Subgroups — #15682141

Order 2944.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625119	C₂	1	2	yes	maximal, complemented
2	#13092495	C₂	1	2	yes	maximal, complemented
3	#13092489	C₂	1	4	yes	maximal, complemented
4	D₄×C₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092384	C₂	1	2	yes	maximal, complemented
7	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#532	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₂₃	C₂²	1	2	yes	—
10	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂₃×#127	C₄	1	1	yes	characteristic
12	C₂₃×#124	C₄	1	2	yes	—
13	C₂₃×#121	C₄	1	2	yes	—
14	C₂₃×#121	C₄	1	2	yes	—
15	C₄×D₄×C₂₃	C₂²	1	2	yes	—
16	C₄×D₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	#10625119	—	2	8	no	—
19	C₂×#67267	C₂²	1	1	yes	characteristic
20	C₂×#67267	C₂²	1	4	yes	—
21	C₂×#67267	C₂²	1	1	yes	characteristic
22	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
23	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
24	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
25	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
26	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
28	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
29	#10624985	C₂²	1	2	yes	—
30	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
31	#41×C₂₃	—	2	4	no	—
32	#41×C₂₃	—	2	4	no	—
33	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
36	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
45	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
46	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	#67267	—	2	2	no	—
51	#67267	—	2	8	no	—
52	#67267	D₄	1	4	yes	complemented
53	C₂×C₂₃⋊C₈	—	2	4	no	—
54	C₂×C₂₃⋊C₈	—	2	8	no	—
55	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
57	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
58	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
59	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
60	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₂₃	—	2	4	no	—
62	D₄×C₂₃	—	2	8	no	—
63	D₄×C₂₃	—	2	4	no	—
64	D₄×C₂₃	—	2	4	no	—
65	D₄×C₂₃	—	4	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	4	no	—
69	C₂×C₄×C₂₃	—	4	1	no	—
70	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₃	#41	1	2	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂₃⋊C₈	—	2	2	no	—
80	C₂₃⋊C₈	—	2	2	no	—
81	C₂₃⋊C₈	—	2	2	no	—
82	C₂₃⋊C₈	C₂×D₄	1	4	yes	—
83	#2935	—	23	1	no	maximal, Sylow
84	C₂²×C₂₃	—	2	1	no	—
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	—	2	4	no	—
87	C₂²×C₂₃	—	2	4	no	—
88	C₂²×C₂₃	—	2	4	no	—
89	C₂²×C₂₃	C₄×D₄	1	2	yes	—
90	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₂₃	—	2	4	no	—
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	—	2	1	no	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	C₄×D₄	1	2	yes	—
96	C₄×C₂₃	#131	1	2	yes	complemented
97	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
99	C₂×#131	—	23	2	no	—
100	#532	D₂₃	1	1	yes	characteristic
101	#436	—	23	2	no	—
102	#435	—	23	4	no	—
103	C₈×D₄	—	23	2	no	—
104	C₂×C₄⋊C₈	—	23	1	no	—
105	#408	—	23	2	no	—
106	C₄×M₁₆	—	23	1	no	—
107	C₂×C₂₃	—	2	2	no	—
108	C₂×C₂₃	—	4	4	no	—
109	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
111	C₂×C₂₃	#568	1	1	yes	characteristic
112	C₂×#41	C₂×D₂₃	1	2	yes	—
113	#131	—	46	8	no	—
114	C₂×M₁₆	—	23	1	no	—
115	C₂×M₁₆	—	23	4	no	—
116	C₂×M₁₆	—	23	1	no	—
117	C₂²×C₈	—	23	2	no	—
118	C₂²×C₈	—	23	4	no	—
119	C₄⋊Q₈	Dic₂₃	1	1	yes	characteristic
120	#127	Dic₂₃	1	1	yes	characteristic
121	#124	Dic₂₃	1	2	yes	—
122	#121	Dic₂₃	1	2	yes	—
123	#121	Dic₂₃	1	2	yes	—
124	C₄×D₄	C₂×D₂₃	1	2	yes	—
125	C₄×D₄	C₂×D₂₃	1	2	yes	—
126	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
127	C₄⋊C₈	—	23	1	no	—
128	C₄⋊C₈	—	23	2	no	—
129	C₄⋊C₈	—	23	1	no	—
130	C₂²⋊C₈	—	23	4	no	—
131	C₂²⋊C₈	—	23	4	no	—
132	#97	—	23	2	no	—
133	C₄×C₈	—	23	2	no	—
134	C₂₃	#2935	1	1	yes	characteristic, complemented, Sylow
135	#41	—	2	4	no	—
136	#41	—	2	4	no	—
137	C₂×Q₈	C₂×Dic₂₃	1	2	yes	—
138	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
139	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
140	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
141	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
142	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
145	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
146	M₁₆	—	23	4	no	—
147	M₁₆	—	46	8	no	—
148	M₁₆	—	46	2	no	—
149	C₂×C₈	—	23	2	no	—
150	C₂×C₈	—	23	2	no	—
151	C₂×C₈	—	23	2	no	—
152	C₂×C₈	—	23	2	no	—
153	C₂×C₈	—	46	8	no	—
154	C₂×C₈	—	46	4	no	—
155	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
156	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
157	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
158	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
159	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
160	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
161	C₄²	C₂²×D₂₃	1	1	yes	characteristic
162	C₄²	C₂²×D₂₃	1	1	yes	characteristic
163	C₂³	C₂²×Dic₂₃	1	4	yes	—
164	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	D₄×D₂₃	1	2	yes	—
175	C₂×C₄	#67290	1	2	yes	—
176	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
180	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
183	C₈	—	23	4	no	—
184	C₈	—	46	2	no	—
185	C₈	—	46	2	no	—
186	C₈	—	46	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	D₄×Dic₂₃	1	2	yes	—
193	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	D₄×Dic₂₃	1	2	yes	—
199	C₄	#10625119	1	2	yes	complemented
200	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
201	C₄	C₂×#67290	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
205	C₂	C₂×#10625119	1	1	yes	characteristic
206	C₂	#13093184	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂₃×#532	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#532	D₂₃
hypocenter	C₂₃	#2935
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#10625119	C₂	semidirect
2. #13092495	C₂	semidirect
3. #13092489	C₂	semidirect
4. D₄×C₂₃⋊C₈	C₂	semidirect
5. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
6. #13092384	C₂	semidirect
7. C₄×#67267	C₂	semidirect
52. #67267	D₄	semidirect
96. C₄×C₂₃	#131	semidirect
134. C₂₃	#2935	semidirect
199. C₄	#10625119	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625119	2
#13092495	2
#13092489	2
D₄×C₂₃⋊C₈	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092384	2
C₄×#67267	2
C₂₃×#532	2
#2935	23