Subgroups — C₂₃×#11932 #19159558

Order 5888.

398 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1362	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₂₃	—	2	1	no	—
6	C₂²×Q₁₆×C₂₃	C₄	1	2	yes	—
7	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₂₃×M₃₂	—	2	1	no	—
9	C₂₃×#494	—	2	1	no	—
10	C₂₃×#494	—	2	1	no	—
11	C₂₃×#494	C₄	1	2	yes	—
12	C₂₃×#494	C₂²	1	2	yes	—
13	C₂₃×#431	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₁₆	—	2	1	no	—
16	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂₃×#335	C₂²	1	2	yes	—
18	C₂₃×#334	—	2	1	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂×Q₁₆×C₂₃	—	2	2	no	—
21	C₂×Q₁₆×C₂₃	—	2	2	no	—
22	C₂×Q₁₆×C₂₃	—	2	2	no	—
23	C₂×Q₁₆×C₂₃	—	4	4	no	—
24	C₂×Q₁₆×C₂₃	—	4	2	no	—
25	C₂×Q₁₆×C₂₃	—	4	2	no	—
26	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
29	C₂₃×C₄⋊Q₈	—	2	2	no	—
30	C₂₃×C₄⋊Q₈	—	2	2	no	—
31	C₂₃×C₄⋊Q₈	—	4	1	no	—
32	C₂₃×C₄⋊Q₈	—	4	1	no	—
33	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
34	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂₃×M₃₂	—	4	2	no	—
37	C₂×C₁₆×C₂₃	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₂²⋊C₈	—	4	1	no	—
40	C₄×C₈×C₂₃	—	4	1	no	—
41	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	—	2	4	no	—
46	C₂×Q₈×C₂₃	—	4	2	no	—
47	C₂×Q₈×C₂₃	—	4	4	no	—
48	C₂×Q₈×C₂₃	—	4	4	no	—
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
53	Q₁₆×C₂₃	—	4	4	no	—
54	Q₁₆×C₂₃	—	4	4	no	—
55	Q₁₆×C₂₃	—	4	4	no	—
56	Q₁₆×C₂₃	—	4	2	no	—
57	Q₁₆×C₂₃	—	4	2	no	—
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₂₃	D₈	1	2	yes	—
63	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	4	2	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₂₃	D₈	1	1	yes	characteristic
72	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
73	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₁₆×C₂₃	—	4	2	no	—
75	#11932	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	#102	1	1	yes	characteristic
77	Q₈×C₂₃	—	4	4	no	—
78	Q₈×C₂₃	—	4	4	no	—
79	Q₈×C₂₃	—	8	4	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	4	no	—
83	C₂×C₄×C₂₃	—	8	4	no	—
84	C₂×C₄×C₂₃	#100	1	2	yes	—
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#102	1	2	yes	—
88	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₂₃	—	2	1	no	—
93	C₈×C₂₃	—	2	1	no	—
94	C₈×C₂₃	—	8	2	no	—
95	C₈×C₂₃	#102	1	2	yes	—
96	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
97	C₈×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
99	#2150	C₂×C₂₃	1	1	yes	characteristic
100	#1564	C₂×C₂₃	1	1	yes	characteristic
101	#1362	C₂×C₂₃	1	4	yes	complemented
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	#331	1	2	yes	—
105	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
106	C₄×C₂₃	—	2	1	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	8	4	no	—
109	C₄×C₂₃	#331	1	2	yes	—
110	C₄×C₂₃	#362	1	2	yes	—
111	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
112	C₄×C₂₃	C₂×#102	1	1	yes	characteristic
113	C₄×C₂₃	#510	1	2	yes	—
114	C₂²×Q₁₆	—	2	1	no	—
115	C₂²×Q₁₆	C₄×C₂₃	1	2	yes	—
116	C₂×C₄⋊Q₈	—	2	1	no	—
117	C₂×M₃₂	—	2	1	no	—
118	#494	—	2	1	no	—
119	#494	—	2	1	no	—
120	#494	C₄×C₂₃	1	2	yes	—
121	#494	C₂²×C₂₃	1	2	yes	—
122	#431	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₄⋊C₁₆	—	2	1	no	—
125	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
126	#335	C₂²×C₂₃	1	2	yes	—
127	#334	—	2	1	no	—
128	C₂×C₂₃	—	2	2	no	—
129	C₂×C₂₃	C₂×#331	1	1	yes	characteristic
130	C₂×C₂₃	#2141	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#362	1	1	yes	characteristic
132	C₂²×Q₈	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	—	4	2	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
140	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₄⋊Q₈	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₂²⋊C₈	—	4	1	no	—
153	C₄×C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
155	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
156	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
157	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
158	C₂₃	#11932	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	Q₁₆	—	4	4	no	—
169	Q₁₆	—	4	4	no	—
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₈	D₈×C₂₃	1	2	yes	—
177	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
185	C₄²	D₈×C₂₃	1	1	yes	characteristic
186	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₂³	C₂₃×#102	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	Q₈	—	4	4	no	—
192	Q₈	—	8	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	C₂₃×#100	1	2	yes	—
198	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
199	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
200	C₂×C₄	C₂₃×#102	1	2	yes	—
201	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
204	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	C₂₃×#102	1	2	yes	—
209	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
210	C₈	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂₃×#331	1	2	yes	—
214	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂₃×#331	1	2	yes	—
219	C₄	C₂₃×#362	1	2	yes	—
220	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
221	C₄	C₂×C₂₃×#102	1	1	yes	characteristic
222	C₄	C₂₃×#510	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₂×C₂₃×#331	1	1	yes	characteristic
225	C₂	C₂₃×#2141	1	1	yes	characteristic
226	C₂	C₂×C₂₃×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
prime radical	C₂³×C₂₃	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#1362	C₂	semidirect
75. #11932	C₂₃	direct (N × H)
101. #1362	C₂×C₂₃	semidirect
158. C₂₃	#11932	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×#494	2
C₂₃×#2150	2
C₂₃×#1564	2
C₂₃×#1362	2
#11932	23

Subgroups — C23×#11932 #19159558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11932 #19159558