Subgroups — C₂×#14521076 #16854270

Order 3968.

512 conjugacy classes of subgroups.

The subgroup lattice has 140 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	1	yes	maximal, characteristic
2	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14521076	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#117	C₄	1	4	yes	—
8	C₃₁×#117	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
10	C₂×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
11	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
12	C₂×#92474	—	2	4	no	—
13	C₂×#92474	C₂²	1	4	yes	—
14	C₂²×C₃₁⋊C₈	—	2	4	no	—
15	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃₁⋊C₈	C₂²	1	4	yes	—
17	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
18	C₄⋊(C₃₁⋊C₈)	C₂²	1	16	yes	complemented
19	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₁	D₄	1	4	yes	—
24	C₂²×C₄×C₃₁	Q₈	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
28	C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
29	C₄²×C₃₁	C₂×C₄	1	4	yes	—
30	C₄²×C₃₁	C₂³	1	4	yes	—
31	#92474	—	2	8	no	—
32	C₂×C₃₁⋊C₈	—	2	4	no	—
33	C₂×C₃₁⋊C₈	—	2	8	no	—
34	C₂×C₃₁⋊C₈	—	2	4	no	—
35	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
36	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
37	C₂³×C₃₁	—	2	4	no	—
38	C₂³×C₃₁	C₄⋊C₄	1	4	yes	—
39	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
40	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	—	2	8	no	—
42	C₂×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
43	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
44	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₃₁⋊C₈	—	2	4	no	—
53	C₃₁⋊C₈	—	2	4	no	—
54	C₂×#424	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	—	2	4	no	—
56	C₂²×C₃₁	—	2	8	no	—
57	C₂²×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
58	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
59	C₂²×C₃₁	#131	1	4	yes	—
60	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₃₁	—	2	8	no	—
62	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
63	C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
64	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×M₁₆	—	31	1	no	—
67	C₂³×C₈	—	31	1	no	—
68	C₂×#117	D₃₁	1	1	yes	characteristic
69	#424	—	31	8	no	—
70	C₂×C₄⋊C₈	—	31	4	no	—
71	C₂×C₃₁	—	2	4	no	—
72	C₂×C₃₁	#424	1	4	yes	complemented
73	C₂×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃₁	C₂×#131	1	2	yes	—
75	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
76	C₂×M₁₆	—	31	4	no	—
77	C₂×M₁₆	—	62	4	no	—
78	C₂²×C₈	—	31	4	no	—
79	C₂²×C₈	—	31	2	no	—
80	C₂²×C₈	—	31	2	no	—
81	C₂²×C₈	—	62	4	no	—
82	#117	Dic₃₁	1	4	yes	—
83	#117	C₂×D₃₁	1	4	yes	—
84	C₂×C₄⋊C₄	Dic₃₁	1	2	yes	—
85	C₂×C₂²⋊C₄	Dic₃₁	1	2	yes	—
86	C₂×C₄²	C₂×D₃₁	1	2	yes	—
87	C₄⋊C₈	—	31	16	no	—
88	C₃₁	C₂×#424	1	1	yes	characteristic, complemented, Sylow
89	C₂⁴	C₂×Dic₃₁	1	1	yes	characteristic
90	C₂²×C₄	Dic₆₂	1	4	yes	—
91	C₂²×C₄	D₁₂₄	1	4	yes	—
92	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
93	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	M₁₆	—	62	8	no	—
98	C₂×C₈	—	31	4	no	—
99	C₂×C₈	—	31	4	no	—
100	C₂×C₈	—	62	4	no	—
101	C₂×C₈	—	62	8	no	—
102	C₂×C₈	—	62	4	no	—
103	C₄⋊C₄	C₂×Dic₃₁	1	8	yes	—
104	C₂²⋊C₄	C₂×Dic₃₁	1	8	yes	—
105	C₄²	C₂×Dic₃₁	1	4	yes	—
106	C₄²	C₂²×D₃₁	1	4	yes	—
107	C₂³	—	2	4	no	—
108	C₂³	C₄⋊Dic₃₁	1	4	yes	—
109	C₂³	C₂²×Dic₃₁	1	2	yes	—
110	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	C₄⋊Dic₃₁	1	8	yes	—
113	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
114	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
115	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
116	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
117	C₂×C₄	C₂×D₁₂₄	1	4	yes	—
118	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
119	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
121	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
122	C₈	—	62	4	no	—
123	C₈	—	62	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×C₄⋊Dic₃₁	1	4	yes	—
127	C₂²	C₂×C₄⋊Dic₃₁	1	2	yes	—
128	C₂²	#11758525	1	4	yes	—
129	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
130	C₄	—	2	8	no	—
131	C₄	C₂×C₄⋊Dic₃₁	1	4	yes	—
132	C₄	C₂×C₄⋊Dic₃₁	1	2	yes	—
133	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
134	C₄	C₂²×D₁₂₄	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#14521076	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊Dic₃₁	1	1	yes	characteristic
138	C₂	C₂×#11758525	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₃₁×#117	C₂
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂⁴×C₃₁	C₂×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#117	D₃₁
hypocenter	C₃₁	C₂×#424
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14521076	C₂	direct (N × H)
4. C₂×C₄⋊(C₃₁⋊C₈)	C₂	semidirect
18. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
72. C₂×C₃₁	#424	semidirect
88. C₃₁	C₂×#424	semidirect
136. C₂	#14521076	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
C₂³×C₃₁⋊C₈	2
#14521076	2
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×C₃₁×#117	2
C₂×#424	31

Subgroups — C2×#14521076 #16854270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521076 #16854270