Finite Group Databasebeta

← C31×#7187 #21463946

Subgroups — C31×#7187 #21463946

Order 7936.

678 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 126 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×#323C212yesmaximal
2C31×C22⋊#95C211yesmaximal, characteristic
3C24×C4×C31C411yescharacteristic
4C22×C22⋊C4×C31C411yescharacteristic
5C22×C22⋊C4×C31C2211yescharacteristic
6C2×C31×C22⋊C8—22no—
7C2×C31×#95—22no—
8C31×#323—22no—
9C31×#323C414yescomplemented
10C25×C31C2×C411yescharacteristic
11C23×C4×C31—21no—
12C23×C4×C31—21no—
13C23×C4×C31—21no—
14C23×C4×C31—44no—
15C23×C4×C31—41no—
16C23×C4×C31—41no—
17C23×C4×C31C812yescomplemented
18C23×C4×C31D412yes—
19C22×C8×C31—42no—
20C2×C22⋊C4×C31—21no—
21C2×C22⋊C4×C31—41no—
22C2×C22⋊C4×C31—41no—
23C2×C22⋊C4×C31—41no—
24C2×C22⋊C4×C31—41no—
25C2×C22⋊C4×C31C812yescomplemented
26C2×C22⋊C4×C31C2×C412yes—
27C2×C22⋊C4×C31D411yescharacteristic
28C2×C22⋊C4×C31Q811yescharacteristic
29C31×C22⋊C8—24no—
30C31×C22⋊C8—42no—
31C31×#95—44no—
32C24×C31—21no—
33C24×C31—21no—
34C24×C31—21no—
35C24×C31—21no—
36C24×C31—42no—
37C24×C31—42no—
38C24×C31C4211yescharacteristic
39C24×C31C22⋊C411yescharacteristic
40C24×C31C4⋊C411yescharacteristic
41C24×C31C2×C811yescharacteristic
42C24×C31C2×C811yescharacteristic
43C24×C31M1611yescharacteristic
44C24×C31M1611yescharacteristic
45C22×C4×C31—24no—
46C22×C4×C31—22no—
47C22×C4×C31—22no—
48C22×C4×C31—22no—
49C22×C4×C31—22no—
50C22×C4×C31—44no—
51C22×C4×C31—44no—
52C22×C4×C31—42no—
53C22×C4×C31—42no—
54C22×C4×C31—48no—
55C22×C4×C31—44no—
56C22×C4×C31—41no—
57C22×C4×C31—41no—
58C22×C4×C31—42no—
59C22×C4×C31—44no—
60C22×C4×C31—42no—
61C22×C4×C31—42no—
62C22×C4×C31—42no—
63C22×C4×C31—42no—
64C22×C4×C31—41no—
65C22×C4×C31—41no—
66C22×C4×C31C22⋊C412yes—
67C22×C4×C31C4⋊C412yes—
68C2×C8×C31—44no—
69C2×C8×C31—82no—
70C2×C8×C31—82no—
71C22⋊C4×C31—42no—
72C22⋊C4×C31—41no—
73C22⋊C4×C31—41no—
74C22⋊C4×C31—42no—
75C22⋊C4×C31—42no—
76#7187C3111yesmaximal, characteristic, direct product factor, Sylow
77C23×C31—22no—
78C23×C31—22no—
79C23×C31—21no—
80C23×C31—21no—
81C23×C31—21no—
82C23×C31—21no—
83C23×C31—21no—
84C23×C31—21no—
85C23×C31—21no—
86C23×C31—21no—
87C23×C31—21no—
88C23×C31—21no—
89C23×C31—21no—
90C23×C31—21no—
91C23×C31—41no—
92C23×C31—41no—
93C23×C31—42no—
94C23×C31—42no—
95C23×C31—41no—
96C23×C31—42no—
97C23×C31—41no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—44no—
102C23×C31—44no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31#9511yescharacteristic
106C23×C31C4×C811yescharacteristic
107C23×C31#9711yescharacteristic
108C23×C31C22⋊C811yescharacteristic
109C23×C31C22⋊C811yescharacteristic
110C23×C31#9911yescharacteristic
111C23×C31#9911yescharacteristic
112C23×C31#10011yescharacteristic
113C23×C31#10111yescharacteristic
114C23×C31C4⋊C811yescharacteristic
115C23×C31C4⋊C811yescharacteristic
116C2×C4×C31—24no—
117C2×C4×C31—22no—
118C2×C4×C31—22no—
119C2×C4×C31—42no—
120C2×C4×C31—44no—
121C2×C4×C31—48no—
122C2×C4×C31—44no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—44no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31—42no—
129C2×C4×C31—44no—
130C2×C4×C31—44no—
131C2×C4×C31—42no—
132C2×C4×C31—82no—
133C2×C4×C31—81no—
134C2×C4×C31—81no—
135C2×C4×C31—82no—
136C8×C31—84no—
137C2×#323C2×C3112yes—
138C22⋊#95C2×C3111yescharacteristic
139C22×C31—21no—
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—21no—
143C22×C31—22no—
144C22×C31—22no—
145C22×C31—22no—
146C22×C31—22no—
147C22×C31—21no—
148C22×C31—21no—
149C22×C31—21no—
150C22×C31—21no—
151C22×C31—41no—
152C22×C31—41no—
153C22×C31—41no—
154C22×C31—41no—
155C22×C31—42no—
156C22×C31—42no—
157C22×C31—42no—
158C22×C31—42no—
159C22×C31—42no—
160C22×C31—42no—
161C22×C31—41no—
162C22×C31—41no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—42no—
166C22×C31—42no—
167C22×C31—42no—
168C22×C31—42no—
169C22×C31#32311yescharacteristic
170C22×C31#32311yescharacteristic
171C22×C31#32411yescharacteristic
172C22×C31#32411yescharacteristic
173C22×C31#33611yescharacteristic
174C22×C31#34211yescharacteristic
175C22×C31#34311yescharacteristic
176C22×C31AΣL(1,𝔽16)12yes—
177C22×C31#35212yes—
178C4×C31—44no—
179C4×C31—42no—
180C4×C31—42no—
181C4×C31—82no—
182C24×C4C4×C3111yescharacteristic
183C22×C22⋊C4C4×C3111yescharacteristic
184C22×C22⋊C4C22×C3111yescharacteristic
185C2×C22⋊C8—22no—
186C2×#95—22no—
187#323—22no—
188#323C4×C3114yescomplemented
189C2×C31—21no—
190C2×C31—21no—
191C2×C31—21no—
192C2×C31—21no—
193C2×C31—41no—
194C2×C31—41no—
195C2×C31—41no—
196C2×C31—41no—
197C2×C31#128011yescharacteristic
198C2×C31#131611yescharacteristic
199C2×C31#131711yescharacteristic
200C2×C31#132612yes—
201C2×C31#139012yes—
202C25C2×C4×C3111yescharacteristic
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—44no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4C8×C3112yescomplemented
210C23×C4D4×C3112yes—
211C22×C8—42no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4C8×C3112yescomplemented
218C2×C22⋊C4C2×C4×C3112yes—
219C2×C22⋊C4D4×C3111yescharacteristic
220C2×C22⋊C4Q8×C3111yescharacteristic
221C22⋊C8—24no—
222C22⋊C8—42no—
223#95—44no—
224C31#718711yescharacteristic, direct product factor, Sylow
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—42no—
230C24—42no—
231C24C42×C3111yescharacteristic
232C24C22⋊C4×C3111yescharacteristic
233C24C4⋊C4×C3111yescharacteristic
234C24C2×C8×C3111yescharacteristic
235C24C2×C8×C3111yescharacteristic
236C24M16×C3111yescharacteristic
237C24M16×C3111yescharacteristic
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4C22⋊C4×C3112yes—
260C22×C4C4⋊C4×C3112yes—
261C2×C8—44no—
262C2×C8—82no—
263C2×C8—82no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C23—22no—
270C23—22no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—41no—
288C23—42no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—42no—
297C23C31×#9511yescharacteristic
298C23C4×C8×C3111yescharacteristic
299C23C31×#9711yescharacteristic
300C23C31×C22⋊C811yescharacteristic
301C23C31×C22⋊C811yescharacteristic
302C23C31×#9911yescharacteristic
303C23C31×#9911yescharacteristic
304C23C31×#10011yescharacteristic
305C23C31×#10111yescharacteristic
306C23C31×C4⋊C811yescharacteristic
307C23C31×C4⋊C811yescharacteristic
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C8—84no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22C31×#32311yescharacteristic
360C22C31×#32311yescharacteristic
361C22C31×#32411yescharacteristic
362C22C31×#32411yescharacteristic
363C22C31×#33611yescharacteristic
364C22C31×#34211yescharacteristic
365C22C31×#34311yescharacteristic
366C22C31×AΣL(1,𝔽16)12yes—
367C22C31×#35212yes—
368C4—44no—
369C4—42no—
370C4—42no—
371C4—82no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2—41no—
379C2—41no—
380C2C31×#128011yescharacteristic
381C2C31×#131611yescharacteristic
382C2C31×#131711yescharacteristic
383C2C31×#132612yes—
384C2C31×#139012yes—
subgroupG/H
center, socleC23×C31#99
commutatorC23C4×C8×C31
Frattini, cosocleC22×C22⋊C4C22×C31
prime radicalC25×C31C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C31 ◁ C24×C31 ◁ C22×C22⋊C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
9. C31×#323C4semidirect
17. C23×C4×C31C8semidirect
25. C2×C22⋊C4×C31C8semidirect
76. #7187C31direct (N × H)
188. #323C4×C31semidirect
209. C23×C4C8×C31semidirect
217. C2×C22⋊C4C8×C31semidirect
224. C31#7187direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#3232
C31×C22⋊#952
#718731