Subgroups — C₂×#14521013 #16854147

Order 3968.

1008 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#92474	C₂	1	2	yes	maximal, complemented
2	#14521013	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊(C₃₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁵×C₃₁	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₃₁	—	2	8	no	—
7	C₂³×C₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₃₁	C₄	1	4	yes	—
9	C₂³×C₄×C₃₁	C₄	1	2	yes	—
10	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₄×C₃₁	C₂²	1	4	yes	—
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×#92474	—	2	8	no	—
14	C₂×#92474	C₂²	1	8	yes	complemented
15	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	complemented
16	C₂²⋊(C₃₁⋊C₈)	C₂²	1	16	yes	complemented
17	C₂⁴×C₃₁	—	2	4	no	—
18	C₂⁴×C₃₁	—	2	8	no	—
19	C₂⁴×C₃₁	C₂×C₄	1	4	yes	—
20	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	—	2	4	no	—
23	C₂²×C₄×C₃₁	—	2	16	no	—
24	C₂²×C₄×C₃₁	—	2	8	no	—
25	C₂²×C₄×C₃₁	—	2	8	no	—
26	C₂²×C₄×C₃₁	—	2	16	no	—
27	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	D₄	1	8	yes	—
34	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	#92474	—	2	16	no	—
38	C₂×C₃₁⋊C₈	—	2	8	no	—
39	C₂×C₃₁⋊C₈	C₂³	1	8	yes	complemented
40	C₂³×C₃₁	—	2	32	no	—
41	C₂³×C₃₁	—	2	8	no	—
42	C₂³×C₃₁	—	2	4	no	—
43	C₂³×C₃₁	—	2	4	no	—
44	C₂³×C₃₁	—	2	4	no	—
45	C₂³×C₃₁	—	2	16	no	—
46	C₂³×C₃₁	C₂²⋊C₄	1	4	yes	—
47	C₂³×C₃₁	M₁₆	1	8	yes	complemented
48	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂³×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	—	2	8	no	—
52	C₂×C₄×C₃₁	—	2	4	no	—
53	C₂×C₄×C₃₁	—	2	8	no	—
54	C₂×C₄×C₃₁	—	2	16	no	—
55	C₂×C₄×C₃₁	—	2	8	no	—
56	C₂×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
57	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₃₁	C₂×D₄	1	8	yes	—
63	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₃₁⋊C₈	—	2	8	no	—
65	C₂×#407	—	31	1	no	maximal, Sylow
66	C₂²×C₃₁	—	2	32	no	—
67	C₂²×C₃₁	—	2	4	no	—
68	C₂²×C₃₁	—	2	4	no	—
69	C₂²×C₃₁	—	2	4	no	—
70	C₂²×C₃₁	—	2	8	no	—
71	C₂²×C₃₁	—	2	16	no	—
72	C₂²×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂²×C₃₁	C₂×M₁₆	1	4	yes	—
75	C₂²×C₃₁	C₂×M₁₆	1	8	yes	complemented
76	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₃₁	—	2	4	no	—
78	C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
79	C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
80	C₄×C₃₁	C₂²×D₄	1	2	yes	—
81	C₂⁴×C₄	D₃₁	1	1	yes	characteristic
82	C₂²×M₁₆	—	31	2	no	—
83	#407	—	31	8	no	—
84	C₂×C₂²⋊C₈	—	31	4	no	—
85	C₂×C₃₁	—	2	4	no	—
86	C₂×C₃₁	—	2	8	no	—
87	C₂×C₃₁	#407	1	4	yes	complemented
88	C₂×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₃₁	C₂²×M₁₆	1	2	yes	—
90	C₂⁵	Dic₃₁	1	1	yes	characteristic
91	C₂³×C₄	—	2	8	no	—
92	C₂³×C₄	Dic₃₁	1	1	yes	characteristic
93	C₂³×C₄	Dic₃₁	1	4	yes	—
94	C₂³×C₄	Dic₃₁	1	2	yes	—
95	C₂³×C₄	C₂×D₃₁	1	4	yes	—
96	C₂³×C₄	C₂×D₃₁	1	2	yes	—
97	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
98	C₂×M₁₆	—	31	8	no	—
99	C₂×M₁₆	—	62	8	no	—
100	C₂²×C₈	—	31	4	no	—
101	C₂²⋊C₈	—	31	16	no	—
102	C₃₁	C₂×#407	1	1	yes	characteristic, complemented, Sylow
103	C₂⁴	—	2	8	no	—
104	C₂⁴	—	2	4	no	—
105	C₂⁴	C₂×Dic₃₁	1	4	yes	—
106	C₂⁴	C₂×Dic₃₁	1	2	yes	—
107	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	16	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	16	no	—
113	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
114	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
115	C₂²×C₄	C₂×Dic₃₁	1	8	yes	—
116	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
117	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
118	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
119	C₂²×C₄	C₂²⋊D₃₁	1	8	yes	—
120	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
121	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
122	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	M₁₆	—	62	16	no	—
124	C₂×C₈	—	31	8	no	—
125	C₂×C₈	—	62	8	no	—
126	C₂³	—	2	32	no	—
127	C₂³	—	2	4	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	16	no	—
130	C₂³	—	2	8	no	—
131	C₂³	—	2	4	no	—
132	C₂³	#92474	1	8	yes	complemented
133	C₂³	C₂²⋊Dic₃₁	1	4	yes	—
134	C₂³	C₂²×Dic₃₁	1	4	yes	—
135	C₂³	C₂²×Dic₃₁	1	2	yes	—
136	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	16	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	C₂²⋊Dic₃₁	1	8	yes	—
143	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
144	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
145	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
146	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
147	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₃₁	1	8	yes	—
149	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
150	C₈	—	62	8	no	—
151	C₂²	—	2	32	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	16	no	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	4	no	—
157	C₂²	C₂×#92474	1	4	yes	—
158	C₂²	C₂×#92474	1	8	yes	complemented
159	C₂²	C₂×C₂²⋊Dic₃₁	1	4	yes	—
160	C₂²	C₂×C₂²⋊Dic₃₁	1	2	yes	—
161	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	C₂×C₂²⋊Dic₃₁	1	4	yes	—
164	C₄	C₂×C₂²⋊Dic₃₁	1	2	yes	—
165	C₄	C₂²×C₂²⋊D₃₁	1	2	yes	—
166	C₂	—	2	8	no	—
167	C₂	—	2	4	no	—
168	C₂	#14521013	1	4	yes	direct product factor
169	C₂	C₂²×C₂²⋊Dic₃₁	1	1	yes	characteristic
170	C₂	C₂²×#92474	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂⁴×C₄×C₃₁	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁵×C₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂⁴×C₄	D₃₁
hypocenter	C₃₁	C₂×#407
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#92474	C₂	semidirect
2. #14521013	C₂	direct (N × H)
3. C₂×C₂²⋊(C₃₁⋊C₈)	C₂	semidirect
14. C₂×#92474	C₂²	semidirect
15. C₂²×C₃₁⋊C₈	C₂²	semidirect
16. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
39. C₂×C₃₁⋊C₈	C₂³	semidirect
47. C₂³×C₃₁	M₁₆	semidirect
75. C₂²×C₃₁	C₂×M₁₆	semidirect
87. C₂×C₃₁	#407	semidirect
102. C₃₁	C₂×#407	semidirect
132. C₂³	#92474	semidirect
158. C₂²	C₂×#92474	semidirect
168. C₂	#14521013	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×#92474	2
#14521013	2
C₂×C₂²⋊(C₃₁⋊C₈)	2
C₂⁴×C₄×C₃₁	2
C₂×#407	31

Subgroups — C2×#14521013 #16854147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521013 #16854147