Subgroups — C₂×C₃₁×H(3,ℤ/4) #16841663

Order 3968.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂×C₄×C₃₁	C₄²
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#95
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×H(3,ℤ/4)	C₂	direct (N × H)
7. C₂×M₁₆×C₃₁	C₄	semidirect
9. C₂²×C₈×C₃₁	C₄	semidirect
12. C₃₁×#117	C₄	semidirect
16. C₂×C₄²×C₃₁	C₄	semidirect
33. C₂×C₈×C₃₁	C₂×C₄	semidirect
39. C₄²×C₃₁	C₂×C₄	semidirect
67. C₂×H(3,ℤ/4)	C₃₁	direct (N × H)
87. H(3,ℤ/4)	C₂×C₃₁	direct (N × H)
92. C₂×C₃₁	H(3,ℤ/4)	direct (N × H)
97. C₂×M₁₆	C₄×C₃₁	semidirect
99. C₂²×C₈	C₄×C₃₁	semidirect
102. #117	C₄×C₃₁	semidirect
106. C₂×C₄²	C₄×C₃₁	semidirect
107. C₃₁	C₂×H(3,ℤ/4)	direct (N × H)
124. C₂×C₈	C₂×C₄×C₃₁	semidirect
130. C₄²	C₂×C₄×C₃₁	semidirect
179. C₂	C₃₁×H(3,ℤ/4)	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×M₁₆×C₃₁	2
C₂×C₃₁×#117	2
C₃₁×H(3,ℤ/4)	2
C₂×H(3,ℤ/4)	31

Subgroups — C2×C31×H(3,ℤ/4) #16841663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×H(3,ℤ/4) #16841663