Subgroups — C₃₁×#12544 #21469303

Order 7936.

678 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	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1374	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	—	2	4	no	—
8	C₂×C₃₁×M₃₂	—	2	4	no	—
9	C₂×C₃₁×M₃₂	C₄	1	8	yes	complemented
10	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₃₁	—	2	4	no	—
12	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₃₁	C₈	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	—	2	2	no	—
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×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	C₂×C₄²×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	—	2	2	no	—
28	C₂×C₄²×C₃₁	C₈	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₃₁×M₃₂	—	2	16	no	—
31	C₂×C₁₆×C₃₁	—	2	4	no	—
32	C₂×C₁₆×C₃₁	—	2	4	no	—
33	C₂×C₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
34	C₄×C₈×C₃₁	—	2	4	no	—
35	C₄×C₈×C₃₁	—	2	4	no	—
36	C₂⁴×C₃₁	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	—	2	4	no	—
38	C₂²×C₄×C₃₁	—	2	8	no	—
39	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
46	C₂²×C₄×C₃₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×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	8	yes	—
54	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
56	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₃₁	—	2	4	no	—
62	C₄²×C₃₁	—	2	2	no	—
63	C₄²×C₃₁	—	2	2	no	—
64	C₁₆×C₃₁	—	2	8	no	—
65	#12544	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	—	2	4	no	—
67	C₂³×C₃₁	C₄×C₈	1	2	yes	—
68	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
69	C₂³×C₃₁	C₄⋊C₈	1	2	yes	—
70	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	—	2	16	no	—
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	#95	1	4	yes	—
75	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
76	C₂×C₄×C₃₁	#97	1	4	yes	—
77	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
78	C₂×C₄×C₃₁	C₂²⋊C₈	1	4	yes	—
79	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₃₁	M₃₂	1	8	yes	complemented
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	C₂×M₁₆	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	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
94	#1374	C₂×C₃₁	1	4	yes	complemented
95	C₂²×C₃₁	—	2	4	no	—
96	C₂²×C₃₁	—	2	4	no	—
97	C₂²×C₃₁	—	2	4	no	—
98	C₂²×C₃₁	#336	1	4	yes	—
99	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×M₃₂	1	4	yes	—
103	C₄×C₃₁	—	2	8	no	—
104	C₄×C₃₁	#336	1	4	yes	—
105	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
106	C₄×C₃₁	C₂×#97	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	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
111	C₂²×C₄²	C₄×C₃₁	1	1	yes	characteristic
112	C₂×M₃₂	—	2	4	no	—
113	C₂×M₃₂	—	2	4	no	—
114	C₂×M₃₂	C₄×C₃₁	1	8	yes	complemented
115	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
116	C₂×C₄×C₈	—	2	4	no	—
117	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
119	C₂×C₃₁	—	2	4	no	—
120	C₂×C₃₁	C₂×#336	1	1	yes	characteristic
121	C₂×C₃₁	C₄×M₃₂	1	2	yes	—
122	C₂×C₃₁	#2112	1	2	yes	—
123	C₂×C₃₁	#2150	1	2	yes	—
124	C₂³×C₄	C₈×C₃₁	1	2	yes	—
125	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	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	C₂×C₄²	—	2	2	no	—
137	C₂×C₄²	—	2	2	no	—
138	C₂×C₄²	C₈×C₃₁	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
140	M₃₂	—	2	16	no	—
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
144	C₄×C₈	—	2	4	no	—
145	C₄×C₈	—	2	4	no	—
146	C₃₁	#12544	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₂×C₈×C₃₁	1	1	yes	characteristic
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
151	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
152	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
153	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
154	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
156	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
157	C₂²×C₄	M₁₆×C₃₁	1	2	yes	—
158	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₈	—	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	8	yes	—
165	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
166	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
167	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
168	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
169	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
171	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
172	C₄²	—	2	4	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₁₆	—	2	8	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄×C₈×C₃₁	1	2	yes	—
178	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
179	C₂³	C₃₁×C₄⋊C₈	1	2	yes	—
180	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	C₃₁×#95	1	4	yes	—
185	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
186	C₂×C₄	C₃₁×#97	1	4	yes	—
187	C₂×C₄	C₃₁×C₂²⋊C₈	1	4	yes	—
188	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
189	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
190	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
191	C₂×C₄	C₃₁×M₃₂	1	8	yes	complemented
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	C₂×M₁₆×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	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	C₃₁×#336	1	4	yes	—
206	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
207	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
208	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
209	C₂²	C₂×C₃₁×M₃₂	1	4	yes	—
210	C₄	—	2	8	no	—
211	C₄	C₃₁×#336	1	4	yes	—
212	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
213	C₄	C₂×C₃₁×#97	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	4	no	—
217	C₂	C₂×C₃₁×#336	1	1	yes	characteristic
218	C₂	C₄×C₃₁×M₃₂	1	2	yes	—
219	C₂	C₃₁×#2112	1	2	yes	—
220	C₂	C₃₁×#2150	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1374	C₂	semidirect
9. C₂×C₃₁×M₃₂	C₄	semidirect
33. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
65. #12544	C₃₁	direct (N × H)
81. C₂×C₄×C₃₁	M₃₂	semidirect
94. #1374	C₂×C₃₁	semidirect
114. C₂×M₃₂	C₄×C₃₁	semidirect
143. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
146. C₃₁	#12544	direct (N × H)
191. C₂×C₄	C₃₁×M₃₂	semidirect

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
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1374	2
#12544	31

Subgroups — C31×#12544 #21469303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12544 #21469303