Subgroups — C₂×#4999 #82203

Order 384.

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	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4999	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
6	C₂×#424	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₃×#117	C₄	1	4	yes	—
9	C₃×#117	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₃×C₄²	C₂²	1	2	yes	—
13	C₂×#208	—	2	4	no	—
14	C₂×#208	C₂²	1	4	yes	—
15	C₂²×C₃⋊C₈	—	2	4	no	—
16	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
18	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
19	C₄⋊(C₃⋊C₈)	C₂²	1	16	yes	complemented
20	C₂²×M₁₆	—	3	1	no	—
21	C₂³×C₈	—	3	1	no	—
22	C₂×#117	D₃	1	1	yes	characteristic
23	#424	—	3	8	no	—
24	C₂×C₄⋊C₈	—	3	4	no	—
25	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	4	yes	—
30	C₂²×C₃×C₄	Q₈	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
34	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
35	C₃×C₄²	C₂×C₄	1	4	yes	—
36	C₃×C₄²	C₂³	1	4	yes	—
37	#208	—	2	8	no	—
38	C₂×C₃⋊C₈	—	2	4	no	—
39	C₂×C₃⋊C₈	—	2	8	no	—
40	C₂×C₃⋊C₈	—	2	4	no	—
41	C₂×C₃⋊C₈	C₂³	1	4	yes	—
42	C₂×C₃⋊C₈	C₂³	1	4	yes	—
43	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×M₁₆	—	3	4	no	—
45	C₂×M₁₆	—	6	4	no	—
46	C₂²×C₈	—	3	4	no	—
47	C₂²×C₈	—	3	2	no	—
48	C₂²×C₈	—	3	2	no	—
49	C₂²×C₈	—	6	4	no	—
50	#117	Dic₃	1	4	yes	—
51	#117	C₂×D₃	1	4	yes	—
52	C₂×C₄⋊C₄	Dic₃	1	2	yes	—
53	C₂×C₂²⋊C₄	Dic₃	1	2	yes	—
54	C₂×C₄²	C₂×D₃	1	2	yes	—
55	C₄⋊C₈	—	3	16	no	—
56	C₂³×C₃	—	2	4	no	—
57	C₂³×C₃	C₄⋊C₄	1	4	yes	—
58	C₂³×C₃	C₂²×C₄	1	2	yes	—
59	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄	—	2	8	no	—
61	C₂×C₃×C₄	C₄⋊C₄	1	8	yes	—
62	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
63	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
64	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
68	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
69	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
71	C₃⋊C₈	—	2	4	no	—
72	C₃⋊C₈	—	2	4	no	—
73	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
74	C₂²×C₄	Dic₆	1	4	yes	—
75	C₂²×C₄	D₁₂	1	4	yes	—
76	C₂²×C₄	C₂×Dic₃	1	4	yes	—
77	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₃	1	2	yes	—
79	C₂²×C₄	C₂²×D₃	1	2	yes	—
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	M₁₆	—	6	8	no	—
82	C₂×C₈	—	3	4	no	—
83	C₂×C₈	—	3	4	no	—
84	C₂×C₈	—	6	4	no	—
85	C₂×C₈	—	6	8	no	—
86	C₂×C₈	—	6	4	no	—
87	C₄⋊C₄	C₂×Dic₃	1	8	yes	—
88	C₂²⋊C₄	C₂×Dic₃	1	8	yes	—
89	C₄²	C₂×Dic₃	1	4	yes	—
90	C₄²	C₂²×D₃	1	4	yes	—
91	C₂²×C₃	—	2	4	no	—
92	C₂²×C₃	—	2	8	no	—
93	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂²×C₃	#131	1	4	yes	—
96	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
97	C₃×C₄	—	2	8	no	—
98	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
99	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
100	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
101	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	C₄⋊Dic₃	1	4	yes	—
104	C₂³	C₂²×Dic₃	1	2	yes	—
105	C₂³	C₂²×Dic₃	1	1	yes	characteristic
106	C₂×C₄	—	2	8	no	—
107	C₂×C₄	C₄⋊Dic₃	1	8	yes	—
108	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
109	C₂×C₄	C₂×Dic₆	1	4	yes	—
110	C₂×C₄	C₂×Dic₆	1	2	yes	—
111	C₂×C₄	C₂×D₁₂	1	2	yes	—
112	C₂×C₄	C₂×D₁₂	1	4	yes	—
113	C₂×C₄	C₂²×Dic₃	1	8	yes	—
114	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×Dic₃	1	2	yes	—
116	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
117	C₈	—	6	4	no	—
118	C₈	—	6	4	no	—
119	C₂×C₃	—	2	4	no	—
120	C₂×C₃	#424	1	4	yes	complemented
121	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₃	C₂×#131	1	2	yes	—
123	C₂²	—	2	8	no	—
124	C₂²	—	2	4	no	—
125	C₂²	C₂×C₄⋊Dic₃	1	4	yes	—
126	C₂²	C₂×C₄⋊Dic₃	1	2	yes	—
127	C₂²	#948	1	4	yes	—
128	C₂²	C₂³×Dic₃	1	1	yes	characteristic
129	C₄	—	2	8	no	—
130	C₄	C₂×C₄⋊Dic₃	1	4	yes	—
131	C₄	C₂×C₄⋊Dic₃	1	2	yes	—
132	C₄	C₂²×Dic₆	1	1	yes	characteristic
133	C₄	C₂²×D₁₂	1	1	yes	characteristic
134	C₃	C₂×#424	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	2	4	no	—
136	C₂	#4999	1	4	yes	direct product factor
137	C₂	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
138	C₂	C₂×#948	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#117	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂×#117	D₃
hypocenter	C₃	C₂×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
4. #4999	C₂	direct (N × H)
5. C₂×C₄⋊(C₃⋊C₈)	C₂	semidirect
19. C₄⋊(C₃⋊C₈)	C₂²	semidirect
120. C₂×C₃	#424	semidirect
134. C₃	C₂×#424	semidirect
136. C₂	#4999	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#117	2
C₂²×#208	2
C₂³×C₃⋊C₈	2
#4999	2
C₂×C₄⋊(C₃⋊C₈)	2
C₂×#424	3

Subgroups — C2×#4999 #82203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4999 #82203