Finite Group Databasebeta

← C31×#7786 #21464545

Subgroups — C31×#7786 #21464545

Order 7936.

682 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2679C211yesmaximal, characteristic
2C31×#2093C211yesmaximal, characteristic, complemented
3C31×#2071C212yesmaximal, complemented
4C31×#1270C211yesmaximal, characteristic, complemented
5C31×#1270C212yesmaximal, complemented
6C2×C31×#120—22no—
7C2×C31×#120—21no—
8C2×C31×#107—21no—
9C2×C31×#102—22no—
10C2×C31×#102—21no—
11C31×#394—21no—
12C31×#394—22no—
13C31×#390C2212yes—
14C31×#390C2211yescharacteristic
15C31×#384C2211yescharacteristic, complemented
16C31×#340—22no—
17C31×#336C2212yescomplemented
18C31×#336C2211yescharacteristic, complemented
19C25×C31—21no—
20C22×D4×C31—21no—
21C22×D4×C31—22no—
22C22×D4×C31D412yes—
23C22×D4×C31D411yescharacteristic
24C22×C8×C31—21no—
25C22×C8×C31—22no—
26C31×#120—44no—
27C31×#120—42no—
28C31×#120—44no—
29C31×#120—42no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31—22no—
32C2×C4⋊C4×C31D411yescharacteristic
33C2×C4⋊C4×C31D412yes—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—21no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C42×C31C2311yescharacteristic
40C31×#107—42no—
41C31×#102—42no—
42C31×#102—44no—
43C31×#95—21no—
44C24×C31—22no—
45C24×C31—21no—
46C24×C31—44no—
47C24×C31—42no—
48C24×C31—81no—
49C24×C31—82no—
50C24×C31—81no—
51C2×D4×C31—22no—
52C2×D4×C31—21no—
53C2×D4×C31—41no—
54C2×D4×C31—42no—
55C2×D4×C31—42no—
56C2×D4×C31—41no—
57C2×D4×C31—44no—
58C2×D4×C31—44no—
59C2×D4×C31—44no—
60C2×D4×C31—81no—
61C2×D4×C31—82no—
62C2×D4×C31D814yes—
63C2×D4×C31SD1612yes—
64C22×C4×C31—21no—
65C22×C4×C31—22no—
66C22×C4×C31—41no—
67C22×C4×C31C2×D412yes—
68C22×C4×C31C2×D411yescharacteristic
69C2×C8×C31—22no—
70C2×C8×C31—41no—
71C2×C8×C31—41no—
72C2×C8×C31—42no—
73C2×C8×C31—42no—
74C2×C8×C31—42no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—42no—
77C4⋊C4×C31—41no—
78C4⋊C4×C31—41no—
79C4⋊C4×C31—42no—
80C4⋊C4×C31—42no—
81C22⋊C4×C31—44no—
82C22⋊C4×C31—42no—
83C22⋊C4×C31—42no—
84C22⋊C4×C31—42no—
85C22⋊C4×C31—42no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C42×C31—41no—
90#7786C3111yesmaximal, characteristic, direct product factor, Sylow
91C23×C31—24no—
92C23×C31—22no—
93C23×C31—42no—
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—41no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—41no—
103C23×C31—84no—
104C23×C31—88no—
105C23×C31—88no—
106C23×C31—84no—
107C23×C31—84no—
108C23×C31#12011yescharacteristic
109D4×C31—44no—
110D4×C31—42no—
111D4×C31—82no—
112D4×C31—81no—
113D4×C31—84no—
114D4×C31—88no—
115C2×C4×C31—21no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—21no—
119C2×C4×C31—42no—
120C2×C4×C31—42no—
121C2×C4×C31—42no—
122C2×C4×C31—41no—
123C2×C4×C31—42no—
124C2×C4×C31—82no—
125C2×C4×C31—81no—
126C2×C4×C31C2×D812yes—
127C2×C4×C31C2×SD1611yescharacteristic
128C2×C4×C31AGL(1,ℤ/8)12yes—
129C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
130C8×C31—42no—
131C8×C31—82no—
132#2679C2×C3111yescharacteristic
133#2093C2×C3111yescharacteristic, complemented
134#2071C2×C3112yescomplemented
135#1270C2×C3111yescharacteristic, complemented
136#1270C2×C3112yescomplemented
137C22×C31—44no—
138C22×C31—42no—
139C22×C31—44no—
140C22×C31—44no—
141C22×C31—44no—
142C22×C31—816no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31—81no—
147C22×C31—82no—
148C22×C31—81no—
149C22×C31—82no—
150C22×C31#44712yes—
151C22×C31#45011yescharacteristic
152C22×C31#45311yescharacteristic
153C22×C31#45512yes—
154C22×C31#45811yescharacteristic
155C4×C31—22no—
156C4×C31—22no—
157C4×C31—21no—
158C4×C31—21no—
159C4×C31—81no—
160C4×C31—82no—
161C4×C31—161no—
162C2×#120—22no—
163C2×#120—21no—
164C2×#107—21no—
165C2×#102—22no—
166C2×#102—21no—
167#394—22no—
168#394—21no—
169#390C22×C3112yes—
170#390C22×C3111yescharacteristic
171#384C22×C3111yescharacteristic, complemented
172#340—22no—
173#336C22×C3112yescomplemented
174#336C22×C3111yescharacteristic, complemented
175C2×C31—84no—
176C2×C31—82no—
177C2×C31#160712yes—
178C2×C31#161311yescharacteristic
179C2×C31#161911yescharacteristic
180C2×C31#163912yes—
181C2×C31#165911yescharacteristic
182C25—21no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4D4×C3112yes—
186C22×D4D4×C3111yescharacteristic
187C22×C8—21no—
188C22×C8—22no—
189#120—44no—
190#120—42no—
191#120—44no—
192#120—42no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4D4×C3112yes—
196C2×C4⋊C4D4×C3111yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42C23×C3111yescharacteristic
203#107—42no—
204#102—42no—
205#102—44no—
206#95—21no—
207C31#778611yescharacteristic, direct product factor, Sylow
208C24—22no—
209C24—21no—
210C24—44no—
211C24—42no—
212C24—81no—
213C24—82no—
214C24—81no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—81no—
225C2×D4—82no—
226C2×D4D8×C3114yes—
227C2×D4SD16×C3112yes—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4C2×D4×C3112yes—
232C22×C4C2×D4×C3111yescharacteristic
233C2×C8—22no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—41no—
266C23—84no—
267C23—88no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23C31×#12011yescharacteristic
272D4—44no—
273D4—42no—
274D4—82no—
275D4—81no—
276D4—84no—
277D4—88no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4C2×D8×C3112yes—
290C2×C4C2×SD16×C3111yescharacteristic
291C2×C4C31×AGL(1,ℤ/8)12yes—
292C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
293C8—42no—
294C8—82no—
295C22—44no—
296C22—42no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—816no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—81no—
305C22—82no—
306C22—81no—
307C22—82no—
308C22C31×#44712yes—
309C22C31×#45011yescharacteristic
310C22C31×#45311yescharacteristic
311C22C31×#45512yes—
312C22C31×#45811yescharacteristic
313C4—22no—
314C4—22no—
315C4—21no—
316C4—21no—
317C4—81no—
318C4—82no—
319C4—161no—
320C2—84no—
321C2—82no—
322C2C31×#160712yes—
323C2C31×#161311yescharacteristic
324C2C31×#161911yescharacteristic
325C2C31×#163912yes—
326C2C31×#165911yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#2679C2
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×#2093C2semidirect
3. C31×#2071C2semidirect
4. C31×#1270C2semidirect
5. C31×#1270C2semidirect
15. C31×#384C22semidirect
17. C31×#336C22semidirect
18. C31×#336C22semidirect
90. #7786C31direct (N × H)
133. #2093C2×C31semidirect
134. #2071C2×C31semidirect
135. #1270C2×C31semidirect
136. #1270C2×C31semidirect
171. #384C22×C31semidirect
173. #336C22×C31semidirect
174. #336C22×C31semidirect
207. C31#7786direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26792
C31×#20932
C31×#20712
C31×#12702
#778631