Subgroups — C₂×#14521526 #16858901

Order 3968.

1488 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	#14521526	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₆₂	C₂	1	4	yes	maximal, complemented
4	C₂×#11758464	C₂	1	4	yes	maximal, complemented
5	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758454	C₂	1	4	yes	maximal
7	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×Dic₃₁	C₂²	1	2	yes	—
10	C₂²×Dic₆₂	C₄	1	16	yes	complemented
11	C₂²×Dic₆₂	C₂²	1	2	yes	—
12	C₂²×Dic₆₂	C₂²	1	2	yes	—
13	C₂²×Dic₆₂	C₂²	1	8	yes	—
14	C₃₁×#117	C₂²	1	8	yes	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₃₁	C₂²	1	2	yes	—
19	#11758503	C₂²	1	8	yes	—
20	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
21	C₂×Dic₃₁⋊C₄	C₂²	1	8	yes	—
22	C₂×C₄×Dic₃₁	C₂²	1	8	yes	—
23	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
24	#11758464	C₂²	1	32	yes	complemented
25	#11758454	C₂²	1	32	yes	—
26	C₄×Dic₆₂	C₂²	1	32	yes	complemented
27	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	—
29	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
33	C₂²×Dic₃₁	C₂×C₄	1	16	yes	—
34	C₂²×Dic₃₁	C₂³	1	4	yes	—
35	C₂²×Dic₃₁	C₂³	1	8	yes	—
36	C₂²×Dic₃₁	C₂³	1	4	yes	—
37	C₂×Dic₆₂	C₂×C₄	1	16	yes	—
38	C₂×Dic₆₂	C₂×C₄	1	16	yes	—
39	C₂×Dic₆₂	C₂×C₄	1	64	yes	complemented
40	C₂×Dic₆₂	C₂³	1	8	yes	—
41	C₂×Dic₆₂	C₂³	1	8	yes	—
42	C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
43	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
44	C₄²×C₃₁	C₂³	1	8	yes	—
45	C₂²⋊Dic₃₁	C₂³	1	8	yes	—
46	C₄⋊Dic₃₁	C₂³	1	8	yes	—
47	Dic₃₁⋊C₄	C₂³	1	32	yes	—
48	C₄×Dic₃₁	C₂³	1	32	yes	—
49	C₄×Dic₃₁	C₂³	1	8	yes	—
50	C₂³×C₃₁	—	2	4	no	—
51	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂³×C₃₁	C₂⁴	1	2	yes	—
53	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₃₁	—	2	8	no	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	16	yes	—
58	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
59	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
60	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
61	C₂×Dic₃₁	—	2	8	no	—
62	C₂×Dic₃₁	C₂²×C₄	1	16	yes	—
63	C₂×Dic₃₁	C₂²×C₄	1	32	yes	—
64	C₂×Dic₃₁	C₂⁴	1	8	yes	—
65	C₂×Dic₃₁	C₂⁴	1	8	yes	—
66	Dic₆₂	C₂²×C₄	1	32	yes	—
67	Dic₆₂	C₂²×C₄	1	32	yes	—
68	C₂×#519	—	31	1	no	maximal, Sylow
69	C₂²×C₃₁	—	2	8	no	—
70	C₂²×C₃₁	—	2	4	no	—
71	C₂²×C₃₁	C₂³×C₄	1	4	yes	—
72	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
73	C₂²×C₃₁	2₋¹⁺⁴	1	4	yes	—
74	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
75	C₄×C₃₁	—	2	8	no	—
76	C₄×C₃₁	C₂³×C₄	1	4	yes	—
77	C₄×C₃₁	C₂³×C₄	1	4	yes	—
78	Dic₃₁	—	2	8	no	—
79	Dic₃₁	C₂³×C₄	1	16	yes	—
80	C₂³×Q₈	—	31	1	no	—
81	#519	—	31	16	no	—
82	C₂×C₄×Q₈	—	31	4	no	—
83	C₂×C₄×Q₈	—	31	4	no	—
84	C₂×#117	—	31	1	no	—
85	C₂×#117	—	31	4	no	—
86	C₂×#117	D₃₁	1	1	yes	characteristic
87	C₂×C₃₁	—	2	4	no	—
88	C₂×C₃₁	#519	1	4	yes	complemented
89	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
90	C₂×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
91	C₂²×Q₈	—	31	16	no	—
92	C₂²×Q₈	—	31	2	no	—
93	C₂²×Q₈	—	31	2	no	—
94	C₂²×Q₈	—	31	8	no	—
95	C₂³×C₄	—	31	2	no	—
96	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
97	C₄×Q₈	—	31	32	no	—
98	C₄×Q₈	—	31	32	no	—
99	#117	—	31	8	no	—
100	#117	—	31	32	no	—
101	#117	C₂×D₃₁	1	8	yes	—
102	C₂×C₄⋊C₄	—	31	8	no	—
103	C₂×C₄⋊C₄	—	31	2	no	—
104	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
105	C₂×C₂²⋊C₄	—	31	2	no	—
106	C₂×C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
107	C₂×C₄²	—	31	8	no	—
108	C₂×C₄²	—	31	2	no	—
109	C₂×C₄²	C₂×D₃₁	1	2	yes	—
110	C₃₁	C₂×#519	1	1	yes	characteristic, complemented, Sylow
111	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
112	C₂×Q₈	—	31	16	no	—
113	C₂×Q₈	—	31	16	no	—
114	C₂×Q₈	—	31	64	no	—
115	C₂×Q₈	—	31	8	no	—
116	C₂×Q₈	—	31	8	no	—
117	C₂²×C₄	—	31	16	no	—
118	C₂²×C₄	—	31	8	no	—
119	C₂²×C₄	—	31	4	no	—
120	C₂²×C₄	—	31	4	no	—
121	C₂²×C₄	C₄×D₃₁	1	8	yes	—
122	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
123	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
124	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	C₄⋊C₄	—	31	32	no	—
127	C₄⋊C₄	—	31	8	no	—
128	C₄⋊C₄	C₂²×D₃₁	1	8	yes	—
129	C₂²⋊C₄	—	31	8	no	—
130	C₂²⋊C₄	C₂²×D₃₁	1	8	yes	—
131	C₄²	—	31	32	no	—
132	C₄²	—	31	8	no	—
133	C₄²	C₂²×D₃₁	1	8	yes	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂×C₄×D₃₁	1	4	yes	—
136	C₂³	C₂³×D₃₁	1	2	yes	—
137	C₂³	C₂³×D₃₁	1	1	yes	characteristic
138	Q₈	—	31	32	no	—
139	Q₈	—	31	32	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	31	32	no	—
142	C₂×C₄	—	31	16	no	—
143	C₂×C₄	—	31	8	no	—
144	C₂×C₄	—	31	8	no	—
145	C₂×C₄	—	62	8	no	—
146	C₂×C₄	C₂×C₄×D₃₁	1	16	yes	—
147	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
149	C₂×C₄	C₂³×D₃₁	1	8	yes	—
150	C₂×C₄	C₂³×D₃₁	1	2	yes	—
151	C₂×C₄	C₂³×D₃₁	1	2	yes	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₂²×C₄×D₃₁	1	4	yes	—
155	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
156	C₂²	#11758567	1	4	yes	—
157	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
158	C₄	—	2	8	no	—
159	C₄	—	31	16	no	—
160	C₄	—	62	8	no	—
161	C₄	C₂²×C₄×D₃₁	1	4	yes	—
162	C₄	C₂²×C₄×D₃₁	1	4	yes	—
163	C₂	—	2	4	no	—
164	C₂	#14521526	1	4	yes	direct product factor
165	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
166	C₂	C₂×#11758567	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
cosocle	C₂²×C₃₁	C₂⁵
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	C₂×#519
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #14521526	C₂	direct (N × H)
3. C₂×C₄×Dic₆₂	C₂	semidirect
4. C₂×#11758464	C₂	semidirect
10. C₂²×Dic₆₂	C₄	semidirect
24. #11758464	C₂²	semidirect
26. C₄×Dic₆₂	C₂²	semidirect
39. C₂×Dic₆₂	C₂×C₄	semidirect
88. C₂×C₃₁	#519	semidirect
110. C₃₁	C₂×#519	semidirect
164. C₂	#14521526	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×Dic₆₂	2
#14521526	2
C₂×C₄×Dic₆₂	2
C₂×#11758464	2
C₂×#11758503	2
C₂×#11758454	2
C₂×C₃₁×#117	2
C₂×#519	31

Subgroups — C2×#14521526 #16858901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521526 #16858901