Subgroups — C₂×#13092474 #15682018

Order 2944.

488 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092474	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#10624985	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₄	1	4	yes	—
11	C₂×C₄²×C₂₃	C₄	1	1	yes	characteristic
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	4	yes	—
15	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
16	C₂²⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
17	C₄⋊(C₂₃⋊C₈)	C₂²	1	8	yes	complemented
18	#10624985	C₂²	1	8	yes	complemented
19	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	4	no	—
21	C₂²×C₄×C₂₃	—	2	4	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₄²×C₂₃	C₂×C₄	1	4	yes	—
34	C₄²×C₂₃	C₂×C₄	1	4	yes	—
35	C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₄²×C₂₃	C₂³	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂×C₂₃⋊C₈	—	2	8	no	—
40	C₂×C₂₃⋊C₈	C₂³	1	8	yes	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	—	2	2	no	—
43	C₂³×C₂₃	M₁₆	1	4	yes	—
44	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
45	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	8	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	—	2	8	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₂₃	#41	1	4	yes	—
62	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₂₃⋊C₈	—	2	8	no	—
64	C₂×#432	—	23	1	no	maximal, Sylow
65	C₂²×C₂₃	—	2	2	no	—
66	C₂²×C₂₃	—	2	2	no	—
67	C₂²×C₂₃	—	2	8	no	—
68	C₂²×C₂₃	#117	1	2	yes	—
69	C₂²×C₂₃	C₂×M₁₆	1	4	yes	—
70	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
71	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
72	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	—	2	2	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	#117	1	2	yes	—
78	C₄×C₂₃	C₂×M₁₆	1	4	yes	—
79	C₄×C₂₃	C₂×#41	1	2	yes	—
80	C₂²×C₄²	D₂₃	1	1	yes	characteristic
81	#432	—	23	8	no	—
82	C₂×C₄⋊C₈	—	23	2	no	—
83	C₂×C₂²⋊C₈	—	23	2	no	—
84	C₂×#97	—	23	2	no	—
85	C₂×C₂₃	—	2	4	no	—
86	C₂×C₂₃	#432	1	4	yes	complemented
87	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
88	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
89	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
90	C₂³×C₄	Dic₂₃	1	1	yes	characteristic
91	C₂³×C₄	Dic₂₃	1	1	yes	characteristic
92	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
93	C₂²×C₈	—	23	4	no	—
94	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
95	C₂×C₄²	Dic₂₃	1	4	yes	—
96	C₂×C₄²	Dic₂₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₂₃	1	4	yes	—
98	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
100	C₄⋊C₈	—	23	8	no	—
101	C₂²⋊C₈	—	23	8	no	—
102	#97	—	23	8	no	—
103	C₂₃	C₂×#432	1	1	yes	characteristic, complemented, Sylow
104	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
110	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
111	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
115	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
117	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
118	C₂×C₈	—	23	8	no	—
119	C₂×C₈	—	46	8	no	—
120	C₄²	C₂×Dic₂₃	1	4	yes	—
121	C₄²	C₂×Dic₂₃	1	4	yes	—
122	C₄²	C₂×Dic₂₃	1	2	yes	—
123	C₄²	C₂×Dic₂₃	1	2	yes	—
124	C₄²	C₂²×D₂₃	1	2	yes	—
125	C₄²	C₂²×D₂₃	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	#67267	1	4	yes	—
129	C₂³	C₂²×Dic₂₃	1	2	yes	—
130	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#67267	1	4	yes	—
140	C₂×C₄	#67288	1	4	yes	—
141	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
142	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
143	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
144	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
145	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
146	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
147	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
148	C₈	—	46	8	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	8	no	—
152	C₂²	C₂×#67267	1	4	yes	—
153	C₂²	C₂×#67267	1	2	yes	—
154	C₂²	C₂×#67267	1	2	yes	—
155	C₂²	#10625097	1	2	yes	—
156	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	C₂×#67267	1	4	yes	—
162	C₄	#10625097	1	2	yes	—
163	C₄	C₂×#67288	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	#13092474	1	4	yes	direct product factor
166	C₂	C₂×#10625097	1	1	yes	characteristic
167	C₂	C₂²×#67267	1	1	yes	characteristic
168	C₂	C₂²×#67267	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle	C₂³×C₂₃	C₂²×C₄
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₂×#432
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
1. #13092474	C₂	direct (N × H)
2. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
4. C₂×#10624985	C₂	semidirect
17. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
18. #10624985	C₂²	semidirect
86. C₂×C₂₃	#432	semidirect
103. C₂₃	C₂×#432	semidirect
165. C₂	#13092474	direct (N × H)

Maximal subgroups

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

subgroup	index
#13092474	2
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
C₂×#10624985	2
C₂²×C₄²×C₂₃	2
C₂×#432	23

Subgroups — C2×#13092474 #15682018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092474 #15682018