Subgroups — #15681815

Order 2944.

564 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67267	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
3	#13092375	C₂	1	4	yes	maximal, complemented
4	#13092372	C₂	1	4	yes	maximal, complemented
5	M₁₆×Dic₂₃	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₃	C₄	1	8	yes	complemented
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₂₃	C₄	1	2	yes	—
13	#10625097	C₄	1	2	yes	—
14	#10625097	C₄	1	2	yes	—
15	#10625097	C₂²	1	2	yes	—
16	#10625097	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₃	C₄	1	2	yes	—
18	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
19	C₂×#67267	C₄	1	8	yes	complemented
20	C₂×#67267	C₂²	1	2	yes	—
21	C₂×#67267	C₂²	1	2	yes	—
22	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
23	#10624997	C₂²	1	8	yes	complemented
24	C₈×Dic₂₃	C₂²	1	8	yes	complemented
25	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
35	M₁₆×C₂₃	C₂×C₄	1	8	yes	—
36	M₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
37	C₂×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
40	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
41	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
42	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
43	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
44	C₄×Dic₂₃	C₂³	1	4	yes	—
45	#67267	C₂×C₄	1	8	yes	—
46	#67267	C₂×C₄	1	8	yes	complemented
47	C₂×C₂₃⋊C₈	C₂×C₄	1	8	yes	complemented
48	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
49	C₂³×C₂₃	—	4	2	no	—
50	C₂³×C₂₃	C₄²	1	4	yes	—
51	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₄²	1	4	yes	—
55	C₂×C₄×C₂₃	C₄²	1	4	yes	—
56	C₂×C₄×C₂₃	C₄²	1	4	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₂×Dic₂₃	—	4	4	no	—
68	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
69	C₈×C₂₃	C₂²×C₄	1	8	yes	—
70	C₂₃⋊C₈	C₂²×C₄	1	8	yes	—
71	#2873	—	23	1	no	maximal, Sylow
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	2	no	—
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	4	4	no	—
76	C₂²×C₂₃	C₂×C₄²	1	2	yes	—
77	C₂²×C₂₃	C₂×C₄²	1	2	yes	—
78	C₂²×C₂₃	C₂×C₄²	1	2	yes	—
79	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₂₃	C₂×C₄²	1	2	yes	—
81	C₄×C₂₃	C₂×C₄²	1	2	yes	—
82	C₄×C₂₃	C₂×C₄²	1	2	yes	—
83	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
84	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
85	Dic₂₃	—	2	8	no	—
86	C₂²×M₁₆	—	23	1	no	—
87	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
88	C₂×#117	—	23	1	no	—
89	#405	—	23	4	no	—
90	#405	—	23	4	no	—
91	C₄×M₁₆	—	23	4	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	—	2	2	no	—
94	C₂×C₂₃	—	2	2	no	—
95	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
96	C₂×C₂₃	#568	1	2	yes	—
97	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
98	C₂×M₁₆	—	23	8	no	—
99	C₂×M₁₆	—	23	2	no	—
100	C₂×M₁₆	—	23	2	no	—
101	C₂×M₁₆	Dic₂₃	1	8	yes	complemented
102	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
103	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
104	C₂²×C₈	—	23	2	no	—
105	C₂²×C₈	C₂×D₂₃	1	2	yes	—
106	#117	—	23	2	no	—
107	#117	—	23	2	no	—
108	#117	—	23	2	no	—
109	#117	—	23	2	no	—
110	C₂×C₄⋊C₄	—	23	2	no	—
111	C₂×C₂²⋊C₄	—	23	2	no	—
112	C₂×C₄²	—	23	2	no	—
113	#97	—	23	8	no	—
114	C₄×C₈	—	23	8	no	—
115	C₂₃	#2873	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
117	C₂²×C₄	—	23	4	no	—
118	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₂₃	1	4	yes	—
122	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
123	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
126	M₁₆	—	23	8	no	—
127	M₁₆	—	23	8	no	—
128	M₁₆	C₂×Dic₂₃	1	8	yes	—
129	M₁₆	C₂×Dic₂₃	1	8	yes	complemented
130	C₂×C₈	—	23	8	no	—
131	C₂×C₈	—	23	4	no	—
132	C₂×C₈	C₂×Dic₂₃	1	8	yes	complemented
133	C₂×C₈	C₂²×D₂₃	1	4	yes	—
134	C₄⋊C₄	—	23	4	no	—
135	C₄⋊C₄	—	23	4	no	—
136	C₂²⋊C₄	—	23	4	no	—
137	C₂²⋊C₄	—	23	4	no	—
138	C₄²	—	23	4	no	—
139	C₄²	—	23	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₄×Dic₂₃	1	4	yes	—
142	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
143	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
144	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
145	C₂×C₄	—	23	8	no	—
146	C₂×C₄	—	92	4	no	—
147	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
148	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
149	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
150	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
155	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
156	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
157	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
158	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
159	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
160	C₈	—	23	8	no	—
161	C₈	C₂²×Dic₂₃	1	8	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	4	4	no	—
166	C₂²	C₂×C₄×Dic₂₃	1	2	yes	—
167	C₂²	C₂×C₄×Dic₂₃	1	2	yes	—
168	C₂²	C₂×C₄×Dic₂₃	1	2	yes	—
169	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
170	C₄	—	46	8	no	—
171	C₄	C₂×C₄×Dic₂₃	1	2	yes	—
172	C₄	C₂×C₄×Dic₂₃	1	2	yes	—
173	C₄	C₂×C₄×Dic₂₃	1	2	yes	—
174	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
175	C₄	C₂³×Dic₂₃	1	1	yes	characteristic
176	C₂	—	2	2	no	—
177	C₂	—	2	2	no	—
178	C₂	—	2	2	no	—
179	C₂	C₂²×C₄×Dic₂₃	1	1	yes	characteristic
180	C₂	#13093183	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂³×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#2873
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13092375	C₂	semidirect
4. #13092372	C₂	semidirect
5. M₁₆×Dic₂₃	C₂	semidirect
8. C₂×M₁₆×C₂₃	C₄	semidirect
19. C₂×#67267	C₄	semidirect
23. #10624997	C₂²	semidirect
24. C₈×Dic₂₃	C₂²	semidirect
36. M₁₆×C₂₃	C₂×C₄	semidirect
37. C₂×C₈×C₂₃	C₂×C₄	semidirect
46. #67267	C₂×C₄	semidirect
47. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
101. C₂×M₁₆	Dic₂₃	semidirect
115. C₂₃	#2873	semidirect
129. M₁₆	C₂×Dic₂₃	semidirect
132. C₂×C₈	C₂×Dic₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67267	2
C₂×#10625097	2
#13092375	2
#13092372	2
M₁₆×Dic₂₃	2
C₂²×M₁₆×C₂₃	2
#2873	23