Subgroups — #15681960

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 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	#13092445	C₂	1	4	yes	maximal, complemented
4	#13092443	C₂	1	4	yes	maximal, complemented
5	#13092440	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	—	2	4	no	—
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	—	2	4	no	—
20	C₂×#67267	C₂²	1	2	yes	—
21	C₂×#67267	C₂²	1	2	yes	—
22	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
23	Dic₂₃⋊C₈	C₂²	1	16	yes	complemented
24	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×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	D₄	1	4	yes	—
29	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
34	M₁₆×C₂₃	—	2	4	no	—
35	M₁₆×C₂₃	—	2	4	no	—
36	C₂×C₈×C₂₃	—	2	4	no	—
37	C₂×C₈×C₂₃	C₂³	1	4	yes	—
38	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
39	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
40	C₄⋊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	C₂³	1	4	yes	—
44	#67267	—	2	4	no	—
45	#67267	—	2	4	no	—
46	C₂×C₂₃⋊C₈	—	2	4	no	—
47	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
48	C₂³×C₂₃	—	4	2	no	—
49	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
50	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₂₃	—	4	4	no	—
67	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
68	C₈×C₂₃	—	2	4	no	—
69	C₂₃⋊C₈	—	2	4	no	—
70	#2905	—	23	1	no	maximal, Sylow
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	2	no	—
74	C₂²×C₂₃	—	4	4	no	—
75	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
79	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
84	Dic₂₃	—	2	8	no	—
85	C₂²×M₁₆	—	23	1	no	—
86	C₂²×M₁₆	D₂₃	1	1	yes	characteristic
87	C₂×#117	—	23	1	no	—
88	#424	—	23	4	no	—
89	#424	—	23	4	no	—
90	#423	—	23	4	no	—
91	C₂×C₂₃	—	2	2	no	—
92	C₂×C₂₃	—	2	2	no	—
93	C₂×C₂₃	—	2	2	no	—
94	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₂₃	#568	1	2	yes	—
96	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
97	C₂×M₁₆	—	2	4	no	—
98	C₂×M₁₆	—	23	2	no	—
99	C₂×M₁₆	—	23	2	no	—
100	C₂×M₁₆	—	46	4	no	—
101	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
102	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
103	C₂²×C₈	—	23	2	no	—
104	C₂²×C₈	C₂×D₂₃	1	2	yes	—
105	#117	—	23	2	no	—
106	#117	—	23	2	no	—
107	#117	—	23	2	no	—
108	#117	—	23	2	no	—
109	C₂×C₄⋊C₄	—	23	2	no	—
110	C₂×C₂²⋊C₄	—	23	2	no	—
111	C₂×C₄²	—	23	2	no	—
112	C₄⋊C₈	—	23	16	no	—
113	C₂₃	#2905	1	1	yes	characteristic, complemented, Sylow
114	C₂⁴	C₄×D₂₃	1	1	yes	characteristic
115	C₂²×C₄	—	23	4	no	—
116	C₂²×C₄	Dic₄₆	1	4	yes	—
117	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
121	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	46	4	no	—
127	M₁₆	—	46	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	23	4	no	—
130	C₂×C₈	—	46	4	no	—
131	C₂×C₈	C₂²×D₂₃	1	4	yes	—
132	C₄⋊C₄	—	23	4	no	—
133	C₄⋊C₄	—	23	4	no	—
134	C₂²⋊C₄	—	23	4	no	—
135	C₂²⋊C₄	—	23	4	no	—
136	C₄²	—	23	4	no	—
137	C₄²	—	23	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	Dic₂₃⋊C₄	1	4	yes	—
140	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
143	C₂×C₄	—	23	8	no	—
144	C₂×C₄	—	92	4	no	—
145	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
146	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
147	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
148	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
149	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
150	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
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	1	yes	characteristic
154	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
156	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
157	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	46	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
165	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
166	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
167	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
168	C₄	—	46	8	no	—
169	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
170	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
171	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
172	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
173	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×Dic₂₃⋊C₄	1	1	yes	characteristic
178	C₂	#13093183	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂²×M₁₆×C₂₃	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#2905
LCS	G ▷ C₂²×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. #13092445	C₂	semidirect
4. #13092443	C₂	semidirect
5. #13092440	C₂	semidirect
23. Dic₂₃⋊C₈	C₂²	semidirect
113. C₂₃	#2905	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
#13092445	2
#13092443	2
#13092440	2
C₂²×M₁₆×C₂₃	2
#2905	23