Finite Group Databasebeta

← C31×#11699 #21468458

Subgroups — C31×#11699 #21468458

Order 7936.

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

normal subgroups only

This group has 552 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×#2619C211yesmaximal, characteristic
2C31×#1742C211yesmaximal, characteristic, complemented
3C31×#1458C211yesmaximal, characteristic, complemented
4C31×#1345C214yesmaximal, complemented
5C22×M16×C31—21no—
6C2×C31×#127C411yescharacteristic
7C2×C31×#121—21no—
8C2×C31×#117C2211yescharacteristic
9C31×#431—21no—
10C31×#394—21no—
11C31×#394C2212yes—
12C31×#392—21no—
13C31×#392C412yes—
14C31×#390—21no—
15C31×#382C411yescharacteristic
16C31×#343—21no—
17C31×#343C2212yes—
18C31×#324C2212yes—
19C31×#323—21no—
20C22×D4×C31—22no—
21C22×D4×C31—21no—
22C22×D4×C31—21no—
23C22×D4×C31C2×C412yes—
24C23×C4×C31D411yescharacteristic
25C2×M16×C31—24no—
26C2×M16×C31—22no—
27C2×M16×C31—41no—
28C2×M16×C31—42no—
29C22×C8×C31—41no—
30C31×#127—41no—
31C31×#127—41no—
32C31×#121—42no—
33C31×#121—42no—
34C31×#117—41no—
35C31×#117—41no—
36C2×C4⋊C4×C31—21no—
37C2×C4⋊C4×C31D411yescharacteristic
38C2×C4⋊C4×C31C2311yescharacteristic
39C2×C22⋊C4×C31—22no—
40C2×C22⋊C4×C31—22no—
41C2×C22⋊C4×C31—22no—
42C2×C22⋊C4×C31C2×C411yescharacteristic
43C2×C22⋊C4×C31D411yescharacteristic
44C2×C42×C31C2×C411yescharacteristic
45C2×C42×C31D411yescharacteristic
46C31×C4⋊C8—41no—
47C31×C22⋊C8—22no—
48C4×C8×C31—41no—
49C31×#95—21no—
50C31×#95C2×C412yes—
51C24×C31—22no—
52C24×C31C22⋊C411yescharacteristic
53C2×D4×C31—42no—
54C2×D4×C31—42no—
55C2×D4×C31—44no—
56C2×D4×C31—41no—
57C2×D4×C31—41no—
58C2×D4×C31—42no—
59C2×D4×C31—42no—
60C2×D4×C31—82no—
61C2×D4×C31—82no—
62C2×D4×C31—82no—
63C2×D4×C31—82no—
64C22×C4×C31—21no—
65C22×C4×C31—21no—
66C22×C4×C31—22no—
67C22×C4×C31—22no—
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31D812yes—
72C22×C4×C31SD1612yes—
73C22×C4×C31C22×C411yescharacteristic
74C22×C4×C31C2×D411yescharacteristic
75C22×C4×C31C2×D411yescharacteristic
76M16×C31—24no—
77M16×C31—44no—
78M16×C31—42no—
79C2×C8×C31—42no—
80C2×C8×C31—44no—
81C2×C8×C31—42no—
82C4⋊C4×C31—41no—
83C4⋊C4×C31—41no—
84C4⋊C4×C31—42no—
85C22⋊C4×C31—41no—
86C22⋊C4×C31—41no—
87C22⋊C4×C31—42no—
88C22⋊C4×C31—42no—
89C22⋊C4×C31—82no—
90C22⋊C4×C31—82no—
91C42×C31—21no—
92C42×C31—21no—
93C42×C31—21no—
94C42×C31—21no—
95#11699C3111yesmaximal, characteristic, direct product factor, Sylow
96C23×C31—21no—
97C23×C31—21no—
98C23×C31—41no—
99C23×C31—41no—
100C23×C31—42no—
101C23×C31—42no—
102C23×C31—42no—
103C23×C31—42no—
104C23×C31—42no—
105C23×C31—42no—
106C23×C31—42no—
107C23×C31#10212yes—
108C23×C31C2×C22⋊C411yescharacteristic
109D4×C31—84no—
110D4×C31—84no—
111D4×C31—88no—
112D4×C31—82no—
113D4×C31—82no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—22no—
117C2×C4×C31—22no—
118C2×C4×C31—22no—
119C2×C4×C31—22no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—41no—
125C2×C4×C31—41no—
126C2×C4×C31—41no—
127C2×C4×C31—41no—
128C2×C4×C31—42no—
129C2×C4×C31—42no—
130C2×C4×C31—42no—
131C2×C4×C31#9911yescharacteristic
132C2×C4×C31#9911yescharacteristic
133C2×C4×C31#10212yes—
134C2×C4×C31#10412yes—
135C2×C4×C31C2×D811yescharacteristic
136C2×C4×C31C2×SD1611yescharacteristic
137C8×C31—44no—
138C8×C31—82no—
139#2619C2×C3111yescharacteristic
140#1742C2×C3111yescharacteristic, complemented
141#1458C2×C3111yescharacteristic, complemented
142#1345C2×C3114yescomplemented
143C22×C31—21no—
144C22×C31—21no—
145C22×C31—22no—
146C22×C31—42no—
147C22×C31—41no—
148C22×C31—41no—
149C22×C31—81no—
150C22×C31—82no—
151C22×C31—82no—
152C22×C31—82no—
153C22×C31—82no—
154C22×C31—82no—
155C22×C31—82no—
156C22×C31—82no—
157C22×C31C2×#9911yescharacteristic
158C22×C31C2×#10211yescharacteristic
159C22×C31C2×#10411yescharacteristic
160C4×C31—21no—
161C4×C31—21no—
162C4×C31—41no—
163C4×C31—41no—
164C4×C31—41no—
165C4×C31—41no—
166C4×C31—82no—
167C4×C31#32712yes—
168C4×C31#32712yes—
169C22×M16—21no—
170C2×#127C4×C3111yescharacteristic
171C2×#121—21no—
172C2×#117C22×C3111yescharacteristic
173#431—21no—
174#394—21no—
175#394C22×C3112yes—
176#392—21no—
177#392C4×C3112yes—
178#390—21no—
179#382C4×C3111yescharacteristic
180#343—21no—
181#343C22×C3112yes—
182#324C22×C3112yes—
183#323—21no—
184C2×C31—21no—
185C2×C31—21no—
186C2×C31—42no—
187C2×C31—162no—
188C2×C31C2×#32711yescharacteristic
189C2×C31#212211yescharacteristic
190C2×C31#213411yescharacteristic
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—21no—
194C22×D4C2×C4×C3112yes—
195C23×C4D4×C3111yescharacteristic
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C22×C8—41no—
201#127—41no—
202#127—41no—
203#121—42no—
204#121—42no—
205#117—41no—
206#117—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4D4×C3111yescharacteristic
209C2×C4⋊C4C23×C3111yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4C2×C4×C3111yescharacteristic
214C2×C22⋊C4D4×C3111yescharacteristic
215C2×C42C2×C4×C3111yescharacteristic
216C2×C42D4×C3111yescharacteristic
217C4⋊C8—41no—
218C22⋊C8—22no—
219C4×C8—41no—
220#95—21no—
221#95C2×C4×C3112yes—
222C31#1169911yescharacteristic, direct product factor, Sylow
223C24—22no—
224C24C22⋊C4×C3111yescharacteristic
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4C22⋊C4×C3111yescharacteristic
241C22×C4C22⋊C4×C3111yescharacteristic
242C22×C4C22⋊C4×C3111yescharacteristic
243C22×C4D8×C3112yes—
244C22×C4SD16×C3112yes—
245C22×C4C22×C4×C3111yescharacteristic
246C22×C4C2×D4×C3111yescharacteristic
247C22×C4C2×D4×C3111yescharacteristic
248M16—24no—
249M16—44no—
250M16—42no—
251C2×C8—42no—
252C2×C8—44no—
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23C31×#10212yes—
279C23C2×C22⋊C4×C3111yescharacteristic
280D4—84no—
281D4—84no—
282D4—88no—
283D4—82no—
284D4—82no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4C31×#9911yescharacteristic
303C2×C4C31×#9911yescharacteristic
304C2×C4C31×#10212yes—
305C2×C4C31×#10412yes—
306C2×C4C2×D8×C3111yescharacteristic
307C2×C4C2×SD16×C3111yescharacteristic
308C8—44no—
309C8—82no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22C2×C31×#9911yescharacteristic
325C22C2×C31×#10211yescharacteristic
326C22C2×C31×#10411yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4C31×#32712yes—
335C4C31×#32712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—162no—
340C2C2×C31×#32711yescharacteristic
341C2C31×#212211yescharacteristic
342C2C31×#213411yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C4⋊C4C23×C31
prime radicalC31×#2619C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C2×C31×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1742C2semidirect
3. C31×#1458C2semidirect
4. C31×#1345C2semidirect
95. #11699C31direct (N × H)
140. #1742C2×C31semidirect
141. #1458C2×C31semidirect
142. #1345C2×C31semidirect
222. C31#11699direct (N × H)

Maximal subgroups

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

subgroupindex
C31×#26192
C31×#17422
C31×#14582
C31×#13452
#1169931