Subgroups — C₂₃×#61704 #19209330

Order 5888.

2022 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3496	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#552	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3343	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3310	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3234	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3193	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3187	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3112	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3094	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#480	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2973	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
13	C₂₃×#552	—	2	4	no	—
14	C₂₃×#552	C₂²	1	8	yes	complemented
15	C₂₃×#547	C₂²	1	4	yes	—
16	C₂₃×#546	—	2	4	no	—
17	D₄²×C₂₃	C₂²	1	4	yes	—
18	C₂₃×#537	—	2	1	no	—
19	C₂₃×#534	—	2	1	no	—
20	C₂₃×#534	—	2	1	no	—
21	C₂₃×#532	C₂²	1	2	yes	—
22	C₂×C₂₃×#126	—	2	1	no	—
23	C₂×C₂₃×#125	—	2	1	no	—
24	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₂₃×#123	—	2	1	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₂₃×#121	C₂²	1	2	yes	—
29	C₂×C₂₃×#121	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#117	—	2	1	no	—
33	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#501	C₂²	1	4	yes	complemented
36	C₂₃×#490	C₂²	1	4	yes	complemented
37	C₂₃×#484	C₂²	1	4	yes	complemented
38	C₂₃×#483	C₂²	1	4	yes	complemented
39	C₂₃×#482	C₂²	1	8	yes	complemented
40	C₂₃×#481	C₂²	1	4	yes	complemented
41	C₂₃×#480	C₂²	1	4	yes	complemented
42	C₂₃×#480	C₂²	1	8	yes	complemented
43	C₂₃×#480	C₂²	1	4	yes	complemented
44	C₂₃×#478	C₂²	1	8	yes	complemented
45	C₂₃×#477	C₂²	1	4	yes	complemented
46	C₂₃×#468	C₂²	1	8	yes	complemented
47	C₂₃×#465	C₂²	1	8	yes	complemented
48	C₂₃×#460	C₂²	1	4	yes	complemented
49	C₂₃×#450	C₂²	1	4	yes	complemented
50	C₂₃×#450	C₂²	1	4	yes	complemented
51	C₂₃×#448	C₂²	1	4	yes	complemented
52	C₂₃×#440	C₂²	1	8	yes	complemented
53	C₂₃×#435	C₂²	1	8	yes	complemented
54	C₂₃×#432	C₂²	1	2	yes	complemented
55	C₂₃×#428	C₂²	1	4	yes	complemented
56	C₂×C₂₃×#107	C₂²	1	4	yes	complemented
57	C₂₃×#418	C₂²	1	4	yes	complemented
58	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
59	C₂₃×#407	C₂²	1	2	yes	complemented
60	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
61	C₂₃×H(5,𝔽₂)	—	4	8	no	—
62	C₂×#41×C₂₃	—	2	4	no	—
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₂₃	—	2	2	no	—
66	C₂²×D₄×C₂₃	—	2	2	no	—
67	C₂²×D₄×C₂₃	—	2	4	no	—
68	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
75	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
76	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
77	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₂₃×#127	C₂³	1	2	yes	—
79	C₂₃×#126	—	2	8	no	—
80	C₂₃×#125	—	2	8	no	—
81	C₂₃×#125	—	2	2	no	—
82	C₂₃×#125	C₂³	1	4	yes	—
83	C₂₃×#124	—	2	2	no	—
84	C₂₃×#123	—	2	8	no	—
85	C₂₃×#123	—	2	4	no	—
86	C₂₃×#123	—	2	4	no	—
87	C₂₃×C₂²⋊Q₈	—	2	2	no	—
88	C₂₃×C₂²⋊Q₈	—	2	2	no	—
89	C₂₃×C₂²⋊Q₈	—	2	4	no	—
90	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
91	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
92	C₂₃×#121	—	2	2	no	—
93	C₂₃×#121	—	2	2	no	—
94	C₂₃×#121	—	2	2	no	—
95	C₂₃×#121	—	2	2	no	—
96	C₂₃×#121	—	2	2	no	—
97	C₂₃×#121	—	2	4	no	—
98	C₂₃×#121	C₂³	1	4	yes	—
99	C₂₃×#121	C₂³	1	4	yes	—
100	C₂₃×#121	C₂³	1	4	yes	—
101	C₂₃×#120	—	2	4	no	—
102	C₂₃×#120	—	2	4	no	—
103	C₄×Q₈×C₂₃	—	2	2	no	—
104	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
105	C₄×D₄×C₂₃	—	2	2	no	—
106	C₄×D₄×C₂₃	D₄	1	4	yes	—
107	C₄×D₄×C₂₃	C₂³	1	4	yes	—
108	C₄×D₄×C₂₃	C₂³	1	4	yes	—
109	C₂₃×#117	—	2	2	no	—
110	C₂₃×#117	—	2	8	no	—
111	C₂₃×#117	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
118	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
119	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
126	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
127	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
128	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
129	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
130	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
131	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
132	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
133	C₂×C₄²×C₂₃	—	2	1	no	—
134	C₂×C₄²×C₂₃	—	2	1	no	—
135	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₂₃×#107	C₂³	1	8	yes	—
137	C₂₃×#107	C₂³	1	8	yes	—
138	C₂₃×#106	C₂³	1	8	yes	—
139	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
140	C₂₃×#103	C₂³	1	8	yes	—
141	C₂₃×#102	C₂³	1	8	yes	—
142	C₂₃×#102	C₂³	1	8	yes	—
143	C₂₃×#102	C₂³	1	8	yes	—
144	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
146	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
147	C₂₃×#97	C₂³	1	4	yes	—
148	C₂⁴×C₂₃	—	2	2	no	—
149	C₂⁴×C₂₃	—	2	2	no	—
150	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
152	#41×C₂₃	—	4	8	no	—
153	#41×C₂₃	—	4	16	no	—
154	C₂×Q₈×C₂₃	—	2	1	no	—
155	C₂×Q₈×C₂₃	—	2	2	no	—
156	C₂×Q₈×C₂₃	—	4	2	no	—
157	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	4	no	—
160	C₂×D₄×C₂₃	—	2	4	no	—
161	C₂×D₄×C₂₃	—	2	4	no	—
162	C₂×D₄×C₂₃	—	2	2	no	—
163	C₂×D₄×C₂₃	—	2	2	no	—
164	C₂×D₄×C₂₃	—	2	2	no	—
165	C₂×D₄×C₂₃	—	2	2	no	—
166	C₂×D₄×C₂₃	—	4	4	no	—
167	C₂×D₄×C₂₃	—	4	4	no	—
168	C₂×D₄×C₂₃	—	4	16	no	—
169	C₂×D₄×C₂₃	—	4	8	no	—
170	C₂×D₄×C₂₃	—	4	8	no	—
171	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
172	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₂₃	—	2	1	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	2	no	—
178	C₂²×C₄×C₂₃	—	2	2	no	—
179	C₂²×C₄×C₂₃	—	2	4	no	—
180	C₂²×C₄×C₂₃	—	2	4	no	—
181	C₂²×C₄×C₂₃	—	4	2	no	—
182	C₂²×C₄×C₂₃	—	4	2	no	—
183	C₂²×C₄×C₂₃	—	4	4	no	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
193	SD₁₆×C₂₃	—	4	8	no	—
194	M₁₆×C₂₃	—	4	4	no	—
195	C₂×C₈×C₂₃	—	4	4	no	—
196	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	2	4	no	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	4	no	—
204	C₄⋊C₄×C₂₃	—	2	4	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	2	2	no	—
207	C₄⋊C₄×C₂₃	—	2	2	no	—
208	C₄⋊C₄×C₂₃	—	2	2	no	—
209	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
210	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
211	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
212	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
213	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
214	C₂²⋊C₄×C₂₃	—	2	4	no	—
215	C₂²⋊C₄×C₂₃	—	2	4	no	—
216	C₂²⋊C₄×C₂₃	—	2	4	no	—
217	C₂²⋊C₄×C₂₃	—	2	4	no	—
218	C₂²⋊C₄×C₂₃	—	2	4	no	—
219	C₂²⋊C₄×C₂₃	—	2	4	no	—
220	C₂²⋊C₄×C₂₃	—	2	4	no	—
221	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
222	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
223	C₄²×C₂₃	—	2	4	no	—
224	C₄²×C₂₃	—	2	4	no	—
225	C₄²×C₂₃	C₂×D₄	1	2	yes	—
226	C₄²×C₂₃	C₂⁴	1	2	yes	—
227	#61704	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₂₃	—	2	2	no	—
229	C₂³×C₂₃	—	2	4	no	—
230	C₂³×C₂₃	—	4	1	no	—
231	C₂³×C₂₃	—	4	1	no	—
232	C₂³×C₂₃	—	4	1	no	—
233	C₂³×C₂₃	—	4	1	no	—
234	C₂³×C₂₃	—	4	4	no	—
235	C₂³×C₂₃	—	4	4	no	—
236	C₂³×C₂₃	—	8	2	no	—
237	C₂³×C₂₃	—	8	2	no	—
238	C₂³×C₂₃	—	8	2	no	—
239	C₂³×C₂₃	—	8	2	no	—
240	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
241	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
242	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
243	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
244	Q₈×C₂₃	—	2	4	no	—
245	Q₈×C₂₃	—	4	2	no	—
246	Q₈×C₂₃	—	4	4	no	—
247	D₄×C₂₃	—	2	8	no	—
248	D₄×C₂₃	—	2	8	no	—
249	D₄×C₂₃	—	4	4	no	—
250	D₄×C₂₃	—	4	8	no	—
251	D₄×C₂₃	—	4	8	no	—
252	D₄×C₂₃	—	4	8	no	—
253	D₄×C₂₃	—	8	2	no	—
254	D₄×C₂₃	—	8	2	no	—
255	D₄×C₂₃	—	8	2	no	—
256	C₂×C₄×C₂₃	—	2	2	no	—
257	C₂×C₄×C₂₃	—	2	4	no	—
258	C₂×C₄×C₂₃	—	2	4	no	—
259	C₂×C₄×C₂₃	—	2	2	no	—
260	C₂×C₄×C₂₃	—	2	2	no	—
261	C₂×C₄×C₂₃	—	2	2	no	—
262	C₂×C₄×C₂₃	—	2	2	no	—
263	C₂×C₄×C₂₃	—	2	4	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	4	2	no	—
266	C₂×C₄×C₂₃	—	4	2	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	8	no	—
271	C₂×C₄×C₂₃	—	4	2	no	—
272	C₂×C₄×C₂₃	—	4	4	no	—
273	C₂×C₄×C₂₃	—	4	2	no	—
274	C₂×C₄×C₂₃	—	4	2	no	—
275	C₂×C₄×C₂₃	—	4	2	no	—
276	C₂×C₄×C₂₃	—	4	8	no	—
277	C₂×C₄×C₂₃	—	4	4	no	—
278	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
280	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
282	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
283	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
284	C₈×C₂₃	—	2	8	no	—
285	#3496	C₂×C₂₃	1	1	yes	characteristic
286	C₂×#552	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#3343	C₂×C₂₃	1	8	yes	complemented
288	#3310	C₂×C₂₃	1	8	yes	complemented
289	#3234	C₂×C₂₃	1	2	yes	complemented
290	#3193	C₂×C₂₃	1	2	yes	complemented
291	#3187	C₂×C₂₃	1	2	yes	complemented
292	#3112	C₂×C₂₃	1	2	yes	complemented
293	#3094	C₂×C₂₃	1	2	yes	complemented
294	C₂×#480	C₂×C₂₃	1	2	yes	complemented
295	#2973	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂²×C₂₃	—	2	1	no	—
297	C₂²×C₂₃	—	2	2	no	—
298	C₂²×C₂₃	—	2	1	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	2	2	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	2	2	no	—
303	C₂²×C₂₃	—	4	4	no	—
304	C₂²×C₂₃	—	4	4	no	—
305	C₂²×C₂₃	—	4	4	no	—
306	C₂²×C₂₃	—	4	4	no	—
307	C₂²×C₂₃	—	8	2	no	—
308	C₂²×C₂₃	—	16	4	no	—
309	C₂²×C₂₃	—	16	4	no	—
310	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
311	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
312	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₂₃	—	2	2	no	—
314	C₄×C₂₃	—	2	1	no	—
315	C₄×C₂₃	—	2	2	no	—
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	2	2	no	—
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	—	4	4	no	—
323	C₄×C₂₃	#585	1	2	yes	—
324	C₂×H(5,𝔽₂)	—	2	1	no	—
325	#552	—	2	4	no	—
326	#552	C₂²×C₂₃	1	8	yes	complemented
327	#547	C₂²×C₂₃	1	4	yes	—
328	#546	—	2	4	no	—
329	D₄²	C₂²×C₂₃	1	4	yes	—
330	#537	—	2	1	no	—
331	#534	—	2	1	no	—
332	#534	—	2	1	no	—
333	#532	C₂²×C₂₃	1	2	yes	—
334	C₂×#126	—	2	1	no	—
335	C₂×#125	—	2	1	no	—
336	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
337	C₂×#123	—	2	1	no	—
338	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
339	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
340	C₂×#121	C₂²×C₂₃	1	2	yes	—
341	C₂×#121	C₂²×C₂₃	1	2	yes	—
342	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
343	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#117	—	2	1	no	—
345	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic, complemented
346	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	#501	C₂²×C₂₃	1	4	yes	complemented
348	#490	C₂²×C₂₃	1	4	yes	complemented
349	#484	C₂²×C₂₃	1	4	yes	complemented
350	#483	C₂²×C₂₃	1	4	yes	complemented
351	#482	C₂²×C₂₃	1	8	yes	complemented
352	#481	C₂²×C₂₃	1	4	yes	complemented
353	#480	C₂²×C₂₃	1	8	yes	complemented
354	#480	C₂²×C₂₃	1	4	yes	complemented
355	#480	C₂²×C₂₃	1	4	yes	complemented
356	#478	C₂²×C₂₃	1	8	yes	complemented
357	#477	C₂²×C₂₃	1	4	yes	complemented
358	#468	C₂²×C₂₃	1	8	yes	complemented
359	#465	C₂²×C₂₃	1	8	yes	complemented
360	#460	C₂²×C₂₃	1	4	yes	complemented
361	#450	C₂²×C₂₃	1	4	yes	complemented
362	#450	C₂²×C₂₃	1	4	yes	complemented
363	#448	C₂²×C₂₃	1	4	yes	complemented
364	#440	C₂²×C₂₃	1	8	yes	complemented
365	#435	C₂²×C₂₃	1	8	yes	complemented
366	#432	C₂²×C₂₃	1	2	yes	complemented
367	#428	C₂²×C₂₃	1	4	yes	complemented
368	C₂×#107	C₂²×C₂₃	1	4	yes	complemented
369	#418	C₂²×C₂₃	1	4	yes	complemented
370	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
371	#407	C₂²×C₂₃	1	2	yes	complemented
372	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
373	C₂×C₂₃	—	2	2	no	—
374	C₂×C₂₃	—	2	4	no	—
375	C₂×C₂₃	—	4	2	no	—
376	C₂×C₂₃	—	8	4	no	—
377	C₂×C₂₃	#3485	1	1	yes	characteristic
378	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
379	C₂×C₂₃	#3586	1	1	yes	characteristic
380	H(5,𝔽₂)	—	4	8	no	—
381	C₂×#41	—	2	4	no	—
382	C₂×#41	—	2	2	no	—
383	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₂²×D₄	—	2	2	no	—
385	C₂²×D₄	—	2	2	no	—
386	C₂²×D₄	—	2	4	no	—
387	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
389	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
394	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
395	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
396	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
397	#127	C₂³×C₂₃	1	2	yes	—
398	#126	—	2	8	no	—
399	#125	—	2	8	no	—
400	#125	—	2	2	no	—
401	#125	C₂³×C₂₃	1	4	yes	—
402	#124	—	2	2	no	—
403	#123	—	2	8	no	—
404	#123	—	2	4	no	—
405	#123	—	2	4	no	—
406	C₂²⋊Q₈	—	2	2	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
410	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
411	#121	—	2	2	no	—
412	#121	—	2	2	no	—
413	#121	—	2	2	no	—
414	#121	—	2	2	no	—
415	#121	—	2	2	no	—
416	#121	—	2	4	no	—
417	#121	C₂³×C₂₃	1	4	yes	—
418	#121	C₂³×C₂₃	1	4	yes	—
419	#121	C₂³×C₂₃	1	4	yes	—
420	#120	—	2	4	no	—
421	#120	—	2	4	no	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	D₄×C₂₃	1	4	yes	—
426	C₄×D₄	C₂³×C₂₃	1	4	yes	—
427	C₄×D₄	C₂³×C₂₃	1	4	yes	—
428	#117	—	2	2	no	—
429	#117	—	2	8	no	—
430	#117	C₂³×C₂₃	1	4	yes	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	2	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
442	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
445	C₂×C₂²⋊C₄	—	2	1	no	—
446	C₂×C₂²⋊C₄	—	2	1	no	—
447	C₂×C₂²⋊C₄	—	2	1	no	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₂²⋊C₄	—	2	2	no	—
450	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
452	C₂×C₄²	—	2	1	no	—
453	C₂×C₄²	—	2	1	no	—
454	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
455	#107	C₂³×C₂₃	1	8	yes	—
456	#107	C₂³×C₂₃	1	8	yes	—
457	#106	C₂³×C₂₃	1	8	yes	—
458	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
459	#103	C₂³×C₂₃	1	8	yes	—
460	#102	C₂³×C₂₃	1	8	yes	—
461	#102	C₂³×C₂₃	1	8	yes	—
462	#102	C₂³×C₂₃	1	8	yes	—
463	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
464	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
465	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
466	#97	C₂³×C₂₃	1	4	yes	—
467	C₂₃	#61704	1	1	yes	characteristic, direct product factor, Sylow
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	2	2	no	—
470	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
472	#41	—	4	8	no	—
473	#41	—	4	16	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	2	no	—
476	C₂×Q₈	—	4	2	no	—
477	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	2	no	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	4	4	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	—	4	16	no	—
489	C₂×D₄	—	4	8	no	—
490	C₂×D₄	—	4	8	no	—
491	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
492	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	4	no	—
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
508	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
509	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
510	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
513	SD₁₆	—	4	8	no	—
514	M₁₆	—	4	4	no	—
515	C₂×C₈	—	4	4	no	—
516	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
530	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
531	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
532	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
533	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	4	no	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
542	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
543	C₄²	—	2	4	no	—
544	C₄²	—	2	4	no	—
545	C₄²	C₂×D₄×C₂₃	1	2	yes	—
546	C₄²	C₂⁴×C₂₃	1	2	yes	—
547	C₂³	—	2	2	no	—
548	C₂³	—	2	4	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	4	no	—
554	C₂³	—	4	4	no	—
555	C₂³	—	8	2	no	—
556	C₂³	—	8	2	no	—
557	C₂³	—	8	2	no	—
558	C₂³	—	8	2	no	—
559	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
560	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
561	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
562	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
563	Q₈	—	2	4	no	—
564	Q₈	—	4	2	no	—
565	Q₈	—	4	4	no	—
566	D₄	—	2	8	no	—
567	D₄	—	2	8	no	—
568	D₄	—	4	4	no	—
569	D₄	—	4	8	no	—
570	D₄	—	4	8	no	—
571	D₄	—	4	8	no	—
572	D₄	—	8	2	no	—
573	D₄	—	8	2	no	—
574	D₄	—	8	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	8	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	8	no	—
596	C₂×C₄	—	4	4	no	—
597	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
598	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
599	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
600	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
601	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
602	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
603	C₈	—	2	8	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	4	4	no	—
614	C₂²	—	4	4	no	—
615	C₂²	—	8	2	no	—
616	C₂²	—	16	4	no	—
617	C₂²	—	16	4	no	—
618	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
619	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
620	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
621	C₄	—	2	1	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	4	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	C₂₃×#585	1	2	yes	—
632	C₂	—	2	2	no	—
633	C₂	—	2	4	no	—
634	C₂	—	4	2	no	—
635	C₂	—	8	4	no	—
636	C₂	C₂₃×#3485	1	1	yes	characteristic
637	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
638	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3496	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#552	C₂	semidirect
3. C₂₃×#3343	C₂	semidirect
4. C₂₃×#3310	C₂	semidirect
5. C₂₃×#3234	C₂	semidirect
6. C₂₃×#3193	C₂	semidirect
7. C₂₃×#3187	C₂	semidirect
8. C₂₃×#3112	C₂	semidirect
9. C₂₃×#3094	C₂	semidirect
10. C₂×C₂₃×#480	C₂	semidirect
11. C₂₃×#2973	C₂	semidirect
14. C₂₃×#552	C₂²	semidirect
24. C₂×C₂₃×#125	C₂²	semidirect
26. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
27. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
30. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
33. C₂×C₂₃×#117	C₂²	semidirect
34. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
35. C₂₃×#501	C₂²	semidirect
36. C₂₃×#490	C₂²	semidirect
37. C₂₃×#484	C₂²	semidirect
38. C₂₃×#483	C₂²	semidirect
39. C₂₃×#482	C₂²	semidirect
40. C₂₃×#481	C₂²	semidirect
41. C₂₃×#480	C₂²	semidirect
42. C₂₃×#480	C₂²	semidirect
43. C₂₃×#480	C₂²	semidirect
44. C₂₃×#478	C₂²	semidirect
45. C₂₃×#477	C₂²	semidirect
46. C₂₃×#468	C₂²	semidirect
47. C₂₃×#465	C₂²	semidirect
48. C₂₃×#460	C₂²	semidirect
49. C₂₃×#450	C₂²	semidirect
50. C₂₃×#450	C₂²	semidirect
51. C₂₃×#448	C₂²	semidirect
52. C₂₃×#440	C₂²	semidirect
53. C₂₃×#435	C₂²	semidirect
54. C₂₃×#432	C₂²	semidirect
55. C₂₃×#428	C₂²	semidirect
56. C₂×C₂₃×#107	C₂²	semidirect
57. C₂₃×#418	C₂²	semidirect
58. C₂×C₂₃×#102	C₂²	semidirect
59. C₂₃×#407	C₂²	semidirect
60. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
227. #61704	C₂₃	direct (N × H)
286. C₂×#552	C₂×C₂₃	semidirect
287. #3343	C₂×C₂₃	semidirect
288. #3310	C₂×C₂₃	semidirect
289. #3234	C₂×C₂₃	semidirect
290. #3193	C₂×C₂₃	semidirect
291. #3187	C₂×C₂₃	semidirect
292. #3112	C₂×C₂₃	semidirect
293. #3094	C₂×C₂₃	semidirect
294. C₂×#480	C₂×C₂₃	semidirect
295. #2973	C₂×C₂₃	semidirect
326. #552	C₂²×C₂₃	semidirect
336. C₂×#125	C₂²×C₂₃	semidirect
338. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
339. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
342. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
345. C₂×#117	C₂²×C₂₃	semidirect
346. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
347. #501	C₂²×C₂₃	semidirect
348. #490	C₂²×C₂₃	semidirect
349. #484	C₂²×C₂₃	semidirect
350. #483	C₂²×C₂₃	semidirect
351. #482	C₂²×C₂₃	semidirect
352. #481	C₂²×C₂₃	semidirect
353. #480	C₂²×C₂₃	semidirect
354. #480	C₂²×C₂₃	semidirect
355. #480	C₂²×C₂₃	semidirect
356. #478	C₂²×C₂₃	semidirect
357. #477	C₂²×C₂₃	semidirect
358. #468	C₂²×C₂₃	semidirect
359. #465	C₂²×C₂₃	semidirect
360. #460	C₂²×C₂₃	semidirect
361. #450	C₂²×C₂₃	semidirect
362. #450	C₂²×C₂₃	semidirect
363. #448	C₂²×C₂₃	semidirect
364. #440	C₂²×C₂₃	semidirect
365. #435	C₂²×C₂₃	semidirect
366. #432	C₂²×C₂₃	semidirect
367. #428	C₂²×C₂₃	semidirect
368. C₂×#107	C₂²×C₂₃	semidirect
369. #418	C₂²×C₂₃	semidirect
370. C₂×#102	C₂²×C₂₃	semidirect
371. #407	C₂²×C₂₃	semidirect
372. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
467. C₂₃	#61704	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×#3496	2
C₂×C₂₃×#552	2
C₂₃×#3343	2
C₂₃×#3310	2
C₂₃×#3234	2
C₂₃×#3193	2
C₂₃×#3187	2
C₂₃×#3112	2
C₂₃×#3094	2
C₂×C₂₃×#480	2
C₂₃×#2973	2
#61704	23

Subgroups — C23×#61704 #19209330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61704 #19209330