Subgroups — C₃₁×#29232 #21485991

Order 7936.

552 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2102	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2050	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2031	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1932	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1906	C₂	1	2	yes	maximal
7	C₃₁×#1841	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1827	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×#104	C₂	1	2	yes	maximal, complemented
10	C₃₁×#575	C₂²	1	1	yes	characteristic
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₄×#41×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#430	C₂²	1	1	yes	characteristic
14	C₃₁×#423	C₂²	1	1	yes	characteristic
15	C₃₁×#421	C₂²	1	2	yes	—
16	C₂×C₃₁×#104	C₂²	1	2	yes	—
17	C₂×C₃₁×#104	C₂²	1	2	yes	—
18	C₃₁×#413	C₂²	1	2	yes	—
19	C₃₁×#410	C₂²	1	2	yes	—
20	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#344	C₂²	1	2	yes	—
23	C₃₁×#344	C₂²	1	2	yes	—
24	C₃₁×#339	C₂²	1	2	yes	—
25	C₃₁×#339	C₂²	1	2	yes	—
26	C₃₁×#339	C₂²	1	2	yes	—
27	C₃₁×#339	C₂²	1	2	yes	—
28	C₃₁×#338	C₂²	1	1	yes	characteristic
29	C₃₁×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
30	C₃₁×H(3,ℤ/4)	C₂²	1	2	yes	—
31	C₂×#41×C₃₁	C₂³	1	2	yes	—
32	C₃₁×#137	—	2	2	no	—
33	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
34	C₃₁×#135	—	4	1	no	—
35	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×#125	—	2	1	no	—
41	C₃₁×C₂²⋊Q₈	—	4	1	no	—
42	C₄×Q₈×C₃₁	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	4	1	no	—
44	C₄×D₄×C₃₁	—	2	2	no	—
45	C₄×D₄×C₃₁	—	4	2	no	—
46	C₃₁×#117	—	2	2	no	—
47	C₃₁×#117	C₂³	1	2	yes	—
48	C₃₁×#117	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	—	4	1	no	—
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#108	—	4	1	no	—
54	C₃₁×C₄⋊C₈	—	4	1	no	—
55	C₃₁×#104	—	2	2	no	—
56	C₃₁×#104	—	2	2	no	—
57	C₃₁×#104	—	2	2	no	—
58	C₃₁×#104	—	2	2	no	—
59	C₃₁×#104	—	4	2	no	—
60	C₃₁×#101	—	2	2	no	—
61	C₃₁×#100	—	2	2	no	—
62	C₃₁×#99	—	4	2	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	#41×C₃₁	—	2	2	no	—
66	#41×C₃₁	—	2	2	no	—
67	#41×C₃₁	—	2	2	no	—
68	#41×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	4	1	no	—
70	C₂×Q₈×C₃₁	#41	1	2	yes	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	#41	1	2	yes	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	4	2	no	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₃₁	—	4	1	no	—
79	SD₁₆×C₃₁	—	4	2	no	—
80	D₈×C₃₁	—	4	1	no	—
81	M₁₆×C₃₁	—	2	2	no	—
82	M₁₆×C₃₁	—	2	2	no	—
83	M₁₆×C₃₁	—	2	2	no	—
84	M₁₆×C₃₁	—	2	2	no	—
85	M₁₆×C₃₁	—	2	2	no	—
86	M₁₆×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	2	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₃₁	—	2	2	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	4	2	no	—
104	C₄²×C₃₁	—	4	4	no	—
105	C₄²×C₃₁	#41	1	2	yes	—
106	C₄²×C₃₁	#41	1	2	yes	—
107	C₄²×C₃₁	#41	1	1	yes	characteristic
108	C₄²×C₃₁	#41	1	1	yes	characteristic
109	#29232	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	2	2	no	—
111	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
112	Q₈×C₃₁	—	2	2	no	—
113	Q₈×C₃₁	—	2	2	no	—
114	Q₈×C₃₁	—	8	2	no	—
115	D₄×C₃₁	—	2	2	no	—
116	D₄×C₃₁	—	2	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	8	4	no	—
131	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
135	C₈×C₃₁	—	4	2	no	—
136	C₈×C₃₁	—	4	1	no	—
137	C₈×C₃₁	—	4	1	no	—
138	C₈×C₃₁	—	4	2	no	—
139	#2102	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#2050	C₂×C₃₁	1	1	yes	characteristic
141	#2031	C₂×C₃₁	1	2	yes	complemented
142	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#1932	C₂×C₃₁	1	2	yes	complemented
144	#1906	C₂×C₃₁	1	2	yes	—
145	#1841	C₂×C₃₁	1	2	yes	complemented
146	#1827	C₂×C₃₁	1	2	yes	complemented
147	C₄×#104	C₂×C₃₁	1	2	yes	complemented
148	C₂²×C₃₁	—	4	1	no	—
149	C₂²×C₃₁	—	4	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	#538	1	1	yes	characteristic
152	C₂²×C₃₁	#551	1	2	yes	—
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	—	4	2	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	#529	1	2	yes	—
161	C₄×C₃₁	#551	1	1	yes	characteristic
162	C₄×C₃₁	#555	1	1	yes	characteristic
163	#575	C₂²×C₃₁	1	1	yes	characteristic
164	#533	C₂²×C₃₁	1	1	yes	characteristic
165	C₄×#41	C₂²×C₃₁	1	2	yes	—
166	#430	C₂²×C₃₁	1	1	yes	characteristic
167	#423	C₂²×C₃₁	1	1	yes	characteristic
168	#421	C₂²×C₃₁	1	2	yes	—
169	C₂×#104	C₂²×C₃₁	1	2	yes	—
170	C₂×#104	C₂²×C₃₁	1	2	yes	—
171	#413	C₂²×C₃₁	1	2	yes	—
172	#410	C₂²×C₃₁	1	2	yes	—
173	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
174	C₄³	C₂²×C₃₁	1	1	yes	characteristic
175	#344	C₂²×C₃₁	1	2	yes	—
176	#344	C₂²×C₃₁	1	2	yes	—
177	#339	C₂²×C₃₁	1	2	yes	—
178	#339	C₂²×C₃₁	1	2	yes	—
179	#339	C₂²×C₃₁	1	2	yes	—
180	#339	C₂²×C₃₁	1	2	yes	—
181	#338	C₂²×C₃₁	1	1	yes	characteristic
182	H(3,ℤ/4)	C₂²×C₃₁	1	1	yes	characteristic
183	H(3,ℤ/4)	C₂²×C₃₁	1	2	yes	—
184	C₂×C₃₁	—	2	1	no	—
185	C₂×C₃₁	—	2	2	no	—
186	C₂×C₃₁	—	8	2	no	—
187	C₂×C₃₁	#2572	1	1	yes	characteristic
188	C₂×#41	C₂³×C₃₁	1	2	yes	—
189	#137	—	2	2	no	—
190	AGL(1,ℤ/8)	—	2	2	no	—
191	#135	—	4	1	no	—
192	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
193	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
194	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
195	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
196	C₄⋊Q₈	—	2	1	no	—
197	#125	—	2	1	no	—
198	C₂²⋊Q₈	—	4	1	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	4	2	no	—
203	#117	—	2	2	no	—
204	#117	C₂³×C₃₁	1	2	yes	—
205	#117	C₂³×C₃₁	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	4	1	no	—
208	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	#108	—	4	1	no	—
211	C₄⋊C₈	—	4	1	no	—
212	#104	—	2	2	no	—
213	#104	—	2	2	no	—
214	#104	—	2	2	no	—
215	#104	—	2	2	no	—
216	#104	—	4	2	no	—
217	#101	—	2	2	no	—
218	#100	—	2	2	no	—
219	#99	—	4	2	no	—
220	#97	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₃₁	#29232	1	1	yes	characteristic, direct product factor, Sylow
223	#41	—	2	2	no	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	#41	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	#41×C₃₁	1	2	yes	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	#41×C₃₁	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	Q₁₆	—	4	1	no	—
237	SD₁₆	—	4	2	no	—
238	D₈	—	4	1	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	4	no	—
263	C₄²	#41×C₃₁	1	2	yes	—
264	C₄²	#41×C₃₁	1	2	yes	—
265	C₄²	#41×C₃₁	1	1	yes	characteristic
266	C₄²	#41×C₃₁	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
269	Q₈	—	2	2	no	—
270	Q₈	—	2	2	no	—
271	Q₈	—	8	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	2	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
291	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
292	C₈	—	4	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	4	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	C₃₁×#538	1	1	yes	characteristic
300	C₂²	C₃₁×#551	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	C₃₁×#529	1	2	yes	—
309	C₄	C₃₁×#551	1	1	yes	characteristic
310	C₄	C₃₁×#555	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	8	2	no	—
314	C₂	C₃₁×#2572	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2572
prime radical	C₃₁×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2102	C₂	semidirect
3. C₃₁×#2031	C₂	semidirect
4. C₃₁×#1994	C₂	semidirect
5. C₃₁×#1932	C₂	semidirect
7. C₃₁×#1841	C₂	semidirect
8. C₃₁×#1827	C₂	semidirect
9. C₄×C₃₁×#104	C₂	semidirect
109. #29232	C₃₁	direct (N × H)
139. #2102	C₂×C₃₁	semidirect
141. #2031	C₂×C₃₁	semidirect
142. #1994	C₂×C₃₁	semidirect
143. #1932	C₂×C₃₁	semidirect
145. #1841	C₂×C₃₁	semidirect
146. #1827	C₂×C₃₁	semidirect
147. C₄×#104	C₂×C₃₁	semidirect
222. C₃₁	#29232	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2102	2
C₃₁×#2050	2
C₃₁×#2031	2
C₃₁×#1994	2
C₃₁×#1932	2
C₃₁×#1906	2
C₃₁×#1841	2
C₃₁×#1827	2
C₄×C₃₁×#104	2
#29232	31

Subgroups — C31×#29232 #21485991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29232 #21485991