Subgroups — #15673123

Order 2944.

372 conjugacy classes of subgroups.

The subgroup lattice has 194 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	C₂²×#67267	C₂	1	1	yes	maximal, characteristic
3	#13092181	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₂₃	—	2	2	no	—
7	C₂×M₁₆×C₂₃	—	2	1	no	—
8	C₂×M₁₆×C₂₃	C₄	1	4	yes	—
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₃	—	2	1	no	—
11	C₂×#67267	—	2	2	no	—
12	C₂×#67267	—	2	2	no	—
13	C₂×#67267	—	2	1	no	—
14	C₂×#67267	—	2	2	no	—
15	C₂×#67267	C₄	1	4	yes	—
16	C₂×#67267	C₄	1	2	yes	—
17	C₂×#67267	C₄	1	2	yes	—
18	C₂×#67267	C₂²	1	2	yes	—
19	C₂²×C₂₃⋊C₈	—	2	1	no	—
20	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
21	C₂⁴×C₂₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃	D₄	1	2	yes	—
27	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₃	—	2	4	no	—
33	M₁₆×C₂₃	—	2	2	no	—
34	M₁₆×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×C₂₃	—	2	4	no	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	#67267	—	2	4	no	—
38	#67267	—	2	4	no	—
39	#67267	—	2	4	no	—
40	#67267	—	2	2	no	—
41	#67267	C₂×C₄	1	4	yes	—
42	C₂×C₂₃⋊C₈	—	2	4	no	—
43	C₂×C₂₃⋊C₈	—	2	2	no	—
44	C₂×C₂₃⋊C₈	—	2	2	no	—
45	C₂×C₂₃⋊C₈	—	2	2	no	—
46	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
47	C₂³×C₂₃	—	4	2	no	—
48	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
51	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₄²	1	2	yes	—
54	C₂×C₄×C₂₃	C₄²	1	2	yes	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₈×C₂₃	—	2	4	no	—
72	C₂₃⋊C₈	—	2	4	no	—
73	C₂₃⋊C₈	—	2	4	no	—
74	#1738	—	23	1	no	maximal, Sylow
75	C₂²×C₂₃	—	2	2	no	—
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	—	2	2	no	—
78	C₂²×C₂₃	—	4	2	no	—
79	C₂²×C₂₃	—	4	2	no	—
80	C₂²×C₂₃	#95	1	2	yes	—
81	C₂²×C₂₃	#95	1	2	yes	—
82	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄×C₂₃	#95	1	2	yes	—
86	C₄×C₂₃	#95	1	2	yes	—
87	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
88	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×M₁₆	—	23	1	no	—
92	C₂²×M₁₆	—	23	1	no	—
93	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
94	#341	—	23	4	no	—
95	C₂×C₂₃	—	2	2	no	—
96	C₂×C₂₃	—	2	2	no	—
97	C₂×C₂₃	—	2	1	no	—
98	C₂×C₂₃	—	2	1	no	—
99	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
100	C₂×C₂₃	#430	1	2	yes	—
101	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	23	2	no	—
105	C₂×M₁₆	—	23	2	no	—
106	C₂×M₁₆	—	23	4	no	—
107	C₂×M₁₆	—	23	2	no	—
108	C₂×M₁₆	—	46	2	no	—
109	C₂×M₁₆	—	46	2	no	—
110	C₂×M₁₆	—	46	2	no	—
111	C₂×M₁₆	—	46	1	no	—
112	C₂×M₁₆	Dic₂₃	1	4	yes	—
113	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	23	2	no	—
116	C₂²×C₈	—	46	1	no	—
117	C₂₃	#1738	1	1	yes	characteristic, complemented, Sylow
118	C₂⁴	Dic₄₆	1	1	yes	characteristic
119	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₂₃	1	2	yes	—
121	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
122	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
123	C₂²×C₄	D₉₂	1	2	yes	—
124	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
125	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
126	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
129	M₁₆	—	2	4	no	—
130	M₁₆	—	2	2	no	—
131	M₁₆	—	23	4	no	—
132	M₁₆	—	46	4	no	—
133	M₁₆	—	46	4	no	—
134	M₁₆	—	46	4	no	—
135	M₁₆	—	46	2	no	—
136	M₁₆	C₂×Dic₂₃	1	4	yes	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	23	4	no	—
140	C₂×C₈	—	46	4	no	—
141	C₂×C₈	—	46	2	no	—
142	C₂×C₈	—	46	2	no	—
143	C₂×C₈	—	46	2	no	—
144	C₂³	—	4	2	no	—
145	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
146	C₂³	Dic₂₃⋊C₄	1	1	yes	characteristic
147	C₂³	Dic₂₃⋊C₄	1	2	yes	—
148	C₂³	C₄⋊Dic₂₃	1	2	yes	—
149	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
150	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
151	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
152	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
153	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
154	C₂×C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
156	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
157	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
158	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
159	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
160	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
161	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
164	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
165	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
167	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
168	C₈	—	2	4	no	—
169	C₈	—	46	4	no	—
170	C₈	—	46	4	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	C₂₃⋊#95	1	2	yes	—
177	C₂²	C₂₃⋊#95	1	2	yes	—
178	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
179	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
180	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
181	C₄	C₂₃⋊#95	1	2	yes	—
182	C₄	C₂₃⋊#95	1	2	yes	—
183	C₄	C₂×C₄×Dic₂₃	1	1	yes	characteristic
184	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
185	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
186	C₄	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	1	no	—
189	C₂	—	2	2	no	—
190	C₂	—	2	2	no	—
191	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
192	C₂	#13092465	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	Q₈
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#1738
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13092181	C₂	semidirect
117. C₂₃	#1738	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
#13092181	2
C₂²×M₁₆×C₂₃	2
#1738	23