Subgroups — C₂₃×#11600 #19159226

Order 5888.

386 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1334	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
7	C₂×C₂₃×M₃₂	—	2	1	no	—
8	C₄×Q₁₆×C₂₃	—	2	1	no	—
9	C₄×Q₁₆×C₂₃	—	2	1	no	—
10	C₄×Q₁₆×C₂₃	C₄	1	2	yes	—
11	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#431	—	2	1	no	—
13	C₂×C₂₃×#107	C₄	1	1	yes	characteristic
14	C₂×C₂₃×#103	—	2	1	no	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#346	C₂²	1	2	yes	—
17	C₄×C₁₆×C₂₃	—	2	1	no	—
18	C₂₃×#335	—	2	1	no	—
19	C₂₃×#334	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂×Q₁₆×C₂₃	—	2	1	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	C₈	1	4	yes	complemented
24	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₂₃	—	2	2	no	—
28	C₄×Q₈×C₂₃	—	2	2	no	—
29	C₄×Q₈×C₂₃	—	4	1	no	—
30	C₄×Q₈×C₂₃	—	4	1	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
33	C₂×C₄²×C₂₃	—	4	1	no	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂₃×M₃₂	—	2	4	no	—
36	C₂×C₁₆×C₂₃	—	2	2	no	—
37	C₂₃×#107	—	2	1	no	—
38	C₂₃×#107	C₂×C₄	1	2	yes	—
39	C₂₃×C₄⋊C₈	—	2	2	no	—
40	C₂₃×#103	—	2	2	no	—
41	C₂₃×#103	—	2	2	no	—
42	C₂₃×C₂²⋊C₈	—	4	1	no	—
43	C₄×C₈×C₂₃	—	4	1	no	—
44	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₂₃	—	2	4	no	—
49	C₂×Q₈×C₂₃	—	2	2	no	—
50	C₂×Q₈×C₂₃	—	4	1	no	—
51	C₂×Q₈×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	4	1	no	—
53	C₂²×C₄×C₂₃	—	4	1	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
57	Q₁₆×C₂₃	—	2	4	no	—
58	Q₁₆×C₂₃	—	2	2	no	—
59	Q₁₆×C₂₃	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₂₃	M₁₆	1	2	yes	—
66	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	—	2	2	no	—
69	C₄⋊C₄×C₂₃	—	4	1	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	—	4	2	no	—
74	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₂₃	D₈	1	1	yes	characteristic
76	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
77	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₂₃	—	2	4	no	—
79	#11600	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	#102	1	1	yes	characteristic
81	Q₈×C₂₃	—	2	4	no	—
82	Q₈×C₂₃	—	4	4	no	—
83	Q₈×C₂₃	—	4	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	4	1	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#104	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	8	2	no	—
101	C₈×C₂₃	C₂²⋊C₈	1	2	yes	—
102	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
103	C₈×C₂₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
105	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic
106	#1564	C₂×C₂₃	1	1	yes	characteristic
107	#1334	C₂×C₂₃	1	4	yes	complemented
108	C₂²×C₂₃	—	2	1	no	—
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	D₄⋊C₈	1	2	yes	—
111	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	4	4	no	—
116	C₄×C₂₃	—	8	2	no	—
117	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
118	C₄×C₂₃	#362	1	2	yes	—
119	C₄×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
121	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	2	1	no	—
123	C₂×M₃₂	—	2	1	no	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	—	2	1	no	—
126	C₄×Q₁₆	C₄×C₂₃	1	2	yes	—
127	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
128	#431	—	2	1	no	—
129	C₂×#107	C₄×C₂₃	1	1	yes	characteristic
130	C₂×#103	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
132	#346	C₂²×C₂₃	1	2	yes	—
133	C₄×C₁₆	—	2	1	no	—
134	#335	—	2	1	no	—
135	#334	C₂²×C₂₃	1	2	yes	—
136	C₂×C₂₃	—	2	2	no	—
137	C₂×C₂₃	C₂×D₄⋊C₈	1	1	yes	characteristic
138	C₂×C₂₃	#2145	1	1	yes	characteristic
139	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
140	C₂²×Q₈	—	2	1	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	C₈×C₂₃	1	4	yes	complemented
144	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	—	4	1	no	—
154	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
155	M₃₂	—	2	4	no	—
156	C₂×C₁₆	—	2	2	no	—
157	#107	—	2	1	no	—
158	#107	C₂×C₄×C₂₃	1	2	yes	—
159	C₄⋊C₈	—	2	2	no	—
160	#103	—	2	2	no	—
161	#103	—	2	2	no	—
162	C₂²⋊C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
166	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
167	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
168	C₂₃	#11600	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	—	4	1	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
178	Q₁₆	—	2	2	no	—
179	Q₁₆	—	2	4	no	—
180	Q₁₆	C₂×C₈×C₂₃	1	4	yes	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₈	C₂×C₈×C₂₃	1	2	yes	—
186	C₂×C₈	M₁₆×C₂₃	1	2	yes	—
187	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
188	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
196	C₄²	D₈×C₂₃	1	1	yes	characteristic
197	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₁₆	—	2	4	no	—
200	C₂³	C₂₃×#102	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
211	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
212	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
214	C₂×C₄	C₂₃×#104	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
217	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	8	2	no	—
221	C₈	C₂₃×C₂²⋊C₈	1	2	yes	—
222	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
223	C₈	C₂×M₁₆×C₂₃	1	1	yes	characteristic
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₂₃×D₄⋊C₈	1	2	yes	—
227	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	8	2	no	—
233	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
234	C₄	C₂₃×#362	1	2	yes	—
235	C₄	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
236	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	C₂×C₂₃×D₄⋊C₈	1	1	yes	characteristic
239	C₂	C₂₃×#2145	1	1	yes	characteristic
240	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#102
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#102
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1334	C₂	semidirect
23. C₂×Q₁₆×C₂₃	C₈	semidirect
79. #11600	C₂₃	direct (N × H)
107. #1334	C₂×C₂₃	semidirect
143. C₂×Q₁₆	C₈×C₂₃	semidirect
168. C₂₃	#11600	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×C₄×Q₁₆×C₂₃	2
C₄×C₂₃×M₃₂	2
C₂₃×#1564	2
C₂₃×#1334	2
#11600	23

Subgroups — C23×#11600 #19159226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11600 #19159226