Finite Group Databasebeta

← C31×#13226 #21469985

Subgroups — C31×#13226 #21469985

Order 7936.

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

normal subgroups only

This group has 540 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×#2548C211yesmaximal, characteristic
2C2×C31×#355C214yesmaximal, complemented
3C31×#1801C211yesmaximal, characteristic, complemented
4C4×C31×#101C211yesmaximal, characteristic, complemented
5C2×C31×#127C411yescharacteristic
6C2×C31×#124C2212yes—
7C2×C4×Q8×C31C2211yescharacteristic
8C31×#423—21no—
9C2×C31×#101C2212yes—
10C2×C31×#101C2212yes—
11C4×M16×C31—21no—
12C31×#394—21no—
13C31×#394—21no—
14C31×#390—21no—
15C31×#390—21no—
16C31×#390C412yes—
17C4×C4⋊C4×C31C411yescharacteristic
18C31×#355—28no—
19C31×#343—21no—
20C31×#343—21no—
21C22×Q8×C31C2311yescharacteristic
22C22×D4×C31—21no—
23C22×D4×C31—21no—
24C22×D4×C31C2×C412yes—
25C2×M16×C31—22no—
26C2×M16×C31—22no—
27C31×#127—41no—
28C31×#127—41no—
29C31×#124—24no—
30C31×#124—24no—
31C4×Q8×C31—41no—
32C4×Q8×C31—41no—
33C2×C4⋊C4×C31—21no—
34C2×C4⋊C4×C31C2×C411yescharacteristic
35C2×C4⋊C4×C31D411yescharacteristic
36C2×C4⋊C4×C31D411yescharacteristic
37C2×C22⋊C4×C31—22no—
38C2×C22⋊C4×C31—22no—
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C42×C31C2×C412yes—
42C2×C42×C31C2×C411yescharacteristic
43C2×C42×C31D411yescharacteristic
44C2×C42×C31D411yescharacteristic
45C31×C4⋊C8—41no—
46C31×C4⋊C8—41no—
47C31×#101—22no—
48C31×#101—24no—
49C31×#101D414yescomplemented
50C31×#97—41no—
51C4×C8×C31—41no—
52C31×#95—21no—
53C24×C31—22no—
54C2×Q8×C31—41no—
55C2×Q8×C31—41no—
56C2×Q8×C31C2×D412yes—
57C2×Q8×C31#4112yes—
58C2×D4×C31—24no—
59C2×D4×C31—42no—
60C2×D4×C31—42no—
61C2×D4×C31—42no—
62C2×D4×C31—42no—
63C2×D4×C31—42no—
64C2×D4×C31—41no—
65C2×D4×C31—41no—
66C2×D4×C31—82no—
67C22×C4×C31—22no—
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22⋊C411yescharacteristic
72C22×C4×C31C22×C411yescharacteristic
73C22×C4×C31C2×D411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75M16×C31—24no—
76M16×C31—42no—
77M16×C31—42no—
78M16×C31—42no—
79C2×C8×C31—42no—
80C2×C8×C31—42no—
81C4⋊C4×C31—41no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—42no—
85C22⋊C4×C31—44no—
86C22⋊C4×C31—42no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—82no—
89C22⋊C4×C31—82no—
90C42×C31—21no—
91C42×C31—21no—
92C42×C31—21no—
93C42×C31—21no—
94C42×C31—21no—
95C42×C31—21no—
96C42×C31—24no—
97#13226C3111yesmaximal, characteristic, direct product factor, Sylow
98C23×C31—42no—
99C23×C31—42no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31C2×C22⋊C411yescharacteristic
106Q8×C31—42no—
107Q8×C31—42no—
108D4×C31—84no—
109D4×C31—82no—
110D4×C31—84no—
111D4×C31—82no—
112C2×C4×C31—21no—
113C2×C4×C31—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—22no—
119C2×C4×C31—21no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—41no—
123C2×C4×C31—41no—
124C2×C4×C31—41no—
125C2×C4×C31—42no—
126C2×C4×C31—42no—
127C2×C4×C31—42no—
128C2×C4×C31#9911yescharacteristic
129C2×C4×C31#9911yescharacteristic
130C2×C4×C31C4×D412yes—
131C2×C4×C31#12112yes—
132C2×C4×C31#12411yescharacteristic
133C2×C4×C31#12711yescharacteristic
134C8×C31—44no—
135C8×C31—82no—
136#2548C2×C3111yescharacteristic
137C2×#355C2×C3114yescomplemented
138#1801C2×C3111yescharacteristic, complemented
139C4×#101C2×C3111yescharacteristic, complemented
140C22×C31—21no—
141C22×C31—21no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31—82no—
147C22×C31—82no—
148C22×C31—82no—
149C22×C31#39011yescharacteristic
150C22×C31C2×#9911yescharacteristic
151C22×C31#41011yescharacteristic
152C4×C31—21no—
153C4×C31—22no—
154C4×C31—21no—
155C4×C31—22no—
156C4×C31—21no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—41no—
160C4×C31—82no—
161C4×C31#35512yescomplemented
162C2×#127C4×C3111yescharacteristic
163C2×#124C22×C3112yes—
164C2×C4×Q8C22×C3111yescharacteristic
165#423—21no—
166C2×#101C22×C3112yes—
167C2×#101C22×C3112yes—
168C4×M16—21no—
169#394—21no—
170#394—21no—
171#390—21no—
172#390—21no—
173#390C4×C3112yes—
174C4×C4⋊C4C4×C3111yescharacteristic
175#355—28no—
176#343—21no—
177#343—21no—
178C2×C31—21no—
179C2×C31—21no—
180C2×C31—162no—
181C2×C31#196411yescharacteristic
182C2×C31C2×#35511yescharacteristic
183C2×C31#213411yescharacteristic
184C22×Q8C23×C3111yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×C4×C3112yes—
188C2×M16—22no—
189C2×M16—22no—
190#127—41no—
191#127—41no—
192#124—24no—
193#124—24no—
194C4×Q8—41no—
195C4×Q8—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C3111yescharacteristic
198C2×C4⋊C4D4×C3111yescharacteristic
199C2×C4⋊C4D4×C3111yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C42C2×C4×C3112yes—
205C2×C42C2×C4×C3111yescharacteristic
206C2×C42D4×C3111yescharacteristic
207C2×C42D4×C3111yescharacteristic
208C4⋊C8—41no—
209C4⋊C8—41no—
210#101—22no—
211#101—24no—
212#101D4×C3114yescomplemented
213#97—41no—
214C4×C8—41no—
215#95—21no—
216C31#1322611yescharacteristic, direct product factor, Sylow
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C2×D4×C3112yes—
221C2×Q8#41×C3112yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊C4×C3111yescharacteristic
233C22×C4C22⋊C4×C3111yescharacteristic
234C22×C4C22⋊C4×C3111yescharacteristic
235C22×C4C22⋊C4×C3111yescharacteristic
236C22×C4C22×C4×C3111yescharacteristic
237C22×C4C2×D4×C3111yescharacteristic
238C22×C4C2×D4×C3111yescharacteristic
239M16—24no—
240M16—42no—
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊C4×C3111yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4C31×#9911yescharacteristic
292C2×C4C31×#9911yescharacteristic
293C2×C4C4×D4×C3112yes—
294C2×C4C31×#12112yes—
295C2×C4C31×#12411yescharacteristic
296C2×C4C31×#12711yescharacteristic
297C8—44no—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C31×#39011yescharacteristic
309C22C2×C31×#9911yescharacteristic
310C22C31×#41011yescharacteristic
311C4—21no—
312C4—22no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4C31×#35512yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2C31×#196411yescharacteristic
325C2C2×C31×#35511yescharacteristic
326C2C31×#213411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC22×Q8C23×C31
prime radicalC2×C31×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C4×Q8×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#355C2semidirect
3. C31×#1801C2semidirect
4. C4×C31×#101C2semidirect
49. C31×#101D4semidirect
97. #13226C31direct (N × H)
137. C2×#355C2×C31semidirect
138. #1801C2×C31semidirect
139. C4×#101C2×C31semidirect
161. C4×C31#355semidirect
212. #101D4×C31semidirect
216. C31#13226direct (N × H)
320. C4C31×#355semidirect

Maximal subgroups

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

subgroupindex
C31×#25482
C2×C31×#3552
C31×#18012
C4×C31×#1012
#1322631