Subgroups — C₂×C₃₁×#1419 #21470382

Order 7936.

326 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1419	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₃₁	—	2	1	no	—
6	C₂×C₃₁×D₁₆	—	2	1	no	—
7	C₂×C₃₁×D₁₆	—	2	1	no	—
8	C₂×C₃₁×D₁₆	—	2	1	no	—
9	C₂×C₃₁×D₁₆	C₄	1	2	yes	—
10	C₂×C₃₁×D₁₆	C₄	1	2	yes	—
11	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#108	—	2	1	no	—
14	C₃₁×M₆₄	—	2	1	no	—
15	C₃₁×M₆₄	C₂²	1	2	yes	complemented
16	C₂×C₃₁×C₃₂	—	2	1	no	—
17	C₃₁×#368	—	2	1	no	—
18	C₃₁×#368	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₃₁	—	4	1	no	—
20	C₂×D₈×C₃₁	—	2	2	no	—
21	C₂×D₈×C₃₁	—	2	2	no	—
22	C₂×D₈×C₃₁	—	2	2	no	—
23	C₂×D₈×C₃₁	—	4	1	no	—
24	C₂×D₈×C₃₁	—	4	1	no	—
25	C₂×D₈×C₃₁	—	4	1	no	—
26	C₂×M₁₆×C₃₁	—	4	1	no	—
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×D₁₆	—	2	2	no	—
29	C₃₁×D₁₆	—	2	2	no	—
30	C₃₁×D₁₆	—	2	2	no	—
31	C₃₁×D₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#108	—	2	2	no	—
39	C₃₁×#108	—	4	1	no	—
40	C₃₁×C₃₂	—	2	2	no	—
41	C₂⁴×C₃₁	—	8	1	no	—
42	C₂×D₄×C₃₁	—	4	2	no	—
43	C₂×D₄×C₃₁	—	4	2	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	—	8	1	no	—
46	C₂×D₄×C₃₁	—	8	1	no	—
47	C₂×D₄×C₃₁	—	8	1	no	—
48	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
49	D₈×C₃₁	—	2	4	no	—
50	D₈×C₃₁	—	4	2	no	—
51	D₈×C₃₁	—	4	2	no	—
52	D₈×C₃₁	—	4	2	no	—
53	M₁₆×C₃₁	—	4	2	no	—
54	M₁₆×C₃₁	—	8	1	no	—
55	C₂×C₈×C₃₁	—	8	1	no	—
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	D₈	1	1	yes	characteristic
59	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
60	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₁₆×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×#1419	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	—	8	2	no	—
68	C₂³×C₃₁	—	8	2	no	—
69	C₂³×C₃₁	—	8	2	no	—
70	C₂³×C₃₁	—	16	4	no	—
71	C₂³×C₃₁	SD₃₂	1	1	yes	characteristic
72	D₄×C₃₁	—	4	4	no	—
73	D₄×C₃₁	—	8	2	no	—
74	D₄×C₃₁	—	8	2	no	—
75	D₄×C₃₁	—	8	2	no	—
76	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₃₁	D₁₆	1	1	yes	characteristic
79	C₂×C₄×C₃₁	D₁₆	1	1	yes	characteristic
80	C₂×C₄×C₃₁	SD₃₂	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
82	C₈×C₃₁	—	8	2	no	—
83	C₈×C₃₁	#102	1	1	yes	characteristic
84	C₈×C₃₁	#102	1	1	yes	characteristic
85	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
88	C₂×M₆₄	C₂×C₃₁	1	1	yes	characteristic, complemented
89	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
90	#1419	C₂×C₃₁	1	4	yes	direct product factor
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	—	8	4	no	—
93	C₂²×C₃₁	—	16	4	no	—
94	C₂²×C₃₁	—	16	4	no	—
95	C₂²×C₃₁	—	16	4	no	—
96	C₂²×C₃₁	#357	1	1	yes	characteristic
97	C₂²×C₃₁	#357	1	1	yes	characteristic
98	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
99	C₄×C₃₁	#357	1	1	yes	characteristic
100	C₄×C₃₁	#357	1	1	yes	characteristic
101	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
102	C₄×C₃₁	C₂×D₁₆	1	1	yes	characteristic
103	C₂²×D₈	—	2	1	no	—
104	C₂×D₁₆	—	2	1	no	—
105	C₂×D₁₆	—	2	1	no	—
106	C₂×D₁₆	—	2	1	no	—
107	C₂×D₁₆	C₄×C₃₁	1	2	yes	—
108	C₂×D₁₆	C₄×C₃₁	1	2	yes	—
109	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
110	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
111	C₂×#108	—	2	1	no	—
112	M₆₄	—	2	1	no	—
113	M₆₄	C₂²×C₃₁	1	2	yes	complemented
114	C₂×C₃₂	—	2	1	no	—
115	#368	—	2	1	no	—
116	#368	C₂²×C₃₁	1	2	yes	complemented
117	C₂×C₃₁	—	2	1	no	—
118	C₂×C₃₁	—	2	1	no	—
119	C₂×C₃₁	—	16	4	no	—
120	C₂×C₃₁	#1419	1	2	yes	direct product factor
121	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
122	C₂²×D₄	—	4	1	no	—
123	C₂×D₈	—	2	2	no	—
124	C₂×D₈	—	2	2	no	—
125	C₂×D₈	—	2	2	no	—
126	C₂×D₈	—	4	1	no	—
127	C₂×D₈	—	4	1	no	—
128	C₂×D₈	—	4	1	no	—
129	C₂×M₁₆	—	4	1	no	—
130	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
131	D₁₆	—	2	2	no	—
132	D₁₆	—	2	2	no	—
133	D₁₆	—	2	2	no	—
134	D₁₆	C₂×C₄×C₃₁	1	4	yes	—
135	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
138	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
139	C₂×C₁₆	D₄×C₃₁	1	1	yes	characteristic
140	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
141	#108	—	2	2	no	—
142	#108	—	4	1	no	—
143	C₃₂	—	2	2	no	—
144	C₃₁	C₂×#1419	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	—	8	1	no	—
146	C₂×D₄	—	4	2	no	—
147	C₂×D₄	—	4	2	no	—
148	C₂×D₄	—	4	2	no	—
149	C₂×D₄	—	8	1	no	—
150	C₂×D₄	—	8	1	no	—
151	C₂×D₄	—	8	1	no	—
152	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
153	D₈	—	2	4	no	—
154	D₈	—	4	2	no	—
155	D₈	—	4	2	no	—
156	D₈	—	4	2	no	—
157	M₁₆	—	4	2	no	—
158	M₁₆	—	8	1	no	—
159	C₂×C₈	—	8	1	no	—
160	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₈	D₈×C₃₁	1	1	yes	characteristic
163	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
164	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
167	C₁₆	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
169	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₂³	—	8	2	no	—
171	C₂³	—	8	2	no	—
172	C₂³	—	8	2	no	—
173	C₂³	—	16	4	no	—
174	C₂³	C₃₁×SD₃₂	1	1	yes	characteristic
175	D₄	—	4	4	no	—
176	D₄	—	8	2	no	—
177	D₄	—	8	2	no	—
178	D₄	—	8	2	no	—
179	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
180	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
181	C₂×C₄	C₃₁×D₁₆	1	1	yes	characteristic
182	C₂×C₄	C₃₁×D₁₆	1	1	yes	characteristic
183	C₂×C₄	C₃₁×SD₃₂	1	1	yes	characteristic
184	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
185	C₈	—	8	2	no	—
186	C₈	C₃₁×#102	1	1	yes	characteristic
187	C₈	C₃₁×#102	1	1	yes	characteristic
188	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
190	C₂²	—	2	2	no	—
191	C₂²	—	8	4	no	—
192	C₂²	—	16	4	no	—
193	C₂²	—	16	4	no	—
194	C₂²	—	16	4	no	—
195	C₂²	C₃₁×#357	1	1	yes	characteristic
196	C₂²	C₃₁×#357	1	1	yes	characteristic
197	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
198	C₄	C₃₁×#357	1	1	yes	characteristic
199	C₄	C₃₁×#357	1	1	yes	characteristic
200	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
201	C₄	C₂×C₃₁×D₁₆	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	—	16	4	no	—
205	C₂	C₃₁×#1419	1	2	yes	direct product factor
206	C₂	C₂×C₃₁×#357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#357
commutator	C₁₆	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ C₂²×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×M₆₄	C₂	semidirect
3. C₂×C₃₁×#368	C₂	semidirect
4. C₃₁×#1419	C₂	direct (N × H)
15. C₃₁×M₆₄	C₂²	semidirect
18. C₃₁×#368	C₂²	semidirect
66. C₂×#1419	C₃₁	direct (N × H)
88. C₂×M₆₄	C₂×C₃₁	semidirect
89. C₂×#368	C₂×C₃₁	semidirect
90. #1419	C₂×C₃₁	direct (N × H)
113. M₆₄	C₂²×C₃₁	semidirect
116. #368	C₂²×C₃₁	semidirect
120. C₂×C₃₁	#1419	direct (N × H)
144. C₃₁	C₂×#1419	direct (N × H)
205. C₂	C₃₁×#1419	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×D₁₆	2
C₂×C₃₁×M₆₄	2
C₂×C₃₁×#368	2
C₃₁×#1419	2
C₂×#1419	31

Subgroups — C2×C31×#1419 #21470382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1419 #21470382