Finite Group Databasebeta

← C31×#7801 #21464560

Subgroups — C31×#7801 #21464560

Order 7936.

446 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2732C211yesmaximal, characteristic
2C31×#2096C211yesmaximal, characteristic, complemented
3C31×#2084C211yesmaximal, characteristic, complemented
4C31×#2071C211yesmaximal, characteristic
5C31×#1272C211yesmaximal, characteristic, complemented
6C31×#1271C211yesmaximal, characteristic, complemented
7C31×#1270C211yesmaximal, characteristic
8C2×C31×C22⋊Q8—21no—
9C2×C31×#106—21no—
10C2×C31×#106—21no—
11C2×C31×#103—21no—
12C2×C31×#102—21no—
13C31×#399—21no—
14C31×#398—21no—
15C31×#396—21no—
16C31×#391C2211yescharacteristic
17C31×#390C2211yescharacteristic
18C31×#389C2211yescharacteristic
19C31×#386—21no—
20C31×#385—21no—
21C31×#384C2211yescharacteristic
22C31×#340—21no—
23C31×#340—21no—
24C31×#336C2211yescharacteristic
25C31×#336C2211yescharacteristic
26C31×#336C2211yescharacteristic
27C22×Q8×C31D411yescharacteristic
28C22×D4×C31D411yescharacteristic
29C23×C4×C31—21no—
30C22×C8×C31—21no—
31C22×C8×C31—21no—
32C22×C8×C31—21no—
33C31×C22⋊Q8—42no—
34C31×C22⋊Q8—42no—
35C2×C4⋊C4×C31—21no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31—21no—
38C2×C4⋊C4×C31—21no—
39C2×C4⋊C4×C31—21no—
40C2×C4⋊C4×C31—21no—
41C2×C4⋊C4×C31—21no—
42C2×C4⋊C4×C31D411yescharacteristic
43C2×C4⋊C4×C31D411yescharacteristic
44C2×C4⋊C4×C31D411yescharacteristic
45C2×C4⋊C4×C31D411yescharacteristic
46C2×C22⋊C4×C31—21no—
47C2×C22⋊C4×C31—21no—
48C2×C22⋊C4×C31—21no—
49C2×C22⋊C4×C31—21no—
50C2×C22⋊C4×C31—21no—
51C2×C42×C31C2311yescharacteristic
52C31×#106—42no—
53C31×#106—42no—
54C31×#103—42no—
55C31×#102—42no—
56C31×#95—21no—
57C31×#95—21no—
58C31×#95—21no—
59C31×#95—21no—
60C24×C31—21no—
61C2×Q8×C31—21no—
62C2×Q8×C31—41no—
63C2×Q8×C31—41no—
64C2×Q8×C31SD1612yes—
65C2×D4×C31—21no—
66C2×D4×C31—41no—
67C2×D4×C31—41no—
68C2×D4×C31SD1612yes—
69C22×C4×C31—21no—
70C22×C4×C31—21no—
71C22×C4×C31—21no—
72C22×C4×C31—21no—
73C22×C4×C31—21no—
74C22×C4×C31—42no—
75C22×C4×C31—41no—
76C22×C4×C31—81no—
77C22×C4×C31—81no—
78C22×C4×C31C2×D411yescharacteristic
79C22×C4×C31C2×D411yescharacteristic
80C22×C4×C31C2×D411yescharacteristic
81C2×C8×C31—22no—
82C2×C8×C31—22no—
83C2×C8×C31—41no—
84C2×C8×C31—41no—
85C2×C8×C31—41no—
86C2×C8×C31—41no—
87C2×C8×C31—41no—
88C2×C8×C31—41no—
89C2×C8×C31—41no—
90C4⋊C4×C31—21no—
91C4⋊C4×C31—21no—
92C4⋊C4×C31—42no—
93C4⋊C4×C31—42no—
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—41no—
104C4⋊C4×C31—41no—
105C4⋊C4×C31—81no—
106C22⋊C4×C31—42no—
107C22⋊C4×C31—41no—
108C22⋊C4×C31—41no—
109C22⋊C4×C31—41no—
110C22⋊C4×C31—41no—
111C22⋊C4×C31—81no—
112C22⋊C4×C31—81no—
113C42×C31—41no—
114#7801C3111yesmaximal, characteristic, direct product factor, Sylow
115C23×C31—22no—
116C23×C31—41no—
117C23×C31—41no—
118C23×C31—41no—
119C23×C31—41no—
120C23×C31—41no—
121C23×C31—41no—
122C23×C31#12011yescharacteristic
123Q8×C31—42no—
124Q8×C31—81no—
125D4×C31—42no—
126D4×C31—81no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—21no—
132C2×C4×C31—21no—
133C2×C4×C31—22no—
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—41no—
143C2×C4×C31—41no—
144C2×C4×C31—41no—
145C2×C4×C31—41no—
146C2×C4×C31—41no—
147C2×C4×C31—41no—
148C2×C4×C31—84no—
149C2×C4×C31—81no—
150C2×C4×C31—81no—
151C2×C4×C31—81no—
152C2×C4×C31C2×SD1611yescharacteristic
153C2×C4×C31C2×SD1611yescharacteristic
154C2×C4×C31#13511yescharacteristic
155C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C31#13711yescharacteristic
158C8×C31—42no—
159C8×C31—42no—
160C8×C31—81no—
161#2732C2×C3111yescharacteristic
162#2096C2×C3111yescharacteristic, complemented
163#2084C2×C3111yescharacteristic, complemented
164#2071C2×C3111yescharacteristic
165#1272C2×C3111yescharacteristic, complemented
166#1271C2×C3111yescharacteristic, complemented
167#1270C2×C3111yescharacteristic
168C22×C31—42no—
169C22×C31—42no—
170C22×C31—42no—
171C22×C31—81no—
172C22×C31—81no—
173C22×C31—81no—
174C22×C31—81no—
175C22×C31#44811yescharacteristic
176C22×C31#44911yescharacteristic
177C22×C31#45011yescharacteristic
178C22×C31#45311yescharacteristic
179C22×C31#45511yescharacteristic
180C22×C31#45611yescharacteristic
181C22×C31#45811yescharacteristic
182C4×C31—21no—
183C4×C31—21no—
184C4×C31—21no—
185C4×C31—21no—
186C4×C31—21no—
187C4×C31—21no—
188C4×C31—42no—
189C4×C31—81no—
190C4×C31—81no—
191C4×C31—81no—
192C4×C31—81no—
193C4×C31—161no—
194C2×C22⋊Q8—21no—
195C2×#106—21no—
196C2×#106—21no—
197C2×#103—21no—
198C2×#102—21no—
199#399—21no—
200#398—21no—
201#396—21no—
202#391C22×C3111yescharacteristic
203#390C22×C3111yescharacteristic
204#389C22×C3111yescharacteristic
205#386—21no—
206#385—21no—
207#384C22×C3111yescharacteristic
208#340—21no—
209#340—21no—
210#336C22×C3111yescharacteristic
211#336C22×C3111yescharacteristic
212#336C22×C3111yescharacteristic
213C2×C31—82no—
214C2×C31#160811yescharacteristic
215C2×C31#160911yescharacteristic
216C2×C31#161311yescharacteristic
217C2×C31#162211yescharacteristic
218C2×C31#164211yescharacteristic
219C2×C31#165111yescharacteristic
220C2×C31#166111yescharacteristic
221C22×Q8D4×C3111yescharacteristic
222C22×D4D4×C3111yescharacteristic
223C23×C4—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22⋊Q8—42no—
228C22⋊Q8—42no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4D4×C3111yescharacteristic
237C2×C4⋊C4D4×C3111yescharacteristic
238C2×C4⋊C4D4×C3111yescharacteristic
239C2×C4⋊C4D4×C3111yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42C23×C3111yescharacteristic
246#106—42no—
247#106—42no—
248#103—42no—
249#102—42no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C31#780111yescharacteristic, direct product factor, Sylow
255C24—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8SD16×C3112yes—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4SD16×C3112yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4C2×D4×C3111yescharacteristic
274C22×C4C2×D4×C3111yescharacteristic
275C22×C4C2×D4×C3111yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—41no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C31×#12011yescharacteristic
317Q8—42no—
318Q8—81no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4C2×SD16×C3111yescharacteristic
347C2×C4C2×SD16×C3111yescharacteristic
348C2×C4C31×#13511yescharacteristic
349C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
350C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
351C2×C4C31×#13711yescharacteristic
352C8—42no—
353C8—42no—
354C8—81no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C31×#44811yescharacteristic
363C22C31×#44911yescharacteristic
364C22C31×#45011yescharacteristic
365C22C31×#45311yescharacteristic
366C22C31×#45511yescharacteristic
367C22C31×#45611yescharacteristic
368C22C31×#45811yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—161no—
381C2—82no—
382C2C31×#160811yescharacteristic
383C2C31×#160911yescharacteristic
384C2C31×#161311yescharacteristic
385C2C31×#162211yescharacteristic
386C2C31×#164211yescharacteristic
387C2C31×#165111yescharacteristic
388C2C31×#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×#2084C2semidirect
5. C31×#1272C2semidirect
6. C31×#1271C2semidirect
114. #7801C31direct (N × H)
162. #2096C2×C31semidirect
163. #2084C2×C31semidirect
165. #1272C2×C31semidirect
166. #1271C2×C31semidirect
254. C31#7801direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#27322
C31×#20962
C31×#20842
C31×#20712
C31×#12722
C31×#12712
C31×#12702
#780131