Subgroups — C₂₃×#22747 #19170373

Order 5888.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₁₆×C₂₃	C₂	1	2	yes	maximal
2	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1950	C₂	1	4	yes	maximal
4	C₂₃×#1950	C₂	1	2	yes	maximal
5	C₂₃×#1838	C₂	1	2	yes	maximal
6	C₂₃×#1837	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1778	C₂	1	2	yes	maximal
8	C₄×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#446	—	2	2	no	—
11	C₂₃×#446	—	2	4	no	—
12	C₂₃×#446	C₄	1	4	yes	—
13	C₂₃×#446	C₄	1	4	yes	—
14	C₈×Q₈×C₂₃	—	2	2	no	—
15	C₂×C₂₃×#108	C₂²	1	2	yes	—
16	C₂×C₂₃×#108	C₂²	1	4	yes	—
17	C₂₃×#423	C₂²	1	2	yes	—
18	C₂₃×#423	C₂²	1	4	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
22	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
23	C₂×C₂₃×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#341	C₂²	1	2	yes	—
26	C₂₃×#341	C₂²	1	4	yes	—
27	C₂₃×#341	C₂²	1	1	yes	characteristic
28	C₂₃×#341	C₂²	1	1	yes	characteristic
29	C₂₃×#341	C₂²	1	2	yes	—
30	C₂²×Q₈×C₂₃	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
32	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
33	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
38	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₂₃	D₄	1	2	yes	—
40	C₄×Q₈×C₂₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	Q₈	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
45	C₂₃×#108	—	2	4	no	—
46	C₂₃×#108	—	2	8	no	—
47	C₂₃×C₄⋊C₈	—	2	2	no	—
48	C₂₃×C₄⋊C₈	—	2	4	no	—
49	C₂₃×C₄⋊C₈	—	2	4	no	—
50	C₂₃×C₄⋊C₈	—	2	2	no	—
51	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×#97	—	2	4	no	—
55	C₂₃×#97	—	2	2	no	—
56	C₂₃×#97	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₂₃	—	2	2	no	—
58	C₄×C₈×C₂₃	—	2	4	no	—
59	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
61	C₂×Q₈×C₂₃	—	4	2	no	—
62	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂×Q₈×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₂₃	—	2	2	no	—
70	M₁₆×C₂₃	—	2	2	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	2	2	no	—
74	M₁₆×C₂₃	#41	1	4	yes	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
84	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
85	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₂₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
91	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₂₃	C₂×D₄	1	4	yes	—
93	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
94	#22747	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₂₃	C₂²×Q₈	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	4	no	—
97	Q₈×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	4	1	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
111	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	2	4	no	—
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₈×C₂₃	—	4	2	no	—
118	Q₈×M₁₆	C₂×C₂₃	1	2	yes	—
119	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
120	#1950	C₂×C₂₃	1	4	yes	—
121	#1950	C₂×C₂₃	1	2	yes	—
122	#1838	C₂×C₂₃	1	2	yes	—
123	#1837	C₂×C₂₃	1	1	yes	characteristic
124	#1778	C₂×C₂₃	1	2	yes	—
125	C₄×#108	C₂×C₂₃	1	1	yes	characteristic
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	2	1	no	—
128	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₂₃	#557	1	2	yes	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	4	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	4	no	—
135	C₄×C₂₃	#430	1	2	yes	—
136	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
137	C₄×C₂₃	#519	1	1	yes	characteristic
138	C₄×C₂₃	#548	1	2	yes	—
139	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
140	#446	—	2	2	no	—
141	#446	—	2	4	no	—
142	#446	C₄×C₂₃	1	4	yes	—
143	#446	C₄×C₂₃	1	4	yes	—
144	C₈×Q₈	—	2	2	no	—
145	C₂×#108	C₂²×C₂₃	1	2	yes	—
146	C₂×#108	C₂²×C₂₃	1	4	yes	—
147	#423	C₂²×C₂₃	1	2	yes	—
148	#423	C₂²×C₂₃	1	4	yes	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
152	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
153	C₂×#97	C₂²×C₂₃	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	#341	C₂²×C₂₃	1	2	yes	—
156	#341	C₂²×C₂₃	1	2	yes	—
157	#341	C₂²×C₂₃	1	1	yes	characteristic
158	#341	C₂²×C₂₃	1	4	yes	—
159	#341	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₂₃	—	2	2	no	—
161	C₂×C₂₃	#2351	1	1	yes	characteristic
162	C₂×C₂₃	C₂×#430	1	1	yes	characteristic
163	C₂×C₂₃	#2916	1	1	yes	characteristic
164	C₂²×Q₈	Q₈×C₂₃	1	1	yes	characteristic
165	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
166	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
167	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
171	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
172	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
173	C₄×Q₈	D₄×C₂₃	1	2	yes	—
174	C₄×Q₈	D₄×C₂₃	1	2	yes	—
175	C₂×C₄⋊C₄	Q₈×C₂₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	Q₈×C₂₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
179	#108	—	2	4	no	—
180	#108	—	2	8	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
186	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
187	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
188	#97	—	2	4	no	—
189	#97	—	2	2	no	—
190	#97	C₂×C₄×C₂₃	1	8	yes	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
194	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
195	C₂₃	#22747	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
198	C₂×Q₈	C₄⋊C₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	#41×C₂₃	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
218	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
219	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
220	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
221	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₄⋊C₄	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
226	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₂₃	1	4	yes	—
228	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
229	C₂³	C₂²×Q₈×C₂₃	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
245	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	4	no	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂²	C₂₃×#557	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	C₂₃×#430	1	2	yes	—
262	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
263	C₄	C₂₃×#519	1	1	yes	characteristic
264	C₄	C₂₃×#548	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	C₂₃×#2351	1	1	yes	characteristic
267	C₂	C₂×C₂₃×#430	1	1	yes	characteristic
268	C₂	C₂₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₂₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
94. #22747	C₂₃	direct (N × H)
195. C₂₃	#22747	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₂₃	2
C₂×C₂₃×#446	2
C₂₃×#1950	2
C₂₃×#1838	2
C₂₃×#1837	2
C₂₃×#1778	2
C₄×C₂₃×#108	2
#22747	23

Subgroups — C23×#22747 #19170373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22747 #19170373