Subgroups — C₂×C₃₁×#1838 #21477526

Order 7936.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	2	yes	maximal
2	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1838	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#341	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
6	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#423	—	2	4	no	—
9	C₃₁×#423	C₂²	1	8	yes	—
10	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
11	C₂×C₃₁×#97	C₄	1	4	yes	—
12	C₂×C₃₁×#97	C₄	1	4	yes	—
13	C₂×C₃₁×#97	C₂²	1	4	yes	—
14	C₃₁×#341	C₂²	1	16	yes	—
15	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
16	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂×M₁₆×C₃₁	—	2	4	no	—
18	C₂×M₁₆×C₃₁	—	2	16	no	—
19	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
20	C₂²×C₈×C₃₁	—	2	8	no	—
21	C₂²×C₈×C₃₁	—	2	4	no	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
25	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₃₁	D₄	1	2	yes	—
28	C₂×C₄²×C₃₁	D₄	1	2	yes	—
29	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
31	C₃₁×C₄⋊C₈	—	2	16	no	—
32	C₃₁×#97	C₂×C₄	1	16	yes	—
33	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	—	4	4	no	—
35	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	#41	1	8	yes	—
44	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
45	M₁₆×C₃₁	—	2	16	no	—
46	M₁₆×C₃₁	—	4	8	no	—
47	C₂×C₈×C₃₁	—	2	8	no	—
48	C₂×C₈×C₃₁	—	2	8	no	—
49	C₂×C₈×C₃₁	—	2	8	no	—
50	C₂×C₈×C₃₁	—	2	8	no	—
51	C₂×C₈×C₃₁	—	4	8	no	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
53	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
54	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
55	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
56	C₄²×C₃₁	C₂×D₄	1	4	yes	—
57	C₂×#1838	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₃₁	—	2	4	no	—
59	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₃₁	#125	1	4	yes	—
62	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₃₁	—	2	8	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
67	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#117	1	4	yes	—
70	C₂×C₄×C₃₁	#117	1	4	yes	—
71	C₂×C₄×C₃₁	#124	1	4	yes	—
72	C₂×C₄×C₃₁	#124	1	4	yes	—
73	C₂×C₄×C₃₁	#125	1	4	yes	—
74	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
77	C₈×C₃₁	—	2	8	no	—
78	C₈×C₃₁	—	4	8	no	—
79	C₂×#423	C₂×C₃₁	1	2	yes	—
80	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
81	#1838	C₂×C₃₁	1	8	yes	direct product factor
82	C₂×#341	C₂×C₃₁	1	4	yes	—
83	C₂²×C₃₁	—	2	8	no	—
84	C₂²×C₃₁	—	2	2	no	—
85	C₂²×C₃₁	—	2	2	no	—
86	C₂²×C₃₁	#382	1	2	yes	—
87	C₂²×C₃₁	#382	1	2	yes	—
88	C₂²×C₃₁	#430	1	4	yes	—
89	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×#125	1	2	yes	—
91	C₄×C₃₁	—	2	8	no	—
92	C₄×C₃₁	#382	1	2	yes	—
93	C₄×C₃₁	#382	1	2	yes	—
94	C₄×C₃₁	C₂×#117	1	2	yes	—
95	C₄×C₃₁	C₂×#124	1	2	yes	—
96	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
97	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
98	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
99	#423	—	2	4	no	—
100	#423	C₂²×C₃₁	1	8	yes	—
101	C₂×C₄⋊C₈	—	2	4	no	—
102	C₂×#97	C₄×C₃₁	1	4	yes	—
103	C₂×#97	C₄×C₃₁	1	4	yes	—
104	C₂×#97	C₂²×C₃₁	1	4	yes	—
105	#341	C₂²×C₃₁	1	16	yes	—
106	C₂×C₃₁	—	2	2	no	—
107	C₂×C₃₁	—	2	2	no	—
108	C₂×C₃₁	#1838	1	4	yes	direct product factor
109	C₂×C₃₁	C₂×#382	1	1	yes	characteristic
110	C₂×C₃₁	C₂×#430	1	2	yes	—
111	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
112	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
113	C₂×M₁₆	—	2	4	no	—
114	C₂×M₁₆	—	2	16	no	—
115	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
116	C₂²×C₈	—	2	8	no	—
117	C₂²×C₈	—	2	4	no	—
118	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
119	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
120	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
121	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
122	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
123	C₂×C₄²	D₄×C₃₁	1	2	yes	—
124	C₂×C₄²	D₄×C₃₁	1	2	yes	—
125	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
126	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
127	C₄⋊C₈	—	2	16	no	—
128	#97	C₂×C₄×C₃₁	1	16	yes	—
129	C₃₁	C₂×#1838	1	1	yes	characteristic, direct product factor, Sylow
130	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
133	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
134	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
135	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
139	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	#41×C₃₁	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
142	M₁₆	—	2	16	no	—
143	M₁₆	—	4	8	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	4	8	no	—
149	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
150	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
151	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
152	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
153	C₄²	C₂×D₄×C₃₁	1	4	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
156	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₂³	C₃₁×#125	1	4	yes	—
158	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#117	1	4	yes	—
166	C₂×C₄	C₃₁×#117	1	4	yes	—
167	C₂×C₄	C₃₁×#124	1	4	yes	—
168	C₂×C₄	C₃₁×#124	1	4	yes	—
169	C₂×C₄	C₃₁×#125	1	4	yes	—
170	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	4	8	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	C₃₁×#382	1	2	yes	—
179	C₂²	C₃₁×#382	1	2	yes	—
180	C₂²	C₃₁×#430	1	4	yes	—
181	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
182	C₂²	C₂×C₃₁×#125	1	2	yes	—
183	C₄	—	2	8	no	—
184	C₄	C₃₁×#382	1	2	yes	—
185	C₄	C₃₁×#382	1	2	yes	—
186	C₄	C₂×C₃₁×#117	1	2	yes	—
187	C₄	C₂×C₃₁×#124	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	C₃₁×#1838	1	4	yes	direct product factor
191	C₂	C₂×C₃₁×#382	1	1	yes	characteristic
192	C₂	C₂×C₃₁×#430	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1838	C₂	direct (N × H)
57. C₂×#1838	C₃₁	direct (N × H)
81. #1838	C₂×C₃₁	direct (N × H)
108. C₂×C₃₁	#1838	direct (N × H)
129. C₃₁	C₂×#1838	direct (N × H)
190. C₂	C₃₁×#1838	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×#423	2
C₂²×C₃₁×#97	2
C₃₁×#1838	2
C₂×C₃₁×#341	2
C₂×#1838	31

Subgroups — C2×C31×#1838 #21477526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1838 #21477526