Subgroups — C₂×C₂₃×#3162 #19208283

Order 5888.

1902 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	2	yes	maximal
2	C₂₃×#3162	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₂₃×#501	C₂	1	8	yes	maximal, complemented
4	C₂×C₂₃×#427	C₂	1	4	yes	maximal
5	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂₃×#533	—	2	8	no	—
8	C₂₃×#533	C₂²	1	16	yes	—
9	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
10	C₂×C₂₃×#125	C₂²	1	4	yes	—
11	C₂×C₂₃×C₂²⋊Q₈	—	2	4	no	—
12	C₂×C₄×Q₈×C₂₃	—	2	4	no	—
13	C₂×C₂₃×#117	C₂²	1	4	yes	—
14	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#501	C₂²	1	64	yes	complemented
16	C₂₃×#427	C₂²	1	32	yes	—
17	C₂×C₂₃×#107	C₂²	1	8	yes	—
18	C₂×C₂₃×#106	C₂²	1	8	yes	—
19	C₂×C₂₃×#97	C₂²	1	4	yes	—
20	C₂×C₂₃×#97	C₂²	1	8	yes	—
21	C₂²×Q₈×C₂₃	—	2	4	no	—
22	C₂³×C₄×C₂₃	D₄	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	Q₈	1	16	yes	complemented
25	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
27	C₂₃×C₄⋊Q₈	—	2	8	no	—
28	C₂₃×C₄⋊Q₈	C₂³	1	16	yes	—
29	C₂₃×#125	—	2	8	no	—
30	C₂₃×#125	C₂³	1	16	yes	—
31	C₂₃×C₂²⋊Q₈	—	2	32	no	—
32	C₄×Q₈×C₂₃	—	2	32	no	—
33	C₂₃×#117	—	2	8	no	—
34	C₂₃×#117	C₂³	1	16	yes	—
35	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
38	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄²×C₂₃	—	2	4	no	—
40	C₂×C₄²×C₂₃	D₄	1	4	yes	—
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
44	C₂₃×#107	C₂³	1	32	yes	—
45	C₂₃×#106	C₂³	1	32	yes	—
46	C₂₃×#97	C₂³	1	16	yes	—
47	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	2	16	no	—
49	C₂×Q₈×C₂₃	—	4	16	no	—
50	C₂²×C₄×C₂₃	—	2	8	no	—
51	C₂²×C₄×C₂₃	—	4	4	no	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
57	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
59	C₂×C₈×C₂₃	C₂×Q₈	1	16	yes	—
60	C₂×C₈×C₂₃	C₂×Q₈	1	32	yes	complemented
61	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
62	C₄⋊C₄×C₂₃	—	2	32	no	—
63	C₄⋊C₄×C₂₃	—	2	16	no	—
64	C₄⋊C₄×C₂₃	—	2	8	no	—
65	C₄⋊C₄×C₂₃	C₂⁴	1	16	yes	—
66	C₂²⋊C₄×C₂₃	—	2	16	no	—
67	C₄²×C₂₃	—	2	16	no	—
68	C₄²×C₂₃	C₂×D₄	1	8	yes	—
69	C₄²×C₂₃	C₂×D₄	1	4	yes	—
70	C₄²×C₂₃	C₂⁴	1	4	yes	—
71	C₂×#3162	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	—	2	4	no	—
73	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
74	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
75	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₂₃	—	4	32	no	—
77	C₂×C₄×C₂₃	—	2	8	no	—
78	C₂×C₄×C₂₃	—	2	16	no	—
79	C₂×C₄×C₂₃	—	4	8	no	—
80	C₂×C₄×C₂₃	—	4	16	no	—
81	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
82	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
83	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
84	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
87	C₂×C₄×C₂₃	C₂²×Q₈	1	8	yes	—
88	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
89	C₈×C₂₃	C₂²×Q₈	1	16	yes	—
90	C₂×#533	C₂×C₂₃	1	2	yes	—
91	#3162	C₂×C₂₃	1	16	yes	direct product factor
92	C₂×#501	C₂×C₂₃	1	8	yes	complemented
93	C₂×#427	C₂×C₂₃	1	4	yes	—
94	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	—	2	8	no	—
96	C₂²×C₂₃	—	2	4	no	—
97	C₂²×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
98	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
99	C₂²×C₂₃	#575	1	4	yes	—
100	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₂₃	—	2	8	no	—
102	C₄×C₂₃	—	4	16	no	—
103	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
104	C₄×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
105	C₄×C₂₃	C₂³×Q₈	1	2	yes	—
106	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
107	#533	—	2	8	no	—
108	#533	C₂²×C₂₃	1	16	yes	—
109	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
110	C₂×#125	C₂²×C₂₃	1	4	yes	—
111	C₂×C₂²⋊Q₈	—	2	4	no	—
112	C₂×C₄×Q₈	—	2	4	no	—
113	C₂×#117	C₂²×C₂₃	1	4	yes	—
114	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
115	#501	C₂²×C₂₃	1	64	yes	complemented
116	#427	C₂²×C₂₃	1	32	yes	—
117	C₂×#107	C₂²×C₂₃	1	8	yes	—
118	C₂×#106	C₂²×C₂₃	1	8	yes	—
119	C₂×#97	C₂²×C₂₃	1	4	yes	—
120	C₂×#97	C₂²×C₂₃	1	8	yes	—
121	C₂×C₂₃	—	2	4	no	—
122	C₂×C₂₃	#3162	1	4	yes	direct product factor
123	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₂₃	C₂×#575	1	2	yes	—
125	C₂²×Q₈	—	2	4	no	—
126	C₂³×C₄	D₄×C₂₃	1	2	yes	—
127	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
128	C₂²×C₈	Q₈×C₂₃	1	16	yes	complemented
129	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
130	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
131	C₄⋊Q₈	—	2	8	no	—
132	C₄⋊Q₈	C₂³×C₂₃	1	16	yes	—
133	#125	—	2	8	no	—
134	#125	C₂³×C₂₃	1	16	yes	—
135	C₂²⋊Q₈	—	2	32	no	—
136	C₄×Q₈	—	2	32	no	—
137	#117	—	2	8	no	—
138	#117	C₂³×C₂₃	1	16	yes	—
139	C₂×C₄⋊C₄	—	2	8	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
142	C₂×C₂²⋊C₄	—	2	4	no	—
143	C₂×C₄²	—	2	4	no	—
144	C₂×C₄²	D₄×C₂₃	1	2	yes	—
145	C₂×C₄²	D₄×C₂₃	1	4	yes	—
146	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
147	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
148	#107	C₂³×C₂₃	1	32	yes	—
149	#106	C₂³×C₂₃	1	32	yes	—
150	#97	C₂³×C₂₃	1	16	yes	—
151	C₂₃	C₂×#3162	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
153	C₂×Q₈	—	2	16	no	—
154	C₂×Q₈	—	4	16	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
158	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
159	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
162	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
164	C₂×C₈	C₂×Q₈×C₂₃	1	16	yes	—
165	C₂×C₈	C₂×Q₈×C₂₃	1	32	yes	complemented
166	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
167	C₄⋊C₄	—	2	32	no	—
168	C₄⋊C₄	—	2	16	no	—
169	C₄⋊C₄	—	2	8	no	—
170	C₄⋊C₄	C₂⁴×C₂₃	1	16	yes	—
171	C₂²⋊C₄	—	2	16	no	—
172	C₄²	—	2	16	no	—
173	C₄²	C₂×D₄×C₂₃	1	8	yes	—
174	C₄²	C₂×D₄×C₂₃	1	4	yes	—
175	C₄²	C₂⁴×C₂₃	1	4	yes	—
176	C₂³	—	2	4	no	—
177	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
178	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
179	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
180	Q₈	—	4	32	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	—	4	16	no	—
185	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
186	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
187	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
188	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
189	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
191	C₂×C₄	C₂²×Q₈×C₂₃	1	8	yes	—
192	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
193	C₈	C₂²×Q₈×C₂₃	1	16	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
197	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
198	C₂²	C₂₃×#575	1	4	yes	—
199	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	4	16	no	—
202	C₄	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
203	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
204	C₄	C₂³×Q₈×C₂₃	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂₃×#3162	1	4	yes	direct product factor
207	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
208	C₂	C₂×C₂₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3162	C₂	direct (N × H)
3. C₂×C₂₃×#501	C₂	semidirect
15. C₂₃×#501	C₂²	semidirect
24. C₂²×C₈×C₂₃	Q₈	semidirect
60. C₂×C₈×C₂₃	C₂×Q₈	semidirect
71. C₂×#3162	C₂₃	direct (N × H)
91. #3162	C₂×C₂₃	direct (N × H)
92. C₂×#501	C₂×C₂₃	semidirect
115. #501	C₂²×C₂₃	semidirect
122. C₂×C₂₃	#3162	direct (N × H)
128. C₂²×C₈	Q₈×C₂₃	semidirect
151. C₂₃	C₂×#3162	direct (N × H)
165. C₂×C₈	C₂×Q₈×C₂₃	semidirect
206. C₂	C₂₃×#3162	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#533	2
C₂₃×#3162	2
C₂×C₂₃×#501	2
C₂×C₂₃×#427	2
C₂²×C₂₃×#97	2
C₂×#3162	23

Subgroups — C2×C23×#3162 #19208283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3162 #19208283