Subgroups — C₂₃×#7170 #19154796

Order 5888.

262 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1770	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×C₂₃×#97	—	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	C₂₃×#346	—	2	1	no	—
13	C₂₃×#346	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	C₂₃×#97	—	4	1	no	—
30	C₂₃×#97	—	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	—	4	2	no	—
54	C₄²×C₂₃	—	4	1	no	—
55	C₄²×C₂₃	—	4	1	no	—
56	C₄²×C₂₃	C₄²	1	1	yes	characteristic
57	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
60	C₁₆×C₂₃	—	2	4	no	—
61	C₁₆×C₂₃	—	4	2	no	—
62	#7170	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	4	2	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	C₈×C₂₃	—	2	1	no	—
80	C₈×C₂₃	—	2	1	no	—
81	C₈×C₂₃	—	4	2	no	—
82	C₈×C₂₃	#102	1	1	yes	characteristic
83	C₈×C₂₃	#103	1	1	yes	characteristic
84	C₈×C₂₃	#106	1	1	yes	characteristic
85	C₈×C₂₃	#107	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
87	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
88	#1770	C₂×C₂₃	1	1	yes	characteristic
89	C₂²×C₂₃	D₄⋊C₈	1	1	yes	characteristic
90	C₂²×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
91	C₂²×C₂₃	#336	1	1	yes	characteristic
92	C₂²×C₂₃	#349	1	2	yes	—
93	C₂²×C₂₃	#350	1	2	yes	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	2	1	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	8	2	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	C₂²×C₁₆	—	2	1	no	—
104	C₂²×C₁₆	—	2	1	no	—
105	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
106	C₂×#97	—	2	1	no	—
107	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
108	C₄×C₄⋊C₄	C₄×C₂₃	1	1	yes	characteristic
109	C₄⋊C₁₆	—	2	1	no	—
110	C₄⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
111	#346	—	2	1	no	—
112	#346	C₄×C₂₃	1	2	yes	complemented
113	#336	—	2	1	no	—
114	C₂×C₂₃	#1279	1	1	yes	characteristic
115	C₂×C₂₃	#1330	1	2	yes	—
116	C₂×C₂₃	#1338	1	2	yes	—
117	C₂×C₂₃	#1377	1	1	yes	characteristic
118	C₂×C₂₃	#1379	1	1	yes	characteristic
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
121	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₈×C₂₃	1	2	yes	—
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	4	1	no	—
130	C₂×C₁₆	—	4	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	C₈×C₂₃	1	2	yes	—
133	#97	—	4	1	no	—
134	#97	—	4	1	no	—
135	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
138	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
139	#95	—	2	1	no	—
140	C₂₃	#7170	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
153	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
155	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
156	C₂×C₈	SD₁₆×C₂₃	1	1	yes	characteristic
157	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	—	4	2	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄²×C₂₃	1	1	yes	characteristic
162	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
163	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
164	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
174	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
175	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
176	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
177	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
178	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
179	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
180	C₂×C₄	C₂₃×#104	1	1	yes	characteristic
181	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
182	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	C₂₃×#102	1	1	yes	characteristic
187	C₈	C₂₃×#103	1	1	yes	characteristic
188	C₈	C₂₃×#106	1	1	yes	characteristic
189	C₈	C₂₃×#107	1	1	yes	characteristic
190	C₂²	C₂₃×D₄⋊C₈	1	1	yes	characteristic
191	C₂²	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
192	C₂²	C₂₃×#336	1	1	yes	characteristic
193	C₂²	C₂₃×#349	1	2	yes	—
194	C₂²	C₂₃×#350	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	C₂₃×D₄⋊C₈	1	1	yes	characteristic
201	C₄	C₂₃×Q₈⋊C₈	1	1	yes	characteristic
202	C₄	C₂₃×#339	1	1	yes	characteristic
203	C₄	C₂₃×#340	1	1	yes	characteristic
204	C₂	C₂₃×#1279	1	1	yes	characteristic
205	C₂	C₂₃×#1330	1	2	yes	—
206	C₂	C₂₃×#1338	1	2	yes	—
207	C₂	C₂₃×#1377	1	1	yes	characteristic
208	C₂	C₂₃×#1379	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₂₃
socle, prime radical	C₂³×C₂₃	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×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. C₂₃×#346	C₄	semidirect
62. #7170	C₂₃	direct (N × H)
110. C₄⋊C₁₆	C₄×C₂₃	semidirect
112. #346	C₄×C₂₃	semidirect
140. C₂₃	#7170	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×C₄⋊C₁₆	2
C₂×C₂₃×#346	2
C₂₃×#1770	2
#7170	23

Subgroups — C23×#7170 #19154796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7170 #19154796