Subgroups — C₂×#13092313 #15675846

Order 2944.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₂₃⋊C₄	C₂	1	2	yes	maximal
3	C₂²×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092313	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₂₃	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×C₄⋊C₄×C₂₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₂₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	C₂×Dic₂₃⋊C₄	—	2	8	no	—
17	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
18	C₂×C₄×Dic₂₃	—	2	4	no	—
19	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
20	C₂₃⋊#95	C₂²	1	8	yes	—
21	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	—	2	4	no	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	D₄	1	4	yes	—
27	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
30	C₂²×Dic₂₃	—	2	2	no	—
31	C₂²×Dic₂₃	—	2	2	no	—
32	C₂²×Dic₂₃	—	2	4	no	—
33	C₂²×Dic₂₃	—	2	2	no	—
34	C₂²×Dic₂₃	—	2	4	no	—
35	C₂²×Dic₂₃	—	2	2	no	—
36	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
37	C₂²×Dic₂₃	D₄	1	4	yes	—
38	C₂²×Dic₂₃	Q₈	1	4	yes	—
39	C₂²×Dic₂₃	C₂³	1	4	yes	—
40	C₂²×Dic₂₃	C₂³	1	2	yes	—
41	C₂²×Dic₂₃	C₂³	1	2	yes	—
42	C₄⋊C₄×C₂₃	—	2	8	no	—
43	C₄⋊Dic₂₃	C₂×C₄	1	16	yes	complemented
44	Dic₂₃⋊C₄	—	2	16	no	—
45	C₄×Dic₂₃	—	2	8	no	—
46	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂³×C₂₃	C₂×D₄	1	2	yes	—
48	C₂³×C₂₃	C₂×D₄	1	2	yes	—
49	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
50	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
51	C₂³×C₂₃	#41	1	2	yes	—
52	C₂³×C₂₃	#41	1	2	yes	—
53	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	4	8	no	—
58	C₂×C₄×C₂₃	C₄⋊C₄	1	16	yes	complemented
59	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
62	C₂×Dic₂₃	—	2	16	no	—
63	C₂×Dic₂₃	—	2	8	no	—
64	C₂×Dic₂₃	—	2	8	no	—
65	C₂×Dic₂₃	—	2	2	no	—
66	C₂×Dic₂₃	—	2	2	no	—
67	C₂×Dic₂₃	—	2	4	no	—
68	C₂×Dic₂₃	—	2	2	no	—
69	C₂×Dic₂₃	—	2	4	no	—
70	C₂×Dic₂₃	—	2	2	no	—
71	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
72	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
73	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
74	C₂×#389	—	23	1	no	maximal, Sylow
75	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂²×C₂₃	C₄×D₄	1	4	yes	—
77	C₂²×C₂₃	C₄×Q₈	1	4	yes	—
78	C₂²×C₂₃	#121	1	4	yes	—
79	C₂²×C₂₃	C₂²⋊Q₈	1	4	yes	—
80	C₂²×C₂₃	#125	1	4	yes	—
81	C₂²×C₂₃	C₄⋊Q₈	1	4	yes	—
82	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
83	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
87	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
89	C₄×C₂₃	—	4	4	no	—
90	C₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	complemented
91	Dic₂₃	—	2	8	no	—
92	Dic₂₃	—	2	4	no	—
93	Dic₂₃	—	2	4	no	—
94	C₂²×C₄⋊C₄	—	23	1	no	—
95	C₂²×C₄⋊C₄	—	23	2	no	—
96	C₂²×C₄⋊C₄	D₂₃	1	1	yes	characteristic
97	C₂²×C₄²	—	23	1	no	—
98	#389	—	23	8	no	—
99	C₂×#95	—	23	2	no	—
100	C₂×C₂₃	#389	1	8	yes	complemented
101	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
104	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
105	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₂₃	C₂×#125	1	1	yes	characteristic
107	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
108	C₂³×C₄	—	23	2	no	—
109	C₂³×C₄	—	23	1	no	—
110	C₂³×C₄	—	23	1	no	—
111	C₂³×C₄	C₂×D₂₃	1	2	yes	—
112	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	23	8	no	—
115	C₂×C₄⋊C₄	—	23	4	no	—
116	C₂×C₄⋊C₄	—	23	8	no	—
117	C₂×C₄⋊C₄	—	46	8	no	—
118	C₂×C₄⋊C₄	C₂×D₂₃	1	4	yes	—
119	C₂×C₄²	—	23	4	no	—
120	C₂×C₄²	—	46	4	no	—
121	#95	—	23	8	no	—
122	C₂₃	C₂×#389	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	2	4	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	23	8	no	—
128	C₂²×C₄	—	23	4	no	—
129	C₂²×C₄	—	23	4	no	—
130	C₂²×C₄	—	23	2	no	—
131	C₂²×C₄	—	23	4	no	—
132	C₂²×C₄	—	23	2	no	—
133	C₂²×C₄	—	46	4	no	—
134	C₂²×C₄	—	46	2	no	—
135	C₂²×C₄	—	46	4	no	—
136	C₂²×C₄	—	46	2	no	—
137	C₂²×C₄	—	46	2	no	—
138	C₂²×C₄	—	46	2	no	—
139	C₂²×C₄	Dic₄₆	1	4	yes	—
140	C₂²×C₄	C₄×D₂₃	1	4	yes	—
141	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
142	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
143	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
144	C₄⋊C₄	—	2	8	no	—
145	C₄⋊C₄	—	23	16	no	—
146	C₄⋊C₄	—	46	16	no	—
147	C₄²	—	46	8	no	—
148	C₂³	C₂×Dic₄₆	1	2	yes	—
149	C₂³	C₂×C₄×D₂₃	1	2	yes	—
150	C₂³	D₄×D₂₃	1	2	yes	—
151	C₂³	#67290	1	2	yes	—
152	C₂³	Q₈×D₂₃	1	2	yes	—
153	C₂³	#67292	1	2	yes	—
154	C₂³	C₂×C₂²⋊D₂₃	1	2	yes	—
155	C₂³	C₂³×D₂₃	1	1	yes	characteristic
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	4	8	no	—
160	C₂×C₄	—	23	8	no	—
161	C₂×C₄	—	23	4	no	—
162	C₂×C₄	—	23	4	no	—
163	C₂×C₄	—	46	16	no	—
164	C₂×C₄	—	46	8	no	—
165	C₂×C₄	—	46	8	no	—
166	C₂×C₄	—	46	4	no	—
167	C₂×C₄	—	46	2	no	—
168	C₂×C₄	—	46	4	no	—
169	C₂×C₄	—	46	2	no	—
170	C₂×C₄	—	46	2	no	—
171	C₂×C₄	—	46	2	no	—
172	C₂×C₄	Dic₂₃⋊C₄	1	16	yes	complemented
173	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
174	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
175	C₂×C₄	C₂×C₂²⋊D₂₃	1	4	yes	—
176	C₂²	#10625058	1	4	yes	—
177	C₂²	#10625059	1	4	yes	—
178	C₂²	#10625061	1	4	yes	—
179	C₂²	#10625064	1	4	yes	—
180	C₂²	C₂×Dic₂₃⋊C₄	1	4	yes	—
181	C₂²	#10625109	1	4	yes	—
182	C₂²	#10625117	1	4	yes	—
183	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
184	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
185	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
186	C₂²	C₂×#67290	1	1	yes	characteristic
187	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
188	C₂²	C₂×#67292	1	1	yes	characteristic
189	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
190	C₄	—	4	4	no	—
191	C₄	—	46	8	no	—
192	C₄	—	46	4	no	—
193	C₄	—	46	4	no	—
194	C₄	C₂×Dic₂₃⋊C₄	1	8	yes	complemented
195	C₂	#13092313	1	8	yes	direct product factor
196	C₂	C₂²×Dic₂₃⋊C₄	1	1	yes	characteristic
197	C₂	C₂×#10625064	1	1	yes	characteristic
198	C₂	C₂×#10625058	1	1	yes	characteristic
199	C₂	C₂×#10625109	1	1	yes	characteristic
200	C₂	C₂×#10625117	1	1	yes	characteristic
201	C₂	C₂×#10625061	1	1	yes	characteristic
202	C₂	C₂×#10625059	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂²×C₄⋊C₄×C₂₃	C₂
socle, prime radical	C₂⁴×C₂₃	C₂³
cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₂₃
hypocenter	C₂₃	C₂×#389
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #13092313	C₂	direct (N × H)
14. C₂×C₄⋊Dic₂₃	C₄	semidirect
43. C₄⋊Dic₂₃	C₂×C₄	semidirect
58. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
90. C₄×C₂₃	C₂×C₄⋊C₄	semidirect
100. C₂×C₂₃	#389	semidirect
122. C₂₃	C₂×#389	semidirect
172. C₂×C₄	Dic₂₃⋊C₄	semidirect
194. C₄	C₂×Dic₂₃⋊C₄	semidirect
195. C₂	#13092313	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×C₄⋊Dic₂₃	2
C₂²×Dic₂₃⋊C₄	2
C₂²×C₄×Dic₂₃	2
#13092313	2
C₂×C₂₃⋊#95	2
C₂²×C₄⋊C₄×C₂₃	2
C₂×#389	23

Subgroups — C2×#13092313 #15675846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092313 #15675846