Finite Group Databasebeta

← C31×#13049 #21469808

Subgroups — C31×#13049 #21469808

Order 7936.

796 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 118 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×#453C211yesmaximal, characteristic
2C31×#2380C211yesmaximal, characteristic, complemented
3C2×C31×#353C211yesmaximal, characteristic, complemented
4C31×#1790C212yesmaximal, complemented
5C31×#1390C212yesmaximal, complemented
6C2×C31×H(5,𝔽2)C2211yescharacteristic
7C2×C31×AGL(1,ℤ/8)—21no—
8C2×C31×#127C2211yescharacteristic
9C31×#453—22no—
10C31×#453C414yescomplemented
11C2×C31×#104—21no—
12C2×C31×#100C2211yescharacteristic
13C2×C31×#99—22no—
14C2×C31×#99C2212yes—
15C31×#386—21no—
16C31×#386—22no—
17C31×#386—22no—
18C31×#386—22no—
19C31×#386C2212yes—
20C31×#383—21no—
21C31×#353—22no—
22C31×H(5,𝔽2)—22no—
23C31×H(5,𝔽2)—21no—
24C31×H(5,𝔽2)—42no—
25C31×H(5,𝔽2)C2×C412yescomplemented
26C2×#41×C31—21no—
27C2×#41×C31—41no—
28C22×D4×C31—22no—
29C22×D4×C31—22no—
30C22×D4×C31—41no—
31C22×D4×C31D411yescharacteristic
32C22×D4×C31C2311yescharacteristic
33C23×C4×C31—21no—
34C23×C4×C31D412yes—
35C31×AGL(1,ℤ/8)—24no—
36C31×AGL(1,ℤ/8)—42no—
37C2×SD16×C31—41no—
38C2×D8×C31—41no—
39C2×M16×C31—21no—
40C31×#127—22no—
41C31×#127C2×C412yes—
42C31×#127D412yes—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—42no—
48C2×C22⋊C4×C31D412yes—
49C2×C42×C31D411yescharacteristic
50C31×#104—24no—
51C31×#100—21no—
52C31×#100C2×C412yescomplemented
53C31×#99—22no—
54C31×#99—22no—
55C31×#99—44no—
56C31×#95—21no—
57C31×#95—22no—
58C31×#95—21no—
59C31×#95—22no—
60C31×#95—42no—
61C24×C31—22no—
62C24×C31—21no—
63C24×C31C2×D412yes—
64#41×C31—22no—
65#41×C31—22no—
66#41×C31—42no—
67#41×C31—41no—
68#41×C31—44no—
69#41×C31—42no—
70#41×C31—41no—
71C2×Q8×C31—21no—
72C2×D4×C31—21no—
73C2×D4×C31—24no—
74C2×D4×C31—24no—
75C2×D4×C31—24no—
76C2×D4×C31—24no—
77C2×D4×C31—21no—
78C2×D4×C31—41no—
79C2×D4×C31—41no—
80C2×D4×C31—41no—
81C2×D4×C31—41no—
82C2×D4×C31—42no—
83C2×D4×C31—42no—
84C2×D4×C31—42no—
85C2×D4×C31—42no—
86C2×D4×C31—44no—
87C2×D4×C31—41no—
88C2×D4×C31—42no—
89C2×D4×C31—82no—
90C2×D4×C31—82no—
91C2×D4×C31C22⋊C412yescomplemented
92C2×D4×C31C22×C411yescharacteristic
93C2×D4×C31C2×D411yescharacteristic
94C2×D4×C31#4112yes—
95C22×C4×C31—22no—
96C22×C4×C31—21no—
97C22×C4×C31—22no—
98C22×C4×C31—22no—
99C22×C4×C31—41no—
100C22×C4×C31—42no—
101C22×C4×C31—42no—
102C22×C4×C31—82no—
103C22×C4×C31—82no—
104C22×C4×C31—81no—
105C22×C4×C31C2×D411yescharacteristic
106SD16×C31—44no—
107D8×C31—44no—
108M16×C31—22no—
109M16×C31—41no—
110C2×C8×C31—41no—
111C22⋊C4×C31—42no—
112C22⋊C4×C31—42no—
113C22⋊C4×C31—42no—
114C22⋊C4×C31—82no—
115C22⋊C4×C31—82no—
116C22⋊C4×C31—81no—
117C22⋊C4×C31—84no—
118C42×C31—21no—
119C42×C31C22⋊C412yescomplemented
120#13049C3111yesmaximal, characteristic, direct product factor, Sylow
121C23×C31—22no—
122C23×C31—22no—
123C23×C31—24no—
124C23×C31—22no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—42no—
129C23×C31—42no—
130C23×C31—41no—
131C23×C31—41no—
132C23×C31—41no—
133C23×C31—82no—
134C23×C31—82no—
135C23×C31C4×D412yes—
136C23×C31#12011yescharacteristic
137C23×C31#12112yes—
138Q8×C31—22no—
139Q8×C31—41no—
140D4×C31—22no—
141D4×C31—41no—
142D4×C31—41no—
143D4×C31—42no—
144D4×C31—42no—
145D4×C31—41no—
146D4×C31—44no—
147D4×C31—44no—
148D4×C31—42no—
149D4×C31—42no—
150D4×C31—81no—
151D4×C31—84no—
152D4×C31—84no—
153C2×C4×C31—21no—
154C2×C4×C31—21no—
155C2×C4×C31—24no—
156C2×C4×C31—22no—
157C2×C4×C31—42no—
158C2×C4×C31—42no—
159C2×C4×C31—42no—
160C2×C4×C31—42no—
161C2×C4×C31—41no—
162C2×C4×C31—42no—
163C2×C4×C31—42no—
164C2×C4×C31—42no—
165C2×C4×C31—42no—
166C2×C4×C31—44no—
167C2×C4×C31—42no—
168C2×C4×C31—42no—
169C2×C4×C31—84no—
170C2×C4×C31—82no—
171C2×C4×C31—82no—
172C2×C4×C31—82no—
173C2×C4×C31—82no—
174C2×C4×C31C2×C22⋊C411yescharacteristic
175C2×C4×C31#12311yescharacteristic
176C8×C31—42no—
177C2×#453C2×C3111yescharacteristic
178#2380C2×C3111yescharacteristic, complemented
179C2×#353C2×C3111yescharacteristic, complemented
180#1790C2×C3112yescomplemented
181#1390C2×C3112yescomplemented
182C22×C31—22no—
183C22×C31—22no—
184C22×C31—22no—
185C22×C31—42no—
186C22×C31—42no—
187C22×C31—44no—
188C22×C31—42no—
189C22×C31—42no—
190C22×C31—42no—
191C22×C31—42no—
192C22×C31—82no—
193C22×C31—81no—
194C22×C31—81no—
195C22×C31—84no—
196C22×C31—82no—
197C22×C31—82no—
198C22×C31—162no—
199C22×C31#38611yescharacteristic
200C22×C31B(4,𝔽2)11yescharacteristic
201C22×C31B(4,𝔽2)11yescharacteristic
202C4×C31—21no—
203C4×C31—21no—
204C4×C31—42no—
205C4×C31—42no—
206C4×C31—82no—
207C4×C31—82no—
208C4×C31—162no—
209C2×H(5,𝔽2)C22×C3111yescharacteristic
210C2×AGL(1,ℤ/8)—21no—
211C2×#127C22×C3111yescharacteristic
212#453—22no—
213#453C4×C3114yescomplemented
214C2×#104—21no—
215C2×#100C22×C3111yescharacteristic
216C2×#99—22no—
217C2×#99C22×C3112yes—
218#386—21no—
219#386—22no—
220#386—22no—
221#386—22no—
222#386C22×C3112yes—
223#383—21no—
224#353—22no—
225C2×C31—21no—
226C2×C31—21no—
227C2×C31—42no—
228C2×C31—42no—
229C2×C31—82no—
230C2×C31—82no—
231C2×C31#188911yescharacteristic
232C2×C31#220012yes—
233H(5,𝔽2)—22no—
234H(5,𝔽2)—21no—
235H(5,𝔽2)—42no—
236H(5,𝔽2)C2×C4×C3112yescomplemented
237C2×#41—21no—
238C2×#41—41no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—41no—
242C22×D4D4×C3111yescharacteristic
243C22×D4C23×C3111yescharacteristic
244C23×C4—21no—
245C23×C4D4×C3112yes—
246AGL(1,ℤ/8)—24no—
247AGL(1,ℤ/8)—42no—
248C2×SD16—41no—
249C2×D8—41no—
250C2×M16—21no—
251#127—22no—
252#127C2×C4×C3112yes—
253#127D4×C3112yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4D4×C3112yes—
260C2×C42D4×C3111yescharacteristic
261#104—24no—
262#100—21no—
263#100C2×C4×C3112yescomplemented
264#99—22no—
265#99—22no—
266#99—44no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—42no—
272C31#1304911yescharacteristic, direct product factor, Sylow
273C24—22no—
274C24—21no—
275C24C2×D4×C3112yes—
276#41—22no—
277#41—22no—
278#41—42no—
279#41—41no—
280#41—44no—
281#41—42no—
282#41—41no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—44no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4C22⋊C4×C3112yescomplemented
304C2×D4C22×C4×C3111yescharacteristic
305C2×D4C2×D4×C3111yescharacteristic
306C2×D4#41×C3112yes—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—81no—
317C22×C4C2×D4×C3111yescharacteristic
318SD16—44no—
319D8—44no—
320M16—22no—
321M16—41no—
322C2×C8—41no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—81no—
329C22⋊C4—84no—
330C42—21no—
331C42C22⋊C4×C3112yescomplemented
332C23—22no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—82no—
345C23—82no—
346C23C4×D4×C3112yes—
347C23C31×#12011yescharacteristic
348C23C31×#12112yes—
349Q8—22no—
350Q8—41no—
351D4—22no—
352D4—41no—
353D4—41no—
354D4—42no—
355D4—42no—
356D4—41no—
357D4—44no—
358D4—44no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—84no—
363D4—84no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4C2×C22⋊C4×C3111yescharacteristic
386C2×C4C31×#12311yescharacteristic
387C8—42no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—81no—
400C22—81no—
401C22—84no—
402C22—82no—
403C22—82no—
404C22—162no—
405C22C31×#38611yescharacteristic
406C22C31×B(4,𝔽2)11yescharacteristic
407C22C31×B(4,𝔽2)11yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—42no—
412C4—82no—
413C4—82no—
414C4—162no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—82no—
420C2—82no—
421C2C31×#188911yescharacteristic
422C2C31×#220012yes—
subgroupG/H
center, socleC22×C31B(4,𝔽2)
commutatorC2×D4C22×C4×C31
Frattini, cosocleC22×D4C23×C31
prime radicalC2×C31×#453C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C31 ◁ C22×D4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2380C2semidirect
3. C2×C31×#353C2semidirect
4. C31×#1790C2semidirect
5. C31×#1390C2semidirect
10. C31×#453C4semidirect
25. C31×H(5,𝔽2)C2×C4semidirect
52. C31×#100C2×C4semidirect
91. C2×D4×C31C22⋊C4semidirect
119. C42×C31C22⋊C4semidirect
120. #13049C31direct (N × H)
178. #2380C2×C31semidirect
179. C2×#353C2×C31semidirect
180. #1790C2×C31semidirect
181. #1390C2×C31semidirect
213. #453C4×C31semidirect
236. H(5,𝔽2)C2×C4×C31semidirect
263. #100C2×C4×C31semidirect
272. C31#13049direct (N × H)
303. C2×D4C22⋊C4×C31semidirect
331. C42C22⋊C4×C31semidirect

Maximal subgroups

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

subgroupindex
C2×C31×#4532
C31×#23802
C2×C31×#3532
C31×#17902
C31×#13902
#1304931