Subgroups — C₂×C₃₁×#1745 #21477121

Order 7936.

1310 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#97	C₂	1	2	yes	maximal
2	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1745	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#336	C₂	1	4	yes	maximal
5	C₂³×C₈×C₃₁	C₂²	1	4	yes	—
6	C₂²×C₄²×C₃₁	C₄	1	4	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#97	C₄	1	16	yes	complemented
10	C₂×C₃₁×#97	C₂²	1	8	yes	—
11	C₄³×C₃₁	C₄	1	4	yes	—
12	C₄³×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#336	C₂²	1	16	yes	—
14	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
15	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	—	2	16	no	—
18	C₂²×C₈×C₃₁	C₂×C₄	1	16	yes	—
19	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
20	C₂×C₄²×C₃₁	—	2	16	no	—
21	C₂×C₄²×C₃₁	—	2	8	no	—
22	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	—
23	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
27	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
28	C₃₁×#97	C₂×C₄	1	32	yes	complemented
29	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	—	2	8	no	—
31	C₂²×C₄×C₃₁	—	2	8	no	—
32	C₂²×C₄×C₃₁	—	2	8	no	—
33	C₂²×C₄×C₃₁	C₄²	1	8	yes	—
34	C₂²×C₄×C₃₁	M₁₆	1	16	yes	—
35	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
36	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₃₁	#41	1	8	yes	—
40	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
41	C₂×C₈×C₃₁	—	2	32	no	—
42	C₂×C₈×C₃₁	—	2	16	no	—
43	C₂×C₈×C₃₁	C₂²×C₄	1	16	yes	—
44	C₄²×C₃₁	—	2	8	no	—
45	C₄²×C₃₁	—	2	32	no	—
46	C₄²×C₃₁	—	2	8	no	—
47	C₄²×C₃₁	C₄²	1	8	yes	—
48	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
49	C₂×#1745	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
51	C₂³×C₃₁	#117	1	4	yes	—
52	C₂³×C₃₁	C₂×M₁₆	1	8	yes	—
53	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	—	2	8	no	—
55	C₂×C₄×C₃₁	—	2	8	no	—
56	C₂×C₄×C₃₁	—	2	8	no	—
57	C₂×C₄×C₃₁	—	2	16	no	—
58	C₂×C₄×C₃₁	—	2	32	no	—
59	C₂×C₄×C₃₁	C₂×C₄²	1	8	yes	—
60	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
61	C₂×C₄×C₃₁	#117	1	16	yes	—
62	C₂×C₄×C₃₁	#117	1	4	yes	—
63	C₂×C₄×C₃₁	C₂×M₁₆	1	16	yes	—
64	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
66	C₈×C₃₁	—	2	16	no	—
67	C₂²×#97	C₂×C₃₁	1	2	yes	—
68	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
69	#1745	C₂×C₃₁	1	8	yes	direct product factor
70	C₂×#336	C₂×C₃₁	1	4	yes	—
71	C₂²×C₃₁	#376	1	4	yes	—
72	C₂²×C₃₁	C₄×M₁₆	1	8	yes	—
73	C₂²×C₃₁	#432	1	16	yes	—
74	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
75	C₂²×C₃₁	C₂×#117	1	2	yes	—
76	C₂²×C₃₁	C₂²×M₁₆	1	4	yes	—
77	C₄×C₃₁	—	2	8	no	—
78	C₄×C₃₁	—	2	16	no	—
79	C₄×C₃₁	#376	1	4	yes	—
80	C₄×C₃₁	C₂×#117	1	4	yes	—
81	C₂³×C₈	C₂²×C₃₁	1	4	yes	—
82	C₂²×C₄²	C₄×C₃₁	1	4	yes	—
83	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
84	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
85	C₂×#97	C₄×C₃₁	1	16	yes	complemented
86	C₂×#97	C₂²×C₃₁	1	8	yes	—
87	C₄³	C₄×C₃₁	1	4	yes	—
88	C₄³	C₂²×C₃₁	1	4	yes	—
89	#336	C₂²×C₃₁	1	16	yes	—
90	C₂×C₃₁	#1745	1	8	yes	direct product factor
91	C₂×C₃₁	C₂×#376	1	1	yes	characteristic
92	C₂×C₃₁	C₂×C₄×M₁₆	1	2	yes	—
93	C₂×C₃₁	C₂×#432	1	4	yes	—
94	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
95	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
96	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
97	C₂²×C₈	—	2	16	no	—
98	C₂²×C₈	C₂×C₄×C₃₁	1	16	yes	—
99	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
100	C₂×C₄²	—	2	16	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
103	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	—
104	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
105	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
106	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
107	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
108	#97	C₂×C₄×C₃₁	1	32	yes	complemented
109	C₃₁	C₂×#1745	1	1	yes	characteristic, direct product factor, Sylow
110	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	2	8	no	—
114	C₂²×C₄	C₄²×C₃₁	1	8	yes	—
115	C₂²×C₄	M₁₆×C₃₁	1	16	yes	—
116	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
117	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
118	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
119	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
120	C₂²×C₄	#41×C₃₁	1	8	yes	—
121	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
122	C₂×C₈	—	2	32	no	—
123	C₂×C₈	—	2	16	no	—
124	C₂×C₈	C₂²×C₄×C₃₁	1	16	yes	—
125	C₄²	—	2	8	no	—
126	C₄²	—	2	32	no	—
127	C₄²	—	2	8	no	—
128	C₄²	C₄²×C₃₁	1	8	yes	—
129	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
130	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
131	C₂³	C₃₁×#117	1	4	yes	—
132	C₂³	C₂×M₁₆×C₃₁	1	8	yes	—
133	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
134	C₂×C₄	—	2	16	no	—
135	C₂×C₄	—	2	32	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	C₂×C₄²×C₃₁	1	8	yes	—
140	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
141	C₂×C₄	C₃₁×#117	1	16	yes	—
142	C₂×C₄	C₃₁×#117	1	4	yes	—
143	C₂×C₄	C₂×M₁₆×C₃₁	1	16	yes	—
144	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
145	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
146	C₈	—	2	16	no	—
147	C₂²	C₃₁×#376	1	4	yes	—
148	C₂²	C₄×M₁₆×C₃₁	1	8	yes	—
149	C₂²	C₃₁×#432	1	16	yes	—
150	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
151	C₂²	C₂×C₃₁×#117	1	2	yes	—
152	C₂²	C₂²×M₁₆×C₃₁	1	4	yes	—
153	C₄	—	2	8	no	—
154	C₄	—	2	16	no	—
155	C₄	C₃₁×#376	1	4	yes	—
156	C₄	C₂×C₃₁×#117	1	4	yes	—
157	C₂	C₃₁×#1745	1	8	yes	direct product factor
158	C₂	C₂×C₃₁×#376	1	1	yes	characteristic
159	C₂	C₂×C₄×M₁₆×C₃₁	1	2	yes	—
160	C₂	C₂×C₃₁×#432	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1745	C₂	direct (N × H)
9. C₂×C₃₁×#97	C₄	semidirect
28. C₃₁×#97	C₂×C₄	semidirect
49. C₂×#1745	C₃₁	direct (N × H)
69. #1745	C₂×C₃₁	direct (N × H)
85. C₂×#97	C₄×C₃₁	semidirect
90. C₂×C₃₁	#1745	direct (N × H)
108. #97	C₂×C₄×C₃₁	semidirect
109. C₃₁	C₂×#1745	direct (N × H)
157. C₂	C₃₁×#1745	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×#97	2
C₂×C₄³×C₃₁	2
C₃₁×#1745	2
C₂×C₃₁×#336	2
C₂×#1745	31

Subgroups — C2×C31×#1745 #21477121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1745 #21477121