Finite Group Databasebeta

← C31×#11800 #21468559

Subgroups — C31×#11800 #21468559

Order 7936.

502 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1820C211yesmaximal, characteristic, complemented
3C31×#1464C211yesmaximal, characteristic, complemented
4C31×#1343C214yesmaximal, complemented
5C2×C31×#127C411yescharacteristic, complemented
6C2×C31×#121—21no—
7C2×C31×#117C2211yescharacteristic
8C31×#432—21no—
9C31×#394—21no—
10C31×#394C412yescomplemented
11C31×#392—21no—
12C31×#392C2212yes—
13C31×#390—21no—
14C31×#382C411yescharacteristic, complemented
15C31×#381—21no—
16C31×#342—21no—
17C31×#342C2212yes—
18C31×#324—21no—
19C31×#323C2212yes—
20C22×D4×C31—22no—
21C22×D4×C31—21no—
22C22×D4×C31—21no—
23C22×D4×C31C2×C412yes—
24C23×C4×C31D411yescharacteristic
25C31×#127—41no—
26C31×#127—41no—
27C31×#121—44no—
28C31×#117—41no—
29C31×#117—41no—
30C2×C4⋊C4×C31—21no—
31C2×C4⋊C4×C31C2×C411yescharacteristic
32C2×C4⋊C4×C31D411yescharacteristic
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31—22no—
35C2×C22⋊C4×C31—22no—
36C2×C22⋊C4×C31—22no—
37C2×C22⋊C4×C31D411yescharacteristic
38C2×C22⋊C4×C31C2311yescharacteristic
39C2×C42×C31C2×C411yescharacteristic
40C2×C42×C31D411yescharacteristic
41C31×C4⋊C8—41no—
42C31×C22⋊C8—22no—
43C31×#97—41no—
44C31×#95—21no—
45C31×#95—41no—
46C31×#95—41no—
47C31×#95C2×C412yes—
48C24×C31—22no—
49C24×C31C2×D411yescharacteristic
50C2×D4×C31—42no—
51C2×D4×C31—42no—
52C2×D4×C31—42no—
53C2×D4×C31—42no—
54C2×D4×C31—44no—
55C2×D4×C31—41no—
56C2×D4×C31—41no—
57C2×D4×C31—82no—
58C2×D4×C31—82no—
59C2×D4×C31—82no—
60C2×D4×C31—82no—
61C22×C4×C31—22no—
62C22×C4×C31—21no—
63C22×C4×C31—21no—
64C22×C4×C31—22no—
65C22×C4×C31—41no—
66C22×C4×C31—42no—
67C22×C4×C31C22⋊C411yescharacteristic
68C22×C4×C31C22⋊C411yescharacteristic
69C22×C4×C31C22⋊C411yescharacteristic
70C22×C4×C31C22⋊C411yescharacteristic
71C22×C4×C31C22×C411yescharacteristic
72C22×C4×C31C2×D411yescharacteristic
73C2×C8×C31—42no—
74C4⋊C4×C31—41no—
75C4⋊C4×C31—41no—
76C4⋊C4×C31—42no—
77C22⋊C4×C31—41no—
78C22⋊C4×C31—41no—
79C22⋊C4×C31—44no—
80C22⋊C4×C31—42no—
81C22⋊C4×C31—42no—
82C22⋊C4×C31—82no—
83C22⋊C4×C31—82no—
84C42×C31—21no—
85C42×C31—21no—
86C42×C31—21no—
87C42×C31—21no—
88#11800C3111yesmaximal, characteristic, direct product factor, Sylow
89C23×C31—21no—
90C23×C31—21no—
91C23×C31—41no—
92C23×C31—41no—
93C23×C31—42no—
94C23×C31—42no—
95C23×C31—42no—
96C23×C31—42no—
97C23×C31—42no—
98C23×C31—42no—
99C23×C31—42no—
100C23×C31C2×C22⋊C411yescharacteristic
101C23×C31AGL(1,ℤ/8)12yes—
102D4×C31—82no—
103D4×C31—82no—
104D4×C31—84no—
105D4×C31—84no—
106D4×C31—88no—
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—21no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—42no—
117C2×C4×C31—42no—
118C2×C4×C31—41no—
119C2×C4×C31—41no—
120C2×C4×C31—41no—
121C2×C4×C31—41no—
122C2×C4×C31—42no—
123C2×C4×C31—42no—
124C2×C4×C31—42no—
125C2×C4×C31—82no—
126C2×C4×C31—82no—
127C2×C4×C31—82no—
128C2×C4×C31#9911yescharacteristic
129C2×C4×C31#9911yescharacteristic
130C8×C31—82no—
131#2619C2×C3111yescharacteristic
132#1820C2×C3111yescharacteristic, complemented
133#1464C2×C3111yescharacteristic, complemented
134#1343C2×C3114yescomplemented
135C22×C31—21no—
136C22×C31—21no—
137C22×C31—22no—
138C22×C31—42no—
139C22×C31—41no—
140C22×C31—41no—
141C22×C31—81no—
142C22×C31—82no—
143C22×C31—82no—
144C22×C31—82no—
145C22×C31—82no—
146C22×C31—82no—
147C22×C31—82no—
148C22×C31—82no—
149C22×C31C2×#9911yescharacteristic
150C22×C31#41811yescharacteristic
151C22×C31#42111yescharacteristic
152C4×C31—21no—
153C4×C31—21no—
154C4×C31—41no—
155C4×C31—41no—
156C4×C31—41no—
157C4×C31—41no—
158C4×C31—41no—
159C4×C31—82no—
160C4×C31—162no—
161C2×#127C4×C3111yescharacteristic, complemented
162C2×#121—21no—
163C2×#117C22×C3111yescharacteristic
164#432—21no—
165#394—21no—
166#394C4×C3112yescomplemented
167#392—21no—
168#392C22×C3112yes—
169#390—21no—
170#382C4×C3111yescharacteristic, complemented
171#381—21no—
172#342—21no—
173#342C22×C3112yes—
174#324—21no—
175#323C22×C3112yes—
176C2×C31—21no—
177C2×C31—21no—
178C2×C31—42no—
179C2×C31—162no—
180C2×C31#151011yescharacteristic
181C2×C31#212211yescharacteristic
182C2×C31#212811yescharacteristic
183C22×D4—22no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4C2×C4×C3112yes—
187C23×C4D4×C3111yescharacteristic
188#127—41no—
189#127—41no—
190#121—44no—
191#117—41no—
192#117—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C2×C4×C3111yescharacteristic
195C2×C4⋊C4D4×C3111yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4D4×C3111yescharacteristic
201C2×C22⋊C4C23×C3111yescharacteristic
202C2×C42C2×C4×C3111yescharacteristic
203C2×C42D4×C3111yescharacteristic
204C4⋊C8—41no—
205C22⋊C8—22no—
206#97—41no—
207#95—21no—
208#95—41no—
209#95—41no—
210#95C2×C4×C3112yes—
211C31#1180011yescharacteristic, direct product factor, Sylow
212C24—22no—
213C24C2×D4×C3111yescharacteristic
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4C22⋊C4×C3111yescharacteristic
232C22×C4C22⋊C4×C3111yescharacteristic
233C22×C4C22⋊C4×C3111yescharacteristic
234C22×C4C22⋊C4×C3111yescharacteristic
235C22×C4C22×C4×C3111yescharacteristic
236C22×C4C2×D4×C3111yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊C4×C3111yescharacteristic
264C23C31×AGL(1,ℤ/8)12yes—
265D4—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4C31×#9911yescharacteristic
292C2×C4C31×#9911yescharacteristic
293C8—82no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×C31×#9911yescharacteristic
309C22C31×#41811yescharacteristic
310C22C31×#42111yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—162no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—162no—
324C2C31×#151011yescharacteristic
325C2C31×#212211yescharacteristic
326C2C31×#212811yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C22⋊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×#1820C2semidirect
3. C31×#1464C2semidirect
4. C31×#1343C2semidirect
5. C2×C31×#127C4semidirect
10. C31×#394C4semidirect
14. C31×#382C4semidirect
88. #11800C31direct (N × H)
132. #1820C2×C31semidirect
133. #1464C2×C31semidirect
134. #1343C2×C31semidirect
161. C2×#127C4×C31semidirect
166. #394C4×C31semidirect
170. #382C4×C31semidirect
211. C31#11800direct (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×#18202
C31×#14642
C31×#13432
#1180031