Subgroups — C₂×#5122 #83746

Order 384.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#926	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#5122	C₂	1	8	yes	maximal, direct product factor
5	C₂×#815	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
7	C₂×#405	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×M₁₆	C₄	1	8	yes	complemented
10	C₂×C₃×M₁₆	C₂²	1	4	yes	—
11	C₂²×C₃×C₈	C₂²	1	2	yes	—
12	C₂×#217	C₄	1	2	yes	—
13	#926	C₄	1	4	yes	—
14	#926	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₃	C₄	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	2	yes	—
17	C₂²×C₃⋊C₈	C₄	1	8	yes	complemented
18	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
21	#815	C₂²	1	8	yes	complemented
22	C₈×Dic₃	C₂²	1	8	yes	complemented
23	C₂²×M₁₆	D₃	1	1	yes	characteristic
24	C₂³×C₈	—	3	1	no	—
25	C₂×#117	—	3	1	no	—
26	#405	—	3	8	no	—
27	C₂×#97	—	3	2	no	—
28	C₂×C₄×C₈	—	3	2	no	—
29	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
33	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
34	C₂²×C₃×C₄	C₂³	1	2	yes	—
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×Dic₃	C₂×C₄	1	4	yes	—
37	C₃×M₁₆	C₂×C₄	1	16	yes	complemented
38	C₂×C₃×C₈	C₂×C₄	1	8	yes	—
39	C₂×C₃×C₈	C₂³	1	4	yes	—
40	#217	C₂×C₄	1	8	yes	—
41	C₄⋊Dic₃	C₂×C₄	1	8	yes	—
42	C₄×Dic₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₃	C₂³	1	4	yes	—
44	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
45	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
46	C₂×C₃⋊C₈	C₂×C₄	1	16	yes	complemented
47	C₂×C₃⋊C₈	C₂³	1	2	yes	—
48	C₂×C₃⋊C₈	C₂³	1	2	yes	—
49	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
50	C₂×M₁₆	Dic₃	1	8	yes	complemented
51	C₂×M₁₆	C₂×D₃	1	4	yes	—
52	C₂²×C₈	—	3	8	no	—
53	C₂²×C₈	—	3	4	no	—
54	C₂²×C₈	—	3	1	no	—
55	C₂²×C₈	—	3	1	no	—
56	C₂²×C₈	C₂×D₃	1	2	yes	—
57	#117	—	3	4	no	—
58	#117	—	3	4	no	—
59	C₂×C₄⋊C₄	—	3	2	no	—
60	C₂×C₂²⋊C₄	—	3	2	no	—
61	C₂×C₄²	—	3	2	no	—
62	#97	—	3	8	no	—
63	C₄×C₈	—	3	8	no	—
64	C₂³×C₃	—	2	4	no	—
65	C₂³×C₃	C₄²	1	4	yes	—
66	C₂³×C₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄	C₄²	1	8	yes	—
69	C₂×C₃×C₄	C₄²	1	4	yes	—
70	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
72	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
74	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₃	—	2	8	no	—
78	C₂×Dic₃	C₂²×C₄	1	8	yes	—
79	C₃×C₈	C₂²×C₄	1	8	yes	—
80	C₃⋊C₈	C₂²×C₄	1	4	yes	—
81	C₃⋊C₈	C₂²×C₄	1	4	yes	—
82	C₂⁴	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₄	—	3	4	no	—
84	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₃	1	2	yes	—
86	C₂²×C₄	C₄×D₃	1	4	yes	—
87	C₂²×C₄	C₂×Dic₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
90	M₁₆	C₂×Dic₃	1	16	yes	complemented
91	C₂×C₈	—	3	4	no	—
92	C₂×C₈	—	3	4	no	—
93	C₂×C₈	—	3	16	no	—
94	C₂×C₈	—	3	2	no	—
95	C₂×C₈	—	3	2	no	—
96	C₂×C₈	C₂×Dic₃	1	8	yes	—
97	C₂×C₈	C₂²×D₃	1	4	yes	—
98	C₄⋊C₄	—	3	8	no	—
99	C₂²⋊C₄	—	3	8	no	—
100	C₄²	—	3	4	no	—
101	C₄²	—	3	4	no	—
102	C₂²×C₃	—	2	4	no	—
103	C₂²×C₃	—	2	8	no	—
104	C₂²×C₃	C₂×C₄²	1	4	yes	—
105	C₂²×C₃	C₂×C₄²	1	2	yes	—
106	C₂²×C₃	#131	1	4	yes	—
107	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
108	C₃×C₄	C₂×C₄²	1	2	yes	—
109	C₃×C₄	C₂×C₄²	1	4	yes	—
110	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
111	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
112	Dic₃	—	2	8	no	—
113	C₂³	—	2	4	no	—
114	C₂³	C₄×Dic₃	1	4	yes	—
115	C₂³	C₂×C₄×D₃	1	2	yes	—
116	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
117	C₂×C₄	—	3	8	no	—
118	C₂×C₄	—	6	8	no	—
119	C₂×C₄	C₄×Dic₃	1	8	yes	—
120	C₂×C₄	C₄×Dic₃	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₃	1	4	yes	—
124	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
125	C₂×C₄	C₂²×Dic₃	1	2	yes	—
126	C₂×C₄	C₂²×Dic₃	1	4	yes	—
127	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
128	C₈	—	3	4	no	—
129	C₈	—	3	4	no	—
130	C₈	C₂²×Dic₃	1	8	yes	—
131	C₂×C₃	—	2	4	no	—
132	C₂×C₃	#405	1	4	yes	complemented
133	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
134	C₂×C₃	C₂×#131	1	2	yes	—
135	C₂²	—	2	8	no	—
136	C₂²	—	2	4	no	—
137	C₂²	#907	1	4	yes	—
138	C₂²	C₂×C₄×Dic₃	1	4	yes	—
139	C₂²	C₂×C₄×Dic₃	1	2	yes	—
140	C₂²	C₂²×C₄×D₃	1	1	yes	characteristic
141	C₄	—	6	8	no	—
142	C₄	C₂×C₄×Dic₃	1	2	yes	—
143	C₄	C₂×C₄×Dic₃	1	4	yes	—
144	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
145	C₄	C₂³×Dic₃	1	1	yes	characteristic
146	C₃	C₂×#405	1	1	yes	characteristic, complemented, Sylow
147	C₂	—	2	4	no	—
148	C₂	#5122	1	4	yes	direct product factor
149	C₂	C₂×#907	1	2	yes	—
150	C₂	C₂²×C₄×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #5122	C₂	direct (N × H)
5. C₂×#815	C₂	semidirect
6. C₂×C₈×Dic₃	C₂	semidirect
9. C₂×C₃×M₁₆	C₄	semidirect
17. C₂²×C₃⋊C₈	C₄	semidirect
21. #815	C₂²	semidirect
22. C₈×Dic₃	C₂²	semidirect
37. C₃×M₁₆	C₂×C₄	semidirect
46. C₂×C₃⋊C₈	C₂×C₄	semidirect
50. C₂×M₁₆	Dic₃	semidirect
90. M₁₆	C₂×Dic₃	semidirect
132. C₂×C₃	#405	semidirect
146. C₃	C₂×#405	semidirect
148. C₂	#5122	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×C₃×M₁₆	2
C₂×#926	2
C₂³×C₃⋊C₈	2
#5122	2
C₂×#815	2
C₂×C₈×Dic₃	2
C₂×#405	3

Subgroups — C2×#5122 #83746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5122 #83746