Subgroups — C₂₃×#7167 #19154793

Order 5888.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₆×C₂₃	—	2	1	no	—
5	C₂²×C₁₆×C₂₃	—	2	1	no	—
6	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₂₃	—	2	1	no	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂₃×C₄⋊C₁₆	—	2	1	no	—
11	C₂₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
12	C₄×C₁₆×C₂₃	—	2	1	no	—
13	C₄×C₁₆×C₂₃	C₄	1	2	yes	complemented
14	C₂₃×#336	—	2	1	no	—
15	C₂²×C₈×C₂₃	—	2	1	no	—
16	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₈	1	2	yes	—
19	C₂×C₄²×C₂₃	—	2	1	no	—
20	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₂₃	—	2	2	no	—
22	C₂×C₁₆×C₂₃	—	2	2	no	—
23	C₂×C₁₆×C₂₃	—	2	2	no	—
24	C₂×C₁₆×C₂₃	—	2	2	no	—
25	C₂×C₁₆×C₂₃	—	4	1	no	—
26	C₂×C₁₆×C₂₃	—	4	1	no	—
27	C₂₃×C₄⋊C₈	—	2	1	no	—
28	C₂₃×C₄⋊C₈	C₈	1	2	yes	—
29	C₄×C₈×C₂₃	—	4	1	no	—
30	C₄×C₈×C₂₃	—	4	1	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
35	C₂₃×#95	—	2	1	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₂₃	—	2	1	no	—
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	—	2	2	no	—
43	C₂×C₈×C₂₃	—	4	1	no	—
44	C₂×C₈×C₂₃	—	4	1	no	—
45	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₂₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	C₁₆	1	4	yes	complemented
55	C₄²×C₂₃	—	4	1	no	—
56	C₄²×C₂₃	—	4	1	no	—
57	C₄²×C₂₃	C₄²	1	1	yes	characteristic
58	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
61	C₁₆×C₂₃	—	2	4	no	—
62	C₁₆×C₂₃	—	4	2	no	—
63	#7167	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	4	2	no	—
68	C₂×C₄×C₂₃	—	4	2	no	—
69	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
73	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
75	C₂×C₄×C₂₃	#104	1	1	yes	characteristic
76	C₂×C₄×C₂₃	#104	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×C₁₆	1	2	yes	—
80	C₂×C₄×C₂₃	M₃₂	1	2	yes	—
81	C₈×C₂₃	—	2	1	no	—
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	4	2	no	—
84	C₈×C₂₃	#102	1	1	yes	characteristic
85	C₈×C₂₃	#103	1	1	yes	characteristic
86	C₈×C₂₃	#106	1	1	yes	characteristic
87	C₈×C₂₃	#107	1	1	yes	characteristic
88	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
89	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
90	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
91	C₂²×C₂₃	D₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
93	C₂²×C₂₃	#336	1	1	yes	characteristic
94	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
95	C₂²×C₂₃	#350	1	2	yes	—
96	C₄×C₂₃	—	2	1	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	4	4	no	—
99	C₄×C₂₃	D₄⋊C₈	1	1	yes	characteristic
100	C₄×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
101	C₄×C₂₃	#339	1	1	yes	characteristic
102	C₄×C₂₃	#340	1	1	yes	characteristic
103	C₄×C₂₃	C₄×C₁₆	1	1	yes	characteristic
104	C₄×C₂₃	#346	1	1	yes	characteristic
105	C₄×C₂₃	C₄⋊C₁₆	1	2	yes	—
106	C₂²×C₁₆	—	2	1	no	—
107	C₂²×C₁₆	—	2	1	no	—
108	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
109	C₂×C₄×C₈	—	2	1	no	—
110	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
112	C₄⋊C₁₆	—	2	1	no	—
113	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
114	C₄×C₁₆	—	2	1	no	—
115	C₄×C₁₆	C₄×C₂₃	1	2	yes	complemented
116	#336	—	2	1	no	—
117	C₂×C₂₃	#1279	1	1	yes	characteristic
118	C₂×C₂₃	D₄⋊C₁₆	1	2	yes	—
119	C₂×C₂₃	Q₈⋊C₁₆	1	2	yes	—
120	C₂×C₂₃	#1374	1	1	yes	characteristic
121	C₂×C₂₃	#1379	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
124	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₈×C₂₃	1	2	yes	—
126	C₂×C₄²	—	2	1	no	—
127	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	4	1	no	—
133	C₂×C₁₆	—	4	1	no	—
134	C₄⋊C₈	—	2	1	no	—
135	C₄⋊C₈	C₈×C₂₃	1	2	yes	—
136	C₄×C₈	—	4	1	no	—
137	C₄×C₈	—	4	1	no	—
138	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
139	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
141	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
142	#95	—	2	1	no	—
143	C₂₃	#7167	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
160	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	C₁₆×C₂₃	1	4	yes	complemented
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₄²×C₂₃	1	1	yes	characteristic
166	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
167	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
168	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
169	C₁₆	—	2	4	no	—
170	C₁₆	—	4	2	no	—
171	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
177	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
179	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
181	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
182	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
183	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
184	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
185	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
186	C₂×C₄	C₂×C₁₆×C₂₃	1	2	yes	—
187	C₂×C₄	C₂₃×M₃₂	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	C₂₃×#102	1	1	yes	characteristic
192	C₈	C₂₃×#103	1	1	yes	characteristic
193	C₈	C₂₃×#106	1	1	yes	characteristic
194	C₈	C₂₃×#107	1	1	yes	characteristic
195	C₂²	C₂₃×D₄⋊C₈	1	1	yes	characteristic
196	C₂²	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
197	C₂²	C₂₃×#336	1	1	yes	characteristic
198	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
199	C₂²	C₂₃×#350	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	C₂₃×D₄⋊C₈	1	1	yes	characteristic
204	C₄	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
205	C₄	C₂₃×#339	1	1	yes	characteristic
206	C₄	C₂₃×#340	1	1	yes	characteristic
207	C₄	C₄×C₁₆×C₂₃	1	1	yes	characteristic
208	C₄	C₂₃×#346	1	1	yes	characteristic
209	C₄	C₂₃×C₄⋊C₁₆	1	2	yes	—
210	C₂	C₂₃×#1279	1	1	yes	characteristic
211	C₂	C₂₃×D₄⋊C₁₆	1	2	yes	—
212	C₂	C₂₃×Q₈⋊C₁₆	1	2	yes	—
213	C₂	C₂₃×#1374	1	1	yes	characteristic
214	C₂	C₂₃×#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₃	D₄
commutator	C₄	C₄×C₁₆×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×C₄⋊C₁₆	C₄	semidirect
13. C₄×C₁₆×C₂₃	C₄	semidirect
54. C₄⋊C₄×C₂₃	C₁₆	semidirect
63. #7167	C₂₃	direct (N × H)
113. C₄⋊C₁₆	C₄×C₂₃	semidirect
115. C₄×C₁₆	C₄×C₂₃	semidirect
143. C₂₃	#7167	direct (N × H)
162. C₄⋊C₄	C₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₄×C₁₆×C₂₃	2
C₈×C₄⋊C₄×C₂₃	2
#7167	23

Subgroups — C23×#7167 #19154793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7167 #19154793