Subgroups — C₂×C₂₃×#3458 #19210310

Order 5888.

7022 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3458	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₂₃×#544	C₂	1	12	yes	maximal, complemented
3	C₂×C₂₃×#543	C₂	1	4	yes	maximal
4	C₂×C₂₃×#533	C₂	1	12	yes	maximal, complemented
5	C₂²×C₂₃×#117	C₂	1	3	yes	maximal
6	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#544	C₂²	1	192	yes	complemented
8	C₂₃×#543	C₂²	1	64	yes	—
9	C₂₃×#533	C₂²	1	192	yes	complemented
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	24	yes	complemented
11	C₂×C₂₃×#125	C₂²	1	24	yes	complemented
12	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	48	yes	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	16	yes	complemented
14	C₂×C₂₃×#117	C₂²	1	12	yes	—
15	C₂×C₂₃×#117	C₂²	1	12	yes	—
16	C₂×C₂₃×#117	C₂²	1	48	yes	—
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	6	yes	—
19	C₂²×C₄²×C₂₃	C₂²	1	6	yes	—
20	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₂₃	C₂³	1	16	yes	—
22	C₂³×C₄×C₂₃	Q₈	1	16	yes	complemented
23	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
27	C₂₃×C₄⋊Q₈	C₂³	1	192	yes	complemented
28	C₂₃×#125	C₂³	1	192	yes	complemented
29	C₂₃×C₂²⋊Q₈	C₂³	1	384	yes	—
30	C₄×Q₈×C₂₃	C₂³	1	128	yes	complemented
31	C₂₃×#117	C₂³	1	96	yes	—
32	C₂₃×#117	C₂³	1	96	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	48	yes	—
36	C₂×C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	48	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
41	C₂⁴×C₂₃	—	4	4	no	—
42	C₂⁴×C₂₃	C₂×Q₈	1	8	yes	—
43	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
44	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
45	C₂×Q₈×C₂₃	—	4	32	no	—
46	C₂×Q₈×C₂₃	C₂⁴	1	64	yes	—
47	C₂²×C₄×C₂₃	—	4	24	no	—
48	C₂²×C₄×C₂₃	C₂×Q₈	1	24	yes	—
49	C₂²×C₄×C₂₃	C₂×Q₈	1	24	yes	—
50	C₂²×C₄×C₂₃	C₂×Q₈	1	64	yes	complemented
51	C₂²×C₄×C₂₃	C₂⁴	1	24	yes	—
52	C₂²×C₄×C₂₃	C₂⁴	1	48	yes	—
53	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
58	C₄⋊C₄×C₂₃	C₂⁴	1	192	yes	—
59	C₂²⋊C₄×C₂₃	C₂⁴	1	96	yes	—
60	C₄²×C₂₃	C₂⁴	1	48	yes	—
61	C₄²×C₂₃	C₂⁴	1	48	yes	—
62	C₂×#3458	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	—	2	12	no	—
64	C₂³×C₂₃	—	4	16	no	—
65	C₂³×C₂₃	—	4	8	no	—
66	C₂³×C₂₃	C₂²×Q₈	1	12	yes	—
67	C₂³×C₂₃	C₂²×Q₈	1	16	yes	—
68	C₂³×C₂₃	C₂⁵	1	6	yes	—
69	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
70	Q₈×C₂₃	—	4	64	no	—
71	C₂×C₄×C₂₃	—	2	48	no	—
72	C₂×C₄×C₂₃	—	4	48	no	—
73	C₂×C₄×C₂₃	C₂²×Q₈	1	12	yes	—
74	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
75	C₂×C₄×C₂₃	C₂²×Q₈	1	48	yes	—
76	C₂×C₄×C₂₃	C₂²×Q₈	1	48	yes	—
77	C₂×C₄×C₂₃	C₂⁵	1	48	yes	—
78	C₂×C₄×C₂₃	C₂⁵	1	2	yes	—
79	C₂×C₄×C₂₃	C₂⁵	1	6	yes	—
80	#3458	C₂×C₂₃	1	32	yes	direct product factor
81	C₂×#544	C₂×C₂₃	1	12	yes	complemented
82	C₂×#543	C₂×C₂₃	1	4	yes	—
83	C₂×#533	C₂×C₂₃	1	12	yes	complemented
84	C₂²×#117	C₂×C₂₃	1	3	yes	—
85	C₂²×C₂₃	—	2	12	no	—
86	C₂²×C₂₃	—	2	24	no	—
87	C₂²×C₂₃	—	4	16	no	—
88	C₂²×C₂₃	C₂³×Q₈	1	12	yes	—
89	C₂²×C₂₃	C₂³×Q₈	1	2	yes	—
90	C₂²×C₂₃	#585	1	4	yes	—
91	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
92	C₄×C₂₃	—	2	48	no	—
93	C₄×C₂₃	C₂³×Q₈	1	12	yes	—
94	C₄×C₂₃	C₂³×Q₈	1	4	yes	—
95	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
96	#544	C₂²×C₂₃	1	192	yes	complemented
97	#543	C₂²×C₂₃	1	64	yes	—
98	#533	C₂²×C₂₃	1	192	yes	complemented
99	C₂×C₄⋊Q₈	C₂²×C₂₃	1	24	yes	complemented
100	C₂×#125	C₂²×C₂₃	1	24	yes	complemented
101	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	48	yes	—
102	C₂×C₄×Q₈	C₂²×C₂₃	1	16	yes	complemented
103	C₂×#117	C₂²×C₂₃	1	12	yes	—
104	C₂×#117	C₂²×C₂₃	1	12	yes	—
105	C₂×#117	C₂²×C₂₃	1	48	yes	—
106	C₂²×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
107	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	6	yes	—
108	C₂²×C₄²	C₂²×C₂₃	1	6	yes	—
109	C₂×C₂₃	—	2	12	no	—
110	C₂×C₂₃	#3458	1	4	yes	direct product factor
111	C₂×C₂₃	C₂⁴×Q₈	1	1	yes	characteristic
112	C₂×C₂₃	C₂×#585	1	2	yes	—
113	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
114	C₂²×Q₈	C₂³×C₂₃	1	16	yes	—
115	C₂³×C₄	Q₈×C₂₃	1	16	yes	complemented
116	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
117	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
118	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
119	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
120	C₄⋊Q₈	C₂³×C₂₃	1	192	yes	complemented
121	#125	C₂³×C₂₃	1	192	yes	complemented
122	C₂²⋊Q₈	C₂³×C₂₃	1	384	yes	—
123	C₄×Q₈	C₂³×C₂₃	1	128	yes	complemented
124	#117	C₂³×C₂₃	1	96	yes	—
125	#117	C₂³×C₂₃	1	96	yes	—
126	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
127	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
128	C₂×C₄⋊C₄	C₂³×C₂₃	1	48	yes	—
129	C₂×C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
130	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
131	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
132	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
133	C₂×C₄²	C₂³×C₂₃	1	48	yes	—
134	C₂₃	C₂×#3458	1	1	yes	characteristic, direct product factor, Sylow
135	C₂⁴	—	4	4	no	—
136	C₂⁴	C₂×Q₈×C₂₃	1	8	yes	—
137	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
138	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
139	C₂×Q₈	—	4	32	no	—
140	C₂×Q₈	C₂⁴×C₂₃	1	64	yes	—
141	C₂²×C₄	—	4	24	no	—
142	C₂²×C₄	C₂×Q₈×C₂₃	1	24	yes	—
143	C₂²×C₄	C₂×Q₈×C₂₃	1	24	yes	—
144	C₂²×C₄	C₂×Q₈×C₂₃	1	64	yes	complemented
145	C₂²×C₄	C₂⁴×C₂₃	1	24	yes	—
146	C₂²×C₄	C₂⁴×C₂₃	1	48	yes	—
147	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
148	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
149	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
150	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
151	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
152	C₄⋊C₄	C₂⁴×C₂₃	1	192	yes	—
153	C₂²⋊C₄	C₂⁴×C₂₃	1	96	yes	—
154	C₄²	C₂⁴×C₂₃	1	48	yes	—
155	C₄²	C₂⁴×C₂₃	1	48	yes	—
156	C₂³	—	2	12	no	—
157	C₂³	—	4	16	no	—
158	C₂³	—	4	8	no	—
159	C₂³	C₂²×Q₈×C₂₃	1	12	yes	—
160	C₂³	C₂²×Q₈×C₂₃	1	16	yes	—
161	C₂³	C₂⁵×C₂₃	1	6	yes	—
162	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
163	Q₈	—	4	64	no	—
164	C₂×C₄	—	2	48	no	—
165	C₂×C₄	—	4	48	no	—
166	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
167	C₂×C₄	C₂²×Q₈×C₂₃	1	12	yes	—
168	C₂×C₄	C₂²×Q₈×C₂₃	1	48	yes	—
169	C₂×C₄	C₂²×Q₈×C₂₃	1	48	yes	—
170	C₂×C₄	C₂⁵×C₂₃	1	48	yes	—
171	C₂×C₄	C₂⁵×C₂₃	1	2	yes	—
172	C₂×C₄	C₂⁵×C₂₃	1	6	yes	—
173	C₂²	—	2	24	no	—
174	C₂²	—	2	12	no	—
175	C₂²	—	4	16	no	—
176	C₂²	C₂³×Q₈×C₂₃	1	12	yes	—
177	C₂²	C₂³×Q₈×C₂₃	1	2	yes	—
178	C₂²	C₂₃×#585	1	4	yes	—
179	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
180	C₄	—	2	48	no	—
181	C₄	C₂³×Q₈×C₂₃	1	4	yes	—
182	C₄	C₂³×Q₈×C₂₃	1	12	yes	—
183	C₂	—	2	12	no	—
184	C₂	C₂₃×#3458	1	4	yes	direct product factor
185	C₂	C₂⁴×Q₈×C₂₃	1	1	yes	characteristic
186	C₂	C₂×C₂₃×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂⁵×C₂₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3458	C₂	direct (N × H)
2. C₂×C₂₃×#544	C₂	semidirect
4. C₂×C₂₃×#533	C₂	semidirect
7. C₂₃×#544	C₂²	semidirect
9. C₂₃×#533	C₂²	semidirect
10. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
11. C₂×C₂₃×#125	C₂²	semidirect
13. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
22. C₂³×C₄×C₂₃	Q₈	semidirect
27. C₂₃×C₄⋊Q₈	C₂³	semidirect
28. C₂₃×#125	C₂³	semidirect
30. C₄×Q₈×C₂₃	C₂³	semidirect
50. C₂²×C₄×C₂₃	C₂×Q₈	semidirect
62. C₂×#3458	C₂₃	direct (N × H)
80. #3458	C₂×C₂₃	direct (N × H)
81. C₂×#544	C₂×C₂₃	semidirect
83. C₂×#533	C₂×C₂₃	semidirect
96. #544	C₂²×C₂₃	semidirect
98. #533	C₂²×C₂₃	semidirect
99. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
100. C₂×#125	C₂²×C₂₃	semidirect
102. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
110. C₂×C₂₃	#3458	direct (N × H)
115. C₂³×C₄	Q₈×C₂₃	semidirect
120. C₄⋊Q₈	C₂³×C₂₃	semidirect
121. #125	C₂³×C₂₃	semidirect
123. C₄×Q₈	C₂³×C₂₃	semidirect
134. C₂₃	C₂×#3458	direct (N × H)
144. C₂²×C₄	C₂×Q₈×C₂₃	semidirect
184. C₂	C₂₃×#3458	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3458	2
C₂×C₂₃×#544	2
C₂×C₂₃×#543	2
C₂×C₂₃×#533	2
C₂²×C₂₃×#117	2
C₂×#3458	23

Subgroups — C2×C23×#3458 #19210310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3458 #19210310