Finite Group Databasebeta

← C31×#7771 #21464530

Subgroups — C31×#7771 #21464530

Order 7936.

392 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2809C211yesmaximal, characteristic
2C31×#2057C211yesmaximal, characteristic
3C31×#2043C211yesmaximal, characteristic
4C31×#2024C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic
6C31×#1271C211yesmaximal, characteristic
7C31×#1271C211yesmaximal, characteristic
8C2×C31×C4⋊Q8C2211yescharacteristic
9C2×C31×#107—21no—
10C2×C31×#103—21no—
11C2×C31×#103—21no—
12C2×C31×#103—21no—
13C31×#400—21no—
14C31×#398—21no—
15C31×#398—21no—
16C31×#391—21no—
17C31×#391C2211yescharacteristic
18C31×#389C2211yescharacteristic
19C31×#389C2211yescharacteristic
20C31×#387—21no—
21C31×#387—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×C8×C31—21no—
30C22×C8×C31—21no—
31C22×C8×C31—21no—
32C31×C4⋊Q8—42no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31—21no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31D411yescharacteristic
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C42×C31—21no—
45C2×C42×C31C2311yescharacteristic
46C31×#107—42no—
47C31×#103—42no—
48C31×#103—42no—
49C31×#103—42no—
50C31×#95—21no—
51C31×#95—21no—
52C31×#95—21no—
53C31×#95—21no—
54C31×#95—21no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C2×Q8×C31—21no—
60C2×Q8×C31—22no—
61C2×Q8×C31—41no—
62C2×Q8×C31—41no—
63C2×Q8×C31—41no—
64C2×Q8×C31—41no—
65C2×Q8×C31—41no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31Q1612yes—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—21no—
75C22×C4×C31—41no—
76C22×C4×C31C2×D411yescharacteristic
77C22×C4×C31C2×D411yescharacteristic
78C22×C4×C31C2×D411yescharacteristic
79C2×C8×C31—22no—
80C2×C8×C31—41no—
81C2×C8×C31—41no—
82C2×C8×C31—41no—
83C2×C8×C31—41no—
84C2×C8×C31—41no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—41no—
88C4⋊C4×C31—22no—
89C4⋊C4×C31—22no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—21no—
93C4⋊C4×C31—21no—
94C4⋊C4×C31—41no—
95C4⋊C4×C31—41no—
96C4⋊C4×C31—41no—
97C4⋊C4×C31—41no—
98C4⋊C4×C31—41no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—81no—
104C4⋊C4×C31—81no—
105C42×C31—41no—
106C42×C31—81no—
107#7771C3111yesmaximal, characteristic, direct product factor, Sylow
108C23×C31#12011yescharacteristic
109Q8×C31—42no—
110Q8×C31—42no—
111Q8×C31—42no—
112Q8×C31—81no—
113Q8×C31—81no—
114Q8×C31—81no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—22no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
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—42no—
140C2×C4×C31—81no—
141C2×C4×C31—81no—
142C2×C4×C31C2×Q1611yescharacteristic
143C2×C4×C31#13511yescharacteristic
144C2×C4×C31#13511yescharacteristic
145C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C31#13711yescharacteristic
147C2×C4×C31#13711yescharacteristic
148C8×C31—42no—
149C8×C31—81no—
150C8×C31—81no—
151#2809C2×C3111yescharacteristic
152#2057C2×C3111yescharacteristic
153#2043C2×C3111yescharacteristic
154#2024C2×C3111yescharacteristic
155#1272C2×C3111yescharacteristic
156#1271C2×C3111yescharacteristic
157#1271C2×C3111yescharacteristic
158C22×C31#44911yescharacteristic
159C22×C31#45111yescharacteristic
160C22×C31#45211yescharacteristic
161C22×C31#45411yescharacteristic
162C22×C31#45511yescharacteristic
163C22×C31#45611yescharacteristic
164C22×C31B(4,𝔽2)11yescharacteristic
165C4×C31—21no—
166C4×C31—21no—
167C4×C31—21no—
168C4×C31—21no—
169C4×C31—21no—
170C4×C31—21no—
171C4×C31—42no—
172C4×C31—81no—
173C4×C31—82no—
174C4×C31—81no—
175C4×C31—81no—
176C4×C31—81no—
177C4×C31—81no—
178C2×C4⋊Q8C22×C3111yescharacteristic
179C2×#107—21no—
180C2×#103—21no—
181C2×#103—21no—
182C2×#103—21no—
183#400—21no—
184#398—21no—
185#398—21no—
186#391—21no—
187#391C22×C3111yescharacteristic
188#389C22×C3111yescharacteristic
189#389C22×C3111yescharacteristic
190#387—21no—
191#387—21no—
192#340—21no—
193#340—21no—
194#336C22×C3111yescharacteristic
195#336C22×C3111yescharacteristic
196#336C22×C3111yescharacteristic
197C2×C31#159711yescharacteristic
198C2×C31#160211yescharacteristic
199C2×C31#160311yescharacteristic
200C2×C31#163611yescharacteristic
201C2×C31#164611yescharacteristic
202C2×C31#164811yescharacteristic
203C2×C31#166411yescharacteristic
204C22×Q8—21no—
205C22×Q8D4×C3111yescharacteristic
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C4⋊Q8—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4D4×C3111yescharacteristic
217C2×C4⋊C4D4×C3111yescharacteristic
218C2×C4⋊C4D4×C3111yescharacteristic
219C2×C4⋊C4D4×C3111yescharacteristic
220C2×C4⋊C4D4×C3111yescharacteristic
221C2×C42—21no—
222C2×C42C23×C3111yescharacteristic
223#107—42no—
224#103—42no—
225#103—42no—
226#103—42no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236C31#777111yescharacteristic, direct product factor, Sylow
237C2×Q8—21no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8Q16×C3112yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4C2×D4×C3111yescharacteristic
255C22×C4C2×D4×C3111yescharacteristic
256C22×C4C2×D4×C3111yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C42—41no—
284C42—81no—
285C23C31×#12011yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—42no—
289Q8—81no—
290Q8—81no—
291Q8—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4C2×Q16×C3111yescharacteristic
320C2×C4C31×#13511yescharacteristic
321C2×C4C31×#13511yescharacteristic
322C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
323C2×C4C31×#13711yescharacteristic
324C2×C4C31×#13711yescharacteristic
325C8—42no—
326C8—81no—
327C8—81no—
328C22C31×#44911yescharacteristic
329C22C31×#45111yescharacteristic
330C22C31×#45211yescharacteristic
331C22C31×#45411yescharacteristic
332C22C31×#45511yescharacteristic
333C22C31×#45611yescharacteristic
334C22C31×B(4,𝔽2)11yescharacteristic
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—81no—
343C4—82no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C2C31×#159711yescharacteristic
349C2C31×#160211yescharacteristic
350C2C31×#160311yescharacteristic
351C2C31×#163611yescharacteristic
352C2C31×#164611yescharacteristic
353C2C31×#164811yescharacteristic
354C2C31×#166411yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
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
107. #7771C31direct (N × H)
236. C31#7771direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28092
C31×#20572
C31×#20432
C31×#20242
C31×#12722
C31×#12712
#777131