Subgroups — C₂×C₂₃×#428 #15670529

Order 2944.

536 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#428	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₃×#107	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#106	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×C₂₃	—	2	1	no	—
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₄	1	8	yes	complemented
10	C₂×M₁₆×C₂₃	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#117	—	2	2	no	—
13	C₂₃×#117	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
21	C₂×C₄²×C₂₃	—	2	1	no	—
22	C₂₃×#107	C₂²	1	8	yes	complemented
23	C₂₃×#106	C₂²	1	8	yes	complemented
24	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄×C₂₃	—	2	2	no	—
29	C₂²×C₄×C₂₃	—	4	4	no	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₃	D₄	1	2	yes	—
32	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
37	C₂×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
38	C₂×C₈×C₂₃	C₂³	1	4	yes	—
39	C₄⋊C₄×C₂₃	—	2	4	no	—
40	C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
44	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
45	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₂₃	—	2	4	no	—
47	C₄²×C₂₃	—	2	4	no	—
48	C₂³×C₂₃	—	2	2	no	—
49	C₂³×C₂₃	—	2	2	no	—
50	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
51	C₂³×C₂₃	C₂×D₄	1	2	yes	—
52	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	2	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	4	8	no	—
57	C₂×C₄×C₂₃	—	4	2	no	—
58	C₂×C₄×C₂₃	—	4	4	no	—
59	C₂×C₄×C₂₃	—	4	2	no	—
60	C₂×C₄×C₂₃	C₄⋊C₄	1	8	yes	—
61	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₈×C₂₃	C₂²×C₄	1	8	yes	complemented
70	C₂×#428	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	8	no	—
74	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂²×C₂₃	#137	1	2	yes	—
78	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₂₃	—	4	2	no	—
80	C₄×C₂₃	—	4	4	no	—
81	C₄×C₂₃	—	4	2	no	—
82	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×M₁₆	C₂×C₂₃	1	1	yes	characteristic
87	C₂×#117	C₂×C₂₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
89	#428	C₂×C₂₃	1	8	yes	direct product factor
90	C₂×#107	C₂×C₂₃	1	2	yes	complemented
91	C₂×#106	C₂×C₂₃	1	2	yes	complemented
92	C₂×C₂₃	—	2	4	no	—
93	C₂×C₂₃	#428	1	4	yes	direct product factor
94	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
97	C₂³×C₄	—	2	1	no	—
98	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
99	C₂×M₁₆	C₄×C₂₃	1	8	yes	complemented
100	C₂×M₁₆	C₂²×C₂₃	1	4	yes	—
101	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
102	#117	—	2	2	no	—
103	#117	C₂²×C₂₃	1	4	yes	—
104	C₂×C₄⋊C₄	—	2	1	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
108	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	—	2	1	no	—
111	C₂×C₄²	—	2	1	no	—
112	#107	C₂²×C₂₃	1	8	yes	complemented
113	#106	C₂²×C₂₃	1	8	yes	complemented
114	C₂₃	C₂×#428	1	1	yes	characteristic, direct product factor, Sylow
115	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	C₂×C₄×C₂₃	1	4	yes	—
122	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
123	C₂²×C₄	D₄×C₂₃	1	2	yes	—
124	C₂²×C₄	Q₈×C₂₃	1	4	yes	—
125	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
127	M₁₆	C₂×C₄×C₂₃	1	16	yes	complemented
128	C₂×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
129	C₂×C₈	C₂³×C₂₃	1	4	yes	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
135	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
136	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₄²	—	2	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	C₄⋊C₄×C₂₃	1	4	yes	—
142	C₂³	C₂×D₄×C₂₃	1	2	yes	—
143	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	C₄⋊C₄×C₂₃	1	8	yes	—
152	C₂×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
153	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
154	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
156	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
158	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
159	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
160	C₈	C₂²×C₄×C₂₃	1	8	yes	complemented
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
165	C₂²	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
166	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
167	C₂²	C₂₃×#137	1	2	yes	—
168	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
169	C₄	—	4	4	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
173	C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
174	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
175	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	C₂₃×#428	1	4	yes	direct product factor
178	C₂	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂	C₂×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#428	C₂	direct (N × H)
5. C₂×C₂₃×#107	C₂	semidirect
6. C₂×C₂₃×#106	C₂	semidirect
9. C₂×M₁₆×C₂₃	C₄	semidirect
22. C₂₃×#107	C₂²	semidirect
23. C₂₃×#106	C₂²	semidirect
36. M₁₆×C₂₃	C₂×C₄	semidirect
37. C₂×C₈×C₂₃	C₂×C₄	semidirect
69. C₈×C₂₃	C₂²×C₄	semidirect
70. C₂×#428	C₂₃	direct (N × H)
89. #428	C₂×C₂₃	direct (N × H)
90. C₂×#107	C₂×C₂₃	semidirect
91. C₂×#106	C₂×C₂₃	semidirect
93. C₂×C₂₃	#428	direct (N × H)
99. C₂×M₁₆	C₄×C₂₃	semidirect
112. #107	C₂²×C₂₃	semidirect
113. #106	C₂²×C₂₃	semidirect
114. C₂₃	C₂×#428	direct (N × H)
127. M₁₆	C₂×C₄×C₂₃	semidirect
128. C₂×C₈	C₂×C₄×C₂₃	semidirect
160. C₈	C₂²×C₄×C₂₃	semidirect
177. C₂	C₂₃×#428	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₂₃	2
C₂×C₂₃×#117	2
C₂²×C₄⋊C₄×C₂₃	2
C₂₃×#428	2
C₂×C₂₃×#107	2
C₂×C₂₃×#106	2
C₂×#428	23

Subgroups — C2×C23×#428 #15670529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#428 #15670529