Subgroups — #15682005

Order 2944.

440 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	#13093183	C₂	1	2	yes	maximal
2	C₂×#10625072	C₂	1	1	yes	maximal, characteristic
3	#13092465	C₂	1	2	yes	maximal, complemented
4	#13092464	C₂	1	4	yes	maximal, complemented
5	C₂×#10625001	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₂₃	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67288	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67267	C₂²	1	2	yes	—
12	C₂×#67267	C₂²	1	1	yes	characteristic
13	#10625078	—	2	4	no	—
14	M₁₆×D₂₃	C₂²	1	8	yes	—
15	#10625072	—	2	4	no	—
16	#10625072	C₄	1	4	yes	—
17	#10625072	C₄	1	4	yes	—
18	C₂×#67262	C₂²	1	2	yes	—
19	C₂×C₈×D₂₃	C₂²	1	2	yes	—
20	C₂₃×#108	C₂²	1	2	yes	—
21	C₂₃×#108	C₂²	1	2	yes	—
22	#10625004	C₂²	1	8	yes	—
23	#10625001	C₂²	1	2	yes	—
24	#10625001	C₂²	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₂₃	Q₈	1	2	yes	—
27	#67288	C₂×C₄	1	2	yes	—
28	#67288	C₂×C₄	1	2	yes	—
29	#67288	D₄	1	2	yes	—
30	#67288	D₄	1	2	yes	—
31	C₂×D₉₂	Q₈	1	1	yes	characteristic
32	C₂×C₄×D₂₃	C₂³	1	2	yes	—
33	C₂×Dic₄₆	Q₈	1	1	yes	characteristic
34	M₁₆×C₂₃	—	2	2	no	—
35	M₁₆×C₂₃	C₂³	1	4	yes	—
36	C₂×C₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	#67267	—	2	2	no	—
42	#67267	C₂×C₄	1	2	yes	—
43	#67267	C₂×C₄	1	2	yes	—
44	#67267	C₂³	1	4	yes	—
45	C₂×C₂₃⋊C₈	—	2	2	no	—
46	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
47	#67262	—	2	8	no	—
48	#67262	C₂×C₄	1	8	yes	—
49	C₈×D₂₃	—	2	8	no	—
50	C₈×D₂₃	C₂×C₄	1	8	yes	—
51	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₂₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
59	C₂²⋊D₂₃	C₄⋊C₄	1	4	yes	—
60	C₂²⋊D₂₃	C₄⋊C₄	1	4	yes	—
61	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
62	D₉₂	C₄⋊C₄	1	2	yes	—
63	D₉₂	C₄⋊C₄	1	2	yes	—
64	C₄×D₂₃	C₂²×C₄	1	4	yes	—
65	C₄×D₂₃	C₂×D₄	1	4	yes	—
66	Dic₄₆	C₄⋊C₄	1	2	yes	—
67	Dic₄₆	C₄⋊C₄	1	2	yes	—
68	C₈×C₂₃	—	2	4	no	—
69	C₈×C₂₃	C₂²×C₄	1	2	yes	—
70	C₈×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂₃⋊C₈	—	2	4	no	—
72	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
73	#2916	—	23	1	no	maximal, Sylow
74	C₂²×C₂₃	—	4	1	no	—
75	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
78	C₂×D₂₃	—	4	2	no	—
79	C₂×D₂₃	C₂×C₄⋊C₄	1	4	yes	—
80	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
84	Dic₂₃	C₂×C₄⋊C₄	1	4	yes	—
85	#568	—	23	2	no	—
86	C₂×#131	—	23	1	no	—
87	#430	—	23	2	no	—
88	#430	—	23	4	no	—
89	C₂×#108	—	23	1	no	—
90	C₂×#108	—	23	4	no	—
91	C₂×#108	D₂₃	1	1	yes	characteristic, complemented
92	C₂×C₂₃	—	2	1	no	—
93	C₂×C₂₃	—	2	1	no	—
94	C₂×C₂₃	—	2	1	no	—
95	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	D₂₃	—	2	4	no	—
97	C₂×#41	—	23	1	no	—
98	#131	—	23	4	no	—
99	#131	—	23	4	no	—
100	#131	—	46	4	no	—
101	#131	—	46	4	no	—
102	C₂×M₁₆	—	23	1	no	—
103	C₂×M₁₆	—	23	2	no	—
104	C₂×M₁₆	—	23	2	no	—
105	C₂×M₁₆	—	23	8	no	—
106	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
107	C₂²×C₈	—	23	2	no	—
108	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
109	#108	—	23	8	no	—
110	#108	—	23	2	no	—
111	#108	—	23	2	no	—
112	#108	C₂×D₂₃	1	2	yes	—
113	#108	C₂×D₂₃	1	2	yes	—
114	C₂₃	#2916	1	1	yes	characteristic, complemented, Sylow
115	#41	—	23	2	no	—
116	#41	—	23	2	no	—
117	#41	—	23	2	no	—
118	#41	—	23	2	no	—
119	C₂×Q₈	—	23	1	no	—
120	C₂×D₄	—	23	2	no	—
121	C₂×D₄	—	23	1	no	—
122	C₂²×C₄	—	23	2	no	—
123	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	23	2	no	—
126	M₁₆	—	23	2	no	—
127	M₁₆	—	23	8	no	—
128	M₁₆	—	23	4	no	—
129	M₁₆	—	46	8	no	—
130	M₁₆	—	46	2	no	—
131	M₁₆	C₂²×D₂₃	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	23	2	no	—
134	C₂×C₈	—	23	8	no	—
135	C₂×C₈	—	46	8	no	—
136	C₂×C₈	—	46	2	no	—
137	C₂×C₈	C₄×D₂₃	1	2	yes	—
138	C₂×C₈	C₄×D₂₃	1	2	yes	—
139	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
141	C₂³	—	23	2	no	—
142	C₂³	Q₈×D₂₃	1	1	yes	characteristic
143	Q₈	—	23	2	no	—
144	Q₈	—	23	2	no	—
145	D₄	—	23	4	no	—
146	D₄	—	23	4	no	—
147	D₄	—	23	2	no	—
148	D₄	—	23	2	no	—
149	C₂×C₄	—	23	2	no	—
150	C₂×C₄	—	23	4	no	—
151	C₂×C₄	—	23	4	no	—
152	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
154	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
155	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	23	4	no	—
160	C₈	—	46	4	no	—
161	C₈	C₂×C₄×D₂₃	1	2	yes	—
162	C₈	C₂×C₄×D₂₃	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	23	4	no	—
165	C₂²	—	92	2	no	—
166	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
169	C₄	—	23	4	no	—
170	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₂₃	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
173	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	46	4	no	—
178	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#108	C₂
socle	C₂×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#108	D₂₃
hypocenter	C₂₃	#2916
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #13092465	C₂	semidirect
4. #13092464	C₂	semidirect
5. C₂×#10625001	C₂	semidirect
6. #108×D₂₃	C₂	semidirect
7. C₂×C₂₃×#108	C₂	semidirect
91. C₂×#108	D₂₃	semidirect
114. C₂₃	#2916	semidirect

Maximal subgroups

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

subgroup	index
#13093183	2
C₂×#10625072	2
#13092465	2
#13092464	2
C₂×#10625001	2
#108×D₂₃	2
C₂×C₂₃×#108	2
#2916	23