Finite Group Databasebeta

← C31×#36600 #21493359

Subgroups — C31×#36600 #21493359

Order 7936.

1574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#2859C211yesmaximal, characteristic
2C31×#2859C212yesmaximal
3C31×#2585C218yesmaximal
4C31×#2570C214yesmaximal
5C31×#2552C212yesmaximal
6C31×#2453C214yesmaximal
7C31×#2370C214yesmaximal
8C31×#2355C212yesmaximal
9C4×C31×#125C212yesmaximal
10C4×C31×#125C211yesmaximal, characteristic
11C42×Q8×C31C211yesmaximal, characteristic
12C2×C31×#125C2212yes—
13C2×C31×#125C2211yescharacteristic
14C2×C31×#125C2214yes—
15C2×C4×Q8×C31C2212yes—
16C2×C4×Q8×C31C2211yescharacteristic
17C31×#400C2218yes—
18C31×#398C2218yes—
19C31×#398C2218yes—
20C31×#391C2214yes—
21C31×#389C2212yes—
22C31×#389C2214yes—
23C31×#389C2214yes—
24C31×#389C2218yes—
25C31×#389C2218yes—
26C31×#389C2214yes—
27C31×#389C2214yes—
28C31×#387C2218yes—
29C31×#387C2218yes—
30C31×#387C2218yes—
31C31×#387C2214yes—
32C31×#387C2218yes—
33C31×#384C2214yes—
34C31×#384C2212yes—
35C31×#382C2212yes—
36C31×#382C2214yes—
37C31×#382C2211yescharacteristic
38C4×C4⋊C4×C31C2212yes—
39C4×C4⋊C4×C31C2214yes—
40C4×C4⋊C4×C31C2214yes—
41C4×C4⋊C4×C31C2218yes—
42C4×C4⋊C4×C31C2214yes—
43C4×C4⋊C4×C31C2212yes—
44C4×C4⋊C4×C31C2212yes—
45C4×C4⋊C4×C31C2214yes—
46C4×C4⋊C4×C31C2211yescharacteristic
47C43×C31C2211yescharacteristic
48C43×C31C2212yes—
49C22×Q8×C31C2311yescharacteristic
50C31×#125—216no—
51C31×#125—28no—
52C31×#125—24no—
53C4×Q8×C31—28no—
54C4×Q8×C31Q818yes—
55C2×C4⋊C4×C31C2314yes—
56C2×C4⋊C4×C31C2318yes—
57C2×C4⋊C4×C31C2318yes—
58C2×C4⋊C4×C31C2314yes—
59C2×C4⋊C4×C31C2312yes—
60C2×C4⋊C4×C31C2314yes—
61C2×C4⋊C4×C31C2318yes—
62C2×C4⋊C4×C31C2314yes—
63C2×C4⋊C4×C31C2312yes—
64C2×C4⋊C4×C31C2314yes—
65C2×C4⋊C4×C31C2314yes—
66C2×C4⋊C4×C31C2318yes—
67C2×C4⋊C4×C31C2311yescharacteristic
68C2×C4⋊C4×C31C2312yes—
69C2×C4⋊C4×C31C2312yes—
70C2×C42×C31C2312yes—
71C2×C42×C31C2314yes—
72C2×C42×C31C2314yes—
73C2×C42×C31C2311yescharacteristic
74C2×C42×C31C2312yes—
75C2×C42×C31C2314yes—
76C2×C42×C31C2312yes—
77C2×C42×C31C2318yes—
78C2×C42×C31C2312yes—
79C2×C42×C31C2311yescharacteristic
80C2×C42×C31C2311yescharacteristic
81C31×#95C2312yes—
82C31×#95C2314yes—
83C31×#95C2318yes—
84C31×#95C2318yes—
85C31×#95C2318yes—
86C31×#95C2318yes—
87C31×#95C2314yes—
88C31×#95C2314yes—
89C31×#95C2314yes—
90C31×#95C2318yes—
91C2×Q8×C31—24no—
92C2×Q8×C31C2×Q814yes—
93C22×C4×C31C2412yes—
94C22×C4×C31C2414yes—
95C22×C4×C31C2414yes—
96C22×C4×C31C2418yes—
97C22×C4×C31C2412yes—
98C22×C4×C31C2412yes—
99C22×C4×C31C2411yescharacteristic
100C22×C4×C31C2414yes—
101C22×C4×C31C2412yes—
102C22×C4×C31C2411yescharacteristic
103C22×C4×C31C2411yescharacteristic
104C4⋊C4×C31—28no—
105C4⋊C4×C31—216no—
106C4⋊C4×C31—216no—
107C4⋊C4×C31—216no—
108C4⋊C4×C31—28no—
109C4⋊C4×C31—28no—
110C4⋊C4×C31—28no—
111C4⋊C4×C31—24no—
112C4⋊C4×C31—24no—
113C4⋊C4×C31—44no—
114C4⋊C4×C31—48no—
115C4⋊C4×C31—44no—
116C4⋊C4×C31C2×Q818yes—
117C4⋊C4×C31C2×Q814yes—
118C4⋊C4×C31#4118yes—
119C42×C31—28no—
120C42×C31—24no—
121C42×C31—28no—
122C42×C31—24no—
123C42×C31—22no—
124C42×C31—44no—
125C42×C31—42no—
126C42×C31—48no—
127C42×C31C2×Q818yes—
128C42×C31C2×Q814yes—
129C42×C31#4114yes—
130#36600C3111yesmaximal, characteristic, direct product factor, Sylow
131C23×C31C2511yescharacteristic
132Q8×C31—28no—
133C2×C4×C31—22no—
134C2×C4×C31—24no—
135C2×C4×C31—24no—
136C2×C4×C31—22no—
137C2×C4×C31—24no—
138C2×C4×C31—22no—
139C2×C4×C31—24no—
140C2×C4×C31—24no—
141C2×C4×C31—22no—
142C2×C4×C31—24no—
143C2×C4×C31—28no—
144C2×C4×C31—24no—
145C2×C4×C31—28no—
146C2×C4×C31—22no—
147C2×C4×C31—24no—
148C2×C4×C31—24no—
149C2×C4×C31—28no—
150C2×C4×C31—22no—
151C2×C4×C31—24no—
152C2×C4×C31C22×Q814yes—
153C2×C4×C31C22×Q812yes—
154C2×C4×C31C22×Q814yes—
155C2×C4×C31C22×Q812yes—
156C2×C4×C31C22×Q812yes—
157C2×C4×C31C2×#4118yes—
158C2×C4×C31C2×#4114yes—
159C2×C4×C31C2×#4114yes—
160C2×C4×C31C2×#4112yes—
161C2×C4×C312−1+412yes—
162#2859C2×C3111yescharacteristic
163#2859C2×C3112yes—
164#2585C2×C3118yes—
165#2570C2×C3114yes—
166#2552C2×C3112yes—
167#2453C2×C3114yes—
168#2370C2×C3114yes—
169#2355C2×C3112yes—
170C4×#125C2×C3112yes—
171C4×#125C2×C3111yescharacteristic
172C42×Q8C2×C3111yescharacteristic
173C22×C31C23×Q811yescharacteristic
174C22×C31C22×#4112yes—
175C22×C31C22×#4111yescharacteristic
176C22×C31C2×2−1+411yescharacteristic
177C22×C31#58512yes—
178C4×C31—24no—
179C4×C31—24no—
180C4×C31—24no—
181C4×C31—22no—
182C4×C31—28no—
183C4×C31—44no—
184C4×C31—48no—
185C4×C31—42no—
186C4×C31—44no—
187C4×C31#53314yes—
188C4×C31#55714yes—
189C2×#125C22×C3112yes—
190C2×#125C22×C3111yescharacteristic
191C2×#125C22×C3114yes—
192C2×C4×Q8C22×C3112yes—
193C2×C4×Q8C22×C3111yescharacteristic
194#400C22×C3118yes—
195#398C22×C3118yes—
196#398C22×C3118yes—
197#391C22×C3114yes—
198#389C22×C3112yes—
199#389C22×C3114yes—
200#389C22×C3114yes—
201#389C22×C3118yes—
202#389C22×C3118yes—
203#389C22×C3114yes—
204#389C22×C3114yes—
205#387C22×C3114yes—
206#387C22×C3118yes—
207#387C22×C3118yes—
208#387C22×C3118yes—
209#387C22×C3118yes—
210#384C22×C3114yes—
211#384C22×C3112yes—
212#382C22×C3111yescharacteristic
213#382C22×C3112yes—
214#382C22×C3114yes—
215C4×C4⋊C4C22×C3114yes—
216C4×C4⋊C4C22×C3118yes—
217C4×C4⋊C4C22×C3112yes—
218C4×C4⋊C4C22×C3114yes—
219C4×C4⋊C4C22×C3112yes—
220C4×C4⋊C4C22×C3114yes—
221C4×C4⋊C4C22×C3114yes—
222C4×C4⋊C4C22×C3112yes—
223C4×C4⋊C4C22×C3111yescharacteristic
224C43C22×C3111yescharacteristic
225C43C22×C3112yes—
226C2×C31C2×#53311yescharacteristic
227C2×C31C2×#55711yescharacteristic
228C2×C31#348012yes—
229C2×C31#350411yescharacteristic
230C2×C31#351512yes—
231C22×Q8C23×C3111yescharacteristic
232#125—216no—
233#125—28no—
234#125—24no—
235C4×Q8—28no—
236C4×Q8Q8×C3118yes—
237C2×C4⋊C4C23×C3114yes—
238C2×C4⋊C4C23×C3114yes—
239C2×C4⋊C4C23×C3112yes—
240C2×C4⋊C4C23×C3114yes—
241C2×C4⋊C4C23×C3118yes—
242C2×C4⋊C4C23×C3118yes—
243C2×C4⋊C4C23×C3114yes—
244C2×C4⋊C4C23×C3118yes—
245C2×C4⋊C4C23×C3112yes—
246C2×C4⋊C4C23×C3114yes—
247C2×C4⋊C4C23×C3114yes—
248C2×C4⋊C4C23×C3118yes—
249C2×C4⋊C4C23×C3111yescharacteristic
250C2×C4⋊C4C23×C3112yes—
251C2×C4⋊C4C23×C3112yes—
252C2×C42C23×C3112yes—
253C2×C42C23×C3114yes—
254C2×C42C23×C3114yes—
255C2×C42C23×C3111yescharacteristic
256C2×C42C23×C3112yes—
257C2×C42C23×C3114yes—
258C2×C42C23×C3112yes—
259C2×C42C23×C3118yes—
260C2×C42C23×C3112yes—
261C2×C42C23×C3111yescharacteristic
262C2×C42C23×C3111yescharacteristic
263#95C23×C3114yes—
264#95C23×C3114yes—
265#95C23×C3112yes—
266#95C23×C3118yes—
267#95C23×C3114yes—
268#95C23×C3114yes—
269#95C23×C3118yes—
270#95C23×C3118yes—
271#95C23×C3118yes—
272#95C23×C3118yes—
273C31#3660011yescharacteristic, direct product factor, Sylow
274C2×Q8—24no—
275C2×Q8C2×Q8×C3114yes—
276C22×C4C24×C3112yes—
277C22×C4C24×C3114yes—
278C22×C4C24×C3114yes—
279C22×C4C24×C3118yes—
280C22×C4C24×C3112yes—
281C22×C4C24×C3112yes—
282C22×C4C24×C3111yescharacteristic
283C22×C4C24×C3114yes—
284C22×C4C24×C3112yes—
285C22×C4C24×C3111yescharacteristic
286C22×C4C24×C3111yescharacteristic
287C4⋊C4—28no—
288C4⋊C4—216no—
289C4⋊C4—216no—
290C4⋊C4—216no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—44no—
299C4⋊C4C2×Q8×C3118yes—
300C4⋊C4C2×Q8×C3114yes—
301C4⋊C4#41×C3118yes—
302C42—28no—
303C42—24no—
304C42—28no—
305C42—24no—
306C42—22no—
307C42—44no—
308C42—42no—
309C42—48no—
310C42C2×Q8×C3118yes—
311C42C2×Q8×C3114yes—
312C42#41×C3114yes—
313C23C25×C3111yescharacteristic
314Q8—28no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4C22×Q8×C3114yes—
335C2×C4C22×Q8×C3112yes—
336C2×C4C22×Q8×C3114yes—
337C2×C4C22×Q8×C3112yes—
338C2×C4C22×Q8×C3112yes—
339C2×C4C2×#41×C3118yes—
340C2×C4C2×#41×C3114yes—
341C2×C4C2×#41×C3114yes—
342C2×C4C2×#41×C3112yes—
343C2×C4C31×2−1+412yes—
344C22C23×Q8×C3111yescharacteristic
345C22C22×#41×C3112yes—
346C22C22×#41×C3111yescharacteristic
347C22C2×C31×2−1+411yescharacteristic
348C22C31×#58512yes—
349C4—24no—
350C4—24no—
351C4—24no—
352C4—22no—
353C4—28no—
354C4—44no—
355C4—48no—
356C4—42no—
357C4—44no—
358C4C31×#53314yes—
359C4C31×#55714yes—
360C2C2×C31×#53311yescharacteristic
361C2C2×C31×#55711yescharacteristic
362C2C31×#348012yes—
363C2C31×#350411yescharacteristic
364C2C31×#351512yes—
subgroupG/H
center, socle, prime radicalC23×C31C25
commutator, Frattini, cosocleC23C25×C31
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ G

Complements

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

NH ≅ G/Ntype
130. #36600C31direct (N × H)
273. C31#36600direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28592
C31×#25852
C31×#25702
C31×#25522
C31×#24532
C31×#23702
C31×#23552
C4×C31×#1252
C42×Q8×C312
#3660031