Finite Group Databasebeta

← C31×#7699 #21464458

Subgroups — C31×#7699 #21464458

Order 7936.

534 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2858C211yesmaximal, characteristic
2C31×#1839C211yesmaximal, characteristic, complemented
3C31×#1745C211yesmaximal, characteristic, complemented
4C31×#1270C214yesmaximal, complemented
5C2×C31×C4⋊Q8C411yescharacteristic
6C2×C31×#127C411yescharacteristic
7C2×C31×#124—21no—
8C2×C31×#124—21no—
9C2×C31×#97—21no—
10C31×#390—21no—
11C31×#390—21no—
12C31×#390C412yes—
13C31×#390C2212yes—
14C43×C31C2211yescharacteristic
15C31×#336—21no—
16C31×#336—21no—
17C31×#336—21no—
18C31×#336C2212yes—
19C31×#336C2212yes—
20C22×Q8×C31—21no—
21C22×D4×C31—21no—
22C22×D4×C31—21no—
23C22×D4×C31C2×C412yes—
24C22×C8×C31—22no—
25C22×C8×C31—22no—
26C31×C4⋊Q8—42no—
27C31×#127—42no—
28C31×#124—44no—
29C31×#124—44no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31C2×C412yes—
32C2×C22⋊C4×C31—22no—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C42×C31C2×C411yescharacteristic
37C2×C42×C31C2×C411yescharacteristic
38C2×C42×C31D411yescharacteristic
39C2×C42×C31D411yescharacteristic
40C2×C42×C31D411yescharacteristic
41C2×C42×C31D411yescharacteristic
42C2×C42×C31C2311yescharacteristic
43C31×#97—24no—
44C24×C31—22no—
45C2×Q8×C31—42no—
46C2×Q8×C31—42no—
47C2×Q8×C31—42no—
48C2×D4×C31—24no—
49C2×D4×C31—42no—
50C2×D4×C31—42no—
51C2×D4×C31—42no—
52C2×D4×C31—42no—
53C2×D4×C31—42no—
54C2×D4×C31—42no—
55C2×D4×C31—42no—
56C2×D4×C31—42no—
57C22×C4×C31—22no—
58C22×C4×C31C22⋊C411yescharacteristic
59C22×C4×C31C22⋊C411yescharacteristic
60C22×C4×C31C22⋊C411yescharacteristic
61C22×C4×C31C22⋊C411yescharacteristic
62C22×C4×C31C22×C411yescharacteristic
63C22×C4×C31C2×D411yescharacteristic
64C22×C4×C31C2×D411yescharacteristic
65C2×C8×C31—24no—
66C2×C8×C31—42no—
67C2×C8×C31—42no—
68C2×C8×C31—42no—
69C2×C8×C31—42no—
70C2×C8×C31—42no—
71C4⋊C4×C31—42no—
72C4⋊C4×C31—42no—
73C22⋊C4×C31—44no—
74C22⋊C4×C31—44no—
75C22⋊C4×C31—44no—
76C22⋊C4×C31—44no—
77C42×C31—22no—
78C42×C31—22no—
79C42×C31—22no—
80C42×C31—22no—
81C42×C31—22no—
82C42×C31—41no—
83C42×C31—41no—
84#7699C3111yesmaximal, characteristic, direct product factor, Sylow
85C23×C31—42no—
86C23×C31—42no—
87C23×C31—42no—
88C23×C31—42no—
89C23×C31—42no—
90C23×C31—42no—
91C23×C31—42no—
92C23×C31C2×C22⋊C411yescharacteristic
93Q8×C31—84no—
94D4×C31—84no—
95D4×C31—84no—
96D4×C31—84no—
97C2×C4×C31—21no—
98C2×C4×C31—21no—
99C2×C4×C31—21no—
100C2×C4×C31—21no—
101C2×C4×C31—21no—
102C2×C4×C31—21no—
103C2×C4×C31—21no—
104C2×C4×C31—21no—
105C2×C4×C31—21no—
106C2×C4×C31—21no—
107C2×C4×C31—21no—
108C2×C4×C31—21no—
109C2×C4×C31—21no—
110C2×C4×C31—21no—
111C2×C4×C31—21no—
112C2×C4×C31—21no—
113C2×C4×C31—42no—
114C2×C4×C31—42no—
115C2×C4×C31—42no—
116C2×C4×C31#10012yes—
117C2×C4×C31#10412yes—
118C2×C4×C31#13112yes—
119C2×C4×C31AGL(1,ℤ/8)12yes—
120C2×C4×C31AGL(1,ℤ/8)12yes—
121C8×C31—44no—
122C8×C31—82no—
123#2858C2×C3111yescharacteristic
124#1839C2×C3111yescharacteristic, complemented
125#1745C2×C3111yescharacteristic, complemented
126#1270C2×C3114yescomplemented
127C22×C31—82no—
128C22×C31—82no—
129C22×C31—82no—
130C22×C31—82no—
131C22×C31—82no—
132C22×C31—82no—
133C22×C31—82no—
134C22×C31#40811yescharacteristic
135C22×C31#41011yescharacteristic
136C22×C31C2×#10011yescharacteristic
137C22×C31#41811yescharacteristic
138C22×C31#41811yescharacteristic
139C22×C31C2×#10411yescharacteristic
140C22×C31#42111yescharacteristic
141C4×C31—22no—
142C4×C31—22no—
143C4×C31—22no—
144C4×C31—22no—
145C4×C31—22no—
146C4×C31—41no—
147C4×C31—41no—
148C4×C31—82no—
149C2×C4⋊Q8C4×C3111yescharacteristic
150C2×#127C4×C3111yescharacteristic
151C2×#124—21no—
152C2×#124—21no—
153C2×#97—21no—
154#390—21no—
155#390—21no—
156#390C4×C3112yes—
157#390C22×C3112yes—
158C43C22×C3111yescharacteristic
159#336—21no—
160#336—21no—
161#336—21no—
162#336C22×C3112yes—
163#336C22×C3112yes—
164C2×C31—162no—
165C2×C31#146911yescharacteristic
166C2×C31#148411yescharacteristic
167C2×C31#149611yescharacteristic
168C2×C31#150511yescharacteristic
169C2×C31#151411yescharacteristic
170C2×C31#152711yescharacteristic
171C2×C31#155511yescharacteristic
172C22×Q8—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C2×C4×C3112yes—
176C22×C8—22no—
177C22×C8—22no—
178C4⋊Q8—42no—
179#127—42no—
180#124—44no—
181#124—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×C4×C3112yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42C2×C4×C3111yescharacteristic
189C2×C42C2×C4×C3111yescharacteristic
190C2×C42D4×C3111yescharacteristic
191C2×C42D4×C3111yescharacteristic
192C2×C42D4×C3111yescharacteristic
193C2×C42D4×C3111yescharacteristic
194C2×C42C23×C3111yescharacteristic
195#97—24no—
196C31#769911yescharacteristic, direct product factor, Sylow
197C24—22no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C22×C4—22no—
211C22×C4C22⋊C4×C3111yescharacteristic
212C22×C4C22⋊C4×C3111yescharacteristic
213C22×C4C22⋊C4×C3111yescharacteristic
214C22×C4C22⋊C4×C3111yescharacteristic
215C22×C4C22×C4×C3111yescharacteristic
216C22×C4C2×D4×C3111yescharacteristic
217C22×C4C2×D4×C3111yescharacteristic
218C2×C8—24no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—41no—
236C42—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23C2×C22⋊C4×C3111yescharacteristic
245Q8—84no—
246D4—84no—
247D4—84no—
248D4—84no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4C31×#10012yes—
269C2×C4C31×#10412yes—
270C2×C4C31×#13112yes—
271C2×C4C31×AGL(1,ℤ/8)12yes—
272C2×C4C31×AGL(1,ℤ/8)12yes—
273C8—44no—
274C8—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22C31×#40811yescharacteristic
283C22C31×#41011yescharacteristic
284C22C2×C31×#10011yescharacteristic
285C22C31×#41811yescharacteristic
286C22C31×#41811yescharacteristic
287C22C2×C31×#10411yescharacteristic
288C22C31×#42111yescharacteristic
289C4—22no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—82no—
297C2—162no—
298C2C31×#146911yescharacteristic
299C2C31×#148411yescharacteristic
300C2C31×#149611yescharacteristic
301C2C31×#150511yescharacteristic
302C2C31×#151411yescharacteristic
303C2C31×#152711yescharacteristic
304C2C31×#155511yescharacteristic
subgroupG/H
center, socleC23×C31C2×C22⋊C4
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC2×C31×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C43×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1839C2semidirect
3. C31×#1745C2semidirect
4. C31×#1270C2semidirect
84. #7699C31direct (N × H)
124. #1839C2×C31semidirect
125. #1745C2×C31semidirect
126. #1270C2×C31semidirect
196. C31#7699direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#28582
C31×#18392
C31×#17452
C31×#12702
#769931