Subgroups — C₂₃×#17991 #19165617

Order 5888.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2985	C₂	1	2	yes	maximal
3	C₂₃×#1587	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1584	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1497	C₂	1	2	yes	maximal
7	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#533	C₂²	1	2	yes	—
9	C₂₃×#500	C₄	1	1	yes	characteristic
10	C₂₃×#494	C₄	1	1	yes	characteristic
11	C₂₃×#487	C₄	1	2	yes	—
12	C₂₃×#467	C₄	1	2	yes	—
13	C₂₃×#467	C₄	1	2	yes	—
14	C₂₃×#446	—	2	4	no	—
15	C₂₃×#446	C₂²	1	4	yes	—
16	C₈×Q₈×C₂₃	C₂²	1	4	yes	—
17	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
18	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#433	—	2	4	no	—
20	C₂₃×#432	C₂²	1	2	yes	—
21	C₂₃×#431	C₂²	1	1	yes	characteristic
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#335	C₂²	1	2	yes	—
25	C₂₃×#334	C₂²	1	2	yes	—
26	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	—
27	C₂₃×#322	C₂²	1	2	yes	—
28	C₂₃×#322	C₂²	1	2	yes	—
29	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂₃×#125	—	2	2	no	—
36	C₂₃×C₂²⋊Q₈	—	2	2	no	—
37	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₂₃	—	2	2	no	—
39	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
40	C₂₃×#117	—	2	2	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#107	C₂×C₄	1	2	yes	—
43	C₂₃×#107	C₂×C₄	1	2	yes	—
44	C₂₃×C₄⋊C₈	—	2	4	no	—
45	C₂₃×C₄⋊C₈	—	2	4	no	—
46	C₂₃×C₄⋊C₈	—	2	1	no	—
47	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
49	C₂₃×#103	C₂×C₄	1	4	yes	—
50	C₂₃×#103	C₂×C₄	1	4	yes	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×C₂²⋊C₈	—	2	1	no	—
53	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
54	C₂₃×#97	—	2	4	no	—
55	C₂₃×#97	—	2	2	no	—
56	C₄×C₈×C₂₃	—	2	4	no	—
57	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂×Q₈×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
68	Q₁₆×C₂₃	—	4	2	no	—
69	Q₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	4	1	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
79	C₄⋊C₄×C₂₃	—	2	2	no	—
80	C₄⋊C₄×C₂₃	—	2	2	no	—
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₄²×C₂₃	—	2	4	no	—
87	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₃	#41	1	1	yes	characteristic
90	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
91	#17991	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	2	4	no	—
95	Q₈×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
108	C₈×C₂₃	—	2	1	no	—
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	4	no	—
113	C₈×C₂₃	—	4	2	no	—
114	C₈×C₂₃	—	4	4	no	—
115	#3170	C₂×C₂₃	1	1	yes	characteristic
116	#2985	C₂×C₂₃	1	2	yes	—
117	#1587	C₂×C₂₃	1	4	yes	complemented
118	#1584	C₂×C₂₃	1	4	yes	complemented
119	#1568	C₂×C₂₃	1	1	yes	characteristic
120	#1497	C₂×C₂₃	1	2	yes	—
121	#1452	C₂×C₂₃	1	1	yes	characteristic
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	1	no	—
124	C₂²×C₂₃	—	2	1	no	—
125	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	4	4	no	—
131	C₄×C₂₃	—	4	4	no	—
132	C₄×C₂₃	#568	1	1	yes	characteristic
133	C₄×C₂₃	#568	1	1	yes	characteristic
134	C₄×C₂₃	#578	1	2	yes	—
135	#533	C₂²×C₂₃	1	2	yes	—
136	#500	C₄×C₂₃	1	1	yes	characteristic
137	#494	C₄×C₂₃	1	1	yes	characteristic
138	#487	C₄×C₂₃	1	2	yes	—
139	#467	C₄×C₂₃	1	2	yes	—
140	#467	C₄×C₂₃	1	2	yes	—
141	#446	—	2	4	no	—
142	#446	C₂²×C₂₃	1	4	yes	—
143	C₈×Q₈	C₂²×C₂₃	1	4	yes	—
144	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
145	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
146	#433	—	2	4	no	—
147	#432	C₂²×C₂₃	1	2	yes	—
148	#431	C₂²×C₂₃	1	1	yes	characteristic
149	#431	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	#335	C₂²×C₂₃	1	2	yes	—
152	#334	C₂²×C₂₃	1	2	yes	—
153	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	—
154	#322	C₂²×C₂₃	1	2	yes	—
155	#322	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	#2932	1	1	yes	characteristic
158	C₂×C₂₃	#2950	1	1	yes	characteristic
159	C₂×C₂₃	#2957	1	1	yes	characteristic
160	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
161	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
162	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
165	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
166	#125	—	2	2	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
171	#117	—	2	2	no	—
172	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
173	#107	C₂×C₄×C₂₃	1	2	yes	—
174	#107	C₂×C₄×C₂₃	1	2	yes	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
179	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
180	#103	C₂×C₄×C₂₃	1	4	yes	—
181	#103	C₂×C₄×C₂₃	1	4	yes	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	1	no	—
184	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
185	#97	—	2	4	no	—
186	#97	—	2	2	no	—
187	C₄×C₈	—	2	4	no	—
188	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₄×C₈	C₂³×C₂₃	1	2	yes	—
191	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₂₃	1	2	yes	—
194	C₂₃	#17991	1	1	yes	characteristic, direct product factor, Sylow
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
200	Q₁₆	—	4	2	no	—
201	Q₁₆	—	4	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₄²	#41×C₂₃	1	1	yes	characteristic
222	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
223	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
235	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	4	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	C₂₃×#568	1	1	yes	characteristic
257	C₄	C₂₃×#568	1	1	yes	characteristic
258	C₄	C₂₃×#578	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	C₂₃×#2932	1	1	yes	characteristic
261	C₂	C₂₃×#2950	1	1	yes	characteristic
262	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂³×C₂₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1587	C₂	semidirect
4. C₂₃×#1584	C₂	semidirect
91. #17991	C₂₃	direct (N × H)
117. #1587	C₂×C₂₃	semidirect
118. #1584	C₂×C₂₃	semidirect
194. C₂₃	#17991	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3170	2
C₂₃×#2985	2
C₂₃×#1587	2
C₂₃×#1584	2
C₂₃×#1568	2
C₂₃×#1497	2
C₂₃×#1452	2
#17991	23

Subgroups — C23×#17991 #19165617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17991 #19165617