Subgroups — C₂×#14521101 #16854326

Order 3968.

528 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521101	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₃₁)⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
8	C₂²×C₄×D₃₁	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₄²×D₃₁	C₄	1	4	yes	—
17	C₄²×D₃₁	C₂²	1	4	yes	—
18	C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
19	(C₂×D₃₁)⋊C₈	C₂²	1	8	yes	—
20	C₈×Dic₃₁	C₂²	1	8	yes	complemented
21	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	complemented
22	C₂³×D₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₃₁	—	2	2	no	—
30	C₂×C₄×D₃₁	—	2	2	no	—
31	C₂×C₄×D₃₁	—	2	4	no	—
32	C₂×C₄×D₃₁	C₈	1	8	yes	complemented
33	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄×D₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄×D₃₁	C₂³	1	2	yes	—
37	C₂×C₈×C₃₁	—	2	4	no	—
38	C₂×C₈×C₃₁	C₂³	1	4	yes	—
39	C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₄²×C₃₁	C₂³	1	2	yes	—
41	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
42	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
43	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
44	C₄×Dic₃₁	C₂³	1	2	yes	—
45	C₂×C₃₁⋊C₈	—	2	4	no	—
46	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
47	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×D₃₁	—	2	2	no	—
49	C₂²×D₃₁	—	2	2	no	—
50	C₂²×D₃₁	C₂×C₈	1	4	yes	—
51	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
54	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	#41	1	2	yes	—
58	C₂×C₄×C₃₁	#41	1	2	yes	—
59	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₃₁	—	2	2	no	—
61	C₂×Dic₃₁	C₂×C₈	1	4	yes	—
62	C₂×Dic₃₁	M₁₆	1	4	yes	—
63	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
64	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
66	C₄×D₃₁	—	2	4	no	—
67	C₄×D₃₁	—	2	2	no	—
68	C₄×D₃₁	—	2	2	no	—
69	C₄×D₃₁	—	2	8	no	—
70	C₄×D₃₁	C₂×C₈	1	16	yes	complemented
71	C₈×C₃₁	—	2	4	no	—
72	C₃₁⋊C₈	—	2	4	no	—
73	C₂×#431	—	31	1	no	maximal, Sylow
74	C₂²×C₃₁	#117	1	2	yes	—
75	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
76	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
77	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
78	C₂×D₃₁	—	2	2	no	—
79	C₂×D₃₁	—	2	2	no	—
80	C₂×D₃₁	—	2	8	no	—
81	C₂×D₃₁	C₂²×C₈	1	4	yes	—
82	C₄×C₃₁	—	2	2	no	—
83	C₄×C₃₁	#117	1	2	yes	—
84	C₄×C₃₁	C₂²×C₈	1	4	yes	—
85	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
86	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
87	Dic₃₁	—	2	2	no	—
88	Dic₃₁	C₂²×C₈	1	4	yes	—
89	Dic₃₁	C₂×M₁₆	1	4	yes	—
90	C₂²×C₄²	—	31	1	no	—
91	#431	—	31	8	no	—
92	C₂×C₄⋊C₈	—	31	1	no	—
93	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic, complemented
94	C₂×C₂²⋊C₈	—	31	2	no	—
95	C₂×C₄×C₈	—	31	2	no	—
96	C₂×C₃₁	#431	1	4	yes	complemented
97	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
98	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
99	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
100	D₃₁	—	2	4	no	—
101	C₂³×C₄	—	31	1	no	—
102	C₂³×C₄	—	31	1	no	—
103	C₂³×C₄	—	31	1	no	—
104	C₂²×C₈	—	31	2	no	—
105	C₂²×C₈	C₂×D₃₁	1	2	yes	—
106	C₂×C₄²	—	31	1	no	—
107	C₂×C₄²	—	31	1	no	—
108	C₂×C₄²	—	31	1	no	—
109	C₂×C₄²	—	31	4	no	—
110	C₂×C₄²	—	31	4	no	—
111	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
112	C₄⋊C₈	—	31	4	no	—
113	C₄⋊C₈	C₂×D₃₁	1	4	yes	complemented
114	C₂²⋊C₈	—	31	8	no	—
115	C₄×C₈	—	31	8	no	—
116	C₃₁	C₂×#431	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴	—	31	1	no	—
118	C₂²×C₄	—	31	1	no	—
119	C₂²×C₄	—	31	1	no	—
120	C₂²×C₄	—	31	8	no	—
121	C₂²×C₄	—	31	4	no	—
122	C₂²×C₄	—	31	1	no	—
123	C₂²×C₄	—	31	4	no	—
124	C₂²×C₄	—	31	2	no	—
125	C₂²×C₄	—	31	2	no	—
126	C₂²×C₄	—	62	4	no	—
127	C₂²×C₄	—	62	2	no	—
128	C₂²×C₄	—	62	2	no	—
129	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
130	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	31	4	no	—
134	C₂×C₈	—	62	4	no	—
135	C₂×C₈	C₂²×D₃₁	1	4	yes	—
136	C₄²	—	31	2	no	—
137	C₄²	—	31	4	no	—
138	C₄²	—	31	4	no	—
139	C₄²	—	31	2	no	—
140	C₄²	C₄×D₃₁	1	2	yes	—
141	C₄²	C₂²×D₃₁	1	2	yes	—
142	C₂³	—	31	4	no	—
143	C₂³	—	31	2	no	—
144	C₂³	—	62	2	no	—
145	C₂³	—	62	2	no	—
146	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	31	4	no	—
149	C₂×C₄	—	31	4	no	—
150	C₂×C₄	—	31	2	no	—
151	C₂×C₄	—	31	2	no	—
152	C₂×C₄	—	31	16	no	—
153	C₂×C₄	—	31	2	no	—
154	C₂×C₄	—	62	8	no	—
155	C₂×C₄	—	62	2	no	—
156	C₂×C₄	—	62	4	no	—
157	C₂×C₄	—	62	2	no	—
158	C₂×C₄	—	62	2	no	—
159	C₂×C₄	C₈×D₃₁	1	4	yes	—
160	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
163	C₂×C₄	#92497	1	2	yes	—
164	C₂×C₄	#92499	1	2	yes	—
165	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	62	4	no	—
168	C₂²	—	31	4	no	—
169	C₂²	—	62	8	no	—
170	C₂²	—	62	2	no	—
171	C₂²	—	62	2	no	—
172	C₂²	#11758469	1	2	yes	—
173	C₂²	C₂×C₈×D₃₁	1	2	yes	—
174	C₂²	M₁₆×D₃₁	1	2	yes	—
175	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	31	4	no	—
178	C₄	—	31	4	no	—
179	C₄	—	62	2	no	—
180	C₄	#11758469	1	2	yes	—
181	C₄	C₂×C₈×D₃₁	1	4	yes	—
182	C₄	C₂×#92497	1	1	yes	characteristic
183	C₄	C₂×#92499	1	1	yes	characteristic
184	C₂	—	62	4	no	—
185	C₂	#14521101	1	4	yes	direct product factor
186	C₂	C₂×#11758469	1	1	yes	characteristic
187	C₂	C₂²×C₈×D₃₁	1	1	yes	characteristic
188	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂³×D₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	C₂×#431
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14521101	C₂	direct (N × H)
3. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
5. C₂×C₈×Dic₃₁	C₂	semidirect
6. C₂×C₃₁×C₄⋊C₈	C₂	semidirect
18. C₃₁×C₄⋊C₈	C₂²	semidirect
20. C₈×Dic₃₁	C₂²	semidirect
21. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
32. C₂×C₄×D₃₁	C₈	semidirect
70. C₄×D₃₁	C₂×C₈	semidirect
93. C₂×C₄⋊C₈	D₃₁	semidirect
96. C₂×C₃₁	#431	semidirect
113. C₄⋊C₈	C₂×D₃₁	semidirect
116. C₃₁	C₂×#431	semidirect
185. C₂	#14521101	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×C₄²×D₃₁	2
#14521101	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×(C₂×D₃₁)⋊C₈	2
C₂×C₈×Dic₃₁	2
C₂×C₃₁×C₄⋊C₈	2
C₂×#431	31

Subgroups — C2×#14521101 #16854326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521101 #16854326