Subgroups — C₂₃×#33608 #19181234

Order 5888.

650 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3059	C₂	1	2	yes	maximal
4	C₂₃×#2219	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2216	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2141	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
10	C₂₃×#536	—	2	2	no	—
11	C₂×C₂₃×Q₃₂	—	2	4	no	—
12	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	—
14	C₂₃×#504	—	2	4	no	—
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#470	—	2	4	no	—
20	C₂₃×#467	C₂²	1	8	yes	—
21	C₂₃×#427	C₂²	1	2	yes	—
22	C₂₃×#419	—	2	2	no	—
23	C₂₃×#367	C₂²	1	2	yes	—
24	C₂₃×#366	C₂²	1	2	yes	—
25	C₂₃×#358	C₂²	1	8	yes	—
26	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₂₃	—	2	2	no	—
28	C₂×Q₁₆×C₂₃	—	2	4	no	—
29	C₂×Q₁₆×C₂₃	—	2	2	no	—
30	C₂×Q₁₆×C₂₃	—	2	4	no	—
31	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
32	C₂₃×#131	—	2	2	no	—
33	C₂₃×#131	—	2	2	no	—
34	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂₃×C₄⋊Q₈	—	4	2	no	—
39	C₂₃×#124	—	4	2	no	—
40	C₂₃×#123	—	4	4	no	—
41	C₂₃×C₂²⋊Q₈	—	2	8	no	—
42	C₂₃×#117	—	2	2	no	—
43	C₂₃×Q₃₂	—	2	4	no	—
44	C₂₃×Q₃₂	—	2	4	no	—
45	C₂₃×M₃₂	—	2	2	no	—
46	C₂₃×M₃₂	—	2	4	no	—
47	C₂×C₁₆×C₂₃	—	2	4	no	—
48	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
49	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	C₂³	1	4	yes	—
53	C₂₃×#106	—	2	2	no	—
54	C₂₃×#103	—	2	8	no	—
55	#41×C₂₃	—	2	2	no	—
56	#41×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	—	4	2	no	—
59	C₂×Q₈×C₂₃	—	4	4	no	—
60	C₂×Q₈×C₂₃	—	4	4	no	—
61	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
62	C₂×D₄×C₂₃	D₈	1	2	yes	—
63	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	4	no	—
68	Q₁₆×C₂₃	—	2	4	no	—
69	Q₁₆×C₂₃	—	4	4	no	—
70	M₁₆×C₂₃	—	2	2	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₂₃	#41	1	2	yes	—
78	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	4	8	no	—
81	C₂²⋊C₄×C₂₃	—	4	2	no	—
82	C₂²⋊C₄×C₂₃	—	4	8	no	—
83	C₄²×C₂₃	—	4	2	no	—
84	C₁₆×C₂₃	—	2	2	no	—
85	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
86	C₁₆×C₂₃	C₂×D₄	1	2	yes	—
87	#33608	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	C₂×D₈	1	2	yes	—
89	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	2	no	—
91	Q₈×C₂₃	—	4	4	no	—
92	Q₈×C₂₃	—	4	4	no	—
93	Q₈×C₂₃	—	8	4	no	—
94	D₄×C₂₃	—	2	4	no	—
95	D₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	—	8	4	no	—
101	C₂×C₄×C₂₃	#121	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#135	1	2	yes	—
105	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	#121	1	2	yes	—
108	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
110	C₂²×Q₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
111	C₂×#504	C₂×C₂₃	1	1	yes	characteristic
112	#3059	C₂×C₂₃	1	2	yes	—
113	#2219	C₂×C₂₃	1	4	yes	complemented
114	#2216	C₂×C₂₃	1	4	yes	complemented
115	#2158	C₂×C₂₃	1	1	yes	characteristic, complemented
116	#2141	C₂×C₂₃	1	2	yes	complemented
117	C₂²×C₂₃	—	2	2	no	—
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	#466	1	2	yes	—
121	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	8	4	no	—
124	C₄×C₂₃	—	8	4	no	—
125	C₄×C₂₃	#466	1	2	yes	—
126	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
127	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
128	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
129	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
130	#536	—	2	2	no	—
131	C₂×Q₃₂	—	2	4	no	—
132	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
133	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	—
134	#504	—	2	4	no	—
135	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
136	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
137	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
138	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
139	#470	—	2	4	no	—
140	#467	C₂²×C₂₃	1	8	yes	—
141	#427	C₂²×C₂₃	1	2	yes	—
142	#419	—	2	2	no	—
143	#367	C₂²×C₂₃	1	2	yes	—
144	#366	C₂²×C₂₃	1	2	yes	—
145	#358	C₂²×C₂₃	1	8	yes	—
146	C₂×C₂₃	—	2	2	no	—
147	C₂×C₂₃	—	4	2	no	—
148	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
149	C₂×C₂₃	#3417	1	2	yes	—
150	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
151	C₂²×Q₈	—	2	2	no	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
156	#131	—	2	2	no	—
157	#131	—	2	2	no	—
158	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
159	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
160	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	C₄⋊Q₈	—	4	2	no	—
163	#124	—	4	2	no	—
164	#123	—	4	4	no	—
165	C₂²⋊Q₈	—	2	8	no	—
166	#117	—	2	2	no	—
167	Q₃₂	—	2	4	no	—
168	Q₃₂	—	2	4	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	4	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
173	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
174	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
176	#107	C₂³×C₂₃	1	4	yes	—
177	#106	—	2	2	no	—
178	#103	—	2	8	no	—
179	C₂₃	#33608	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
187	C₂×D₄	D₈×C₂₃	1	2	yes	—
188	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	4	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
202	C₂×C₈	#41×C₂₃	1	2	yes	—
203	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	8	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	8	no	—
208	C₄²	—	4	2	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
211	C₁₆	C₂×D₄×C₂₃	1	2	yes	—
212	C₂³	C₂×D₈×C₂₃	1	2	yes	—
213	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	4	4	no	—
216	Q₈	—	4	4	no	—
217	Q₈	—	8	4	no	—
218	D₄	—	2	4	no	—
219	D₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	8	4	no	—
225	C₂×C₄	C₂₃×#121	1	2	yes	—
226	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
227	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#135	1	2	yes	—
229	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	C₂₃×#121	1	2	yes	—
232	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
233	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	C₂₃×#466	1	2	yes	—
238	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	8	4	no	—
241	C₄	—	8	4	no	—
242	C₄	C₂₃×#466	1	2	yes	—
243	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
244	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	4	2	no	—
247	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
248	C₂	C₂₃×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×#41×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×Q₃₂	C₂	semidirect
4. C₂₃×#2219	C₂	semidirect
5. C₂₃×#2216	C₂	semidirect
6. C₂₃×#2158	C₂	semidirect
7. C₂₃×#2141	C₂	semidirect
87. #33608	C₂₃	direct (N × H)
110. C₂²×Q₃₂	C₂×C₂₃	semidirect
113. #2219	C₂×C₂₃	semidirect
114. #2216	C₂×C₂₃	semidirect
115. #2158	C₂×C₂₃	semidirect
116. #2141	C₂×C₂₃	semidirect
179. C₂₃	#33608	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×Q₃₂	2
C₂×C₂₃×#504	2
C₂₃×#3059	2
C₂₃×#2219	2
C₂₃×#2216	2
C₂₃×#2158	2
C₂₃×#2141	2
#33608	23

Subgroups — C23×#33608 #19181234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33608 #19181234