Subgroups — C₂×#13092268 #15675731

Order 2944.

928 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₃⋊C₄	C₂	1	2	yes	maximal
2	C₂²×C₄×Dic₂₃	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092268	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₃	C₂²	1	2	yes	—
10	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂³×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₃⋊C₄	C₄	1	16	yes	complemented
13	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₂₃	—	2	8	no	—
15	C₂×C₄×Dic₂₃	C₄	1	8	yes	complemented
16	C₂×C₄×Dic₂₃	C₂²	1	8	yes	—
17	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
18	C₂₃×#95	C₂²	1	4	yes	—
19	C₂₃⋊#95	C₂²	1	4	yes	—
20	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	—	2	4	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×Dic₂₃	—	2	4	no	—
29	C₂²×Dic₂₃	—	2	4	no	—
30	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
31	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
32	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
33	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
34	C₂²×Dic₂₃	D₄	1	4	yes	—
35	C₂²×Dic₂₃	Q₈	1	4	yes	—
36	C₂²×Dic₂₃	C₂³	1	4	yes	—
37	C₂²×Dic₂₃	C₂³	1	2	yes	—
38	C₂²×Dic₂₃	C₂³	1	2	yes	—
39	Dic₂₃⋊C₄	C₂×C₄	1	32	yes	complemented
40	C₄×Dic₂₃	—	2	16	no	—
41	C₄×Dic₂₃	C₂×C₄	1	16	yes	complemented
42	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
44	C₂³×C₂₃	C₂×D₄	1	2	yes	—
45	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂³×C₂₃	#41	1	4	yes	—
47	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	2	8	no	—
49	C₂×C₄×C₂₃	—	2	16	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
55	C₂×Dic₂₃	—	2	16	no	—
56	C₂×Dic₂₃	—	2	4	no	—
57	C₂×Dic₂₃	—	2	4	no	—
58	C₂×Dic₂₃	C₄²	1	16	yes	complemented
59	C₂×Dic₂₃	C₄⋊C₄	1	16	yes	complemented
60	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
61	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
62	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
63	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
64	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
65	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
66	C₂×C₄×C₄⋊C₄	—	23	1	no	maximal, Sylow
67	C₂²×C₂₃	C₂×C₄²	1	4	yes	—
68	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
69	C₂²×C₂₃	#117	1	4	yes	—
70	C₂²×C₂₃	C₄×D₄	1	8	yes	—
71	C₂²×C₂₃	C₄×Q₈	1	8	yes	—
72	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
74	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
75	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×C₂₃	C₂×#41	1	2	yes	—
77	C₄×C₂₃	—	2	4	no	—
78	C₄×C₂₃	—	2	8	no	—
79	Dic₂₃	—	2	8	no	—
80	Dic₂₃	C₂×C₄²	1	8	yes	complemented
81	Dic₂₃	C₂×C₄⋊C₄	1	8	yes	complemented
82	C₂²×C₄⋊C₄	—	23	2	no	—
83	C₂²×C₄²	—	23	1	no	—
84	C₂²×C₄²	—	23	2	no	—
85	C₄×C₄⋊C₄	—	23	8	no	—
86	C₂×#95	—	23	1	no	—
87	C₂×#95	D₂₃	1	1	yes	characteristic
88	C₂×C₂₃	C₄×C₄⋊C₄	1	8	yes	complemented
89	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
90	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₃	C₂×C₄×D₄	1	2	yes	—
93	C₂×C₂₃	C₂×C₄×Q₈	1	2	yes	—
94	C₂³×C₄	—	23	1	no	—
95	C₂³×C₄	—	23	1	no	—
96	C₂³×C₄	—	23	2	no	—
97	C₂³×C₄	C₂×D₂₃	1	2	yes	—
98	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
99	C₂×C₄⋊C₄	—	23	16	no	—
100	C₂×C₄⋊C₄	—	23	8	no	—
101	C₂×C₄²	—	23	8	no	—
102	C₂×C₄²	—	23	4	no	—
103	C₂×C₄²	—	23	8	no	—
104	C₂×C₄²	—	46	8	no	—
105	#95	—	23	4	no	—
106	#95	C₂×D₂₃	1	4	yes	—
107	C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	23	4	no	—
113	C₂²×C₄	—	23	4	no	—
114	C₂²×C₄	—	23	8	no	—
115	C₂²×C₄	—	23	4	no	—
116	C₂²×C₄	—	23	4	no	—
117	C₂²×C₄	—	23	2	no	—
118	C₂²×C₄	—	23	2	no	—
119	C₂²×C₄	—	23	8	no	—
120	C₂²×C₄	—	23	4	no	—
121	C₂²×C₄	—	46	4	no	—
122	C₂²×C₄	—	46	4	no	—
123	C₂²×C₄	C₄×D₂₃	1	4	yes	—
124	C₂²×C₄	C₄×D₂₃	1	8	yes	—
125	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
127	C₄⋊C₄	—	23	32	no	—
128	C₄²	—	23	16	no	—
129	C₄²	—	46	16	no	—
130	C₂³	C₂×C₄×D₂₃	1	2	yes	—
131	C₂³	C₂×C₄×D₂₃	1	4	yes	—
132	C₂³	D₄×D₂₃	1	2	yes	—
133	C₂³	#67290	1	4	yes	—
134	C₂³	Q₈×D₂₃	1	2	yes	—
135	C₂³	C₂³×D₂₃	1	1	yes	characteristic
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	16	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	23	16	no	—
142	C₂×C₄	—	23	16	no	—
143	C₂×C₄	—	23	4	no	—
144	C₂×C₄	—	23	4	no	—
145	C₂×C₄	—	23	8	no	—
146	C₂×C₄	—	23	4	no	—
147	C₂×C₄	—	23	4	no	—
148	C₂×C₄	—	23	8	no	—
149	C₂×C₄	—	46	16	no	—
150	C₂×C₄	—	46	4	no	—
151	C₂×C₄	—	46	4	no	—
152	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
153	C₂×C₄	C₂×C₄×D₂₃	1	8	yes	—
154	C₂²	C₄²×D₂₃	1	4	yes	—
155	C₂²	#10625048	1	4	yes	—
156	C₂²	(C₂²⋊D₂₃)⋊C₄	1	8	yes	—
157	C₂²	#10625058	1	8	yes	—
158	C₂²	C₄⋊C₄×D₂₃	1	4	yes	—
159	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
160	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
161	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
162	C₂²	C₂×#67290	1	2	yes	—
163	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
164	C₄	—	2	4	no	—
165	C₄	—	2	8	no	—
166	C₄	—	23	8	no	—
167	C₄	—	23	8	no	—
168	C₄	—	46	8	no	—
169	C₂	#13092268	1	8	yes	direct product factor
170	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
171	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
172	C₂	C₂×#10625048	1	1	yes	characteristic
173	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
174	C₂	C₂×#10625058	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂×C₂₃×#95	C₂
socle, prime radical	C₂⁴×C₂₃	C₂³
cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×#95	D₂₃
hypocenter	C₂₃	C₂×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
4. #13092268	C₂	direct (N × H)
12. C₂×Dic₂₃⋊C₄	C₄	semidirect
15. C₂×C₄×Dic₂₃	C₄	semidirect
39. Dic₂₃⋊C₄	C₂×C₄	semidirect
41. C₄×Dic₂₃	C₂×C₄	semidirect
58. C₂×Dic₂₃	C₄²	semidirect
59. C₂×Dic₂₃	C₄⋊C₄	semidirect
80. Dic₂₃	C₂×C₄²	semidirect
81. Dic₂₃	C₂×C₄⋊C₄	semidirect
88. C₂×C₂₃	C₄×C₄⋊C₄	semidirect
107. C₂₃	C₂×C₄×C₄⋊C₄	semidirect
169. C₂	#13092268	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×Dic₂₃⋊C₄	2
C₂²×C₄×Dic₂₃	2
#13092268	2
C₂×C₂₃⋊#95	2
C₂×C₂₃×#95	2
C₂×C₄×C₄⋊C₄	23

Subgroups — C2×#13092268 #15675731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092268 #15675731