Finite Group Databasebeta

← C2×C31×#2365 #21490897

Subgroups — C2×C31×#2365 #21490897

Order 7936.

3158 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
1C22×C31×C4⋊Q8C211yesmaximal, characteristic
2C22×C4×Q8×C31C212yesmaximal
3C31×#2365C2116yesmaximal, direct product factor
4C2×C31×#391C212yesmaximal
5C2×C31×#389C212yesmaximal
6C2×C31×#387C214yesmaximal
7C2×C4×C4⋊C4×C31C212yesmaximal
8C2×C4×C4⋊C4×C31C211yesmaximal, characteristic
9C2×C4×C4⋊C4×C31C211yesmaximal, characteristic
10C23×Q8×C31C2212yes—
11C2×C31×C4⋊Q8C4116yescomplemented
12C2×C31×C4⋊Q8C2218yes—
13C2×C4×Q8×C31—216no—
14C2×C4×Q8×C31C22116yes—
15C22×C4⋊C4×C31C2214yes—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2212yes—
19C22×C4⋊C4×C31C2212yes—
20C22×C4⋊C4×C31C2211yescharacteristic
21C22×C4⋊C4×C31C2211yescharacteristic
22C22×C42×C31C2214yes—
23C22×C42×C31C2212yes—
24C22×C42×C31C2212yes—
25C22×C42×C31C2211yescharacteristic
26C31×#391C22116yes—
27C31×#389C22116yes—
28C31×#387C22132yes—
29C4×C4⋊C4×C31C2218yes—
30C4×C4⋊C4×C31C2218yes—
31C4×C4⋊C4×C31C22116yes—
32C2×C31×#95C2212yes—
33C2×C31×#95C2212yes—
34C2×C31×#95C2214yes—
35C2×C31×#95C2214yes—
36C22×Q8×C31—28no—
37C22×Q8×C31—28no—
38C22×Q8×C31C2×C4116yes—
39C22×Q8×C31C2318yes—
40C23×C4×C31C2312yes—
41C23×C4×C31C2312yes—
42C23×C4×C31C2314yes—
43C23×C4×C31C2311yescharacteristic
44C23×C4×C31C2312yes—
45C23×C4×C31C2312yes—
46C23×C4×C31C2311yescharacteristic
47C23×C4×C31C2311yescharacteristic
48C31×C4⋊Q8C2×C4164yescomplemented
49C4×Q8×C31—264no—
50C2×C4⋊C4×C31—216no—
51C2×C4⋊C4×C31—24no—
52C2×C4⋊C4×C31—24no—
53C2×C4⋊C4×C31C2×C4116yes—
54C2×C4⋊C4×C31C2×C4116yes—
55C2×C4⋊C4×C31D418yes—
56C2×C4⋊C4×C31Q818yescomplemented
57C2×C4⋊C4×C31C23116yes—
58C2×C4⋊C4×C31C2314yes—
59C2×C4⋊C4×C31C2314yes—
60C2×C4⋊C4×C31C2318yes—
61C2×C4⋊C4×C31C2318yes—
62C2×C4⋊C4×C31C2314yes—
63C2×C4⋊C4×C31C2314yes—
64C2×C42×C31—216no—
65C2×C42×C31—28no—
66C2×C42×C31—28no—
67C2×C42×C31C2×C418yes—
68C2×C42×C31C23116yes—
69C2×C42×C31C2318yes—
70C2×C42×C31C2318yes—
71C2×C42×C31C2314yes—
72C31×#95C2318yes—
73C31×#95C2318yes—
74C31×#95C23116yes—
75C31×#95C23116yes—
76C24×C31C2411yescharacteristic
77C2×Q8×C31—264no—
78C2×Q8×C31—216no—
79C2×Q8×C31—216no—
80C2×Q8×C31C22×C4132yes—
81C22×C4×C31—24no—
82C22×C4×C31—28no—
83C22×C4×C31—24no—
84C22×C4×C31—28no—
85C22×C4×C31—22no—
86C22×C4×C31—22no—
87C22×C4×C31—24no—
88C22×C4×C31—24no—
89C22×C4×C31—24no—
90C22×C4×C31—24no—
91C22×C4×C31—24no—
92C22×C4×C31—24no—
93C22×C4×C31—24no—
94C22×C4×C31—24no—
95C22×C4×C31C22×C414yes—
96C22×C4×C31C22×C4116yes—
97C22×C4×C31C22×C414yes—
98C22×C4×C31C22×C414yes—
99C22×C4×C31C2×D418yes—
100C22×C4×C31C2×D414yes—
101C22×C4×C31C2×Q818yes—
102C22×C4×C31C2×Q814yes—
103C22×C4×C31#4114yes—
104C22×C4×C31#4114yes—
105C22×C4×C31C2414yes—
106C22×C4×C31C2414yes—
107C22×C4×C31C2412yes—
108C22×C4×C31C2418yes—
109C22×C4×C31C2414yes—
110C22×C4×C31C2414yes—
111C22×C4×C31C2412yes—
112C22×C4×C31C2412yes—
113C4⋊C4×C31—232no—
114C4⋊C4×C31—28no—
115C4⋊C4×C31—28no—
116C4⋊C4×C31C22×C4132yes—
117C4⋊C4×C31C22×C4132yes—
118C4⋊C4×C31C2×D4116yes—
119C4⋊C4×C31C2×Q8116yescomplemented
120C42×C31—232no—
121C42×C31—216no—
122C42×C31—216no—
123C42×C31C22×C4116yes—
124C2×#2365C3111yesmaximal, characteristic, direct product factor, Sylow
125C23×C31C23×C412yes—
126C23×C31C22×D412yes—
127C23×C31C22×Q812yes—
128C23×C31C2×#4112yes—
129C23×C31C2×#4112yes—
130C23×C312−1+412yes—
131C23×C312−1+412yes—
132C23×C31C2511yescharacteristic
133Q8×C31—264no—
134C2×C4×C31—232no—
135C2×C4×C31—28no—
136C2×C4×C31—216no—
137C2×C4×C31—216no—
138C2×C4×C31—24no—
139C2×C4×C31—28no—
140C2×C4×C31—24no—
141C2×C4×C31—28no—
142C2×C4×C31—22no—
143C2×C4×C31—22no—
144C2×C4×C31—24no—
145C2×C4×C31—24no—
146C2×C4×C31—24no—
147C2×C4×C31—24no—
148C2×C4×C31—24no—
149C2×C4×C31—24no—
150C2×C4×C31—24no—
151C2×C4×C31—24no—
152C2×C4×C31—48no—
153C2×C4×C31—48no—
154C2×C4×C31C4×D4116yes—
155C2×C4×C31C4×Q8116yescomplemented
156C2×C4×C31C23×C414yes—
157C2×C4×C31C23×C4116yes—
158C2×C4×C31C23×C414yes—
159C2×C4×C31C23×C414yes—
160C2×C4×C31C22×D418yes—
161C2×C4×C31C22×D414yes—
162C2×C4×C31C22×Q818yes—
163C2×C4×C31C22×Q814yes—
164C2×C4×C31C2×#4114yes—
165C2×C4×C31C2×#4114yes—
166C22×C4⋊Q8C2×C3111yescharacteristic
167C22×C4×Q8C2×C3112yes—
168#2365C2×C31116yesdirect product factor
169C2×#391C2×C3112yes—
170C2×#389C2×C3112yes—
171C2×#387C2×C3114yes—
172C2×C4×C4⋊C4C2×C3111yescharacteristic
173C2×C4×C4⋊C4C2×C3111yescharacteristic
174C2×C4×C4⋊C4C2×C3112yes—
175C22×C31C2×C4×D414yes—
176C22×C31C2×C4×Q814yes—
177C22×C31#51914yes—
178C22×C31#54714yes—
179C22×C31D4×Q814yes—
180C22×C31#55614yes—
181C22×C31#55714yes—
182C22×C31C24×C411yescharacteristic
183C22×C31C23×D411yescharacteristic
184C22×C31C23×Q811yescharacteristic
185C22×C31C22×#4111yescharacteristic
186C22×C31C22×#4111yescharacteristic
187C22×C31C2×2−1+411yescharacteristic
188C22×C31C2×2−1+411yescharacteristic
189C4×C31—216no—
190C4×C31—24no—
191C4×C31—28no—
192C4×C31—28no—
193C4×C31—44no—
194C4×C31—44no—
195C4×C31C2×C4×D418yes—
196C4×C31C2×C4×Q818yescomplemented
197C23×Q8C22×C3112yes—
198C2×C4⋊Q8C4×C31116yescomplemented
199C2×C4⋊Q8C22×C3118yes—
200C2×C4×Q8—216no—
201C2×C4×Q8C22×C31116yes—
202C22×C4⋊C4C22×C3114yes—
203C22×C4⋊C4C22×C3111yescharacteristic
204C22×C4⋊C4C22×C3111yescharacteristic
205C22×C4⋊C4C22×C3112yes—
206C22×C4⋊C4C22×C3112yes—
207C22×C4⋊C4C22×C3111yescharacteristic
208C22×C4⋊C4C22×C3111yescharacteristic
209C22×C42C22×C3114yes—
210C22×C42C22×C3112yes—
211C22×C42C22×C3112yes—
212C22×C42C22×C3111yescharacteristic
213#391C22×C31116yes—
214#389C22×C31116yes—
215#387C22×C31132yes—
216C4×C4⋊C4C22×C3118yes—
217C4×C4⋊C4C22×C3118yes—
218C4×C4⋊C4C22×C31116yes—
219C2×#95C22×C3112yes—
220C2×#95C22×C3112yes—
221C2×#95C22×C3114yes—
222C2×#95C22×C3114yes—
223C2×C31#236518yesdirect product factor
224C2×C31C22×C4×D411yescharacteristic
225C2×C31C22×C4×Q811yescharacteristic
226C2×C31C2×#51911yescharacteristic
227C2×C31C2×#54711yescharacteristic
228C2×C31C2×D4×Q811yescharacteristic
229C2×C31C2×#55611yescharacteristic
230C2×C31C2×#55711yescharacteristic
231C22×Q8—28no—
232C22×Q8—28no—
233C22×Q8C2×C4×C31116yes—
234C22×Q8C23×C3118yes—
235C23×C4C23×C3112yes—
236C23×C4C23×C3112yes—
237C23×C4C23×C3114yes—
238C23×C4C23×C3111yescharacteristic
239C23×C4C23×C3112yes—
240C23×C4C23×C3112yes—
241C23×C4C23×C3111yescharacteristic
242C23×C4C23×C3111yescharacteristic
243C4⋊Q8C2×C4×C31164yescomplemented
244C4×Q8—264no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4C2×C4×C31116yes—
249C2×C4⋊C4C2×C4×C31116yes—
250C2×C4⋊C4D4×C3118yes—
251C2×C4⋊C4Q8×C3118yescomplemented
252C2×C4⋊C4C23×C31116yes—
253C2×C4⋊C4C23×C3114yes—
254C2×C4⋊C4C23×C3114yes—
255C2×C4⋊C4C23×C3118yes—
256C2×C4⋊C4C23×C3118yes—
257C2×C4⋊C4C23×C3114yes—
258C2×C4⋊C4C23×C3114yes—
259C2×C42—216no—
260C2×C42—28no—
261C2×C42—28no—
262C2×C42C2×C4×C3118yes—
263C2×C42C23×C31116yes—
264C2×C42C23×C3118yes—
265C2×C42C23×C3118yes—
266C2×C42C23×C3114yes—
267#95C23×C3118yes—
268#95C23×C3118yes—
269#95C23×C31116yes—
270#95C23×C31116yes—
271C31C2×#236511yescharacteristic, direct product factor, Sylow
272C24C24×C3111yescharacteristic
273C2×Q8—264no—
274C2×Q8—216no—
275C2×Q8—216no—
276C2×Q8C22×C4×C31132yes—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4C22×C4×C3114yes—
292C22×C4C22×C4×C31116yes—
293C22×C4C22×C4×C3114yes—
294C22×C4C22×C4×C3114yes—
295C22×C4C2×D4×C3118yes—
296C22×C4C2×D4×C3114yes—
297C22×C4C2×Q8×C3118yes—
298C22×C4C2×Q8×C3114yes—
299C22×C4#41×C3114yes—
300C22×C4#41×C3114yes—
301C22×C4C24×C3114yes—
302C22×C4C24×C3114yes—
303C22×C4C24×C3112yes—
304C22×C4C24×C3118yes—
305C22×C4C24×C3114yes—
306C22×C4C24×C3114yes—
307C22×C4C24×C3112yes—
308C22×C4C24×C3112yes—
309C4⋊C4—232no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4C22×C4×C31132yes—
313C4⋊C4C22×C4×C31132yes—
314C4⋊C4C2×D4×C31116yes—
315C4⋊C4C2×Q8×C31116yescomplemented
316C42—232no—
317C42—216no—
318C42—216no—
319C42C22×C4×C31116yes—
320C23C23×C4×C3112yes—
321C23C22×D4×C3112yes—
322C23C22×Q8×C3112yes—
323C23C2×#41×C3112yes—
324C23C2×#41×C3112yes—
325C23C31×2−1+412yes—
326C23C31×2−1+412yes—
327C23C25×C3111yescharacteristic
328Q8—264no—
329C2×C4—232no—
330C2×C4—28no—
331C2×C4—216no—
332C2×C4—216no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4C4×D4×C31116yes—
350C2×C4C4×Q8×C31116yescomplemented
351C2×C4C23×C4×C3114yes—
352C2×C4C23×C4×C31116yes—
353C2×C4C23×C4×C3114yes—
354C2×C4C23×C4×C3114yes—
355C2×C4C22×D4×C3118yes—
356C2×C4C22×D4×C3114yes—
357C2×C4C22×Q8×C3118yes—
358C2×C4C22×Q8×C3114yes—
359C2×C4C2×#41×C3114yes—
360C2×C4C2×#41×C3114yes—
361C22C2×C4×D4×C3114yes—
362C22C2×C4×Q8×C3114yes—
363C22C31×#51914yes—
364C22C31×#54714yes—
365C22D4×Q8×C3114yes—
366C22C31×#55614yes—
367C22C31×#55714yes—
368C22C24×C4×C3111yescharacteristic
369C22C23×D4×C3111yescharacteristic
370C22C23×Q8×C3111yescharacteristic
371C22C22×#41×C3111yescharacteristic
372C22C22×#41×C3111yescharacteristic
373C22C2×C31×2−1+411yescharacteristic
374C22C2×C31×2−1+411yescharacteristic
375C4—216no—
376C4—24no—
377C4—28no—
378C4—28no—
379C4—44no—
380C4—44no—
381C4C2×C4×D4×C3118yes—
382C4C2×C4×Q8×C3118yescomplemented
383C2C31×#236518yesdirect product factor
384C2C22×C4×D4×C3111yescharacteristic
385C2C22×C4×Q8×C3111yescharacteristic
386C2C2×C31×#51911yescharacteristic
387C2C2×C31×#54711yescharacteristic
388C2C2×D4×Q8×C3111yescharacteristic
389C2C2×C31×#55611yescharacteristic
390C2C2×C31×#55711yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C31C24
commutatorC22C24×C4×C31
Frattini, cosocleC23C25×C31
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C31 ◁ G

Complements

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

NH ≅ G/Ntype
3. C31×#2365C2direct (N × H)
11. C2×C31×C4⋊Q8C4semidirect
48. C31×C4⋊Q8C2×C4semidirect
56. C2×C4⋊C4×C31Q8semidirect
119. C4⋊C4×C31C2×Q8semidirect
124. C2×#2365C31direct (N × H)
155. C2×C4×C31C4×Q8semidirect
168. #2365C2×C31direct (N × H)
196. C4×C31C2×C4×Q8semidirect
198. C2×C4⋊Q8C4×C31semidirect
223. C2×C31#2365direct (N × H)
243. C4⋊Q8C2×C4×C31semidirect
251. C2×C4⋊C4Q8×C31semidirect
271. C31C2×#2365direct (N × H)
315. C4⋊C4C2×Q8×C31semidirect
350. C2×C4C4×Q8×C31semidirect
382. C4C2×C4×Q8×C31semidirect
383. C2C31×#2365direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×C4⋊Q82
C22×C4×Q8×C312
C31×#23652
C2×C31×#3912
C2×C31×#3892
C2×C31×#3872
C2×C4×C4⋊C4×C312
C2×#236531