Finite Group Databasebeta

← C31×#8553 #21465312

Subgroups — C31×#8553 #21465312

Order 7936.

1166 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C31×#123C211yesmaximal, characteristic
2C31×#1821C212yesmaximal, complemented
3C31×#1462C212yesmaximal, complemented
4C31×#1462C212yesmaximal, complemented
5C23×D4×C31C411yescharacteristic
6C24×C4×C31C411yescharacteristic
7C2×C31×#123—24no—
8C2×C31×#123—24no—
9C2×C31×#123—24no—
10C22×C4⋊C4×C31C411yescharacteristic
11C22×C4⋊C4×C31C2211yescharacteristic
12C22×C22⋊C4×C31C411yescharacteristic
13C22×C22⋊C4×C31C2211yescharacteristic
14C22×C22⋊C4×C31C2211yescharacteristic
15C31×#407C2214yes—
16C31×#343—24no—
17C31×#323—24no—
18C31×#323—24no—
19C25×C31C2×C411yescharacteristic
20C25×C31C2×C411yescharacteristic
21C22×D4×C31—22no—
22C22×D4×C31—22no—
23C22×D4×C31—22no—
24C22×D4×C31—44no—
25C23×C4×C31—22no—
26C23×C4×C31—22no—
27C23×C4×C31—22no—
28C23×C4×C31—44no—
29C23×C4×C31C2×C411yescharacteristic
30C23×C4×C31C2×C411yescharacteristic
31C23×C4×C31C2×C411yescharacteristic
32C23×C4×C31C2×C411yescharacteristic
33C23×C4×C31C2311yescharacteristic
34C2×M16×C31—24no—
35C31×#123—232no—
36C2×C4⋊C4×C31—22no—
37C2×C4⋊C4×C31—22no—
38C2×C4⋊C4×C31—22no—
39C2×C4⋊C4×C31—22no—
40C2×C4⋊C4×C31—22no—
41C2×C4⋊C4×C31D414yes—
42C2×C22⋊C4×C31—22no—
43C2×C22⋊C4×C31—22no—
44C2×C22⋊C4×C31—22no—
45C2×C22⋊C4×C31—22no—
46C2×C22⋊C4×C31—22no—
47C2×C22⋊C4×C31—22no—
48C2×C22⋊C4×C31—22no—
49C2×C22⋊C4×C31D414yes—
50C2×C22⋊C4×C31D414yes—
51C31×C22⋊C8—24no—
52C31×C22⋊C8—24no—
53C24×C31—24no—
54C24×C31—21no—
55C24×C31—21no—
56C24×C31—21no—
57C24×C31—21no—
58C24×C31—42no—
59C24×C31—42no—
60C24×C31—42no—
61C24×C31—42no—
62C24×C31—42no—
63C24×C31C22⋊C412yes—
64C24×C31C22⋊C412yes—
65C24×C31C22×C411yescharacteristic
66C2×D4×C31—28no—
67C2×D4×C31—48no—
68C2×D4×C31—44no—
69C2×D4×C31—48no—
70C2×D4×C31—44no—
71C22×C4×C31—28no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—21no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—42no—
81C22×C4×C31—48no—
82C22×C4×C31—48no—
83C22×C4×C31—44no—
84C22×C4×C31—44no—
85C22×C4×C31—42no—
86C22×C4×C31—42no—
87C22×C4×C31—42no—
88C22×C4×C31—42no—
89C22×C4×C31C22⋊C412yes—
90C22×C4×C31C22⋊C412yes—
91C22×C4×C31C22⋊C412yes—
92C22×C4×C31C22⋊C412yes—
93C22×C4×C31C2×D412yes—
94C22×C4×C31C2×D412yes—
95C22×C4×C31C2×D412yes—
96M16×C31—48no—
97C2×C8×C31—44no—
98C4⋊C4×C31—28no—
99C4⋊C4×C31—28no—
100C22⋊C4×C31—28no—
101C22⋊C4×C31—28no—
102C22⋊C4×C31—28no—
103#8553C3111yesmaximal, characteristic, direct product factor, Sylow
104C23×C31—22no—
105C23×C31—22no—
106C23×C31—22no—
107C23×C31—24no—
108C23×C31—24no—
109C23×C31—22no—
110C23×C31—22no—
111C23×C31—22no—
112C23×C31—24no—
113C23×C31—42no—
114C23×C31—42no—
115C23×C31—416no—
116C23×C31—42no—
117C23×C31—42no—
118C23×C31—42no—
119C23×C31—42no—
120C23×C31—42no—
121C23×C31—42no—
122C23×C31—42no—
123C23×C31—42no—
124C23×C31—42no—
125C23×C31—42no—
126C23×C31—42no—
127C23×C31—42no—
128C23×C31—88no—
129C23×C31#10014yes—
130C23×C31C2×C22⋊C411yescharacteristic
131C23×C31C2×C22⋊C411yescharacteristic
132C23×C31C2×C22⋊C411yescharacteristic
133D4×C31—416no—
134C2×C4×C31—24no—
135C2×C4×C31—22no—
136C2×C4×C31—22no—
137C2×C4×C31—22no—
138C2×C4×C31—22no—
139C2×C4×C31—42no—
140C2×C4×C31—42no—
141C2×C4×C31—42no—
142C2×C4×C31—42no—
143C2×C4×C31—42no—
144C2×C4×C31—416no—
145C2×C4×C31—42no—
146C2×C4×C31—42no—
147C2×C4×C31—42no—
148C2×C4×C31—42no—
149C2×C4×C31—42no—
150C2×C4×C31—42no—
151C2×C4×C31—42no—
152C2×C4×C31—42no—
153C2×C4×C31—42no—
154C2×C4×C31#12014yes—
155C8×C31—84no—
156C22×#123C2×C3111yescharacteristic
157#1821C2×C3112yescomplemented
158#1462C2×C3112yescomplemented
159#1462C2×C3112yescomplemented
160C22×C31—28no—
161C22×C31—21no—
162C22×C31—21no—
163C22×C31—21no—
164C22×C31—21no—
165C22×C31—21no—
166C22×C31—21no—
167C22×C31—24no—
168C22×C31—44no—
169C22×C31—48no—
170C22×C31—44no—
171C22×C31—48no—
172C22×C31—42no—
173C22×C31—42no—
174C22×C31—42no—
175C22×C31—42no—
176C22×C31—42no—
177C22×C31—84no—
178C22×C31—84no—
179C22×C31—84no—
180C22×C31#37911yescharacteristic
181C22×C31#41012yes—
182C22×C31C2×#10012yes—
183C22×C31#41312yes—
184C4×C31—24no—
185C4×C31—42no—
186C4×C31—42no—
187C4×C31—42no—
188C4×C31—42no—
189C23×D4C4×C3111yescharacteristic
190C24×C4C4×C3111yescharacteristic
191C2×#123—24no—
192C2×#123—24no—
193C2×#123—24no—
194C22×C4⋊C4C4×C3111yescharacteristic
195C22×C4⋊C4C22×C3111yescharacteristic
196C22×C22⋊C4C4×C3111yescharacteristic
197C22×C22⋊C4C22×C3111yescharacteristic
198C22×C22⋊C4C22×C3111yescharacteristic
199#407C22×C3114yes—
200#343—24no—
201#323—24no—
202#323—24no—
203C2×C31—22no—
204C2×C31—22no—
205C2×C31—22no—
206C2×C31—44no—
207C2×C31—82no—
208C2×C31#178211yescharacteristic
209C2×C31#178311yescharacteristic
210C2×C31#178511yescharacteristic
211C25C2×C4×C3111yescharacteristic
212C25C2×C4×C3111yescharacteristic
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—44no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—44no—
221C23×C4C2×C4×C3111yescharacteristic
222C23×C4C2×C4×C3111yescharacteristic
223C23×C4C2×C4×C3111yescharacteristic
224C23×C4C2×C4×C3111yescharacteristic
225C23×C4C23×C3111yescharacteristic
226C2×M16—24no—
227#123—232no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4D4×C3114yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4D4×C3114yes—
242C2×C22⋊C4D4×C3114yes—
243C22⋊C8—24no—
244C22⋊C8—24no—
245C31#855311yescharacteristic, direct product factor, Sylow
246C24—24no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C22⋊C4×C3112yes—
257C24C22⋊C4×C3112yes—
258C24C22×C4×C3111yescharacteristic
259C2×D4—28no—
260C2×D4—48no—
261C2×D4—44no—
262C2×D4—48no—
263C2×D4—44no—
264C22×C4—28no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C22⋊C4×C3112yes—
283C22×C4C22⋊C4×C3112yes—
284C22×C4C22⋊C4×C3112yes—
285C22×C4C22⋊C4×C3112yes—
286C22×C4C2×D4×C3112yes—
287C22×C4C2×D4×C3112yes—
288C22×C4C2×D4×C3112yes—
289M16—48no—
290C2×C8—44no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—28no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23C31×#10014yes—
322C23C2×C22⋊C4×C3111yescharacteristic
323C23C2×C22⋊C4×C3111yescharacteristic
324C23C2×C22⋊C4×C3111yescharacteristic
325D4—416no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—416no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4C31×#12014yes—
347C8—84no—
348C22—28no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—24no—
356C22—44no—
357C22—48no—
358C22—44no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22C31×#37911yescharacteristic
369C22C31×#41012yes—
370C22C2×C31×#10012yes—
371C22C31×#41312yes—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—42no—
376C4—42no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—44no—
381C2—82no—
382C2C31×#178211yescharacteristic
383C2C31×#178311yescharacteristic
384C2C31×#178511yescharacteristic
subgroupG/H
center, socleC22×C31#379
commutatorC24C22×C4×C31
Frattini, cosocleC23×C4C23×C31
prime radicalC23×D4×C31C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1821C2semidirect
3. C31×#1462C2semidirect
4. C31×#1462C2semidirect
103. #8553C31direct (N × H)
157. #1821C2×C31semidirect
158. #1462C2×C31semidirect
159. #1462C2×C31semidirect
245. C31#8553direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1232
C31×#18212
C31×#14622
#855331