Subgroups — C₂×#14521180 #16854639

Order 3968.

400 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521180	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	2	yes	maximal
3	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758391	C₂	1	2	yes	maximal
5	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₃₁	C₄	1	4	yes	—
9	C₄×Q₈×C₃₁	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃₁⋊C₈)	C₂²	1	8	yes	—
18	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
19	#11758391	C₂²	1	8	yes	—
20	C₄×C₃₁⋊C₈	C₂²	1	4	yes	—
21	C₂×Q₈×C₃₁	—	2	4	no	—
22	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
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	2	yes	—
26	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
29	C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₄²×C₃₁	C₂³	1	2	yes	—
33	C₄²×C₃₁	C₂³	1	4	yes	—
34	C₂×C₃₁⋊C₈	—	2	2	no	—
35	C₂×C₃₁⋊C₈	—	2	4	no	—
36	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
37	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
38	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
39	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
40	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	Q₈×C₃₁	—	2	8	no	—
42	C₂×C₄×C₃₁	—	2	4	no	—
43	C₂×C₄×C₃₁	—	2	2	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	M₁₆	1	4	yes	—
46	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₁	#41	1	2	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₃₁⋊C₈	—	2	2	no	—
55	C₃₁⋊C₈	—	2	4	no	—
56	C₃₁⋊C₈	C₂×Q₈	1	4	yes	complemented
57	C₂×#446	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	C₄×Q₈	1	2	yes	—
59	C₂²×C₃₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₃₁	#131	1	2	yes	—
61	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₃₁	—	2	4	no	—
63	C₄×C₃₁	—	2	2	no	—
64	C₄×C₃₁	—	2	4	no	—
65	C₄×C₃₁	C₄×Q₈	1	2	yes	—
66	C₄×C₃₁	C₂×M₁₆	1	4	yes	—
67	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₄×Q₈	D₃₁	1	1	yes	characteristic
70	#446	—	31	8	no	—
71	C₂×C₄⋊C₈	—	31	1	no	—
72	C₂×C₄⋊C₈	—	31	2	no	—
73	C₂×#97	—	31	2	no	—
74	C₂×C₄×C₈	—	31	1	no	—
75	C₂×C₃₁	#446	1	4	yes	complemented
76	C₂×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
77	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
78	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
79	C₂²×Q₈	Dic₃₁	1	1	yes	characteristic
80	C₂²×C₈	—	31	1	no	—
81	C₂²×C₈	—	31	1	no	—
82	C₂²×C₈	—	31	2	no	—
83	C₄×Q₈	Dic₃₁	1	4	yes	—
84	C₄×Q₈	C₂×D₃₁	1	4	yes	—
85	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
86	C₂×C₄⋊C₄	Dic₃₁	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₃₁	1	2	yes	—
89	C₄⋊C₈	—	31	4	no	—
90	C₄⋊C₈	—	31	8	no	—
91	#97	—	31	8	no	—
92	C₄×C₈	—	31	4	no	—
93	C₃₁	C₂×#446	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	2	4	no	—
95	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
96	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
98	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
99	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
101	C₂×C₈	—	31	4	no	—
102	C₂×C₈	—	31	2	no	—
103	C₂×C₈	—	31	2	no	—
104	C₂×C₈	—	31	4	no	—
105	C₂×C₈	—	62	2	no	—
106	C₂×C₈	—	62	4	no	—
107	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
108	C₄⋊C₄	C₂×Dic₃₁	1	4	yes	—
109	C₄²	C₂×Dic₃₁	1	4	yes	—
110	C₄²	C₂×Dic₃₁	1	2	yes	—
111	C₄²	C₂²×D₃₁	1	2	yes	—
112	C₄²	C₂²×D₃₁	1	4	yes	—
113	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
114	Q₈	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	#92474	1	4	yes	—
119	C₂×C₄	Q₈×D₃₁	1	2	yes	—
120	C₂×C₄	#92499	1	2	yes	—
121	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
122	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
123	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
124	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
125	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
127	C₈	—	31	4	no	—
128	C₈	—	62	2	no	—
129	C₈	—	62	4	no	—
130	C₂²	C₂×#92474	1	2	yes	—
131	C₂²	Q₈×Dic₃₁	1	2	yes	—
132	C₂²	#11758525	1	2	yes	—
133	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	4	no	—
137	C₄	C₂×#92474	1	4	yes	—
138	C₄	Q₈×Dic₃₁	1	2	yes	—
139	C₄	C₂×Q₈×D₃₁	1	1	yes	characteristic
140	C₄	C₂×#92499	1	1	yes	characteristic
141	C₂	#14521180	1	4	yes	direct product factor
142	C₂	C₂×Q₈×Dic₃₁	1	1	yes	characteristic
143	C₂	C₂²×#92474	1	1	yes	characteristic
144	C₂	C₂×#11758525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×Q₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×C₄×Q₈	D₃₁
hypocenter	C₃₁	C₂×#446
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14521180	C₂	direct (N × H)
36. C₂×C₃₁⋊C₈	Q₈	semidirect
56. C₃₁⋊C₈	C₂×Q₈	semidirect
75. C₂×C₃₁	#446	semidirect
93. C₃₁	C₂×#446	semidirect
141. C₂	#14521180	direct (N × H)

Maximal subgroups

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

subgroup	index
#14521180	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×#11758391	2
C₂×C₄×C₃₁⋊C₈	2
C₂×C₄×Q₈×C₃₁	2
C₂×#446	31

Subgroups — C2×#14521180 #16854639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521180 #16854639