Subgroups — C₂₃×#61587 #19209213

Order 5888.

2430 conjugacy classes of subgroups.

The subgroup lattice has 516 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, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3332	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3274	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3205	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3192	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3103	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3006	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
11	C₂²×#41×C₂₃	C₂²	1	2	yes	complemented
12	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
13	C₂₃×#550	—	2	4	no	—
14	C₂₃×#550	C₂²	1	8	yes	complemented
15	C₂₃×#547	C₂²	1	4	yes	complemented
16	C₂₃×#546	—	2	4	no	—
17	D₄²×C₂₃	C₂²	1	4	yes	complemented
18	C₂₃×#537	—	2	1	no	—
19	C₂₃×#534	—	2	2	no	—
20	C₂₃×#532	C₂²	1	2	yes	—
21	C₂×C₂₃×#127	—	2	1	no	—
22	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂₃×#121	—	2	2	no	—
24	C₂×C₂₃×#121	C₂²	1	4	yes	—
25	C₂×C₂₃×#121	C₂²	1	2	yes	complemented
26	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄×D₄×C₂₃	—	2	1	no	—
28	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#493	C₂²	1	4	yes	complemented
30	C₂₃×#486	C₂²	1	4	yes	complemented
31	C₂₃×#481	C₂²	1	4	yes	complemented
32	C₂₃×#481	C₂²	1	8	yes	complemented
33	C₂₃×#475	C₂²	1	4	yes	complemented
34	C₂₃×#469	C₂²	1	16	yes	complemented
35	C₂₃×#460	C₂²	1	4	yes	complemented
36	C₂₃×#459	C₂²	1	8	yes	complemented
37	C₂₃×#449	C₂²	1	16	yes	complemented
38	C₂₃×#448	C₂²	1	4	yes	complemented
39	C₂₃×#447	C₂²	1	8	yes	complemented
40	C₂₃×#447	C₂²	1	8	yes	complemented
41	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
42	C₂₃×#436	C₂²	1	8	yes	complemented
43	C₂₃×#431	C₂²	1	2	yes	complemented
44	C₂₃×#417	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	complemented
46	C₂₃×H(5,𝔽₂)	—	4	8	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	2	no	—
50	C₂×#41×C₂₃	—	4	8	no	—
51	C₂×#41×C₂₃	C₂³	1	8	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	—	2	2	no	—
54	C₂²×D₄×C₂₃	—	2	4	no	—
55	C₂²×D₄×C₂₃	—	2	2	no	—
56	C₂²×D₄×C₂₃	—	2	4	no	—
57	C₂²×D₄×C₂₃	—	2	1	no	—
58	C₂²×D₄×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	—	2	2	no	—
63	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
64	C₂₃×AGL(1,ℤ/8)	—	4	16	no	—
65	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
66	C₂×D₈×C₂₃	C₂³	1	16	yes	—
67	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
68	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₂₃×#127	—	2	8	no	—
70	C₂₃×#127	—	2	2	no	—
71	C₂₃×#127	C₂³	1	4	yes	—
72	C₂₃×#127	C₂³	1	2	yes	—
73	C₂₃×#124	—	2	2	no	—
74	C₂₃×#123	—	2	8	no	—
75	C₂₃×C₂²⋊Q₈	—	2	4	no	—
76	C₂₃×#121	—	2	4	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	4	no	—
79	C₂₃×#121	—	2	2	no	—
80	C₂₃×#121	—	2	4	no	—
81	C₂₃×#121	—	2	16	no	—
82	C₂₃×#121	C₂³	1	8	yes	—
83	C₂₃×#121	C₂³	1	8	yes	—
84	C₂₃×#121	C₂³	1	4	yes	—
85	C₂₃×#120	—	2	8	no	—
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
88	C₄×D₄×C₂₃	—	2	8	no	—
89	C₄×D₄×C₂₃	—	2	2	no	—
90	C₄×D₄×C₂₃	D₄	1	4	yes	—
91	C₄×D₄×C₂₃	C₂³	1	4	yes	—
92	C₄×D₄×C₂₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
100	C₂×C₄²×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×#106	C₂³	1	8	yes	—
103	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
104	C₂₃×#103	C₂³	1	8	yes	—
105	C₂₃×#102	C₂³	1	16	yes	—
106	C₂₃×#102	C₂³	1	8	yes	—
107	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
108	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₄×C₈×C₂₃	C₂³	1	4	yes	—
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	4	no	—
112	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
113	#41×C₂₃	—	2	16	no	—
114	#41×C₂₃	—	4	8	no	—
115	#41×C₂₃	—	4	16	no	—
116	#41×C₂₃	—	4	16	no	—
117	#41×C₂₃	—	4	8	no	—
118	C₂×Q₈×C₂₃	—	2	4	no	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
122	C₂×D₄×C₂₃	—	2	8	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	2	8	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	—	2	2	no	—
131	C₂×D₄×C₂₃	—	2	4	no	—
132	C₂×D₄×C₂₃	—	2	4	no	—
133	C₂×D₄×C₂₃	—	4	4	no	—
134	C₂×D₄×C₂₃	—	4	8	no	—
135	C₂×D₄×C₂₃	—	4	8	no	—
136	C₂×D₄×C₂₃	—	4	16	no	—
137	C₂×D₄×C₂₃	—	4	16	no	—
138	C₂×D₄×C₂₃	—	4	4	no	—
139	C₂×D₄×C₂₃	—	8	2	no	—
140	C₂×D₄×C₂₃	—	8	4	no	—
141	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
143	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	8	no	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	—	4	4	no	—
150	C₂²×C₄×C₂₃	—	4	8	no	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
157	SD₁₆×C₂₃	—	2	16	no	—
158	D₈×C₂₃	—	4	16	no	—
159	M₁₆×C₂₃	—	4	8	no	—
160	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
166	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	—	2	8	no	—
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
173	C₄²×C₂₃	—	2	4	no	—
174	C₄²×C₂₃	C₂×D₄	1	2	yes	—
175	C₄²×C₂₃	C₂⁴	1	2	yes	—
176	#61587	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	2	4	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	4	no	—
183	C₂³×C₂₃	—	4	4	no	—
184	C₂³×C₂₃	—	4	8	no	—
185	C₂³×C₂₃	—	4	4	no	—
186	C₂³×C₂₃	—	8	4	no	—
187	C₂³×C₂₃	—	8	4	no	—
188	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
189	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₂₃	—	2	4	no	—
191	Q₈×C₂₃	—	2	2	no	—
192	Q₈×C₂₃	—	4	4	no	—
193	Q₈×C₂₃	H(5,𝔽₂)	1	4	yes	complemented
194	D₄×C₂₃	—	2	8	no	—
195	D₄×C₂₃	—	2	8	no	—
196	D₄×C₂₃	—	2	8	no	—
197	D₄×C₂₃	—	4	16	no	—
198	D₄×C₂₃	—	4	4	no	—
199	D₄×C₂₃	—	4	16	no	—
200	D₄×C₂₃	—	4	8	no	—
201	D₄×C₂₃	—	4	8	no	—
202	D₄×C₂₃	—	4	4	no	—
203	D₄×C₂₃	—	8	2	no	—
204	D₄×C₂₃	—	8	4	no	—
205	D₄×C₂₃	—	8	4	no	—
206	D₄×C₂₃	—	8	8	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	4	2	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	4	16	no	—
217	C₂×C₄×C₂₃	—	4	8	no	—
218	C₂×C₄×C₂₃	—	4	2	no	—
219	C₂×C₄×C₂₃	—	4	8	no	—
220	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	complemented
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
226	C₈×C₂₃	—	2	8	no	—
227	#3496	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂×#550	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#3332	C₂×C₂₃	1	8	yes	complemented
230	#3274	C₂×C₂₃	1	8	yes	complemented
231	#3205	C₂×C₂₃	1	2	yes	complemented
232	#3192	C₂×C₂₃	1	4	yes	complemented
233	#3103	C₂×C₂₃	1	2	yes	complemented
234	#3006	C₂×C₂₃	1	4	yes	complemented
235	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	—	4	4	no	—
243	C₂²×C₂₃	—	4	4	no	—
244	C₂²×C₂₃	—	4	4	no	—
245	C₂²×C₂₃	—	4	4	no	—
246	C₂²×C₂₃	—	4	4	no	—
247	C₂²×C₂₃	—	8	4	no	—
248	C₂²×C₂₃	—	8	8	no	—
249	C₂²×C₂₃	—	16	8	no	—
250	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
251	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₂₃	—	2	2	no	—
253	C₄×C₂₃	—	2	4	no	—
254	C₄×C₂₃	—	2	4	no	—
255	C₄×C₂₃	—	2	1	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
259	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
260	C₂×H(5,𝔽₂)	—	2	1	no	—
261	C₂²×#41	C₂²×C₂₃	1	2	yes	complemented
262	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
263	#550	—	2	4	no	—
264	#550	C₂²×C₂₃	1	8	yes	complemented
265	#547	C₂²×C₂₃	1	4	yes	complemented
266	#546	—	2	4	no	—
267	D₄²	C₂²×C₂₃	1	4	yes	complemented
268	#537	—	2	1	no	—
269	#534	—	2	2	no	—
270	#532	C₂²×C₂₃	1	2	yes	—
271	C₂×#127	—	2	1	no	—
272	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
273	C₂×#121	—	2	2	no	—
274	C₂×#121	C₂²×C₂₃	1	4	yes	—
275	C₂×#121	C₂²×C₂₃	1	2	yes	complemented
276	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
277	C₂×C₄×D₄	—	2	1	no	—
278	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
279	#493	C₂²×C₂₃	1	4	yes	complemented
280	#486	C₂²×C₂₃	1	4	yes	complemented
281	#481	C₂²×C₂₃	1	8	yes	complemented
282	#481	C₂²×C₂₃	1	4	yes	complemented
283	#475	C₂²×C₂₃	1	4	yes	complemented
284	#469	C₂²×C₂₃	1	16	yes	complemented
285	#460	C₂²×C₂₃	1	4	yes	complemented
286	#459	C₂²×C₂₃	1	8	yes	complemented
287	#449	C₂²×C₂₃	1	16	yes	complemented
288	#448	C₂²×C₂₃	1	4	yes	complemented
289	#447	C₂²×C₂₃	1	8	yes	complemented
290	#447	C₂²×C₂₃	1	8	yes	complemented
291	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
292	#436	C₂²×C₂₃	1	8	yes	complemented
293	#431	C₂²×C₂₃	1	2	yes	complemented
294	#417	C₂²×C₂₃	1	8	yes	complemented
295	#407	C₂²×C₂₃	1	4	yes	complemented
296	C₂×C₂₃	—	2	2	no	—
297	C₂×C₂₃	—	4	4	no	—
298	C₂×C₂₃	—	8	4	no	—
299	C₂×C₂₃	—	8	4	no	—
300	C₂×C₂₃	#3484	1	1	yes	characteristic
301	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₂×C₂₃	#3585	1	1	yes	characteristic
303	H(5,𝔽₂)	—	4	8	no	—
304	C₂×#41	—	2	4	no	—
305	C₂×#41	—	2	2	no	—
306	C₂×#41	—	2	4	no	—
307	C₂×#41	—	4	8	no	—
308	C₂×#41	C₂³×C₂₃	1	8	yes	—
309	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
310	C₂²×D₄	—	2	2	no	—
311	C₂²×D₄	—	2	4	no	—
312	C₂²×D₄	—	2	2	no	—
313	C₂²×D₄	—	2	4	no	—
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
317	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
318	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
319	C₂³×C₄	—	2	2	no	—
320	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
321	AGL(1,ℤ/8)	—	4	16	no	—
322	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
323	C₂×D₈	C₂³×C₂₃	1	16	yes	—
324	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
325	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
326	#127	—	2	8	no	—
327	#127	—	2	2	no	—
328	#127	C₂³×C₂₃	1	4	yes	—
329	#127	C₂³×C₂₃	1	2	yes	—
330	#124	—	2	2	no	—
331	#123	—	2	8	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	#121	—	2	4	no	—
334	#121	—	2	4	no	—
335	#121	—	2	4	no	—
336	#121	—	2	2	no	—
337	#121	—	2	4	no	—
338	#121	—	2	16	no	—
339	#121	C₂³×C₂₃	1	4	yes	—
340	#121	C₂³×C₂₃	1	8	yes	—
341	#121	C₂³×C₂₃	1	8	yes	—
342	#120	—	2	8	no	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	D₄×C₂₃	1	4	yes	—
348	C₄×D₄	C₂³×C₂₃	1	4	yes	—
349	C₄×D₄	C₂³×C₂₃	1	4	yes	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
353	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	4	no	—
356	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
359	#106	C₂³×C₂₃	1	8	yes	—
360	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
361	#103	C₂³×C₂₃	1	8	yes	—
362	#102	C₂³×C₂₃	1	16	yes	—
363	#102	C₂³×C₂₃	1	8	yes	—
364	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
365	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
366	C₄×C₈	C₂³×C₂₃	1	4	yes	—
367	C₂₃	#61587	1	1	yes	characteristic, direct product factor, Sylow
368	C₂⁴	—	2	2	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
371	#41	—	2	16	no	—
372	#41	—	4	8	no	—
373	#41	—	4	16	no	—
374	#41	—	4	16	no	—
375	#41	—	4	8	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	8	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	4	no	—
390	C₂×D₄	—	2	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	8	no	—
393	C₂×D₄	—	4	8	no	—
394	C₂×D₄	—	4	16	no	—
395	C₂×D₄	—	4	16	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂×D₄	—	8	4	no	—
399	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
400	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
401	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	4	4	no	—
408	C₂²×C₄	—	4	8	no	—
409	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
410	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
412	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
415	SD₁₆	—	2	16	no	—
416	D₈	—	4	16	no	—
417	M₁₆	—	4	8	no	—
418	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
424	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
425	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
431	C₄²	—	2	4	no	—
432	C₄²	C₂×D₄×C₂₃	1	2	yes	—
433	C₄²	C₂⁴×C₂₃	1	2	yes	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	4	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	4	no	—
443	C₂³	—	8	4	no	—
444	C₂³	—	8	4	no	—
445	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
446	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
447	Q₈	—	2	4	no	—
448	Q₈	—	2	2	no	—
449	Q₈	—	4	4	no	—
450	Q₈	C₂₃×H(5,𝔽₂)	1	4	yes	complemented
451	D₄	—	2	8	no	—
452	D₄	—	2	8	no	—
453	D₄	—	2	8	no	—
454	D₄	—	4	16	no	—
455	D₄	—	4	4	no	—
456	D₄	—	4	16	no	—
457	D₄	—	4	8	no	—
458	D₄	—	4	4	no	—
459	D₄	—	4	8	no	—
460	D₄	—	8	4	no	—
461	D₄	—	8	4	no	—
462	D₄	—	8	2	no	—
463	D₄	—	8	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	16	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	complemented
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
481	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
482	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
483	C₈	—	2	8	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	4	no	—
496	C₂²	—	8	8	no	—
497	C₂²	—	16	8	no	—
498	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
499	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	4	no	—
502	C₄	—	2	1	no	—
503	C₄	—	2	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
507	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
508	C₂	—	2	2	no	—
509	C₂	—	4	4	no	—
510	C₂	—	8	4	no	—
511	C₂	—	8	4	no	—
512	C₂	C₂₃×#3484	1	1	yes	characteristic
513	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
514	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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
1. C₂₃×#3496	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂₃×#3332	C₂	semidirect
4. C₂₃×#3274	C₂	semidirect
5. C₂₃×#3205	C₂	semidirect
6. C₂₃×#3192	C₂	semidirect
7. C₂₃×#3103	C₂	semidirect
8. C₂₃×#3006	C₂	semidirect
9. C₂₃×#2975	C₂	semidirect
11. C₂²×#41×C₂₃	C₂²	semidirect
12. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂₃×#550	C₂²	semidirect
15. C₂₃×#547	C₂²	semidirect
17. D₄²×C₂₃	C₂²	semidirect
22. C₂×C₂₃×#127	C₂²	semidirect
25. C₂×C₂₃×#121	C₂²	semidirect
26. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#493	C₂²	semidirect
30. C₂₃×#486	C₂²	semidirect
31. C₂₃×#481	C₂²	semidirect
32. C₂₃×#481	C₂²	semidirect
33. C₂₃×#475	C₂²	semidirect
34. C₂₃×#469	C₂²	semidirect
35. C₂₃×#460	C₂²	semidirect
36. C₂₃×#459	C₂²	semidirect
37. C₂₃×#449	C₂²	semidirect
38. C₂₃×#448	C₂²	semidirect
39. C₂₃×#447	C₂²	semidirect
40. C₂₃×#447	C₂²	semidirect
41. C₄×SD₁₆×C₂₃	C₂²	semidirect
42. C₂₃×#436	C₂²	semidirect
43. C₂₃×#431	C₂²	semidirect
44. C₂₃×#417	C₂²	semidirect
45. C₂₃×#407	C₂²	semidirect
176. #61587	C₂₃	direct (N × H)
193. Q₈×C₂₃	H(5,𝔽₂)	semidirect
220. C₂×C₄×C₂₃	AGL(1,ℤ/8)	semidirect
227. #3496	C₂×C₂₃	semidirect
228. C₂×#550	C₂×C₂₃	semidirect
229. #3332	C₂×C₂₃	semidirect
230. #3274	C₂×C₂₃	semidirect
231. #3205	C₂×C₂₃	semidirect
232. #3192	C₂×C₂₃	semidirect
233. #3103	C₂×C₂₃	semidirect
234. #3006	C₂×C₂₃	semidirect
235. #2975	C₂×C₂₃	semidirect
261. C₂²×#41	C₂²×C₂₃	semidirect
262. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
264. #550	C₂²×C₂₃	semidirect
265. #547	C₂²×C₂₃	semidirect
267. D₄²	C₂²×C₂₃	semidirect
272. C₂×#127	C₂²×C₂₃	semidirect
275. C₂×#121	C₂²×C₂₃	semidirect
276. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
279. #493	C₂²×C₂₃	semidirect
280. #486	C₂²×C₂₃	semidirect
281. #481	C₂²×C₂₃	semidirect
282. #481	C₂²×C₂₃	semidirect
283. #475	C₂²×C₂₃	semidirect
284. #469	C₂²×C₂₃	semidirect
285. #460	C₂²×C₂₃	semidirect
286. #459	C₂²×C₂₃	semidirect
287. #449	C₂²×C₂₃	semidirect
288. #448	C₂²×C₂₃	semidirect
289. #447	C₂²×C₂₃	semidirect
290. #447	C₂²×C₂₃	semidirect
291. C₄×SD₁₆	C₂²×C₂₃	semidirect
292. #436	C₂²×C₂₃	semidirect
293. #431	C₂²×C₂₃	semidirect
294. #417	C₂²×C₂₃	semidirect
295. #407	C₂²×C₂₃	semidirect
367. C₂₃	#61587	direct (N × H)
450. Q₈	C₂₃×H(5,𝔽₂)	semidirect
477. C₂×C₄	C₂₃×AGL(1,ℤ/8)	semidirect

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×#550	2
C₂₃×#3332	2
C₂₃×#3274	2
C₂₃×#3205	2
C₂₃×#3192	2
C₂₃×#3103	2
C₂₃×#3006	2
C₂₃×#2975	2
#61587	23

Subgroups — C23×#61587 #19209213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61587 #19209213