Finite Group Databasebeta

← C31×#36700 #21493459

Subgroups — C31×#36700 #21493459

Order 7936.

1356 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2839C214yesmaximal, complemented
2C31×#2836C214yesmaximal
3C31×#2833C214yesmaximal, complemented
4C31×#2831C214yesmaximal
5C31×#2402C211yesmaximal, characteristic, complemented
6C31×#2402C214yesmaximal, complemented
7C31×#2401C214yesmaximal
8C31×#2399C211yesmaximal, characteristic, complemented
9C31×#2396C212yesmaximal
10C31×#2298C211yesmaximal, characteristic
11C31×#2295C212yesmaximal, complemented
12C2×C4×Q8×C31C2212yes—
13C2×C31×#117C2211yescharacteristic
14C31×#401C2214yes—
15C31×#400C2214yes—
16C31×#400C2214yes—
17C31×#400C2214yes—
18C31×#400C2214yes—
19C31×#400C2214yes—
20C31×#400C2214yes—
21C31×#400C2214yes—
22C31×#400C2212yes—
23C31×#400C2214yes—
24C31×#399C2212yes—
25C31×#398C2214yes—
26C31×#398C2214yes—
27C31×#398C2214yes—
28C31×#398C2214yes—
29C31×#398C2214yes—
30C31×#398C2214yes—
31C31×#398C2214yes—
32C31×#397C2214yes—
33C31×#397C2214yes—
34C31×#397C2214yes—
35C31×#396C2214yes—
36C31×#396C2214yes—
37C31×#395C2214yes—
38C31×#395C2212yes—
39C31×#393C2212yes—
40C31×#391C2214yes—
41C31×#389C2214yes—
42C31×#388C2214yes—
43C31×#387C2214yes—
44C31×#387C2214yes—
45C31×#387C2214yes—
46C4×C4⋊C4×C31C2214yes—
47C4×C4⋊C4×C31C2212yes—
48C4×C4⋊C4×C31C2214yes—
49C4×C4⋊C4×C31C2214yes—
50C4×C4⋊C4×C31C2214yes—
51C4×C4⋊C4×C31C2212yes—
52C4×C22⋊C4×C31C2214yes—
53C4×C22⋊C4×C31C2212yes—
54C31×#376C2211yescharacteristic
55C31×#376C2214yes—
56C43×C31C2211yescharacteristic
57C22×Q8×C31C2312yes—
58C23×C4×C31C2311yescharacteristic
59C4×Q8×C31—28no—
60C31×#117—24no—
61C2×C4⋊C4×C31C2314yes—
62C2×C4⋊C4×C31C2312yes—
63C2×C4⋊C4×C31C2312yes—
64C2×C4⋊C4×C31C2314yes—
65C2×C4⋊C4×C31C2314yes—
66C2×C4⋊C4×C31C2314yes—
67C2×C4⋊C4×C31C2314yes—
68C2×C4⋊C4×C31C2314yes—
69C2×C4⋊C4×C31C2314yes—
70C2×C4⋊C4×C31C2314yes—
71C2×C4⋊C4×C31C2314yes—
72C2×C4⋊C4×C31C2314yes—
73C2×C4⋊C4×C31C2314yes—
74C2×C22⋊C4×C31C2312yes—
75C2×C22⋊C4×C31C2314yes—
76C2×C22⋊C4×C31C2314yes—
77C2×C22⋊C4×C31C2314yes—
78C2×C42×C31C2311yescharacteristic
79C2×C42×C31C2314yes—
80C2×C42×C31C2312yes—
81C2×C42×C31C2312yes—
82C2×C42×C31C2314yes—
83C2×C42×C31C2314yes—
84C2×C42×C31C2311yescharacteristic
85C31×#95C2314yes—
86C31×#95C2314yes—
87C31×#95C2314yes—
88C31×#95C2312yes—
89C31×#95C2314yes—
90C31×#95C2314yes—
91C31×#95C2314yes—
92C31×#95C2314yes—
93C31×#95C2314yes—
94C31×#95C2314yes—
95C31×#95C2314yes—
96C31×#95C2314yes—
97C31×#95C2314yes—
98C31×#95C2314yes—
99C31×#95C2314yes—
100C31×#95C2312yes—
101C31×#95C2314yes—
102C31×#95C2314yes—
103C31×#95C2314yes—
104C24×C31C2411yescharacteristic
105C2×Q8×C31—24no—
106C2×Q8×C31—44no—
107C22×C4×C31—22no—
108C22×C4×C31—24no—
109C22×C4×C31C2411yescharacteristic
110C22×C4×C31C2414yes—
111C22×C4×C31C2414yes—
112C22×C4×C31C2414yes—
113C22×C4×C31C2414yes—
114C22×C4×C31C2414yes—
115C22×C4×C31C2412yes—
116C22×C4×C31C2414yes—
117C22×C4×C31C2412yes—
118C22×C4×C31C2411yescharacteristic
119C4⋊C4×C31—24no—
120C4⋊C4×C31—24no—
121C4⋊C4×C31—28no—
122C4⋊C4×C31—44no—
123C4⋊C4×C31—44no—
124C4⋊C4×C31—44no—
125C4⋊C4×C31—44no—
126C4⋊C4×C31—44no—
127C4⋊C4×C31—44no—
128C4⋊C4×C31—44no—
129C4⋊C4×C31—44no—
130C4⋊C4×C31—44no—
131C4⋊C4×C31—44no—
132C22⋊C4×C31—24no—
133C22⋊C4×C31—44no—
134C22⋊C4×C31—44no—
135C22⋊C4×C31—44no—
136C42×C31—22no—
137C42×C31—28no—
138C42×C31—24no—
139C42×C31—24no—
140C42×C31—28no—
141C42×C31—44no—
142C42×C31#4114yes—
143#36700C3111yesmaximal, characteristic, direct product factor, Sylow
144C23×C31—21no—
145C23×C31—22no—
146C23×C31—24no—
147C23×C31C2511yescharacteristic
148Q8×C31—48no—
149C2×C4×C31—24no—
150C2×C4×C31—22no—
151C2×C4×C31—24no—
152C2×C4×C31—24no—
153C2×C4×C31—21no—
154C2×C4×C31—24no—
155C2×C4×C31—24no—
156C2×C4×C31—24no—
157C2×C4×C31—24no—
158C2×C4×C31—22no—
159C2×C4×C31—24no—
160C2×C4×C31—24no—
161C2×C4×C31—24no—
162C2×C4×C31—24no—
163C2×C4×C31—24no—
164C2×C4×C31—24no—
165C2×C4×C31—24no—
166C2×C4×C31—24no—
167C2×C4×C31—24no—
168C2×C4×C31—24no—
169C2×C4×C31—24no—
170C2×C4×C31—24no—
171C2×C4×C31—24no—
172C2×C4×C31—44no—
173C2×C4×C31C2×#4114yes—
174C2×C4×C31C2×#4112yes—
175C2×C4×C312−1+414yes—
176#2839C2×C3114yescomplemented
177#2836C2×C3114yes—
178#2833C2×C3114yescomplemented
179#2831C2×C3114yes—
180#2402C2×C3111yescharacteristic, complemented
181#2402C2×C3114yescomplemented
182#2401C2×C3114yes—
183#2399C2×C3111yescharacteristic, complemented
184#2396C2×C3112yes—
185#2298C2×C3111yescharacteristic
186#2295C2×C3112yescomplemented
187C22×C31—41no—
188C22×C31—42no—
189C22×C31—44no—
190C22×C31C22×#4111yescharacteristic
191C22×C31C2×2−1+412yes—
192C22×C31#58514yes—
193C4×C31—24no—
194C4×C31—41no—
195C4×C31—42no—
196C4×C31—44no—
197C4×C31—44no—
198C4×C31—44no—
199C4×C31—44no—
200C4×C31—44no—
201C4×C31—44no—
202C4×C31#54114yes—
203C2×C4×Q8C22×C3112yes—
204C2×#117C22×C3111yescharacteristic
205#401C22×C3114yes—
206#400C22×C3114yes—
207#400C22×C3114yes—
208#400C22×C3114yes—
209#400C22×C3114yes—
210#400C22×C3114yes—
211#400C22×C3114yes—
212#400C22×C3114yes—
213#400C22×C3112yes—
214#400C22×C3114yes—
215#399C22×C3112yes—
216#398C22×C3114yes—
217#398C22×C3114yes—
218#398C22×C3114yes—
219#398C22×C3114yes—
220#398C22×C3114yes—
221#398C22×C3114yes—
222#398C22×C3114yes—
223#397C22×C3114yes—
224#397C22×C3114yes—
225#397C22×C3114yes—
226#396C22×C3114yes—
227#396C22×C3114yes—
228#395C22×C3114yes—
229#395C22×C3112yes—
230#393C22×C3112yes—
231#391C22×C3114yes—
232#389C22×C3114yes—
233#388C22×C3114yes—
234#387C22×C3114yes—
235#387C22×C3114yes—
236#387C22×C3114yes—
237C4×C4⋊C4C22×C3114yes—
238C4×C4⋊C4C22×C3112yes—
239C4×C4⋊C4C22×C3114yes—
240C4×C4⋊C4C22×C3114yes—
241C4×C4⋊C4C22×C3114yes—
242C4×C4⋊C4C22×C3112yes—
243C4×C22⋊C4C22×C3114yes—
244C4×C22⋊C4C22×C3112yes—
245#376C22×C3111yescharacteristic
246#376C22×C3114yes—
247C43C22×C3111yescharacteristic
248C2×C31—81no—
249C2×C31C2×#54111yescharacteristic
250C2×C31#345512yes—
251C2×C31#353114yes—
252C22×Q8C23×C3112yes—
253C23×C4C23×C3111yescharacteristic
254C4×Q8—28no—
255#117—24no—
256C2×C4⋊C4C23×C3112yes—
257C2×C4⋊C4C23×C3114yes—
258C2×C4⋊C4C23×C3112yes—
259C2×C4⋊C4C23×C3114yes—
260C2×C4⋊C4C23×C3114yes—
261C2×C4⋊C4C23×C3114yes—
262C2×C4⋊C4C23×C3114yes—
263C2×C4⋊C4C23×C3114yes—
264C2×C4⋊C4C23×C3114yes—
265C2×C4⋊C4C23×C3114yes—
266C2×C4⋊C4C23×C3114yes—
267C2×C4⋊C4C23×C3114yes—
268C2×C4⋊C4C23×C3114yes—
269C2×C22⋊C4C23×C3112yes—
270C2×C22⋊C4C23×C3114yes—
271C2×C22⋊C4C23×C3114yes—
272C2×C22⋊C4C23×C3114yes—
273C2×C42C23×C3111yescharacteristic
274C2×C42C23×C3114yes—
275C2×C42C23×C3112yes—
276C2×C42C23×C3112yes—
277C2×C42C23×C3114yes—
278C2×C42C23×C3114yes—
279C2×C42C23×C3111yescharacteristic
280#95C23×C3114yes—
281#95C23×C3114yes—
282#95C23×C3114yes—
283#95C23×C3114yes—
284#95C23×C3114yes—
285#95C23×C3114yes—
286#95C23×C3114yes—
287#95C23×C3114yes—
288#95C23×C3112yes—
289#95C23×C3112yes—
290#95C23×C3114yes—
291#95C23×C3114yes—
292#95C23×C3114yes—
293#95C23×C3114yes—
294#95C23×C3114yes—
295#95C23×C3114yes—
296#95C23×C3114yes—
297#95C23×C3114yes—
298#95C23×C3114yes—
299C31#3670011yescharacteristic, direct product factor, Sylow
300C24C24×C3111yescharacteristic
301C2×Q8—24no—
302C2×Q8—44no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4C24×C3111yescharacteristic
306C22×C4C24×C3114yes—
307C22×C4C24×C3114yes—
308C22×C4C24×C3114yes—
309C22×C4C24×C3114yes—
310C22×C4C24×C3114yes—
311C22×C4C24×C3112yes—
312C22×C4C24×C3114yes—
313C22×C4C24×C3112yes—
314C22×C4C24×C3111yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—24no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C42—22no—
333C42—28no—
334C42—24no—
335C42—24no—
336C42—28no—
337C42—44no—
338C42#41×C3114yes—
339C23—21no—
340C23—22no—
341C23—24no—
342C23C25×C3111yescharacteristic
343Q8—48no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—44no—
368C2×C4C2×#41×C3114yes—
369C2×C4C2×#41×C3112yes—
370C2×C4C31×2−1+414yes—
371C22—41no—
372C22—42no—
373C22—44no—
374C22C22×#41×C3111yescharacteristic
375C22C2×C31×2−1+412yes—
376C22C31×#58514yes—
377C4—24no—
378C4—41no—
379C4—44no—
380C4—42no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—44no—
385C4—44no—
386C4C31×#54114yes—
387C2—81no—
388C2C2×C31×#54111yescharacteristic
389C2C31×#345512yes—
390C2C31×#353114yes—
subgroupG/H
centerC22×C4×C31C24
commutator, Frattini, cosocleC23C25×C31
socleC23×C31C25
prime radicalC24×C31C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
1. C31×#2839C2semidirect
3. C31×#2833C2semidirect
5. C31×#2402C2semidirect
6. C31×#2402C2semidirect
8. C31×#2399C2semidirect
11. C31×#2295C2semidirect
143. #36700C31direct (N × H)
176. #2839C2×C31semidirect
178. #2833C2×C31semidirect
180. #2402C2×C31semidirect
181. #2402C2×C31semidirect
183. #2399C2×C31semidirect
186. #2295C2×C31semidirect
299. C31#36700direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28392
C31×#28362
C31×#28332
C31×#28312
C31×#24022
C31×#24012
C31×#23992
C31×#23962
C31×#22982
C31×#22952
#3670031