Subgroups — C₂×C₄×Q₁₆×C₂₂₃ #43251953

Order 28544.

488 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₄×Q₁₆×C₂₂₃	C₂	direct (N × H)
8. C₂×Q₁₆×C₂₂₃	C₄	direct (N × H)
30. Q₁₆×C₂₂₃	C₂×C₄	direct (N × H)
50. C₂×C₄×C₂₂₃	Q₁₆	direct (N × H)
67. C₄×C₂₂₃	C₂×Q₁₆	direct (N × H)
70. C₂×C₂₂₃	C₄×Q₁₆	direct (N × H)
74. C₂₂₃	C₂×C₄×Q₁₆	direct (N × H)
75. C₂×C₄×Q₁₆	C₂₂₃	direct (N × H)
78. C₄×Q₁₆	C₂×C₂₂₃	direct (N × H)
83. C₂×Q₁₆	C₄×C₂₂₃	direct (N × H)
105. Q₁₆	C₂×C₄×C₂₂₃	direct (N × H)
125. C₂×C₄	Q₁₆×C₂₂₃	direct (N × H)
142. C₄	C₂×Q₁₆×C₂₂₃	direct (N × H)
145. C₂	C₄×Q₁₆×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
C₂²×Q₁₆×C₂₂₃	2
C₂×C₄×Q₈×C₂₂₃	2
C₄×Q₁₆×C₂₂₃	2
C₂×#107×C₂₂₃	2
C₂×#103×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
C₂×C₄×Q₁₆	223

Subgroups — C2×C4×Q16×C223 #43251953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₁₆×C₂₂₃ #43251953