Subgroups — #15675804

Order 2944.

656 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625097	C₂	1	2	yes	maximal
2	#13092287	C₂	1	2	yes	maximal, complemented
3	#13092283	C₂	1	2	yes	maximal, complemented
4	#13092277	C₂	1	4	yes	maximal
5	C₄×C₄⋊Dic₂₃	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₂₃	C₄	1	8	yes	complemented
10	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
14	#10625097	—	2	8	no	—
15	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₂₃	C₂²	1	8	yes	—
17	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
18	C₂₃⋊#95	C₂²	1	8	yes	—
19	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	8	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₂₃	D₄	1	4	yes	—
24	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₃	C₂³	1	8	yes	—
30	C₄²×C₂₃	C₂×C₄	1	4	yes	—
31	C₄²×C₂₃	C₂×C₄	1	8	yes	—
32	C₄²×C₂₃	C₂×C₄	1	4	yes	—
33	C₂²⋊Dic₂₃	—	2	8	no	—
34	C₄⋊Dic₂₃	—	2	8	no	—
35	C₄⋊Dic₂₃	—	2	16	no	—
36	C₄×Dic₂₃	—	2	8	no	—
37	C₂³×C₂₃	—	2	4	no	—
38	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
39	C₂³×C₂₃	C₂×D₄	1	2	yes	—
40	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
41	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	2	16	no	—
44	C₂×C₄×C₂₃	C₄⋊C₄	1	16	yes	complemented
45	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₃	#41	1	8	yes	—
54	C₂×Dic₂₃	—	2	8	no	—
55	C₂×Dic₂₃	—	2	16	no	—
56	#2285	—	23	1	no	maximal, Sylow
57	C₂²×C₂₃	—	2	4	no	—
58	C₂²×C₂₃	—	2	4	no	—
59	C₂²×C₂₃	—	2	4	no	—
60	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
61	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
62	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
63	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
64	C₂²×C₂₃	C₂×#41	1	4	yes	—
65	C₄×C₂₃	—	2	4	no	—
66	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₂₃	#117	1	8	yes	complemented
69	Dic₂₃	—	4	8	no	—
70	C₂×#117	—	23	2	no	—
71	C₂²×C₄²	D₂₃	1	1	yes	characteristic
72	#384	—	23	2	no	—
73	#382	—	23	2	no	—
74	#380	—	23	4	no	—
75	C₄×C₄⋊C₄	—	23	4	no	—
76	C₂×C₂₃	—	2	4	no	—
77	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₂₃	C₂×#117	1	2	yes	—
79	C₂×C₂₃	#532	1	2	yes	—
80	C₂×C₂₃	#533	1	2	yes	—
81	C₂³×C₄	C₂×D₂₃	1	2	yes	—
82	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
83	#117	—	46	8	no	—
84	C₂×C₄⋊C₄	—	23	4	no	—
85	C₂×C₄⋊C₄	—	23	8	no	—
86	C₂×C₂²⋊C₄	—	23	4	no	—
87	C₂×C₄²	—	23	4	no	—
88	C₂×C₄²	Dic₂₃	1	8	yes	complemented
89	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
90	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₂₃	1	2	yes	—
92	#95	—	23	8	no	—
93	C₂₃	#2285	1	1	yes	characteristic, complemented, Sylow
94	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
95	C₂²×C₄	—	2	8	no	—
96	C₂²×C₄	—	23	8	no	—
97	C₂²×C₄	Dic₄₆	1	4	yes	—
98	C₂²×C₄	D₉₂	1	4	yes	—
99	C₂²×C₄	C₂×Dic₂₃	1	8	yes	—
100	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
101	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	C₄⋊C₄	—	46	8	no	—
106	C₄⋊C₄	—	46	16	no	—
107	C₂²⋊C₄	—	46	8	no	—
108	C₄²	—	46	8	no	—
109	C₄²	C₂×Dic₂₃	1	4	yes	—
110	C₄²	C₂×Dic₂₃	1	8	yes	—
111	C₄²	C₂×Dic₂₃	1	4	yes	—
112	C₂³	—	2	4	no	—
113	C₂³	C₂×Dic₄₆	1	2	yes	—
114	C₂³	C₂×D₉₂	1	2	yes	—
115	C₂³	C₂²×Dic₂₃	1	2	yes	—
116	C₂³	C₂³×D₂₃	1	1	yes	characteristic
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	2	16	no	—
119	C₂×C₄	—	46	8	no	—
120	C₂×C₄	—	46	16	no	—
121	C₂×C₄	C₄⋊Dic₂₃	1	16	yes	complemented
122	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
123	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
124	C₂×C₄	C₂×D₉₂	1	2	yes	—
125	C₂×C₄	C₂×D₉₂	1	2	yes	—
126	C₂×C₄	#67288	1	8	yes	—
127	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
128	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
129	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
130	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
131	C₂²	—	2	4	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂×C₄⋊Dic₂₃	1	4	yes	—
135	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
136	C₂²	C₂²×D₉₂	1	1	yes	characteristic
137	C₂²	C₂×#67288	1	4	yes	—
138	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	92	8	no	—
141	C₄	C₂×C₄⋊Dic₂₃	1	4	yes	—
142	C₄	C₂×C₄⋊Dic₂₃	1	4	yes	—
143	C₄	#10625097	1	8	yes	complemented
144	C₂	—	2	4	no	—
145	C₂	C₂²×C₄⋊Dic₂₃	1	1	yes	characteristic
146	C₂	C₂×#10625097	1	2	yes	—
147	C₂	#13092967	1	2	yes	—
148	C₂	#13092974	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	#2285
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. #13092287	C₂	semidirect
3. #13092283	C₂	semidirect
5. C₄×C₄⋊Dic₂₃	C₂	semidirect
9. C₂×C₄²×C₂₃	C₄	semidirect
44. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
68. C₄×C₂₃	#117	semidirect
88. C₂×C₄²	Dic₂₃	semidirect
93. C₂₃	#2285	semidirect
121. C₂×C₄	C₄⋊Dic₂₃	semidirect
143. C₄	#10625097	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625097	2
#13092287	2
#13092283	2
#13092277	2
C₄×C₄⋊Dic₂₃	2
C₂²×C₄²×C₂₃	2
#2285	23