Subgroups — #15686499

Order 2944.

360 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625072	C₂	1	1	yes	maximal, characteristic
2	#13092835	C₂	1	4	yes	maximal, complemented
3	#13092833	C₂	1	4	yes	maximal, complemented
4	C₂×#10624994	C₂	1	1	yes	maximal, characteristic, complemented
5	M₃₂×D₂₃	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#67288	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₄	1	1	yes	characteristic
10	#10625072	C₄	1	2	yes	—
11	#10625072	C₄	1	2	yes	—
12	#10625072	C₂²	1	2	yes	—
13	#10625072	C₂²	1	2	yes	—
14	C₂×#67262	C₄	1	2	yes	—
15	C₂×C₈×D₂₃	C₂²	1	2	yes	—
16	C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂₃×M₃₂	C₂²	1	2	yes	complemented
18	C₂×C₁₆×C₂₃	C₂²	1	2	yes	complemented
19	#10624994	C₂²	1	2	yes	—
20	#10624994	C₂²	1	2	yes	complemented
21	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	complemented
22	#10624980	C₂²	1	8	yes	complemented
23	C₁₆×D₂₃	C₂²	1	8	yes	complemented
24	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₂²⋊D₂₃	C₈	1	2	yes	—
26	#67288	C₈	1	2	yes	—
27	#67288	C₈	1	2	yes	—
28	#67288	C₂×C₄	1	2	yes	—
29	#67288	C₂×C₄	1	2	yes	—
30	C₂×D₉₂	C₈	1	1	yes	characteristic
31	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₄₆	C₈	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	#67267	C₂×C₄	1	2	yes	—
40	#67267	C₂×C₄	1	2	yes	—
41	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
42	#67262	C₂×C₄	1	4	yes	—
43	#67262	C₂×C₄	1	4	yes	—
44	C₈×D₂₃	C₂×C₄	1	4	yes	—
45	C₈×D₂₃	C₂³	1	4	yes	—
46	C₁₆×C₂₃	C₂³	1	4	yes	—
47	C₂₃⋊C₁₆	C₂³	1	4	yes	—
48	C₂³×C₂₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₂₃	C₂×C₈	1	2	yes	—
50	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²⋊D₂₃	C₂×C₈	1	4	yes	—
57	C₂²⋊D₂₃	C₂×C₈	1	4	yes	—
58	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
59	D₉₂	C₂×C₈	1	2	yes	—
60	D₉₂	C₂×C₈	1	2	yes	—
61	C₄×D₂₃	C₂×C₈	1	4	yes	—
62	C₄×D₂₃	C₂²×C₄	1	4	yes	—
63	Dic₄₆	C₂×C₈	1	2	yes	—
64	Dic₄₆	C₂×C₈	1	2	yes	—
65	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₈×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
70	#3407	—	23	1	no	maximal, Sylow
71	C₂²×C₂₃	—	4	1	no	—
72	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
73	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
74	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×D₂₃	—	4	2	no	—
76	C₂×D₂₃	C₂²×C₈	1	4	yes	—
77	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
78	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
80	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
81	Dic₂₃	C₂²×C₈	1	4	yes	—
82	C₂×#131	—	23	1	no	—
83	#504	—	23	4	no	—
84	#504	—	23	4	no	—
85	C₂×M₃₂	—	23	1	no	—
86	C₂×M₃₂	—	23	4	no	—
87	C₂×M₃₂	D₂₃	1	1	yes	characteristic, complemented
88	C₂×C₂₃	—	2	1	no	—
89	C₂×C₂₃	—	2	1	no	—
90	C₂×C₂₃	—	2	1	no	—
91	C₂×C₂₃	C₂³×C₈	1	1	yes	characteristic
92	D₂₃	—	2	4	no	—
93	C₂×#41	—	23	1	no	—
94	#131	—	23	2	no	—
95	#131	—	23	2	no	—
96	#131	—	23	2	no	—
97	#131	—	23	2	no	—
98	C₂×M₁₆	—	23	1	no	—
99	C₂×M₁₆	—	23	2	no	—
100	C₂²×C₈	—	23	2	no	—
101	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
102	M₃₂	—	23	2	no	—
103	M₃₂	—	23	2	no	—
104	M₃₂	—	23	8	no	—
105	M₃₂	C₂×D₂₃	1	2	yes	—
106	M₃₂	C₂×D₂₃	1	2	yes	complemented
107	C₂×C₁₆	—	23	2	no	—
108	C₂×C₁₆	—	23	8	no	—
109	C₂×C₁₆	C₂×D₂₃	1	2	yes	complemented
110	C₂₃	#3407	1	1	yes	characteristic, complemented, Sylow
111	#41	—	23	2	no	—
112	#41	—	23	2	no	—
113	#41	—	23	2	no	—
114	#41	—	23	2	no	—
115	C₂×Q₈	—	23	1	no	—
116	C₂×D₄	—	23	2	no	—
117	C₂×D₄	—	23	1	no	—
118	C₂²×C₄	—	23	2	no	—
119	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
120	M₁₆	—	23	2	no	—
121	M₁₆	—	23	2	no	—
122	M₁₆	—	23	4	no	—
123	M₁₆	—	23	4	no	—
124	C₂×C₈	—	23	2	no	—
125	C₂×C₈	—	23	4	no	—
126	C₂×C₈	—	23	4	no	—
127	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
128	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
129	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
130	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
132	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
133	C₁₆	—	23	4	no	—
134	C₁₆	C₂²×D₂₃	1	4	yes	—
135	C₂³	—	23	2	no	—
136	C₂³	C₈×D₂₃	1	1	yes	characteristic
137	Q₈	—	23	2	no	—
138	Q₈	—	23	2	no	—
139	D₄	—	23	4	no	—
140	D₄	—	23	4	no	—
141	D₄	—	23	2	no	—
142	D₄	—	23	2	no	—
143	C₂×C₄	—	23	2	no	—
144	C₂×C₄	—	23	4	no	—
145	C₂×C₄	—	23	4	no	—
146	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
147	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
148	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
152	C₈	—	23	4	no	—
153	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
154	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
155	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
156	C₈	C₂³×D₂₃	1	1	yes	characteristic
157	C₂²	—	4	1	no	—
158	C₂²	—	23	4	no	—
159	C₂²	—	92	2	no	—
160	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
161	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
162	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
163	C₄	—	23	4	no	—
164	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
165	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
166	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
167	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	—	2	1	no	—
171	C₂	—	46	4	no	—
172	C₂	C₂²×C₈×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₂₃
commutator, socle	C₂×C₂₃	C₂³×C₈
Fitting	C₂×C₂₃×M₃₂	C₂
prime radical	C₂×#67288	C₄
cosocle	C₈×C₂₃	C₂⁴
hypercenter	C₂×M₃₂	D₂₃
hypocenter	C₂₃	#3407
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #13092835	C₂	semidirect
3. #13092833	C₂	semidirect
4. C₂×#10624994	C₂	semidirect
5. M₃₂×D₂₃	C₂	semidirect
6. C₂×C₂₃×M₃₂	C₂	semidirect
17. C₂₃×M₃₂	C₂²	semidirect
18. C₂×C₁₆×C₂₃	C₂²	semidirect
20. #10624994	C₂²	semidirect
21. C₂×C₂₃⋊C₁₆	C₂²	semidirect
22. #10624980	C₂²	semidirect
23. C₁₆×D₂₃	C₂²	semidirect
87. C₂×M₃₂	D₂₃	semidirect
106. M₃₂	C₂×D₂₃	semidirect
109. C₂×C₁₆	C₂×D₂₃	semidirect
110. C₂₃	#3407	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625072	2
#13092835	2
#13092833	2
C₂×#10624994	2
M₃₂×D₂₃	2
C₂×C₂₃×M₃₂	2
#3407	23