Finite Group Databasebeta

← C31×#7805 #21464564

Subgroups — C31×#7805 #21464564

Order 7936.

396 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C31×#2808C211yesmaximal, characteristic
2C31×#2096C211yesmaximal, characteristic, complemented
3C31×#2083C211yesmaximal, characteristic, complemented
4C31×#2071C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1271C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic
8C2×C31×#107—21no—
9C2×C31×#106—21no—
10C2×C31×#103—21no—
11C2×C31×#102—21no—
12C31×#397—21no—
13C31×#394—21no—
14C31×#393—21no—
15C31×#391C2211yescharacteristic
16C31×#390C2211yescharacteristic
17C31×#389C2211yescharacteristic
18C31×#388—21no—
19C31×#388—21no—
20C31×#387—21no—
21C31×#384C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31D411yescharacteristic
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C22×C8×C31—21no—
32C2×C4⋊C4×C31—21no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C4⋊C4×C31D411yescharacteristic
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C22⋊C4×C31—21no—
41C2×C22⋊C4×C31—21no—
42C2×C22⋊C4×C31—21no—
43C2×C22⋊C4×C31—21no—
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C42×C31—21no—
47C2×C42×C31C2311yescharacteristic
48C31×#107—42no—
49C31×#106—42no—
50C31×#103—42no—
51C31×#102—42no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C24×C31—21no—
59C2×Q8×C31—21no—
60C2×Q8×C31—21no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×D4×C31—21no—
64C2×D4×C31—21no—
65C2×D4×C31—41no—
66C2×D4×C31—41no—
67C22×C4×C31—21no—
68C22×C4×C31—21no—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—41no—
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76C2×C8×C31—22no—
77C2×C8×C31—22no—
78C2×C8×C31—41no—
79C2×C8×C31—41no—
80C2×C8×C31—41no—
81C2×C8×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31—41no—
85C4⋊C4×C31—21no—
86C4⋊C4×C31—21no—
87C4⋊C4×C31—41no—
88C4⋊C4×C31—41no—
89C4⋊C4×C31—41no—
90C4⋊C4×C31—41no—
91C4⋊C4×C31—41no—
92C4⋊C4×C31—41no—
93C4⋊C4×C31—41no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C22⋊C4×C31—41no—
99C22⋊C4×C31—41no—
100C22⋊C4×C31—41no—
101C22⋊C4×C31—41no—
102C22⋊C4×C31—81no—
103C22⋊C4×C31—81no—
104C22⋊C4×C31—81no—
105C22⋊C4×C31—81no—
106C42×C31—41no—
107C42×C31—81no—
108#7805C3111yesmaximal, characteristic, direct product factor, Sylow
109C23×C31—41no—
110C23×C31—41no—
111C23×C31—41no—
112C23×C31—41no—
113C23×C31—41no—
114C23×C31—41no—
115C23×C31—41no—
116C23×C31#12011yescharacteristic
117Q8×C31—42no—
118Q8×C31—81no—
119D4×C31—42no—
120D4×C31—81no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—41no—
128C2×C4×C31—41no—
129C2×C4×C31—41no—
130C2×C4×C31—41no—
131C2×C4×C31—41no—
132C2×C4×C31—41no—
133C2×C4×C31—41no—
134C2×C4×C31—41no—
135C2×C4×C31—41no—
136C2×C4×C31—41no—
137C2×C4×C31—41no—
138C2×C4×C31—41no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—81no—
143C2×C4×C31—81no—
144C2×C4×C31—81no—
145C2×C4×C31#13511yescharacteristic
146C2×C4×C31#13511yescharacteristic
147C2×C4×C31#13511yescharacteristic
148C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C31#13711yescharacteristic
151C8×C31—42no—
152C8×C31—42no—
153C8×C31—81no—
154#2808C2×C3111yescharacteristic
155#2096C2×C3111yescharacteristic, complemented
156#2083C2×C3111yescharacteristic, complemented
157#2071C2×C3111yescharacteristic
158#1272C2×C3111yescharacteristic, complemented
159#1271C2×C3111yescharacteristic, complemented
160#1270C2×C3111yescharacteristic
161C22×C31—81no—
162C22×C31—81no—
163C22×C31—81no—
164C22×C31—81no—
165C22×C31—81no—
166C22×C31—81no—
167C22×C31—81no—
168C22×C31#44911yescharacteristic
169C22×C31#44911yescharacteristic
170C22×C31#45211yescharacteristic
171C22×C31#45311yescharacteristic
172C22×C31#45511yescharacteristic
173C22×C31#45611yescharacteristic
174C22×C31#45811yescharacteristic
175C4×C31—21no—
176C4×C31—21no—
177C4×C31—21no—
178C4×C31—21no—
179C4×C31—21no—
180C4×C31—21no—
181C4×C31—81no—
182C4×C31—81no—
183C4×C31—81no—
184C4×C31—81no—
185C4×C31—81no—
186C4×C31—161no—
187C2×#107—21no—
188C2×#106—21no—
189C2×#103—21no—
190C2×#102—21no—
191#397—21no—
192#394—21no—
193#393—21no—
194#391C22×C3111yescharacteristic
195#390C22×C3111yescharacteristic
196#389C22×C3111yescharacteristic
197#388—21no—
198#388—21no—
199#387—21no—
200#384C22×C3111yescharacteristic
201#340—21no—
202#340—21no—
203#336C22×C3111yescharacteristic
204#336C22×C3111yescharacteristic
205#336C22×C3111yescharacteristic
206C2×C31—161no—
207C2×C31#160911yescharacteristic
208C2×C31#161211yescharacteristic
209C2×C31#161811yescharacteristic
210C2×C31#162411yescharacteristic
211C2×C31#164011yescharacteristic
212C2×C31#165011yescharacteristic
213C2×C31#166111yescharacteristic
214C22×Q8D4×C3111yescharacteristic
215C22×D4D4×C3111yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4D4×C3111yescharacteristic
224C2×C4⋊C4D4×C3111yescharacteristic
225C2×C4⋊C4D4×C3111yescharacteristic
226C2×C4⋊C4D4×C3111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42C23×C3111yescharacteristic
235#107—42no—
236#106—42no—
237#103—42no—
238#102—42no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245C31#780511yescharacteristic, direct product factor, Sylow
246C24—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4C2×D4×C3111yescharacteristic
262C22×C4C2×D4×C3111yescharacteristic
263C22×C4C2×D4×C3111yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C42—41no—
295C42—81no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C31×#12011yescharacteristic
304Q8—42no—
305Q8—81no—
306D4—42no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4C31×#13511yescharacteristic
333C2×C4C31×#13511yescharacteristic
334C2×C4C31×#13511yescharacteristic
335C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
336C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
337C2×C4C31×#13711yescharacteristic
338C8—42no—
339C8—42no—
340C8—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22C31×#44911yescharacteristic
349C22C31×#44911yescharacteristic
350C22C31×#45211yescharacteristic
351C22C31×#45311yescharacteristic
352C22C31×#45511yescharacteristic
353C22C31×#45611yescharacteristic
354C22C31×#45811yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—161no—
367C2—161no—
368C2C31×#160911yescharacteristic
369C2C31×#161211yescharacteristic
370C2C31×#161811yescharacteristic
371C2C31×#162411yescharacteristic
372C2C31×#164011yescharacteristic
373C2C31×#165011yescharacteristic
374C2C31×#166111yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC22×D4×C31D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#2096C2semidirect
3. C31×#2083C2semidirect
5. C31×#1272C2semidirect
6. C31×#1271C2semidirect
108. #7805C31direct (N × H)
155. #2096C2×C31semidirect
156. #2083C2×C31semidirect
158. #1272C2×C31semidirect
159. #1271C2×C31semidirect
245. C31#7805direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28082
C31×#20962
C31×#20832
C31×#20712
C31×#12722
C31×#12712
C31×#12702
#780531