Finite Group Databasebeta

← C31×#7785 #21464544

Subgroups — C31×#7785 #21464544

Order 7936.

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

normal subgroups only

This group has 454 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×#2058C211yesmaximal, characteristic, complemented
3C31×#2023C211yesmaximal, characteristic
4C31×#2016C211yesmaximal, characteristic, complemented
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1271C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic
8C22×Q16×C31—21no—
9C2×C31×C4⋊Q8C2211yescharacteristic
10C2×C31×#106—21no—
11C2×C31×#103—21no—
12C2×C31×#102—21no—
13C31×#397—21no—
14C31×#394—21no—
15C31×#393—21no—
16C31×#391C2211yescharacteristic
17C31×#390C2211yescharacteristic
18C31×#389C2211yescharacteristic
19C31×#388—21no—
20C31×#388—21no—
21C31×#387—21no—
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31—21no—
28C22×Q8×C31D411yescharacteristic
29C22×D4×C31D411yescharacteristic
30C2×Q16×C31—42no—
31C2×Q16×C31—42no—
32C2×Q16×C31—42no—
33C22×C8×C31—21no—
34C22×C8×C31—21no—
35C22×C8×C31—21no—
36C31×C4⋊Q8—42no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31D411yescharacteristic
41C2×C4⋊C4×C31D411yescharacteristic
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C22⋊C4×C31—21no—
45C2×C22⋊C4×C31—21no—
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C42×C31—21no—
51C2×C42×C31C2311yescharacteristic
52C31×#106—42no—
53C31×#103—42no—
54C31×#102—42no—
55C31×#95—21no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C31×#95—21no—
61C24×C31—21no—
62C2×Q8×C31—22no—
63C2×Q8×C31—21no—
64C2×Q8×C31—21no—
65C2×Q8×C31—41no—
66C2×Q8×C31—41no—
67C2×Q8×C31—41no—
68C2×Q8×C31—41no—
69C2×Q8×C31—41no—
70C2×Q8×C31—41no—
71C2×Q8×C31—41no—
72C2×D4×C31—21no—
73C2×D4×C31—21no—
74C2×D4×C31—41no—
75C2×D4×C31—41no—
76C22×C4×C31—21no—
77C22×C4×C31—21no—
78C22×C4×C31—21no—
79C22×C4×C31—21no—
80C22×C4×C31—21no—
81C22×C4×C31—41no—
82C22×C4×C31C2×D411yescharacteristic
83C22×C4×C31C2×D411yescharacteristic
84C22×C4×C31C2×D411yescharacteristic
85Q16×C31—48no—
86C2×C8×C31—22no—
87C2×C8×C31—22no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C2×C8×C31—41no—
91C2×C8×C31—41no—
92C2×C8×C31—41no—
93C2×C8×C31—41no—
94C2×C8×C31—41no—
95C4⋊C4×C31—22no—
96C4⋊C4×C31—22no—
97C4⋊C4×C31—21no—
98C4⋊C4×C31—21no—
99C4⋊C4×C31—41no—
100C4⋊C4×C31—41no—
101C4⋊C4×C31—41no—
102C4⋊C4×C31—41no—
103C4⋊C4×C31—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—41no—
106C22⋊C4×C31—41no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—81no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C22⋊C4×C31—81no—
114C42×C31—41no—
115C42×C31—81no—
116#7785C3111yesmaximal, characteristic, direct product factor, Sylow
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31—41no—
123C23×C31—41no—
124C23×C31#12011yescharacteristic
125Q8×C31—42no—
126Q8×C31—42no—
127Q8×C31—42no—
128Q8×C31—81no—
129Q8×C31—81no—
130Q8×C31—81no—
131D4×C31—42no—
132D4×C31—81no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—41no—
140C2×C4×C31—41no—
141C2×C4×C31—41no—
142C2×C4×C31—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—41no—
149C2×C4×C31—41no—
150C2×C4×C31—41no—
151C2×C4×C31—41no—
152C2×C4×C31—41no—
153C2×C4×C31—41no—
154C2×C4×C31—42no—
155C2×C4×C31—81no—
156C2×C4×C31—81no—
157C2×C4×C31#13511yescharacteristic
158C2×C4×C31#13511yescharacteristic
159C2×C4×C31#13511yescharacteristic
160C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C31#13711yescharacteristic
162C2×C4×C31#13711yescharacteristic
163C8×C31—42no—
164C8×C31—42no—
165C8×C31—81no—
166#2808C2×C3111yescharacteristic
167#2058C2×C3111yescharacteristic, complemented
168#2023C2×C3111yescharacteristic
169#2016C2×C3111yescharacteristic, complemented
170#1272C2×C3111yescharacteristic, complemented
171#1271C2×C3111yescharacteristic, complemented
172#1270C2×C3111yescharacteristic
173C22×C31—81no—
174C22×C31—81no—
175C22×C31—81no—
176C22×C31—81no—
177C22×C31—81no—
178C22×C31—81no—
179C22×C31—81no—
180C22×C31#44911yescharacteristic
181C22×C31#45211yescharacteristic
182C22×C31#45211yescharacteristic
183C22×C31#45311yescharacteristic
184C22×C31#45511yescharacteristic
185C22×C31#45611yescharacteristic
186C22×C31B(4,𝔽2)11yescharacteristic
187C4×C31—21no—
188C4×C31—21no—
189C4×C31—21no—
190C4×C31—21no—
191C4×C31—21no—
192C4×C31—21no—
193C4×C31—81no—
194C4×C31—82no—
195C4×C31—81no—
196C4×C31—81no—
197C4×C31—81no—
198C4×C31—81no—
199C22×Q16—21no—
200C2×C4⋊Q8C22×C3111yescharacteristic
201C2×#106—21no—
202C2×#103—21no—
203C2×#102—21no—
204#397—21no—
205#394—21no—
206#393—21no—
207#391C22×C3111yescharacteristic
208#390C22×C3111yescharacteristic
209#389C22×C3111yescharacteristic
210#388—21no—
211#388—21no—
212#387—21no—
213#340—21no—
214#340—21no—
215#336C22×C3111yescharacteristic
216#336C22×C3111yescharacteristic
217#336C22×C3111yescharacteristic
218C2×C31—161no—
219C2×C31#160011yescharacteristic
220C2×C31#160311yescharacteristic
221C2×C31#160611yescharacteristic
222C2×C31#163411yescharacteristic
223C2×C31#164011yescharacteristic
224C2×C31#165211yescharacteristic
225C2×C31#166111yescharacteristic
226C22×Q8—21no—
227C22×Q8D4×C3111yescharacteristic
228C22×D4D4×C3111yescharacteristic
229C2×Q16—42no—
230C2×Q16—42no—
231C2×Q16—42no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C4⋊Q8—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C3111yescharacteristic
240C2×C4⋊C4D4×C3111yescharacteristic
241C2×C4⋊C4D4×C3111yescharacteristic
242C2×C4⋊C4D4×C3111yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42C23×C3111yescharacteristic
251#106—42no—
252#103—42no—
253#102—42no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C31#778511yescharacteristic, direct product factor, Sylow
261C24—21no—
262C2×Q8—22no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4C2×D4×C3111yescharacteristic
283C22×C4C2×D4×C3111yescharacteristic
284C22×C4C2×D4×C3111yescharacteristic
285Q16—48no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—81no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C31×#12011yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—42no—
327Q8—81no—
328Q8—81no—
329Q8—81no—
330D4—42no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4C31×#13511yescharacteristic
357C2×C4C31×#13511yescharacteristic
358C2×C4C31×#13511yescharacteristic
359C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
360C2×C4C31×#13711yescharacteristic
361C2×C4C31×#13711yescharacteristic
362C8—42no—
363C8—42no—
364C8—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C31×#44911yescharacteristic
373C22C31×#45211yescharacteristic
374C22C31×#45211yescharacteristic
375C22C31×#45311yescharacteristic
376C22C31×#45511yescharacteristic
377C22C31×#45611yescharacteristic
378C22C31×B(4,𝔽2)11yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—161no—
392C2C31×#160011yescharacteristic
393C2C31×#160311yescharacteristic
394C2C31×#160611yescharacteristic
395C2C31×#163411yescharacteristic
396C2C31×#164011yescharacteristic
397C2C31×#165211yescharacteristic
398C2C31×#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×#2058C2semidirect
4. C31×#2016C2semidirect
5. C31×#1272C2semidirect
6. C31×#1271C2semidirect
116. #7785C31direct (N × H)
167. #2058C2×C31semidirect
169. #2016C2×C31semidirect
170. #1272C2×C31semidirect
171. #1271C2×C31semidirect
260. C31#7785direct (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×#20582
C31×#20232
C31×#20162
C31×#12722
C31×#12712
C31×#12702
#778531