Subgroups — C₂×#340×C₂₂₃ #43250749

Order 28544.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₂₂₃	C₂	1	2	yes	maximal
3	#340×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₂₂₃	—	2	2	no	—
5	C₂³×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₂₂₃	—	2	4	no	—
7	C₂²×C₈×C₂₂₃	C₄	1	4	yes	complemented
8	C₂²×C₈×C₂₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₂₃	—	2	4	no	—
11	C₂×C₄⋊C₄×C₂₂₃	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
13	C₂⁴×C₂₂₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₂₃	—	2	8	no	—
15	C₂²×C₄×C₂₂₃	—	2	4	no	—
16	C₂²×C₄×C₂₂₃	—	4	4	no	—
17	C₂²×C₄×C₂₂₃	—	4	4	no	—
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₂₃	D₄	1	4	yes	—
22	C₂²×C₄×C₂₂₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₂₂₃	—	2	8	no	—
25	C₂×C₈×C₂₂₃	—	2	4	no	—
26	C₂×C₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
27	C₄⋊C₄×C₂₂₃	—	2	8	no	—
28	C₄⋊C₄×C₂₂₃	—	2	4	no	—
29	C₄⋊C₄×C₂₂₃	C₂×C₄	1	8	yes	complemented
30	C₂³×C₂₂₃	C₂²⋊C₄	1	4	yes	—
31	C₂³×C₂₂₃	C₄⋊C₄	1	2	yes	—
32	C₂³×C₂₂₃	D₈	1	2	yes	—
33	C₂³×C₂₂₃	SD₁₆	1	4	yes	—
34	C₂³×C₂₂₃	Q₁₆	1	2	yes	—
35	C₂³×C₂₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₂₃	—	2	8	no	—
37	C₂×C₄×C₂₂₃	—	4	16	no	—
38	C₂×C₄×C₂₂₃	—	4	4	no	—
39	C₂×C₄×C₂₂₃	—	4	4	no	—
40	C₂×C₄×C₂₂₃	C₄²	1	4	yes	—
41	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₂₃	C₄⋊C₄	1	8	yes	—
45	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₂₂₃	—	2	4	no	—
50	C₂²×C₂₂₃	#95	1	4	yes	—
51	C₂²×C₂₂₃	#102	1	8	yes	—
52	C₂²×C₂₂₃	#103	1	8	yes	—
53	C₂²×C₂₂₃	#106	1	4	yes	—
54	C₂²×C₂₂₃	#107	1	4	yes	—
55	C₂²×C₂₂₃	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₂₂₃	C₂×SD₁₆	1	2	yes	—
59	C₂²×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₂₂₃	—	4	8	no	—
61	C₄×C₂₂₃	#95	1	4	yes	—
62	C₄×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
63	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₂₂₃	C₂×C₄⋊C₄	1	2	yes	—
65	C₂×C₂₂₃	#340	1	8	yes	direct product factor
66	C₂×C₂₂₃	C₂×#95	1	1	yes	characteristic
67	C₂×C₂₂₃	C₂×#102	1	2	yes	—
68	C₂×C₂₂₃	C₂×#103	1	2	yes	—
69	C₂×C₂₂₃	C₂×#106	1	1	yes	characteristic
70	C₂×C₂₂₃	C₂×#107	1	1	yes	characteristic
71	C₂₂₃	C₂×#340	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×#340	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₈	C₂×C₂₂₃	1	1	yes	characteristic
74	C₂²×C₄⋊C₄	C₂×C₂₂₃	1	2	yes	—
75	#340	C₂×C₂₂₃	1	4	yes	direct product factor
76	C₂³×C₄	—	2	2	no	—
77	C₂³×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	C₄×C₂₂₃	1	4	yes	complemented
80	C₂²×C₈	C₂²×C₂₂₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	—	2	4	no	—
83	C₂×C₄⋊C₄	C₄×C₂₂₃	1	8	yes	complemented
84	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
85	C₂⁴	D₄×C₂₂₃	1	1	yes	characteristic
86	C₂²×C₄	—	2	8	no	—
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	4	4	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	C₂×C₄×C₂₂₃	1	2	yes	—
91	C₂²×C₄	C₂×C₄×C₂₂₃	1	4	yes	—
92	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
93	C₂²×C₄	D₄×C₂₂₃	1	4	yes	—
94	C₂²×C₄	Q₈×C₂₂₃	1	2	yes	—
95	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	C₂×C₄×C₂₂₃	1	4	yes	complemented
99	C₄⋊C₄	—	2	8	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	C₂×C₄×C₂₂₃	1	8	yes	complemented
102	C₂³	C₂²⋊C₄×C₂₂₃	1	4	yes	—
103	C₂³	C₄⋊C₄×C₂₂₃	1	2	yes	—
104	C₂³	D₈×C₂₂₃	1	2	yes	—
105	C₂³	SD₁₆×C₂₂₃	1	4	yes	—
106	C₂³	Q₁₆×C₂₂₃	1	2	yes	—
107	C₂³	C₂×D₄×C₂₂₃	1	1	yes	characteristic
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	—	4	16	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	C₄²×C₂₂₃	1	4	yes	—
113	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	4	yes	—
114	C₂×C₄	C₂²⋊C₄×C₂₂₃	1	4	yes	—
115	C₂×C₄	C₄⋊C₄×C₂₂₃	1	2	yes	—
116	C₂×C₄	C₄⋊C₄×C₂₂₃	1	8	yes	—
117	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
119	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
120	C₂×C₄	C₂×Q₈×C₂₂₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	#95×C₂₂₃	1	4	yes	—
123	C₂²	#102×C₂₂₃	1	8	yes	—
124	C₂²	#103×C₂₂₃	1	8	yes	—
125	C₂²	#106×C₂₂₃	1	4	yes	—
126	C₂²	#107×C₂₂₃	1	4	yes	—
127	C₂²	C₂×C₂²⋊C₄×C₂₂₃	1	2	yes	—
128	C₂²	C₂×C₄⋊C₄×C₂₂₃	1	1	yes	characteristic
129	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
130	C₂²	C₂×SD₁₆×C₂₂₃	1	2	yes	—
131	C₂²	C₂×Q₁₆×C₂₂₃	1	1	yes	characteristic
132	C₄	—	4	8	no	—
133	C₄	#95×C₂₂₃	1	4	yes	—
134	C₄	C₂×C₄²×C₂₂₃	1	1	yes	characteristic
135	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic
136	C₄	C₂×C₄⋊C₄×C₂₂₃	1	2	yes	—
137	C₂	#340×C₂₂₃	1	8	yes	direct product factor
138	C₂	C₂×#95×C₂₂₃	1	1	yes	characteristic
139	C₂	C₂×#102×C₂₂₃	1	2	yes	—
140	C₂	C₂×#103×C₂₂₃	1	2	yes	—
141	C₂	C₂×#106×C₂₂₃	1	1	yes	characteristic
142	C₂	C₂×#107×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₂₂₃	D₄
commutator	C₄	C₂×C₄²×C₂₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₂₃ ◁ C₂³×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. #340×C₂₂₃	C₂	direct (N × H)
7. C₂²×C₈×C₂₂₃	C₄	semidirect
11. C₂×C₄⋊C₄×C₂₂₃	C₄	semidirect
26. C₂×C₈×C₂₂₃	C₂×C₄	semidirect
29. C₄⋊C₄×C₂₂₃	C₂×C₄	semidirect
65. C₂×C₂₂₃	#340	direct (N × H)
71. C₂₂₃	C₂×#340	direct (N × H)
72. C₂×#340	C₂₂₃	direct (N × H)
75. #340	C₂×C₂₂₃	direct (N × H)
79. C₂²×C₈	C₄×C₂₂₃	semidirect
83. C₂×C₄⋊C₄	C₄×C₂₂₃	semidirect
98. C₂×C₈	C₂×C₄×C₂₂₃	semidirect
101. C₄⋊C₄	C₂×C₄×C₂₂₃	semidirect
137. C₂	#340×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₂₂₃	2
C₂²×C₄⋊C₄×C₂₂₃	2
#340×C₂₂₃	2
C₂×#340	223

Subgroups — C2×#340×C223 #43250749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#340×C₂₂₃ #43250749