Subgroups — C₂₃×#33436 #19181062

Order 5888.

692 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2169	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2168	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2168	C₂	1	2	yes	maximal, complemented
7	D₄×C₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#525	C₄	1	1	yes	characteristic
15	C₂₃×#504	—	2	4	no	—
16	C₂×C₂₃×M₃₂	—	2	4	no	—
17	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
19	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	complemented
25	C₂₃×#435	C₄	1	2	yes	—
26	C₂₃×#435	C₄	1	2	yes	—
27	C₈×D₄×C₂₃	C₂²	1	2	yes	—
28	C₈×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#424	C₄	1	1	yes	characteristic
30	C₂₃×#408	C₄	1	1	yes	characteristic
31	C₂₃×#407	C₄	1	1	yes	characteristic
32	C₂₃×#405	C₂²	1	1	yes	characteristic
33	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
35	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
41	C₂₃×#346	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₂₃	C₂²	1	2	yes	complemented
43	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂₃×#131	—	2	2	no	—
46	C₂₃×#131	—	2	2	no	—
47	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₂₃	—	2	2	no	—
51	C₂²×C₈×C₂₃	—	2	2	no	—
52	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂₃×#123	C₈	1	2	yes	—
60	C₂₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
61	C₂₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
62	C₂₃×#121	C₈	1	1	yes	characteristic
63	C₂₃×#121	C₈	1	1	yes	characteristic
64	C₂₃×#120	C₈	1	2	yes	—
65	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
68	C₂₃×M₃₂	—	2	2	no	—
69	C₂₃×M₃₂	—	2	4	no	—
70	C₂₃×M₃₂	—	2	4	no	—
71	C₂₃×M₃₂	D₄	1	4	yes	complemented
72	C₂×C₁₆×C₂₃	—	2	2	no	—
73	C₂×C₁₆×C₂₃	—	2	2	no	—
74	C₂×C₁₆×C₂₃	—	2	4	no	—
75	C₂×C₁₆×C₂₃	—	2	4	no	—
76	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
81	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
82	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
83	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
84	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
85	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
86	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
87	C₂₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
88	C₂₃×#97	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₂₃	C₂³	1	2	yes	—
90	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
91	#41×C₂₃	—	2	2	no	—
92	#41×C₂₃	—	2	2	no	—
93	C₂×Q₈×C₂₃	C₂×C₈	1	1	yes	characteristic
94	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
95	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
96	C₂×D₄×C₂₃	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₂₃	—	2	2	no	—
108	M₁₆×C₂₃	—	2	2	no	—
109	M₁₆×C₂₃	—	2	4	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	2	no	—
113	C₂×C₈×C₂₃	—	2	1	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	1	no	—
116	C₂×C₈×C₂₃	—	2	1	no	—
117	C₂×C₈×C₂₃	—	2	1	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₂₃	#41	1	2	yes	—
126	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
128	C₄⋊C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
129	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
130	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
131	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
132	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
133	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
134	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
135	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
136	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
137	C₁₆×C₂₃	—	2	1	no	—
138	C₁₆×C₂₃	—	2	1	no	—
139	C₁₆×C₂₃	—	2	2	no	—
140	C₁₆×C₂₃	—	2	1	no	—
141	C₁₆×C₂₃	—	2	1	no	—
142	C₁₆×C₂₃	C₂×D₄	1	4	yes	—
143	#33436	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₂₃	—	2	2	no	—
145	C₂³×C₂₃	—	2	2	no	—
146	C₂³×C₂₃	—	4	1	no	—
147	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
148	C₂³×C₂₃	C₂²×C₈	1	2	yes	—
149	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
150	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
151	Q₈×C₂₃	—	2	2	no	—
152	D₄×C₂₃	—	2	4	no	—
153	D₄×C₂₃	—	2	4	no	—
154	D₄×C₂₃	—	2	2	no	—
155	D₄×C₂₃	—	4	2	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	1	no	—
158	C₂×C₄×C₂₃	—	2	1	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	1	no	—
161	C₂×C₄×C₂₃	—	2	1	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
167	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
168	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
170	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
171	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₄×C₂₃	#131	1	2	yes	—
173	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
174	C₈×C₂₃	—	2	1	no	—
175	C₈×C₂₃	—	2	1	no	—
176	C₈×C₂₃	—	2	2	no	—
177	C₈×C₂₃	—	2	2	no	—
178	C₈×C₂₃	C₄×D₄	1	2	yes	—
179	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
180	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
181	C₂×#504	C₂×C₂₃	1	1	yes	characteristic, complemented
182	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
183	#2929	C₂×C₂₃	1	1	yes	characteristic
184	#2169	C₂×C₂₃	1	2	yes	complemented
185	#2168	C₂×C₂₃	1	2	yes	complemented
186	#2168	C₂×C₂₃	1	2	yes	complemented
187	D₄×C₁₆	C₂×C₂₃	1	2	yes	complemented
188	#2151	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#2113	C₂×C₂₃	1	1	yes	characteristic, complemented
190	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
191	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
192	C₂²×C₂₃	—	2	1	no	—
193	C₂²×C₂₃	—	2	1	no	—
194	C₂²×C₂₃	—	2	1	no	—
195	C₂²×C₂₃	—	2	1	no	—
196	C₂²×C₂₃	—	2	2	no	—
197	C₂²×C₂₃	—	2	1	no	—
198	C₂²×C₂₃	—	2	1	no	—
199	C₂²×C₂₃	—	2	2	no	—
200	C₂²×C₂₃	—	2	1	no	—
201	C₂²×C₂₃	—	2	2	no	—
202	C₂²×C₂₃	—	4	4	no	—
203	C₂²×C₂₃	C₈×D₄	1	2	yes	—
204	C₂²×C₂₃	#504	1	2	yes	complemented
205	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
206	C₄×C₂₃	—	2	2	no	—
207	C₄×C₂₃	—	2	2	no	—
208	C₄×C₂₃	—	2	1	no	—
209	C₄×C₂₃	—	2	1	no	—
210	C₄×C₂₃	—	2	2	no	—
211	C₄×C₂₃	C₈×D₄	1	2	yes	—
212	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
213	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
214	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
215	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
216	#525	C₄×C₂₃	1	1	yes	characteristic
217	#504	—	2	4	no	—
218	C₂×M₃₂	—	2	4	no	—
219	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
221	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×M₃₂	C₂²×C₂₃	1	2	yes	complemented
223	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
224	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
225	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
226	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	complemented
227	#435	C₄×C₂₃	1	2	yes	—
228	#435	C₄×C₂₃	1	2	yes	—
229	C₈×D₄	C₂²×C₂₃	1	2	yes	—
230	C₈×D₄	C₂²×C₂₃	1	2	yes	—
231	#424	C₄×C₂₃	1	1	yes	characteristic
232	#408	C₄×C₂₃	1	1	yes	characteristic
233	#407	C₄×C₂₃	1	1	yes	characteristic
234	#405	C₂²×C₂₃	1	1	yes	characteristic
235	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
236	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
237	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
238	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
239	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
240	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
241	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
242	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
243	#346	C₂²×C₂₃	1	2	yes	complemented
244	C₄×C₁₆	C₂²×C₂₃	1	2	yes	complemented
245	C₂×C₂₃	—	2	2	no	—
246	C₂×C₂₃	—	2	2	no	—
247	C₂×C₂₃	—	4	2	no	—
248	C₂×C₂₃	—	4	1	no	—
249	C₂×C₂₃	C₂×C₈×D₄	1	1	yes	characteristic
250	C₂×C₂₃	C₂×#504	1	1	yes	characteristic
251	C₂×C₂₃	#3407	1	1	yes	characteristic
252	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
253	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
254	#131	—	2	2	no	—
255	#131	—	2	2	no	—
256	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
257	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
258	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
259	C₂²×C₈	—	2	2	no	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
262	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
267	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
268	#123	C₈×C₂₃	1	2	yes	—
269	C₂²⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
270	C₂²⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
271	#121	C₈×C₂₃	1	1	yes	characteristic
272	#121	C₈×C₂₃	1	1	yes	characteristic
273	#120	C₈×C₂₃	1	2	yes	—
274	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
275	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
276	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
277	M₃₂	—	2	2	no	—
278	M₃₂	—	2	4	no	—
279	M₃₂	—	2	4	no	—
280	M₃₂	D₄×C₂₃	1	4	yes	complemented
281	C₂×C₁₆	—	2	2	no	—
282	C₂×C₁₆	—	2	2	no	—
283	C₂×C₁₆	—	2	4	no	—
284	C₂×C₁₆	—	2	4	no	—
285	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
286	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
287	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
288	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
289	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
290	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
291	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
292	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
293	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
294	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
295	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
296	C₂²⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
297	#97	C₂×C₄×C₂₃	1	2	yes	—
298	C₄×C₈	C₂³×C₂₃	1	2	yes	—
299	C₂₃	#33436	1	1	yes	characteristic, direct product factor, Sylow
300	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₂×C₈×C₂₃	1	1	yes	characteristic
304	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
305	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
306	C₂×D₄	C₂×C₈×C₂₃	1	1	yes	characteristic
307	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
312	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
330	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
331	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
332	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
333	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
335	C₂×C₈	#41×C₂₃	1	2	yes	—
336	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
337	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
339	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
340	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
341	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
342	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
343	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
344	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
345	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
346	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
347	C₁₆	—	2	1	no	—
348	C₁₆	—	2	1	no	—
349	C₁₆	—	2	2	no	—
350	C₁₆	—	2	1	no	—
351	C₁₆	—	2	1	no	—
352	C₁₆	C₂×D₄×C₂₃	1	4	yes	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
357	C₂³	C₂²×C₈×C₂₃	1	2	yes	—
358	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
359	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	4	no	—
363	D₄	—	2	2	no	—
364	D₄	—	4	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
375	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
376	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
377	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
379	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
380	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₂₃×#131	1	2	yes	—
382	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
383	C₈	—	2	2	no	—
384	C₈	—	2	1	no	—
385	C₈	—	2	1	no	—
386	C₈	—	2	2	no	—
387	C₈	C₄×D₄×C₂₃	1	2	yes	—
388	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
389	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	2	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	2	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	C₈×D₄×C₂₃	1	2	yes	—
402	C₂²	C₂₃×#504	1	2	yes	complemented
403	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
404	C₄	—	2	1	no	—
405	C₄	—	2	1	no	—
406	C₄	—	2	2	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	C₈×D₄×C₂₃	1	2	yes	—
410	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
411	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	2	2	no	—
414	C₂	—	4	1	no	—
415	C₂	—	4	2	no	—
416	C₂	C₂×C₈×D₄×C₂₃	1	1	yes	characteristic
417	C₂	C₂×C₂₃×#504	1	1	yes	characteristic
418	C₂	C₂₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂₃×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#504	C₂	semidirect
2. C₂²×C₂₃×M₃₂	C₂	semidirect
4. C₂₃×#2169	C₂	semidirect
5. C₂₃×#2168	C₂	semidirect
6. C₂₃×#2168	C₂	semidirect
7. D₄×C₁₆×C₂₃	C₂	semidirect
8. C₂₃×#2151	C₂	semidirect
9. C₂₃×#2113	C₂	semidirect
10. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
11. C₂₃×#2108	C₂	semidirect
20. C₂×C₂₃×M₃₂	C₂²	semidirect
24. C₂²×C₁₆×C₂₃	C₂²	semidirect
33. C₂₃×C₄⋊C₁₆	C₂²	semidirect
34. C₂₃×C₄⋊C₁₆	C₂²	semidirect
39. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
40. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
41. C₂₃×#346	C₂²	semidirect
42. C₄×C₁₆×C₂₃	C₂²	semidirect
71. C₂₃×M₃₂	D₄	semidirect
143. #33436	C₂₃	direct (N × H)
181. C₂×#504	C₂×C₂₃	semidirect
182. C₂²×M₃₂	C₂×C₂₃	semidirect
184. #2169	C₂×C₂₃	semidirect
185. #2168	C₂×C₂₃	semidirect
186. #2168	C₂×C₂₃	semidirect
187. D₄×C₁₆	C₂×C₂₃	semidirect
188. #2151	C₂×C₂₃	semidirect
189. #2113	C₂×C₂₃	semidirect
190. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
191. #2108	C₂×C₂₃	semidirect
204. C₂²×C₂₃	#504	semidirect
222. C₂×M₃₂	C₂²×C₂₃	semidirect
226. C₂²×C₁₆	C₂²×C₂₃	semidirect
235. C₄⋊C₁₆	C₂²×C₂₃	semidirect
236. C₄⋊C₁₆	C₂²×C₂₃	semidirect
241. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
242. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
243. #346	C₂²×C₂₃	semidirect
244. C₄×C₁₆	C₂²×C₂₃	semidirect
280. M₃₂	D₄×C₂₃	semidirect
299. C₂₃	#33436	direct (N × H)
402. C₂²	C₂₃×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#504	2
C₂²×C₂₃×M₃₂	2
C₂₃×#2929	2
C₂₃×#2169	2
C₂₃×#2168	2
D₄×C₁₆×C₂₃	2
C₂₃×#2151	2
C₂₃×#2113	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂₃×#2108	2
#33436	23

Subgroups — C23×#33436 #19181062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33436 #19181062