Subgroups — C₂×#5125 #83759

Order 384.

460 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	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	2	yes	maximal
3	#5125	C₂	1	8	yes	maximal, direct product factor
4	C₂×#819	C₂	1	4	yes	maximal, complemented
5	C₂×#430	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×M₁₆	C₄	1	4	yes	—
8	C₂×C₃×M₁₆	C₄	1	4	yes	—
9	C₂×C₃×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₃×C₈	C₂²	1	2	yes	—
11	C₂×#208	—	2	2	no	—
12	C₂×#208	—	2	4	no	—
13	C₂×#208	C₂²	1	8	yes	—
14	C₂×#208	C₂²	1	4	yes	—
15	C₂²×C₃⋊C₈	—	2	2	no	—
16	#819	C₂²	1	16	yes	complemented
17	C₂²×M₁₆	—	3	2	no	—
18	C₂²×M₁₆	D₃	1	1	yes	characteristic
19	#430	—	3	8	no	—
20	C₂×#108	—	3	4	no	—
21	C₂⁴×C₃	Q₈	1	1	yes	characteristic
22	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄	D₄	1	2	yes	—
25	C₂²×C₃×C₄	D₄	1	2	yes	—
26	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
27	C₂²×C₃×C₄	Q₈	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂³	1	2	yes	—
30	C₃×M₁₆	C₂×C₄	1	16	yes	—
31	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₈	C₂³	1	4	yes	—
34	#208	—	2	8	no	—
35	#208	—	2	4	no	—
36	#208	C₂³	1	8	yes	—
37	C₂×C₃⋊C₈	—	2	8	no	—
38	C₂×C₃⋊C₈	—	2	4	no	—
39	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂×M₁₆	—	3	4	no	—
41	C₂×M₁₆	—	3	8	no	—
42	C₂×M₁₆	—	6	4	no	—
43	C₂×M₁₆	—	6	2	no	—
44	C₂×M₁₆	Dic₃	1	4	yes	—
45	C₂×M₁₆	Dic₃	1	4	yes	—
46	C₂×M₁₆	C₂×D₃	1	4	yes	—
47	C₂²×C₈	—	6	2	no	—
48	C₂²×C₈	C₂×D₃	1	2	yes	—
49	#108	—	3	16	no	—
50	C₂³×C₃	—	4	2	no	—
51	C₂³×C₃	C₄⋊C₄	1	2	yes	—
52	C₂³×C₃	C₄⋊C₄	1	2	yes	—
53	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
54	C₂³×C₃	C₂×Q₈	1	2	yes	—
55	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
56	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
57	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
62	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
68	C₃×C₈	C₂²×C₄	1	8	yes	—
69	C₃⋊C₈	—	2	8	no	—
70	C₂⁴	Dic₆	1	1	yes	characteristic
71	C₂²×C₄	Dic₆	1	1	yes	characteristic
72	C₂²×C₄	Dic₆	1	2	yes	—
73	C₂²×C₄	D₁₂	1	2	yes	—
74	C₂²×C₄	D₁₂	1	2	yes	—
75	C₂²×C₄	C₂×Dic₃	1	2	yes	—
76	C₂²×C₄	C₂×Dic₃	1	2	yes	—
77	C₂²×C₄	C₂²×D₃	1	2	yes	—
78	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
79	M₁₆	—	3	8	no	—
80	M₁₆	—	6	8	no	—
81	M₁₆	—	6	4	no	—
82	M₁₆	C₂×Dic₃	1	16	yes	—
83	C₂×C₈	—	6	8	no	—
84	C₂×C₈	—	6	4	no	—
85	C₂×C₈	C₂×Dic₃	1	4	yes	—
86	C₂×C₈	C₂×Dic₃	1	4	yes	—
87	C₂×C₈	C₂²×D₃	1	4	yes	—
88	C₂²×C₃	—	2	4	no	—
89	C₂²×C₃	—	2	2	no	—
90	C₂²×C₃	—	4	2	no	—
91	C₂²×C₃	—	4	2	no	—
92	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
96	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
98	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
100	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₂³	—	4	2	no	—
102	C₂³	C₄⋊Dic₃	1	2	yes	—
103	C₂³	C₄⋊Dic₃	1	2	yes	—
104	C₂³	C₂×Dic₆	1	1	yes	characteristic
105	C₂³	C₂×Dic₆	1	2	yes	—
106	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
107	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
108	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
109	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
110	C₂×C₄	C₂×Dic₆	1	2	yes	—
111	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
112	C₂×C₄	C₂×D₁₂	1	4	yes	—
113	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
114	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₃	1	4	yes	—
118	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
119	C₈	—	6	8	no	—
120	C₈	C₂²×Dic₃	1	8	yes	—
121	C₂×C₃	—	2	4	no	—
122	C₂×C₃	—	2	1	no	—
123	C₂×C₃	—	2	1	no	—
124	C₂×C₃	#430	1	2	yes	complemented
125	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²	—	2	4	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	2	no	—
130	C₂²	C₂×C₄⋊Dic₃	1	4	yes	—
131	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
132	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
133	C₂²	C₂²×Dic₆	1	1	yes	characteristic
134	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
135	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
136	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
137	C₄	C₂²×D₁₂	1	1	yes	characteristic
138	C₄	C₂³×Dic₃	1	1	yes	characteristic
139	C₃	C₂×#430	1	1	yes	characteristic, complemented, Sylow
140	C₂	—	2	4	no	—
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	#5125	1	2	yes	direct product factor
144	C₂	C₂²×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #5125	C₂	direct (N × H)
4. C₂×#819	C₂	semidirect
16. #819	C₂²	semidirect
124. C₂×C₃	#430	semidirect
139. C₃	C₂×#430	semidirect
143. C₂	#5125	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
C₂²×#208	2
#5125	2
C₂×#819	2
C₂×#430	3

Subgroups — C2×#5125 #83759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5125 #83759