Subgroups — C₃₁×#33919 #21490678

Order 7936.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3336	C₂	1	2	yes	maximal
2	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2237	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2236	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2219	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2159	C₂	1	2	yes	maximal, complemented
10	C₃₁×#552	—	2	2	no	—
11	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
13	C₃₁×#499	C₂²	1	2	yes	—
14	C₃₁×#485	—	2	2	no	—
15	C₃₁×#483	—	2	2	no	—
16	C₃₁×#479	—	2	2	no	—
17	C₃₁×#467	C₂²	1	2	yes	—
18	C₃₁×#467	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#427	C₂²	1	2	yes	—
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₃₁×#367	C₂²	1	2	yes	—
24	C₃₁×#367	C₂²	1	2	yes	—
25	C₃₁×#366	C₂²	1	1	yes	characteristic
26	C₃₁×#366	C₂²	1	2	yes	—
27	C₃₁×#366	C₂²	1	1	yes	characteristic
28	C₃₁×#366	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₃₁×#358	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#126	—	4	2	no	—
38	C₃₁×#125	—	2	2	no	—
39	C₃₁×#125	—	4	2	no	—
40	C₃₁×#123	—	4	2	no	—
41	C₃₁×C₂²⋊Q₈	—	2	2	no	—
42	C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
45	C₃₁×#117	—	2	2	no	—
46	C₃₁×#117	—	4	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₃₁×Q₃₂	—	2	2	no	—
49	C₃₁×M₃₂	—	2	4	no	—
50	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
51	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#107	—	2	2	no	—
54	C₃₁×#107	C₂³	1	2	yes	—
55	C₃₁×#107	C₂³	1	2	yes	—
56	C₃₁×#107	C₂³	1	2	yes	—
57	C₃₁×#106	—	2	2	no	—
58	C₃₁×#106	—	2	2	no	—
59	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₃₁×#103	—	2	2	no	—
61	C₃₁×#103	—	2	2	no	—
62	C₃₁×#103	—	2	2	no	—
63	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	—	4	2	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	Q₁₆×C₃₁	—	2	2	no	—
70	Q₁₆×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	2	no	—
88	C₂²⋊C₄×C₃₁	—	4	2	no	—
89	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
90	C₄²×C₃₁	—	4	2	no	—
91	C₄²×C₃₁	—	4	2	no	—
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₃₁	—	2	2	no	—
94	C₁₆×C₃₁	—	2	1	no	—
95	C₁₆×C₃₁	#41	1	2	yes	—
96	#33919	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	C₂×D₈	1	2	yes	—
98	Q₈×C₃₁	—	4	2	no	—
99	Q₈×C₃₁	—	4	2	no	—
100	D₄×C₃₁	—	4	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	8	4	no	—
108	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	2	1	no	—
114	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
115	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
116	#3336	C₂×C₃₁	1	2	yes	—
117	#2253	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#2237	C₂×C₃₁	1	2	yes	complemented
119	#2236	C₂×C₃₁	1	2	yes	complemented
120	#2229	C₂×C₃₁	1	2	yes	complemented
121	#2219	C₂×C₃₁	1	2	yes	complemented
122	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
123	#2169	C₂×C₃₁	1	1	yes	characteristic
124	#2159	C₂×C₃₁	1	2	yes	complemented
125	C₂²×C₃₁	—	2	2	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	8	2	no	—
133	C₄×C₃₁	—	8	2	no	—
134	C₄×C₃₁	—	8	2	no	—
135	C₄×C₃₁	—	8	2	no	—
136	C₄×C₃₁	#547	1	1	yes	characteristic
137	C₄×C₃₁	#578	1	1	yes	characteristic
138	#552	—	2	2	no	—
139	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
141	#499	C₂²×C₃₁	1	2	yes	—
142	#485	—	2	2	no	—
143	#483	—	2	2	no	—
144	#479	—	2	2	no	—
145	#467	C₂²×C₃₁	1	2	yes	—
146	#467	C₂²×C₃₁	1	2	yes	—
147	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
148	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
149	#427	C₂²×C₃₁	1	2	yes	—
150	C₂×#107	C₂²×C₃₁	1	2	yes	—
151	#367	C₂²×C₃₁	1	2	yes	—
152	#367	C₂²×C₃₁	1	2	yes	—
153	#366	C₂²×C₃₁	1	1	yes	characteristic
154	#366	C₂²×C₃₁	1	2	yes	—
155	#366	C₂²×C₃₁	1	1	yes	characteristic
156	#366	C₂²×C₃₁	1	1	yes	characteristic
157	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
158	#358	C₂²×C₃₁	1	2	yes	—
159	#358	C₂²×C₃₁	1	2	yes	—
160	#358	C₂²×C₃₁	1	2	yes	—
161	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
162	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₃₁	—	4	2	no	—
164	C₂×C₃₁	#3334	1	1	yes	characteristic
165	C₂×C₃₁	#3414	1	1	yes	characteristic
166	C₂×C₃₁	#3417	1	1	yes	characteristic
167	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
168	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
169	#126	—	4	2	no	—
170	#125	—	2	2	no	—
171	#125	—	4	2	no	—
172	#123	—	4	2	no	—
173	C₂²⋊Q₈	—	2	2	no	—
174	C₂²⋊Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
177	#117	—	2	2	no	—
178	#117	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	Q₃₂	—	2	2	no	—
181	M₃₂	—	2	4	no	—
182	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
184	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
185	#107	—	2	2	no	—
186	#107	C₂³×C₃₁	1	2	yes	—
187	#107	C₂³×C₃₁	1	2	yes	—
188	#107	C₂³×C₃₁	1	2	yes	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
192	#103	—	2	2	no	—
193	#103	—	2	2	no	—
194	#103	—	2	2	no	—
195	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
196	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₃₁	#33919	1	1	yes	characteristic, direct product factor, Sylow
198	C₂×Q₈	—	2	2	no	—
199	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
202	Q₁₆	—	2	2	no	—
203	Q₁₆	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
208	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
223	C₄²	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₁₆	—	2	2	no	—
227	C₁₆	—	2	1	no	—
228	C₁₆	#41×C₃₁	1	2	yes	—
229	C₂³	C₂×D₈×C₃₁	1	2	yes	—
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	D₄	—	4	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
242	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
247	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	—	8	2	no	—
259	C₄	C₃₁×#547	1	1	yes	characteristic
260	C₄	C₃₁×#578	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	C₃₁×#3334	1	1	yes	characteristic
263	C₂	C₃₁×#3414	1	1	yes	characteristic
264	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2237	C₂	semidirect
4. C₃₁×#2236	C₂	semidirect
5. C₃₁×#2229	C₂	semidirect
6. C₃₁×#2219	C₂	semidirect
7. C₄×C₃₁×Q₃₂	C₂	semidirect
9. C₃₁×#2159	C₂	semidirect
96. #33919	C₃₁	direct (N × H)
117. #2253	C₂×C₃₁	semidirect
118. #2237	C₂×C₃₁	semidirect
119. #2236	C₂×C₃₁	semidirect
120. #2229	C₂×C₃₁	semidirect
121. #2219	C₂×C₃₁	semidirect
122. C₄×Q₃₂	C₂×C₃₁	semidirect
124. #2159	C₂×C₃₁	semidirect
197. C₃₁	#33919	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×#3336	2
C₃₁×#2253	2
C₃₁×#2237	2
C₃₁×#2236	2
C₃₁×#2229	2
C₃₁×#2219	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2169	2
C₃₁×#2159	2
#33919	31

Subgroups — C31×#33919 #21490678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33919 #21490678