C2×C31—21no—
215C2×C31—21no—
216C2×C31—161no—
217C2×C31—161no—
218C2×C31#160011yescharacteristic
219C2×C31#219811yescharacteristic
220C2×C31#220411yescharacteristic
221C2×#41—21no—
222C22×D4D4×C3111yescharacteristic
223AGL(1,ℤ/8)—42no—
224AGL(1,ℤ/8)—42no—
225C2×SD16—41no—
226C2×D8—41no—
227C2×M16—21no—
228C22⋊Q8—41no—
229#121—21no—
230#121—21no—
231#121—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4D4×C3111yescharacteristic
235C2×C4⋊C4D4×C3111yescharacteristic
236C2×C4⋊C4C23×C3111yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42D4×C3111yescharacteristic
246#107—41no—
247#106—41no—
248#104—42no—
249#102—41no—
250#102—41no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95D4×C3111yescharacteristic
258#95D4×C3111yescharacteristic
259C31#1219611yescharacteristic, direct product factor, Sylow
260C24—21no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4C2×D4×C3111yescharacteristic
281C22×C4C2×D4×C3111yescharacteristic
282C22×C4C2×D4×C3111yescharacteristic
283SD16—44no—
284D8—44no—
285M16—22no—
286M16—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C42—41no—
307C42—42no—
308C42—81no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23C31×#12011yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—41no—
321D4—42no—
322D4—41no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C31×#13511yescharacteristic
353C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
354C8—42no—
355C8—81no—
356C8—81no—
357C22—21no—
358C22—21no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C31×#44911yescharacteristic
370C22C31×#45311yescharacteristic
371C22C31×B(4,𝔽2)11yescharacteristic
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C2—21no—
383C2—21no—
384C2—161no—
385C2—161no—
386C2C31×#160011yescharacteristic
387C2C31×#219811yescharacteristic
388C2C31×#220411yescharacteristic
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2006C2
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×#2593C2semidirect
3. C31×#1616C2semidirect
4. C31×#1610C2semidirect
5. C31×#1353C2semidirect
6. C31×#1346C2semidirect
7. C31×#1318C2semidirect
114. #12196C31direct (N × H)
163. #2593C2×C31semidirect
165. #1616C2×C31semidirect
166. #1610C2×C31semidirect
167. #1353C2×C31semidirect
168. #1346C2×C31semidirect
169. #1318C2×C31semidirect
259. C31#12196direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#25932
C31×#20062
C31×#16162
C31×#16102
C31×#13532
C31×#13462
C31×#13182
#1219631