Subgroups — C₂×C₃₁×#1709 #21476356

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#486	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1709	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₃₁	—	2	2	no	—
7	C₂²×D₈×C₃₁	—	2	2	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃₁×#127	C₂²	1	2	yes	—
10	C₃₁×#494	—	2	2	no	—
11	C₃₁×#494	C₂²	1	4	yes	complemented
12	C₃₁×#493	—	2	2	no	—
13	C₃₁×#493	C₂²	1	4	yes	—
14	C₃₁×#486	—	2	8	no	—
15	C₃₁×#486	C₂²	1	16	yes	complemented
16	C₂×C₃₁×#102	—	2	8	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₈²×C₃₁	C₂²	1	4	yes	complemented
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×D₄×C₃₁	—	2	4	no	—
22	C₂²×D₄×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	8	no	—
24	C₂×Q₁₆×C₃₁	—	4	8	no	—
25	C₂×D₈×C₃₁	—	2	8	no	—
26	C₂×D₈×C₃₁	—	4	8	no	—
27	C₂²×C₈×C₃₁	D₄	1	4	yes	—
28	C₂²×C₈×C₃₁	D₄	1	2	yes	—
29	C₃₁×C₄⋊Q₈	—	2	4	no	—
30	C₃₁×C₄⋊Q₈	—	2	2	no	—
31	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
32	C₃₁×#127	—	2	4	no	—
33	C₃₁×#127	—	2	2	no	—
34	C₃₁×#127	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#102	—	2	32	no	—
38	C₄×C₈×C₃₁	D₄	1	4	yes	—
39	C₄×C₈×C₃₁	D₄	1	2	yes	—
40	C₄×C₈×C₃₁	C₂³	1	4	yes	—
41	C₄×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂⁴×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂×Q₈×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	—	4	8	no	—
46	C₂×D₄×C₃₁	—	2	8	no	—
47	C₂×D₄×C₃₁	—	2	4	no	—
48	C₂×D₄×C₃₁	—	4	2	no	—
49	C₂×D₄×C₃₁	—	4	4	no	—
50	C₂×D₄×C₃₁	—	4	8	no	—
51	C₂×D₄×C₃₁	—	4	16	no	—
52	C₂²×C₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	Q₁₆×C₃₁	—	4	16	no	—
56	D₈×C₃₁	—	4	16	no	—
57	C₂×C₈×C₃₁	—	2	8	no	—
58	C₂×C₈×C₃₁	D₈	1	8	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
61	C₄⋊C₄×C₃₁	—	2	8	no	—
62	C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₄²×C₃₁	C₂×D₄	1	2	yes	—
64	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
66	C₂×#1709	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	—	4	4	no	—
68	C₂³×C₃₁	—	8	4	no	—
69	C₂³×C₃₁	—	8	8	no	—
70	C₂³×C₃₁	#127	1	1	yes	characteristic
71	Q₈×C₃₁	—	4	8	no	—
72	Q₈×C₃₁	—	8	4	no	—
73	D₄×C₃₁	—	4	8	no	—
74	D₄×C₃₁	—	4	16	no	—
75	D₄×C₃₁	—	8	4	no	—
76	D₄×C₃₁	—	8	8	no	—
77	C₂×C₄×C₃₁	—	4	4	no	—
78	C₂×C₄×C₃₁	—	8	4	no	—
79	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
80	C₂×C₄×C₃₁	#127	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
82	C₂×C₄×C₃₁	#135	1	8	yes	—
83	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
85	C₈×C₃₁	—	2	8	no	—
86	C₈×C₃₁	C₂×D₈	1	8	yes	—
87	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
88	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
89	C₂×#486	C₂×C₃₁	1	4	yes	complemented
90	#1709	C₂×C₃₁	1	8	yes	direct product factor
91	C₂×C₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
92	C₂²×C₃₁	—	8	4	no	—
93	C₂²×C₃₁	—	8	8	no	—
94	C₂²×C₃₁	—	16	8	no	—
95	C₂²×C₃₁	#493	1	2	yes	—
96	C₂²×C₃₁	#495	1	4	yes	—
97	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
98	C₄×C₃₁	—	8	4	no	—
99	C₄×C₃₁	#493	1	2	yes	—
100	C₄×C₃₁	#495	1	4	yes	—
101	C₄×C₃₁	C₂²×D₈	1	2	yes	—
102	C₄×C₃₁	C₂×#135	1	4	yes	—
103	C₂²×Q₁₆	—	2	2	no	—
104	C₂²×D₈	—	2	2	no	—
105	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
106	C₂×#127	C₂²×C₃₁	1	2	yes	—
107	#494	—	2	2	no	—
108	#494	C₂²×C₃₁	1	4	yes	complemented
109	#493	—	2	2	no	—
110	#493	C₂²×C₃₁	1	4	yes	—
111	#486	—	2	8	no	—
112	#486	C₂²×C₃₁	1	16	yes	complemented
113	C₂×#102	—	2	8	no	—
114	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
116	C₈²	C₂²×C₃₁	1	4	yes	complemented
117	C₂×C₃₁	—	16	4	no	—
118	C₂×C₃₁	#1709	1	4	yes	direct product factor
119	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
120	C₂×C₃₁	C₂×#495	1	2	yes	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂²×D₄	—	2	2	no	—
123	C₂²×D₄	—	2	4	no	—
124	C₂×Q₁₆	—	2	8	no	—
125	C₂×Q₁₆	—	4	8	no	—
126	C₂×D₈	—	2	8	no	—
127	C₂×D₈	—	4	8	no	—
128	C₂²×C₈	D₄×C₃₁	1	4	yes	—
129	C₂²×C₈	D₄×C₃₁	1	2	yes	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
133	#127	—	2	4	no	—
134	#127	—	2	2	no	—
135	#127	C₂³×C₃₁	1	4	yes	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
138	#102	—	2	32	no	—
139	C₄×C₈	D₄×C₃₁	1	4	yes	—
140	C₄×C₈	D₄×C₃₁	1	2	yes	—
141	C₄×C₈	C₂³×C₃₁	1	4	yes	—
142	C₄×C₈	C₂³×C₃₁	1	2	yes	—
143	C₃₁	C₂×#1709	1	1	yes	characteristic, direct product factor, Sylow
144	C₂⁴	—	4	2	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂×D₄	—	2	8	no	—
149	C₂×D₄	—	2	4	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂×D₄	—	4	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	16	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
157	Q₁₆	—	4	16	no	—
158	D₈	—	4	16	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	D₈×C₃₁	1	8	yes	—
161	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
162	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂×D₄×C₃₁	1	2	yes	—
166	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
168	C₂³	—	4	4	no	—
169	C₂³	—	8	4	no	—
170	C₂³	—	8	8	no	—
171	C₂³	C₃₁×#127	1	1	yes	characteristic
172	Q₈	—	4	8	no	—
173	Q₈	—	8	4	no	—
174	D₄	—	4	8	no	—
175	D₄	—	4	16	no	—
176	D₄	—	8	4	no	—
177	D₄	—	8	8	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
181	C₂×C₄	C₃₁×#127	1	2	yes	—
182	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
183	C₂×C₄	C₃₁×#135	1	8	yes	—
184	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
186	C₈	—	2	8	no	—
187	C₈	C₂×D₈×C₃₁	1	8	yes	—
188	C₂²	—	8	4	no	—
189	C₂²	—	8	8	no	—
190	C₂²	—	16	8	no	—
191	C₂²	C₃₁×#493	1	2	yes	—
192	C₂²	C₃₁×#495	1	4	yes	—
193	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	C₃₁×#493	1	2	yes	—
196	C₄	C₃₁×#495	1	4	yes	—
197	C₄	C₂²×D₈×C₃₁	1	2	yes	—
198	C₄	C₂×C₃₁×#135	1	4	yes	—
199	C₂	—	16	4	no	—
200	C₂	C₃₁×#1709	1	4	yes	direct product factor
201	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
202	C₂	C₂×C₃₁×#495	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#494	C₂	semidirect
3. C₂×C₃₁×#486	C₂	semidirect
4. C₃₁×#1709	C₂	direct (N × H)
5. C₂×C₈²×C₃₁	C₂	semidirect
11. C₃₁×#494	C₂²	semidirect
15. C₃₁×#486	C₂²	semidirect
19. C₈²×C₃₁	C₂²	semidirect
66. C₂×#1709	C₃₁	direct (N × H)
87. C₂×#494	C₂×C₃₁	semidirect
89. C₂×#486	C₂×C₃₁	semidirect
90. #1709	C₂×C₃₁	direct (N × H)
91. C₂×C₈²	C₂×C₃₁	semidirect
108. #494	C₂²×C₃₁	semidirect
112. #486	C₂²×C₃₁	semidirect
116. C₈²	C₂²×C₃₁	semidirect
118. C₂×C₃₁	#1709	direct (N × H)
143. C₃₁	C₂×#1709	direct (N × H)
200. C₂	C₃₁×#1709	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×#494	2
C₂×C₃₁×#493	2
C₂×C₃₁×#486	2
C₃₁×#1709	2
C₂×C₈²×C₃₁	2
C₂×#1709	31

Subgroups — C2×C31×#1709 #21476356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1709 #21476356