Subgroups — C₃₁×#12549 #21469308

Order 7936.

586 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	C₂²×C₃₁×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1376	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₃₁×#117	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	—	2	4	no	—
8	C₂×C₃₁×M₃₂	—	2	2	no	—
9	C₂×C₃₁×M₃₂	C₄	1	8	yes	complemented
10	C₂×C₃₁×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₃₁	—	2	2	no	—
12	C₃₁×#405	—	2	4	no	—
13	C₂×C₃₁×#97	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₃₁	—	2	2	no	—
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂×M₁₆×C₃₁	C₈	1	4	yes	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₃₁	D₄	1	2	yes	—
23	C₂²×C₈×C₃₁	D₄	1	4	yes	—
24	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×#117	—	2	2	no	—
27	C₃₁×#117	C₈	1	4	yes	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₃₁×M₃₂	—	2	8	no	—
32	C₃₁×M₃₂	—	2	4	no	—
33	C₃₁×M₃₂	C₂×C₄	1	8	yes	—
34	C₂×C₁₆×C₃₁	—	2	8	no	—
35	C₂×C₁₆×C₃₁	—	2	4	no	—
36	C₃₁×#97	—	2	4	no	—
37	C₄×C₈×C₃₁	—	2	4	no	—
38	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₃₁	—	2	8	no	—
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	2	2	no	—
51	C₂×C₈×C₃₁	C₄²	1	2	yes	—
52	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
54	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
57	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₂²⋊C₄×C₃₁	—	2	4	no	—
64	C₄²×C₃₁	—	2	2	no	—
65	C₄²×C₃₁	C₂×C₈	1	4	yes	—
66	C₁₆×C₃₁	—	2	8	no	—
67	#12549	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	—	4	2	no	—
69	C₂³×C₃₁	#97	1	2	yes	—
70	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
71	C₂³×C₃₁	C₄⋊C₈	1	2	yes	—
72	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	4	4	no	—
75	C₂×C₄×C₃₁	#95	1	4	yes	—
76	C₂×C₄×C₃₁	C₄×C₈	1	4	yes	—
77	C₂×C₄×C₃₁	#97	1	2	yes	—
78	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
80	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
81	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
82	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
88	C₈×C₃₁	—	2	4	no	—
89	C₈×C₃₁	#95	1	4	yes	—
90	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
91	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×M₃₂	C₂×C₃₁	1	2	yes	—
93	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
94	#1376	C₂×C₃₁	1	4	yes	complemented
95	C₂²×C₃₁	—	2	2	no	—
96	C₂²×C₃₁	—	2	4	no	—
97	C₂²×C₃₁	—	4	2	no	—
98	C₂²×C₃₁	—	4	2	no	—
99	C₂²×C₃₁	#336	1	4	yes	—
100	C₂²×C₃₁	C₂×#97	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₄×C₃₁	—	4	4	no	—
104	C₄×C₃₁	#336	1	4	yes	—
105	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
106	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
108	C₄×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
111	C₂×#117	C₄×C₃₁	1	1	yes	characteristic
112	C₂×M₃₂	—	2	4	no	—
113	C₂×M₃₂	—	2	2	no	—
114	C₂×M₃₂	C₄×C₃₁	1	8	yes	complemented
115	C₂×M₃₂	C₂²×C₃₁	1	4	yes	—
116	C₂²×C₁₆	—	2	2	no	—
117	#405	—	2	4	no	—
118	C₂×#97	C₄×C₃₁	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
120	C₂×C₃₁	—	2	1	no	—
121	C₂×C₃₁	—	2	4	no	—
122	C₂×C₃₁	—	2	1	no	—
123	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
124	C₂×C₃₁	#2153	1	2	yes	—
125	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂×M₁₆	—	2	2	no	—
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	C₈×C₃₁	1	4	yes	—
129	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
132	C₂²×C₈	D₄×C₃₁	1	2	yes	—
133	C₂²×C₈	D₄×C₃₁	1	4	yes	—
134	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
135	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
136	#117	—	2	2	no	—
137	#117	C₈×C₃₁	1	4	yes	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
141	M₃₂	—	2	8	no	—
142	M₃₂	—	2	4	no	—
143	M₃₂	C₂×C₄×C₃₁	1	8	yes	—
144	C₂×C₁₆	—	2	8	no	—
145	C₂×C₁₆	—	2	4	no	—
146	#97	—	2	4	no	—
147	C₄×C₈	—	2	4	no	—
148	C₃₁	#12549	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
152	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
153	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
154	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
156	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
157	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
159	M₁₆	—	2	8	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₄²×C₃₁	1	2	yes	—
163	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
164	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
165	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
166	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
167	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
168	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
169	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
171	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
172	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×C₈×C₃₁	1	4	yes	—
177	C₁₆	—	2	8	no	—
178	C₂³	—	4	2	no	—
179	C₂³	C₃₁×#97	1	2	yes	—
180	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
181	C₂³	C₃₁×C₄⋊C₈	1	2	yes	—
182	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₃₁×#95	1	4	yes	—
186	C₂×C₄	C₄×C₈×C₃₁	1	4	yes	—
187	C₂×C₄	C₃₁×#97	1	2	yes	—
188	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
189	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
190	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
191	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
192	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	C₃₁×#95	1	4	yes	—
200	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
201	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₃₁×#336	1	4	yes	—
207	C₂²	C₂×C₃₁×#97	1	1	yes	characteristic
208	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
209	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
210	C₄	—	4	4	no	—
211	C₄	C₃₁×#336	1	4	yes	—
212	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
213	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
214	C₄	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
215	C₄	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	4	no	—
219	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
220	C₂	C₃₁×#2153	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₄⋊C₈
prime radical	C₂⁴×C₃₁	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1376	C₂	semidirect
9. C₂×C₃₁×M₃₂	C₄	semidirect
67. #12549	C₃₁	direct (N × H)
94. #1376	C₂×C₃₁	semidirect
114. C₂×M₃₂	C₄×C₃₁	semidirect
148. C₃₁	#12549	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×M₃₂	2
C₂×C₃₁×#405	2
C₃₁×#1376	2
#12549	31

Subgroups — C31×#12549 #21469308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12549 #21469308