Subgroups — #15673118

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 188 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×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092179	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₃	—	2	1	no	—
7	C₂₃×#117	—	2	2	no	—
8	C₂₃×#117	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
11	C₂×C₄²×C₂₃	—	2	1	no	—
12	C₂×C₄⋊Dic₂₃	—	2	2	no	—
13	C₂×C₄⋊Dic₂₃	—	2	1	no	—
14	C₂×C₄⋊Dic₂₃	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
16	C₂×#67267	—	2	2	no	—
17	C₂×#67267	—	2	2	no	—
18	C₂×#67267	C₄	1	4	yes	complemented
19	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
20	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	D₄	1	2	yes	—
28	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×Dic₂₃	—	2	4	no	—
31	C₂²×Dic₂₃	—	2	2	no	—
32	C₂²×Dic₂₃	—	4	4	no	—
33	C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
35	C₂²⋊C₄×C₂₃	—	2	4	no	—
36	C₄²×C₂₃	—	2	4	no	—
37	C₄⋊Dic₂₃	—	2	4	no	—
38	C₄⋊Dic₂₃	—	2	2	no	—
39	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
40	#67267	—	2	8	no	—
41	C₂×C₂₃⋊C₈	—	2	2	no	—
42	C₂×C₂₃⋊C₈	—	2	2	no	—
43	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₂³×C₂₃	—	2	2	no	—
45	C₂³×C₂₃	—	2	2	no	—
46	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	—	4	2	no	—
53	C₂×C₄×C₂₃	C₄²	1	4	yes	—
54	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
60	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×Dic₂₃	—	2	4	no	—
67	C₂×Dic₂₃	—	4	8	no	—
68	C₂×Dic₂₃	—	4	4	no	—
69	C₂₃⋊C₈	—	2	4	no	—
70	#1736	—	23	1	no	maximal, Sylow
71	C₂²×C₂₃	—	2	4	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	2	no	—
74	C₂²×C₂₃	—	2	4	no	—
75	C₂²×C₂₃	#95	1	4	yes	—
76	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
80	C₂²×C₂₃	#137	1	2	yes	—
81	C₄×C₂₃	—	4	4	no	—
82	C₄×C₂₃	#95	1	4	yes	—
83	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
84	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	Dic₂₃	—	4	4	no	—
88	C₂²×M₁₆	—	23	1	no	—
89	C₂×#117	D₂₃	1	1	yes	characteristic
90	C₂²×C₄⋊C₄	—	23	1	no	—
91	#340	—	23	4	no	—
92	C₂×C₂₃	—	2	4	no	—
93	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
94	C₂×C₂₃	#417	1	2	yes	—
95	C₂×C₂₃	#418	1	1	yes	characteristic
96	C₂×C₂₃	#419	1	1	yes	characteristic
97	C₂×C₂₃	#428	1	2	yes	—
98	C₂³×C₄	—	46	1	no	—
99	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
100	C₂×M₁₆	—	23	4	no	—
101	C₂×M₁₆	—	46	2	no	—
102	C₂×M₁₆	—	46	2	no	—
103	C₂²×C₈	—	23	2	no	—
104	#117	—	2	2	no	—
105	#117	Dic₂₃	1	4	yes	complemented
106	C₂×C₄⋊C₄	—	23	4	no	—
107	C₂×C₄⋊C₄	—	23	2	no	—
108	C₂×C₄⋊C₄	—	46	2	no	—
109	C₂×C₄⋊C₄	—	46	1	no	—
110	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
111	C₂×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²	—	2	1	no	—
113	C₂₃	#1736	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	46	4	no	—
117	C₂²×C₄	—	46	2	no	—
118	C₂²×C₄	—	92	4	no	—
119	C₂²×C₄	Dic₄₆	1	2	yes	—
120	C₂²×C₄	C₄×D₂₃	1	2	yes	—
121	C₂²×C₄	C₄×D₂₃	1	2	yes	—
122	C₂²×C₄	D₉₂	1	2	yes	—
123	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
124	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
126	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
127	M₁₆	—	46	8	no	—
128	C₂×C₈	—	23	4	no	—
129	C₂×C₈	—	46	2	no	—
130	C₂×C₈	—	46	2	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	23	4	no	—
133	C₄⋊C₄	—	46	4	no	—
134	C₄⋊C₄	—	46	2	no	—
135	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₄²	—	2	4	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	Dic₂₃⋊C₄	1	2	yes	—
141	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
142	C₂³	C₂²⋊Dic₂₃	1	2	yes	—
143	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	46	4	no	—
148	C₂×C₄	—	92	8	no	—
149	C₂×C₄	—	92	4	no	—
150	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
151	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
152	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
153	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
154	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
155	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
156	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
157	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
160	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
163	C₈	—	46	4	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	C₂₃⋊#95	1	4	yes	—
169	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
170	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
171	C₂²	#10625120	1	2	yes	—
172	C₂²	#10625122	1	2	yes	—
173	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
174	C₄	—	4	4	no	—
175	C₄	—	92	4	no	—
176	C₄	C₂₃⋊#95	1	4	yes	—
177	C₄	C₂×C₄×Dic₂₃	1	1	yes	characteristic
178	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
179	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
180	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
183	C₂	#13092418	1	2	yes	—
184	C₂	#13092421	1	1	yes	characteristic
185	C₂	#13092425	1	1	yes	characteristic
186	C₂	#13092460	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#1736
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #13092179	C₂	semidirect
8. C₂₃×#117	C₄	semidirect
14. C₂×C₄⋊Dic₂₃	C₄	semidirect
18. C₂×#67267	C₄	semidirect
34. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
43. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
105. #117	Dic₂₃	semidirect
113. C₂₃	#1736	semidirect
135. C₄⋊C₄	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
C₂²×C₄⋊Dic₂₃	2
#13092179	2
C₂×C₂₃×#117	2
#1736	23