Subgroups — C₂×#14521102 #16854327

Order 3968.

528 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521102	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal
4	C₂×C₄×C₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₄	1	4	yes	—
11	C₂×C₄²×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	4	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
16	C₂²⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
17	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	complemented
18	C₄×C₃₁⋊C₈	C₂²	1	8	yes	complemented
19	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	—	2	4	no	—
21	C₂²×C₄×C₃₁	—	2	4	no	—
22	C₂²×C₄×C₃₁	C₈	1	8	yes	complemented
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	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₄²×C₃₁	C₂×C₄	1	4	yes	—
34	C₄²×C₃₁	C₂×C₄	1	4	yes	—
35	C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₄²×C₃₁	C₂³	1	2	yes	—
38	C₄²×C₃₁	C₂³	1	2	yes	—
39	C₂×C₃₁⋊C₈	—	2	8	no	—
40	C₂×C₃₁⋊C₈	C₂³	1	8	yes	—
41	C₂³×C₃₁	—	2	2	no	—
42	C₂³×C₃₁	—	2	2	no	—
43	C₂³×C₃₁	C₂×C₈	1	4	yes	—
44	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	—	2	4	no	—
49	C₂×C₄×C₃₁	—	2	8	no	—
50	C₂×C₄×C₃₁	—	2	4	no	—
51	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
52	C₂×C₄×C₃₁	C₂×C₈	1	4	yes	—
53	C₂×C₄×C₃₁	C₂×C₈	1	16	yes	complemented
54	C₂×C₄×C₃₁	M₁₆	1	4	yes	—
55	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₃₁	#41	1	4	yes	—
62	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₃₁⋊C₈	—	2	8	no	—
64	C₂×#431	—	31	1	no	maximal, Sylow
65	C₂²×C₃₁	—	2	2	no	—
66	C₂²×C₃₁	—	2	2	no	—
67	C₂²×C₃₁	—	2	8	no	—
68	C₂²×C₃₁	#117	1	2	yes	—
69	C₂²×C₃₁	C₂²×C₈	1	4	yes	—
70	C₂²×C₃₁	C₂²×C₈	1	2	yes	—
71	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
72	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₃₁	—	2	2	no	—
74	C₄×C₃₁	—	2	2	no	—
75	C₄×C₃₁	#117	1	2	yes	—
76	C₄×C₃₁	C₂²×C₈	1	4	yes	—
77	C₄×C₃₁	C₂²×C₈	1	4	yes	—
78	C₄×C₃₁	C₂×M₁₆	1	4	yes	—
79	C₄×C₃₁	C₂×#41	1	2	yes	—
80	C₂²×C₄²	D₃₁	1	1	yes	characteristic
81	#431	—	31	8	no	—
82	C₂×C₄⋊C₈	—	31	2	no	—
83	C₂×C₂²⋊C₈	—	31	2	no	—
84	C₂×C₄×C₈	—	31	2	no	—
85	C₂×C₃₁	—	2	4	no	—
86	C₂×C₃₁	#431	1	4	yes	complemented
87	C₂×C₃₁	C₂×#117	1	1	yes	characteristic
88	C₂×C₃₁	C₂³×C₈	1	1	yes	characteristic
89	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
90	C₂³×C₄	Dic₃₁	1	1	yes	characteristic
91	C₂³×C₄	Dic₃₁	1	1	yes	characteristic
92	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
93	C₂²×C₈	—	31	4	no	—
94	C₂×C₄²	Dic₃₁	1	1	yes	characteristic
95	C₂×C₄²	Dic₃₁	1	1	yes	characteristic
96	C₂×C₄²	Dic₃₁	1	4	yes	—
97	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₃₁	1	4	yes	—
99	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
100	C₄⋊C₈	—	31	8	no	—
101	C₂²⋊C₈	—	31	8	no	—
102	C₄×C₈	—	31	8	no	—
103	C₃₁	C₂×#431	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	C₃₁⋊C₈	1	8	yes	complemented
108	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
110	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
112	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
115	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
117	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
118	C₂×C₈	—	31	8	no	—
119	C₂×C₈	—	62	8	no	—
120	C₄²	C₂×Dic₃₁	1	4	yes	—
121	C₄²	C₂×Dic₃₁	1	4	yes	—
122	C₄²	C₂×Dic₃₁	1	2	yes	—
123	C₄²	C₂×Dic₃₁	1	2	yes	—
124	C₄²	C₂²×D₃₁	1	2	yes	—
125	C₄²	C₂²×D₃₁	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	C₂×C₃₁⋊C₈	1	4	yes	—
129	C₂³	C₂²×Dic₃₁	1	2	yes	—
130	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
137	C₂×C₄	C₂×C₃₁⋊C₈	1	4	yes	—
138	C₂×C₄	C₂×C₃₁⋊C₈	1	16	yes	complemented
139	C₂×C₄	#92474	1	4	yes	—
140	C₂×C₄	#92495	1	4	yes	—
141	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
142	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
143	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
144	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
145	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
146	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
147	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
148	C₈	—	62	8	no	—
149	C₂²	—	2	8	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	C₂²×C₃₁⋊C₈	1	4	yes	—
153	C₂²	C₂²×C₃₁⋊C₈	1	2	yes	—
154	C₂²	C₂×#92474	1	2	yes	—
155	C₂²	#11758503	1	2	yes	—
156	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	C₂²×C₃₁⋊C₈	1	4	yes	—
160	C₄	C₂²×C₃₁⋊C₈	1	4	yes	—
161	C₄	C₂×#92474	1	4	yes	—
162	C₄	#11758503	1	2	yes	—
163	C₄	C₂×#92495	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	#14521102	1	4	yes	direct product factor
166	C₂	C₂×#11758503	1	1	yes	characteristic
167	C₂	C₂³×C₃₁⋊C₈	1	1	yes	characteristic
168	C₂	C₂²×#92474	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₃₁
commutator	C₂×C₃₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	C₂×#431
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. #14521102	C₂	direct (N × H)
2. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
4. C₂×C₄×C₃₁⋊C₈	C₂	semidirect
17. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
18. C₄×C₃₁⋊C₈	C₂²	semidirect
22. C₂²×C₄×C₃₁	C₈	semidirect
53. C₂×C₄×C₃₁	C₂×C₈	semidirect
86. C₂×C₃₁	#431	semidirect
103. C₃₁	C₂×#431	semidirect
107. C₂²×C₄	C₃₁⋊C₈	semidirect
138. C₂×C₄	C₂×C₃₁⋊C₈	semidirect
165. C₂	#14521102	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521102	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×C₂²⋊(C₃₁⋊C₈)	2
C₂×C₄×C₃₁⋊C₈	2
C₂²×C₄²×C₃₁	2
C₂×#431	31

Subgroups — C2×#14521102 #16854327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521102 #16854327