Subgroups — C₂×C₄×#92474 #16854120

Order 3968.

656 conjugacy classes of subgroups.

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

	subgroup	G/H
center	C₂×C₄²	C₂×D₃₁
commutator	C₂×C₃₁	C₂²×C₄²
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×C₄×M₁₆
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
2. C₄×#92474	C₂	direct (N × H)
3. C₂×#11758391	C₂	semidirect
4. C₂×C₄×C₃₁⋊C₈	C₂	semidirect
13. C₂×#92474	C₄	direct (N × H)
16. #11758391	C₂²	semidirect
17. C₄×C₃₁⋊C₈	C₂²	semidirect
33. #92474	C₂×C₄	direct (N × H)
34. C₂×C₃₁⋊C₈	C₂×C₄	semidirect
44. C₂×C₄×C₃₁	M₁₆	semidirect
52. C₃₁⋊C₈	C₂²×C₄	semidirect
63. C₄×C₃₁	C₂×M₁₆	semidirect
71. C₂×C₃₁	C₄×M₁₆	semidirect
86. C₃₁	C₂×C₄×M₁₆	semidirect
111. C₂×C₄	#92474	direct (N × H)
129. C₄	C₂×#92474	direct (N × H)
134. C₂	C₄×#92474	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
C₄×#92474	2
C₂×#11758391	2
C₂×C₄×C₃₁⋊C₈	2
C₂²×C₄²×C₃₁	2
C₂×C₄×M₁₆	31

Subgroups — C2×C4×#92474 #16854120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#92474 #16854120