Subgroups — C₃₁×#33419 #21490178

Order 7936.

590 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2164	C₂	1	8	yes	maximal, complemented
3	C₃₁×#2150	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2112	C₂	1	2	yes	maximal
5	C₄×C₃₁×M₃₂	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
9	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
10	C₃₁×#405	C₄	1	4	yes	—
11	C₃₁×#405	C₂²	1	4	yes	—
12	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
13	C₂×C₃₁×#97	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	complemented
17	C₃₁×C₂²⋊C₁₆	C₂²	1	8	yes	—
18	C₃₁×#346	C₂²	1	4	yes	complemented
19	C₄×C₁₆×C₃₁	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₃₁	—	2	4	no	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₃₁×#117	C₈	1	4	yes	—
31	C₃₁×#117	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₃₁×M₃₂	—	2	8	no	—
37	C₂×C₁₆×C₃₁	C₂³	1	8	yes	—
38	C₃₁×#97	C₂×C₄	1	4	yes	—
39	C₃₁×#97	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₃₁	C₂³	1	2	yes	—
43	C₄×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	M₁₆×C₃₁	—	2	8	no	—
54	C₂×C₈×C₃₁	—	2	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₃₁	#41	1	4	yes	—
64	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
67	C₄²×C₃₁	C₂×C₈	1	2	yes	—
68	C₄²×C₃₁	C₂×C₈	1	2	yes	—
69	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
70	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
71	C₁₆×C₃₁	—	2	8	no	—
72	#33419	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	4	2	no	—
75	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
76	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	#117	1	2	yes	—
84	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
85	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
86	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₃₁	—	2	2	no	—
91	C₈×C₃₁	—	2	2	no	—
92	C₈×C₃₁	#117	1	2	yes	—
93	C₈×C₃₁	C₂×M₁₆	1	4	yes	—
94	C₈×C₃₁	C₂×#41	1	2	yes	—
95	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
96	#2164	C₂×C₃₁	1	8	yes	complemented
97	#2150	C₂×C₃₁	1	2	yes	complemented
98	#2112	C₂×C₃₁	1	2	yes	—
99	C₄×M₃₂	C₂×C₃₁	1	2	yes	complemented
100	C₂²×C₃₁	—	2	2	no	—
101	C₂²×C₃₁	—	2	4	no	—
102	C₂²×C₃₁	—	2	2	no	—
103	C₂²×C₃₁	—	4	4	no	—
104	C₂²×C₃₁	#431	1	2	yes	—
105	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₃₁	—	2	4	no	—
107	C₄×C₃₁	—	2	4	no	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	#431	1	2	yes	—
110	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
111	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
112	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
113	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
114	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
115	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
116	#405	C₄×C₃₁	1	4	yes	—
117	#405	C₂²×C₃₁	1	4	yes	—
118	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
119	C₂×#97	C₄×C₃₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	complemented
123	C₂²⋊C₁₆	C₂²×C₃₁	1	8	yes	—
124	#346	C₂²×C₃₁	1	4	yes	complemented
125	C₄×C₁₆	C₂²×C₃₁	1	4	yes	complemented
126	C₂×C₃₁	—	2	2	no	—
127	C₂×C₃₁	—	4	2	no	—
128	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
129	C₂×C₃₁	#3407	1	2	yes	—
130	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
133	C₂²×C₈	—	2	4	no	—
134	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
136	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
139	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
140	#117	C₈×C₃₁	1	4	yes	—
141	#117	C₂×C₄×C₃₁	1	4	yes	—
142	C₂×C₄⋊C₄	C₈×C₃₁	1	2	yes	—
143	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
144	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
146	M₃₂	—	2	8	no	—
147	C₂×C₁₆	C₂³×C₃₁	1	8	yes	—
148	#97	C₂×C₄×C₃₁	1	4	yes	—
149	#97	C₂×C₄×C₃₁	1	4	yes	—
150	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
151	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
152	C₄×C₈	C₂³×C₃₁	1	2	yes	—
153	C₄×C₈	C₂³×C₃₁	1	2	yes	—
154	C₃₁	#33419	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
160	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
164	M₁₆	—	2	8	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
170	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
171	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
172	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
174	C₂×C₈	#41×C₃₁	1	4	yes	—
175	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
176	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
177	C₂²⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
178	C₄²	C₂×C₈×C₃₁	1	2	yes	—
179	C₄²	C₂×C₈×C₃₁	1	2	yes	—
180	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
181	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
182	C₁₆	—	2	8	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	4	2	no	—
185	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
186	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₃₁×#117	1	2	yes	—
194	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
195	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
196	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
197	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
199	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
200	C₈	—	2	2	no	—
201	C₈	—	2	2	no	—
202	C₈	C₃₁×#117	1	2	yes	—
203	C₈	C₂×M₁₆×C₃₁	1	4	yes	—
204	C₈	C₂×#41×C₃₁	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	4	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	C₃₁×#431	1	2	yes	—
210	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	C₃₁×#431	1	2	yes	—
215	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
216	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	4	2	no	—
219	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
220	C₂	C₃₁×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂⁴×C₃₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2164	C₂	semidirect
3. C₃₁×#2150	C₂	semidirect
5. C₄×C₃₁×M₃₂	C₂	semidirect
16. C₃₁×C₄⋊C₁₆	C₂²	semidirect
18. C₃₁×#346	C₂²	semidirect
19. C₄×C₁₆×C₃₁	C₂²	semidirect
72. #33419	C₃₁	direct (N × H)
96. #2164	C₂×C₃₁	semidirect
97. #2150	C₂×C₃₁	semidirect
99. C₄×M₃₂	C₂×C₃₁	semidirect
122. C₄⋊C₁₆	C₂²×C₃₁	semidirect
124. #346	C₂²×C₃₁	semidirect
125. C₄×C₁₆	C₂²×C₃₁	semidirect
154. C₃₁	#33419	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#405	2
C₃₁×#2164	2
C₃₁×#2150	2
C₃₁×#2112	2
C₄×C₃₁×M₃₂	2
#33419	31

Subgroups — C31×#33419 #21490178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33419 #21490178