Subgroups — #15675753

Order 2944.

696 conjugacy classes of subgroups.

The subgroup lattice has 172 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	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₂₃	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊Dic₂₃	C₂	1	4	yes	maximal, complemented
5	C₂²⋊C₄×Dic₂₃	C₂	1	4	yes	maximal
6	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₃	C₂²	1	2	yes	—
9	C₂₃×#117	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
14	#10625097	C₄	1	8	yes	complemented
15	C₂×C₄⋊Dic₂₃	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₃	C₂²	1	8	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₂₃⋊#95	C₂²	1	8	yes	—
22	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₃	—	4	4	no	—
30	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
31	C₂²×Dic₂₃	C₂³	1	4	yes	—
32	C₂²×Dic₂₃	C₂³	1	4	yes	—
33	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
34	C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
35	C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
36	C₂²⋊Dic₂₃	C₂×C₄	1	8	yes	—
37	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	—
38	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	—
39	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	complemented
40	C₄×Dic₂₃	—	4	8	no	—
41	C₄×Dic₂₃	C₂×C₄	1	8	yes	complemented
42	C₂³×C₂₃	—	2	2	no	—
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
47	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	C₄²	1	16	yes	complemented
51	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×Dic₂₃	—	2	4	no	—
59	C₂×Dic₂₃	—	2	8	no	—
60	C₂×Dic₂₃	—	2	4	no	—
61	C₂×Dic₂₃	—	4	8	no	—
62	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
63	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
64	#2274	—	23	1	no	maximal, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	—	2	4	no	—
68	C₂²×C₂₃	—	2	4	no	—
69	C₂²×C₂₃	C₂×C₄²	1	4	yes	—
70	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
71	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
74	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
75	C₄×C₂₃	—	2	8	no	—
76	C₄×C₂₃	C₂×C₄²	1	4	yes	—
77	C₄×C₂₃	C₂×C₄²	1	4	yes	—
78	Dic₂₃	—	2	8	no	—
79	Dic₂₃	—	2	8	no	—
80	C₂×#117	—	23	1	no	—
81	C₂×#117	D₂₃	1	1	yes	characteristic
82	C₂²×C₄⋊C₄	—	23	1	no	—
83	C₄×C₄⋊C₄	—	23	4	no	—
84	C₄×C₄⋊C₄	—	23	4	no	—
85	C₄×C₂²⋊C₄	—	23	4	no	—
86	C₂×C₂₃	—	2	4	no	—
87	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₂₃	#518	1	1	yes	characteristic
89	C₂×C₂₃	#519	1	1	yes	characteristic
90	C₂×C₂₃	#520	1	2	yes	—
91	C₂×C₂₃	#520	1	2	yes	—
92	C₂³×C₄	—	23	2	no	—
93	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
94	#117	—	23	8	no	—
95	#117	Dic₂₃	1	8	yes	complemented
96	C₂×C₄⋊C₄	—	23	8	no	—
97	C₂×C₄⋊C₄	—	23	2	no	—
98	C₂×C₄⋊C₄	—	23	2	no	—
99	C₂×C₄⋊C₄	—	23	2	no	—
100	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
101	C₂×C₂²⋊C₄	—	23	2	no	—
102	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
103	C₂×C₄²	—	23	2	no	—
104	C₂×C₄²	—	23	8	no	—
105	C₂×C₄²	C₂×D₂₃	1	2	yes	—
106	#95	—	23	8	no	—
107	C₂₃	#2274	1	1	yes	characteristic, complemented, Sylow
108	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
109	C₂²×C₄	—	23	4	no	—
110	C₂²×C₄	—	23	4	no	—
111	C₂²×C₄	—	23	8	no	—
112	C₂²×C₄	—	92	4	no	—
113	C₂²×C₄	C₄×D₂₃	1	4	yes	—
114	C₂²×C₄	C₄×D₂₃	1	4	yes	—
115	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
116	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
117	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
119	C₄⋊C₄	—	23	8	no	—
120	C₄⋊C₄	—	23	8	no	—
121	C₄⋊C₄	—	23	8	no	—
122	C₄⋊C₄	C₂×Dic₂₃	1	8	yes	complemented
123	C₂²⋊C₄	—	23	8	no	—
124	C₂²⋊C₄	C₂×Dic₂₃	1	8	yes	—
125	C₄²	—	23	8	no	—
126	C₄²	—	92	8	no	—
127	C₄²	C₂×Dic₂₃	1	8	yes	complemented
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	C₂×C₄×D₂₃	1	2	yes	—
131	C₂³	C₂×C₄×D₂₃	1	2	yes	—
132	C₂³	C₂²×Dic₂₃	1	2	yes	—
133	C₂³	C₂³×D₂₃	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	23	8	no	—
137	C₂×C₄	—	23	8	no	—
138	C₂×C₄	—	46	4	no	—
139	C₂×C₄	—	46	8	no	—
140	C₂×C₄	—	46	4	no	—
141	C₂×C₄	—	92	8	no	—
142	C₂×C₄	C₄×Dic₂₃	1	16	yes	complemented
143	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
144	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
146	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
147	C₂×C₄	C₂²×Dic₂₃	1	8	yes	—
148	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
149	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	C₂×C₄×Dic₂₃	1	4	yes	—
155	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
156	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
157	C₂²	#10625160	1	2	yes	—
158	C₂²	#10625161	1	2	yes	—
159	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
160	C₄	—	2	8	no	—
161	C₄	—	46	8	no	—
162	C₄	—	46	8	no	—
163	C₄	C₂×C₄×Dic₂₃	1	4	yes	—
164	C₄	C₂×C₄×Dic₂₃	1	4	yes	—
165	C₂	—	2	4	no	—
166	C₂	C₂²×C₄×Dic₂₃	1	1	yes	characteristic
167	C₂	#13092891	1	1	yes	characteristic
168	C₂	#13092894	1	1	yes	characteristic
169	C₂	#13092897	1	2	yes	—
170	C₂	#13092900	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Fitting	C₂×C₂₃×#117	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#2274
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₄×Dic₂₃	C₂	semidirect
4. C₄×C₄⋊Dic₂₃	C₂	semidirect
9. C₂₃×#117	C₄	semidirect
14. #10625097	C₄	semidirect
15. C₂×C₄⋊Dic₂₃	C₄	semidirect
33. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
35. C₄²×C₂₃	C₂×C₄	semidirect
39. C₄⋊Dic₂₃	C₂×C₄	semidirect
41. C₄×Dic₂₃	C₂×C₄	semidirect
50. C₂×C₄×C₂₃	C₄²	semidirect
95. #117	Dic₂₃	semidirect
107. C₂₃	#2274	semidirect
122. C₄⋊C₄	C₂×Dic₂₃	semidirect
127. C₄²	C₂×Dic₂₃	semidirect
142. C₂×C₄	C₄×Dic₂₃	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
C₂²×C₄⋊Dic₂₃	2
C₄⋊C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₂²⋊C₄×Dic₂₃	2
C₂×C₂₃×#117	2
#2274	23