Subgroups — C₂×#14521007 #16854126

Order 3968.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758503	C₂	1	1	yes	maximal, characteristic
3	#14521007	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11758403	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×Dic₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₃₁	C₄	1	8	yes	complemented
9	C₂×M₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₃₁	C₄	1	2	yes	—
12	#11758503	C₄	1	4	yes	—
13	#11758503	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₃₁	C₄	1	2	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
16	C₂²×C₃₁⋊C₈	C₄	1	8	yes	complemented
17	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
20	#11758403	C₂²	1	8	yes	complemented
21	C₈×Dic₃₁	C₂²	1	8	yes	complemented
22	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₁	C₂×C₄	1	4	yes	—
30	M₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
31	C₂×C₈×C₃₁	C₂×C₄	1	8	yes	—
32	C₂×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂²⋊Dic₃₁	C₂×C₄	1	8	yes	—
34	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	—
35	C₄×Dic₃₁	C₂×C₄	1	4	yes	—
36	C₄×Dic₃₁	C₂³	1	4	yes	—
37	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
39	C₂×C₃₁⋊C₈	C₂×C₄	1	16	yes	complemented
40	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
41	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
42	C₂³×C₃₁	—	2	4	no	—
43	C₂³×C₃₁	C₄²	1	4	yes	—
44	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₄²	1	8	yes	—
47	C₂×C₄×C₃₁	C₄²	1	4	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₃₁	—	2	8	no	—
56	C₂×Dic₃₁	C₂²×C₄	1	8	yes	—
57	C₈×C₃₁	C₂²×C₄	1	8	yes	—
58	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
59	C₃₁⋊C₈	C₂²×C₄	1	4	yes	—
60	C₂×#405	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	—	2	4	no	—
62	C₂²×C₃₁	—	2	8	no	—
63	C₂²×C₃₁	C₂×C₄²	1	4	yes	—
64	C₂²×C₃₁	C₂×C₄²	1	2	yes	—
65	C₂²×C₃₁	#131	1	4	yes	—
66	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₃₁	C₂×C₄²	1	2	yes	—
68	C₄×C₃₁	C₂×C₄²	1	4	yes	—
69	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
71	Dic₃₁	—	2	8	no	—
72	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
73	C₂³×C₈	—	31	1	no	—
74	C₂×#117	—	31	1	no	—
75	#405	—	31	8	no	—
76	C₂×#97	—	31	2	no	—
77	C₂×C₄×C₈	—	31	2	no	—
78	C₂×C₃₁	—	2	4	no	—
79	C₂×C₃₁	#405	1	4	yes	complemented
80	C₂×C₃₁	C₂²×C₄²	1	1	yes	characteristic
81	C₂×C₃₁	C₂×#131	1	2	yes	—
82	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
83	C₂×M₁₆	Dic₃₁	1	8	yes	complemented
84	C₂×M₁₆	C₂×D₃₁	1	4	yes	—
85	C₂²×C₈	—	31	8	no	—
86	C₂²×C₈	—	31	4	no	—
87	C₂²×C₈	—	31	1	no	—
88	C₂²×C₈	—	31	1	no	—
89	C₂²×C₈	C₂×D₃₁	1	2	yes	—
90	#117	—	31	4	no	—
91	#117	—	31	4	no	—
92	C₂×C₄⋊C₄	—	31	2	no	—
93	C₂×C₂²⋊C₄	—	31	2	no	—
94	C₂×C₄²	—	31	2	no	—
95	#97	—	31	8	no	—
96	C₄×C₈	—	31	8	no	—
97	C₃₁	C₂×#405	1	1	yes	characteristic, complemented, Sylow
98	C₂⁴	C₄×D₃₁	1	1	yes	characteristic
99	C₂²×C₄	—	31	4	no	—
100	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
101	C₂²×C₄	C₄×D₃₁	1	2	yes	—
102	C₂²×C₄	C₄×D₃₁	1	4	yes	—
103	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
104	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
105	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
106	M₁₆	C₂×Dic₃₁	1	16	yes	complemented
107	C₂×C₈	—	31	4	no	—
108	C₂×C₈	—	31	4	no	—
109	C₂×C₈	—	31	16	no	—
110	C₂×C₈	—	31	2	no	—
111	C₂×C₈	—	31	2	no	—
112	C₂×C₈	C₂×Dic₃₁	1	8	yes	—
113	C₂×C₈	C₂²×D₃₁	1	4	yes	—
114	C₄⋊C₄	—	31	8	no	—
115	C₂²⋊C₄	—	31	8	no	—
116	C₄²	—	31	4	no	—
117	C₄²	—	31	4	no	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄×Dic₃₁	1	4	yes	—
120	C₂³	C₂×C₄×D₃₁	1	2	yes	—
121	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
122	C₂×C₄	—	31	8	no	—
123	C₂×C₄	—	62	8	no	—
124	C₂×C₄	C₄×Dic₃₁	1	8	yes	—
125	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	—
129	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
130	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
131	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
132	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
133	C₈	—	31	4	no	—
134	C₈	—	31	4	no	—
135	C₈	C₂²×Dic₃₁	1	8	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	4	no	—
138	C₂²	#11758484	1	4	yes	—
139	C₂²	C₂×C₄×Dic₃₁	1	4	yes	—
140	C₂²	C₂×C₄×Dic₃₁	1	2	yes	—
141	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
142	C₄	—	62	8	no	—
143	C₄	C₂×C₄×Dic₃₁	1	2	yes	—
144	C₄	C₂×C₄×Dic₃₁	1	4	yes	—
145	C₄	C₂²×C₄×D₃₁	1	1	yes	characteristic
146	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#14521007	1	4	yes	direct product factor
149	C₂	C₂²×C₄×Dic₃₁	1	1	yes	characteristic
150	C₂	C₂×#11758484	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	C₂×#405
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14521007	C₂	direct (N × H)
4. C₂×#11758403	C₂	semidirect
5. C₂×C₈×Dic₃₁	C₂	semidirect
8. C₂×M₁₆×C₃₁	C₄	semidirect
16. C₂²×C₃₁⋊C₈	C₄	semidirect
20. #11758403	C₂²	semidirect
21. C₈×Dic₃₁	C₂²	semidirect
30. M₁₆×C₃₁	C₂×C₄	semidirect
39. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
79. C₂×C₃₁	#405	semidirect
83. C₂×M₁₆	Dic₃₁	semidirect
97. C₃₁	C₂×#405	semidirect
106. M₁₆	C₂×Dic₃₁	semidirect
148. C₂	#14521007	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×C₃₁⋊C₈	2
C₂×#11758503	2
#14521007	2
C₂×#11758403	2
C₂×C₈×Dic₃₁	2
C₂²×M₁₆×C₃₁	2
C₂×#405	31

Subgroups — C2×#14521007 #16854126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521007 #16854126