Subgroups — #15676017

Order 2944.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#13092315	C₂	1	4	yes	maximal, complemented
4	#13092289	C₂	1	4	yes	maximal
5	#13092286	C₂	1	2	yes	maximal, complemented
6	#13092282	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₂₃	C₂²	1	2	yes	—
10	C₂₃×#117	—	2	4	no	—
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
16	#10625097	C₄	1	8	yes	complemented
17	C₂×C₄⋊Dic₂₃	—	2	4	no	—
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
21	C₂×Dic₂₃⋊C₄	C₂²	1	8	yes	—
22	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
23	C₂₃⋊#95	C₂²	1	8	yes	—
24	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	D₄	1	4	yes	—
27	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₂₃	—	2	4	no	—
33	C₂²×Dic₂₃	—	4	4	no	—
34	C₂²×Dic₂₃	C₂³	1	4	yes	—
35	C₂²×Dic₂₃	C₂³	1	4	yes	—
36	C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂²⋊C₄×C₂₃	—	2	4	no	—
39	C₄²×C₂₃	—	2	4	no	—
40	C₂²⋊Dic₂₃	C₂×C₄	1	8	yes	—
41	C₄⋊Dic₂₃	—	2	4	no	—
42	C₄⋊Dic₂₃	—	2	4	no	—
43	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	complemented
44	Dic₂₃⋊C₄	—	4	8	no	—
45	C₄×Dic₂₃	C₂×C₄	1	8	yes	complemented
46	C₂³×C₂₃	—	2	2	no	—
47	C₂³×C₂₃	—	2	2	no	—
48	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₃	C₂×D₄	1	2	yes	—
50	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
51	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	C₄⋊C₄	1	16	yes	complemented
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
63	C₂×Dic₂₃	—	2	4	no	—
64	C₂×Dic₂₃	—	2	4	no	—
65	C₂×Dic₂₃	—	2	4	no	—
66	C₂×Dic₂₃	—	2	8	no	—
67	C₂×Dic₂₃	—	4	8	no	—
68	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
69	#2317	—	23	1	no	maximal, Sylow
70	C₂²×C₂₃	—	2	4	no	—
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	4	no	—
74	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
76	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
79	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
80	C₄×C₂₃	—	4	2	no	—
81	C₄×C₂₃	—	4	2	no	—
82	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
83	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
84	Dic₂₃	—	2	8	no	—
85	Dic₂₃	—	4	4	no	—
86	C₂×#117	—	23	1	no	—
87	C₂×#117	D₂₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	23	1	no	—
89	#389	—	23	4	no	—
90	#385	—	23	4	no	—
91	#384	—	23	2	no	—
92	#382	—	23	2	no	—
93	C₂×C₂₃	—	2	4	no	—
94	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₂₃	#520	1	2	yes	—
96	C₂×C₂₃	#535	1	1	yes	characteristic
97	C₂×C₂₃	#536	1	1	yes	characteristic
98	C₂×C₂₃	#544	1	2	yes	—
99	C₂³×C₄	—	23	2	no	—
100	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
101	#117	—	2	4	no	—
102	#117	—	23	8	no	—
103	C₂×C₄⋊C₄	—	23	2	no	—
104	C₂×C₄⋊C₄	—	23	2	no	—
105	C₂×C₄⋊C₄	—	23	2	no	—
106	C₂×C₄⋊C₄	—	23	8	no	—
107	C₂×C₄⋊C₄	—	46	4	no	—
108	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	23	2	no	—
111	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
112	C₂×C₄²	—	23	2	no	—
113	C₂×C₄²	C₂×D₂₃	1	2	yes	—
114	#95	—	23	8	no	—
115	C₂₃	#2317	1	1	yes	characteristic, complemented, Sylow
116	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
117	C₂²×C₄	—	23	4	no	—
118	C₂²×C₄	—	23	4	no	—
119	C₂²×C₄	—	46	4	no	—
120	C₂²×C₄	—	92	4	no	—
121	C₂²×C₄	Dic₄₆	1	4	yes	—
122	C₂²×C₄	C₄×D₂₃	1	4	yes	—
123	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
124	C₂²×C₄	C₂²×D₂₃	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	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	23	8	no	—
131	C₄⋊C₄	—	46	4	no	—
132	C₄⋊C₄	—	46	4	no	—
133	C₄⋊C₄	—	92	8	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	23	8	no	—
136	C₄²	—	2	4	no	—
137	C₄²	—	23	8	no	—
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Dic₄₆	1	2	yes	—
141	C₂³	C₂×C₄×D₂₃	1	2	yes	—
142	C₂³	C₂×C₂²⋊D₂₃	1	2	yes	—
143	C₂³	C₂³×D₂₃	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	23	8	no	—
149	C₂×C₄	—	46	4	no	—
150	C₂×C₄	—	46	4	no	—
151	C₂×C₄	—	46	8	no	—
152	C₂×C₄	—	46	4	no	—
153	C₂×C₄	—	92	8	no	—
154	C₂×C₄	Dic₂₃⋊C₄	1	16	yes	complemented
155	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
156	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
157	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
159	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
160	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×Dic₂₃⋊C₄	1	4	yes	—
166	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
167	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
168	C₂²	#10625160	1	2	yes	—
169	C₂²	#10625161	1	2	yes	—
170	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	4	2	no	—
173	C₄	—	46	8	no	—
174	C₄	—	92	4	no	—
175	C₄	C₂×Dic₂₃⋊C₄	1	4	yes	—
176	C₄	C₂×Dic₂₃⋊C₄	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×Dic₂₃⋊C₄	1	1	yes	characteristic
179	C₂	#13092897	1	2	yes	—
180	C₂	#13092981	1	1	yes	characteristic
181	C₂	#13092988	1	1	yes	characteristic
182	C₂	#13093045	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂×C₂₃×#117	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#2317
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #13092315	C₂	semidirect
5. #13092286	C₂	semidirect
6. #13092282	C₂	semidirect
16. #10625097	C₄	semidirect
43. C₄⋊Dic₂₃	C₂×C₄	semidirect
45. C₄×Dic₂₃	C₂×C₄	semidirect
56. C₂×C₄×C₂₃	C₄⋊C₄	semidirect
115. C₂₃	#2317	semidirect
154. C₂×C₄	Dic₂₃⋊C₄	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
#13092315	2
#13092289	2
#13092286	2
#13092282	2
C₂×C₂₃×#117	2
#2317	23