Subgroups — C₂×C₂₃×#2192 #19181160

Order 5888.

1150 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#509	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2192	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#350	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×H(5,𝔽₂)	—	2	1	no	—
10	C₂₃×#576	—	2	2	no	—
11	C₂₃×#576	—	2	2	no	—
12	C₂₃×#576	—	2	2	no	—
13	C₂₃×#576	C₂²	1	4	yes	complemented
14	C₂×C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
15	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
16	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂×C₂₃×#135	—	2	1	no	—
18	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
19	C₂²×D₈×C₂₃	—	2	1	no	—
20	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
23	C₂₃×#509	—	2	2	no	—
24	C₂₃×#509	C₂²	1	4	yes	complemented
25	C₂×C₂₃×SD₃₂	—	2	1	no	—
26	C₂×C₂₃×D₁₆	—	2	2	no	—
27	C₂×C₂₃×D₁₆	—	2	2	no	—
28	C₂×C₂₃×D₁₆	—	2	1	no	—
29	C₂×C₂₃×D₁₆	C₂²	1	4	yes	complemented
30	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#472	—	2	2	no	—
33	C₂₃×#472	C₂²	1	4	yes	complemented
34	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₂₃×#100	—	2	1	no	—
36	C₂₃×#361	C₂²	1	4	yes	complemented
37	C₂₃×#359	C₂²	1	4	yes	complemented
38	C₂₃×#350	C₂²	1	4	yes	complemented
39	C₂₃×H(5,𝔽₂)	—	2	4	no	—
40	C₂₃×H(5,𝔽₂)	—	2	4	no	—
41	C₂₃×H(5,𝔽₂)	—	4	2	no	—
42	C₂₃×H(5,𝔽₂)	—	4	2	no	—
43	C₂×#41×C₂₃	—	2	2	no	—
44	C₂×#41×C₂₃	—	2	1	no	—
45	C₂×#41×C₂₃	—	4	1	no	—
46	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	—	4	2	no	—
51	C₂²×D₄×C₂₃	—	4	1	no	—
52	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
53	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
54	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
55	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
56	C₂₃×AGL(1,ℤ/8)	—	2	4	no	—
57	C₂₃×AGL(1,ℤ/8)	—	4	2	no	—
58	C₂₃×#135	—	2	4	no	—
59	C₂₃×#135	—	2	4	no	—
60	C₂₃×#135	—	2	1	no	—
61	C₂₃×#135	—	2	1	no	—
62	C₂₃×#135	D₄	1	2	yes	—
63	C₂₃×#135	C₂³	1	2	yes	—
64	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
65	C₂×SD₁₆×C₂₃	—	2	2	no	—
66	C₂×SD₁₆×C₂₃	—	2	1	no	—
67	C₂×SD₁₆×C₂₃	—	4	1	no	—
68	C₂×D₈×C₂₃	—	2	2	no	—
69	C₂×D₈×C₂₃	—	2	4	no	—
70	C₂×D₈×C₂₃	—	2	4	no	—
71	C₂×D₈×C₂₃	—	2	1	no	—
72	C₂×D₈×C₂₃	—	2	1	no	—
73	C₂×D₈×C₂₃	—	2	1	no	—
74	C₂×D₈×C₂₃	—	2	1	no	—
75	C₂×D₈×C₂₃	—	2	1	no	—
76	C₂×D₈×C₂₃	—	2	1	no	—
77	C₂×D₈×C₂₃	—	2	1	no	—
78	C₂×D₈×C₂₃	—	2	1	no	—
79	C₂×D₈×C₂₃	—	4	1	no	—
80	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
81	C₂×D₈×C₂₃	D₄	1	2	yes	—
82	C₂×D₈×C₂₃	D₄	1	2	yes	—
83	C₂×D₈×C₂₃	C₂³	1	2	yes	—
84	C₂×D₈×C₂₃	C₂³	1	2	yes	complemented
85	C₂₃×#131	—	2	1	no	—
86	C₂₃×#131	—	2	1	no	—
87	C₂₃×#131	D₄	1	2	yes	—
88	C₂₃×#131	C₂³	1	2	yes	—
89	C₂×M₁₆×C₂₃	—	2	1	no	—
90	C₂×M₁₆×C₂₃	—	2	1	no	—
91	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
92	C₂²×C₈×C₂₃	—	2	1	no	—
93	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
94	C₂₃×SD₃₂	—	2	4	no	—
95	C₂₃×D₁₆	—	2	8	no	—
96	C₂₃×D₁₆	—	2	4	no	—
97	C₂₃×M₃₂	—	2	1	no	—
98	C₂₃×M₃₂	C₂³	1	2	yes	complemented
99	C₂×C₁₆×C₂₃	—	2	1	no	—
100	C₂×C₁₆×C₂₃	—	2	1	no	—
101	C₂×C₁₆×C₂₃	—	2	1	no	—
102	C₂×C₁₆×C₂₃	C₂³	1	2	yes	complemented
103	C₂₃×#108	—	2	1	no	—
104	C₂₃×#108	C₂³	1	2	yes	complemented
105	C₂₃×#100	—	2	4	no	—
106	C₂⁴×C₂₃	—	4	2	no	—
107	C₂⁴×C₂₃	—	4	1	no	—
108	C₂⁴×C₂₃	—	4	1	no	—
109	#41×C₂₃	—	2	1	no	—
110	#41×C₂₃	—	2	1	no	—
111	#41×C₂₃	—	2	8	no	—
112	#41×C₂₃	—	2	2	no	—
113	#41×C₂₃	—	2	2	no	—
114	#41×C₂₃	—	4	4	no	—
115	#41×C₂₃	—	4	4	no	—
116	#41×C₂₃	—	4	4	no	—
117	#41×C₂₃	—	4	1	no	—
118	#41×C₂₃	—	4	1	no	—
119	#41×C₂₃	D₈	1	2	yes	—
120	#41×C₂₃	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₂₃	—	2	1	no	—
122	C₂×Q₈×C₂₃	D₈	1	1	yes	characteristic
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	2	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	—	4	2	no	—
131	C₂×D₄×C₂₃	—	4	4	no	—
132	C₂×D₄×C₂₃	—	4	8	no	—
133	C₂×D₄×C₂₃	—	4	8	no	—
134	C₂×D₄×C₂₃	—	4	4	no	—
135	C₂×D₄×C₂₃	—	4	2	no	—
136	C₂×D₄×C₂₃	—	4	1	no	—
137	C₂×D₄×C₂₃	—	4	4	no	—
138	C₂×D₄×C₂₃	—	4	1	no	—
139	C₂×D₄×C₂₃	—	4	1	no	—
140	C₂×D₄×C₂₃	—	4	1	no	—
141	C₂×D₄×C₂₃	—	4	1	no	—
142	C₂×D₄×C₂₃	—	4	1	no	—
143	C₂×D₄×C₂₃	—	4	1	no	—
144	C₂×D₄×C₂₃	—	4	1	no	—
145	C₂×D₄×C₂₃	—	4	1	no	—
146	C₂×D₄×C₂₃	D₈	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	4	2	no	—
149	C₂²×C₄×C₂₃	—	4	1	no	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	Q₁₆×C₂₃	—	2	1	no	—
152	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
153	SD₁₆×C₂₃	—	2	8	no	—
154	SD₁₆×C₂₃	—	2	2	no	—
155	SD₁₆×C₂₃	—	2	2	no	—
156	SD₁₆×C₂₃	—	4	4	no	—
157	D₈×C₂₃	—	2	8	no	—
158	D₈×C₂₃	—	2	1	no	—
159	D₈×C₂₃	—	2	2	no	—
160	D₈×C₂₃	—	2	2	no	—
161	D₈×C₂₃	—	2	2	no	—
162	D₈×C₂₃	—	2	2	no	—
163	D₈×C₂₃	—	2	2	no	—
164	D₈×C₂₃	—	4	4	no	—
165	D₈×C₂₃	—	4	1	no	—
166	D₈×C₂₃	—	4	1	no	—
167	D₈×C₂₃	C₂×D₄	1	2	yes	—
168	D₈×C₂₃	C₂×D₄	1	4	yes	—
169	M₁₆×C₂₃	—	2	1	no	—
170	M₁₆×C₂₃	—	2	2	no	—
171	M₁₆×C₂₃	—	2	2	no	—
172	M₁₆×C₂₃	—	2	2	no	—
173	M₁₆×C₂₃	—	4	1	no	—
174	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	2	no	—
177	C₂×C₈×C₂₃	—	2	2	no	—
178	C₂×C₈×C₂₃	—	4	1	no	—
179	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
185	C₁₆×C₂₃	—	2	2	no	—
186	C₁₆×C₂₃	—	2	2	no	—
187	C₂×#2192	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	1	no	—
189	C₂³×C₂₃	—	4	8	no	—
190	C₂³×C₂₃	—	4	2	no	—
191	C₂³×C₂₃	—	4	1	no	—
192	C₂³×C₂₃	—	4	2	no	—
193	C₂³×C₂₃	—	4	2	no	—
194	C₂³×C₂₃	—	4	2	no	—
195	C₂³×C₂₃	—	8	1	no	—
196	C₂³×C₂₃	—	8	1	no	—
197	C₂³×C₂₃	—	16	2	no	—
198	C₂³×C₂₃	—	16	2	no	—
199	C₂³×C₂₃	—	16	4	no	—
200	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
201	Q₈×C₂₃	—	2	1	no	—
202	Q₈×C₂₃	—	2	2	no	—
203	Q₈×C₂₃	—	4	1	no	—
204	Q₈×C₂₃	C₂×D₈	1	2	yes	—
205	D₄×C₂₃	—	2	2	no	—
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	2	2	no	—
208	D₄×C₂₃	—	2	4	no	—
209	D₄×C₂₃	—	2	1	no	—
210	D₄×C₂₃	—	4	8	no	—
211	D₄×C₂₃	—	4	8	no	—
212	D₄×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	4	2	no	—
214	D₄×C₂₃	—	4	2	no	—
215	D₄×C₂₃	—	4	2	no	—
216	D₄×C₂₃	—	4	2	no	—
217	D₄×C₂₃	—	4	1	no	—
218	D₄×C₂₃	—	4	2	no	—
219	D₄×C₂₃	—	4	2	no	—
220	D₄×C₂₃	—	8	1	no	—
221	D₄×C₂₃	—	8	1	no	—
222	D₄×C₂₃	C₂×D₈	1	2	yes	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	4	8	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	2	no	—
229	C₂×C₄×C₂₃	—	4	2	no	—
230	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
231	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
234	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
236	C₈×C₂₃	—	2	2	no	—
237	C₈×C₂₃	—	4	2	no	—
238	C₈×C₂₃	#120	1	1	yes	characteristic
239	C₈×C₂₃	#120	1	1	yes	characteristic
240	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
241	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	C₂×#576	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#509	C₂×C₂₃	1	1	yes	characteristic, complemented
244	C₂²×D₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂×#472	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#2192	C₂×C₂₃	1	8	yes	direct product factor
247	C₂×#361	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂×#359	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×#350	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂²×C₂₃	—	2	2	no	—
251	C₂²×C₂₃	—	2	2	no	—
252	C₂²×C₂₃	—	4	2	no	—
253	C₂²×C₂₃	—	4	4	no	—
254	C₂²×C₂₃	—	4	2	no	—
255	C₂²×C₂₃	—	8	4	no	—
256	C₂²×C₂₃	—	8	2	no	—
257	C₂²×C₂₃	—	8	2	no	—
258	C₂²×C₂₃	—	16	2	no	—
259	C₂²×C₂₃	—	16	2	no	—
260	C₂²×C₂₃	—	16	2	no	—
261	C₂²×C₂₃	—	16	2	no	—
262	C₂²×C₂₃	—	16	8	no	—
263	C₂²×C₂₃	—	16	2	no	—
264	C₂²×C₂₃	#447	1	1	yes	characteristic
265	C₂²×C₂₃	#447	1	1	yes	characteristic
266	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₄×C₂₃	—	2	2	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	#447	1	1	yes	characteristic
270	C₄×C₂₃	#447	1	1	yes	characteristic
271	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
272	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
273	C₂×H(5,𝔽₂)	—	2	1	no	—
274	#576	—	2	2	no	—
275	#576	—	2	2	no	—
276	#576	—	2	2	no	—
277	#576	C₂²×C₂₃	1	4	yes	complemented
278	C₂×AGL(1,ℤ/8)	—	2	2	no	—
279	C₂×AGL(1,ℤ/8)	—	2	1	no	—
280	C₂×AGL(1,ℤ/8)	—	2	1	no	—
281	C₂×#135	—	2	1	no	—
282	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
283	C₂²×D₈	—	2	1	no	—
284	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
285	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
286	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
287	#509	—	2	2	no	—
288	#509	C₂²×C₂₃	1	4	yes	complemented
289	C₂×SD₃₂	—	2	1	no	—
290	C₂×D₁₆	—	2	2	no	—
291	C₂×D₁₆	—	2	2	no	—
292	C₂×D₁₆	—	2	1	no	—
293	C₂×D₁₆	C₂²×C₂₃	1	4	yes	complemented
294	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic, complemented
295	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
296	#472	—	2	2	no	—
297	#472	C₂²×C₂₃	1	4	yes	complemented
298	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic, complemented
299	C₂×#100	—	2	1	no	—
300	#361	C₂²×C₂₃	1	4	yes	complemented
301	#359	C₂²×C₂₃	1	4	yes	complemented
302	#350	C₂²×C₂₃	1	4	yes	complemented
303	C₂×C₂₃	—	2	1	no	—
304	C₂×C₂₃	—	2	1	no	—
305	C₂×C₂₃	—	4	2	no	—
306	C₂×C₂₃	—	8	4	no	—
307	C₂×C₂₃	—	8	2	no	—
308	C₂×C₂₃	—	16	2	no	—
309	C₂×C₂₃	#2192	1	2	yes	direct product factor
310	C₂×C₂₃	C₂×#447	1	1	yes	characteristic
311	H(5,𝔽₂)	—	2	4	no	—
312	H(5,𝔽₂)	—	2	4	no	—
313	H(5,𝔽₂)	—	4	2	no	—
314	H(5,𝔽₂)	—	4	2	no	—
315	C₂×#41	—	2	2	no	—
316	C₂×#41	—	2	1	no	—
317	C₂×#41	—	4	1	no	—
318	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
319	C₂²×D₄	—	2	2	no	—
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	—	4	2	no	—
323	C₂²×D₄	—	4	1	no	—
324	AGL(1,ℤ/8)	—	2	4	no	—
325	AGL(1,ℤ/8)	—	2	8	no	—
326	AGL(1,ℤ/8)	—	2	8	no	—
327	AGL(1,ℤ/8)	—	2	4	no	—
328	AGL(1,ℤ/8)	—	2	4	no	—
329	AGL(1,ℤ/8)	—	4	2	no	—
330	#135	—	2	4	no	—
331	#135	—	2	4	no	—
332	#135	—	2	1	no	—
333	#135	—	2	1	no	—
334	#135	D₄×C₂₃	1	2	yes	—
335	#135	C₂³×C₂₃	1	2	yes	—
336	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
337	C₂×SD₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	4	1	no	—
340	C₂×D₈	—	2	2	no	—
341	C₂×D₈	—	2	4	no	—
342	C₂×D₈	—	2	4	no	—
343	C₂×D₈	—	2	1	no	—
344	C₂×D₈	—	2	1	no	—
345	C₂×D₈	—	2	1	no	—
346	C₂×D₈	—	2	1	no	—
347	C₂×D₈	—	2	1	no	—
348	C₂×D₈	—	2	1	no	—
349	C₂×D₈	—	2	1	no	—
350	C₂×D₈	—	2	1	no	—
351	C₂×D₈	—	4	1	no	—
352	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
353	C₂×D₈	D₄×C₂₃	1	2	yes	—
354	C₂×D₈	D₄×C₂₃	1	2	yes	—
355	C₂×D₈	C₂³×C₂₃	1	2	yes	—
356	C₂×D₈	C₂³×C₂₃	1	2	yes	complemented
357	#131	—	2	1	no	—
358	#131	—	2	1	no	—
359	#131	D₄×C₂₃	1	2	yes	—
360	#131	C₂³×C₂₃	1	2	yes	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
364	C₂²×C₈	—	2	1	no	—
365	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	SD₃₂	—	2	4	no	—
367	D₁₆	—	2	8	no	—
368	D₁₆	—	2	4	no	—
369	M₃₂	—	2	1	no	—
370	M₃₂	C₂³×C₂₃	1	2	yes	complemented
371	C₂×C₁₆	—	2	1	no	—
372	C₂×C₁₆	—	2	1	no	—
373	C₂×C₁₆	—	2	1	no	—
374	C₂×C₁₆	C₂³×C₂₃	1	2	yes	complemented
375	#108	—	2	1	no	—
376	#108	C₂³×C₂₃	1	2	yes	complemented
377	#100	—	2	4	no	—
378	C₂₃	C₂×#2192	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	4	2	no	—
380	C₂⁴	—	4	1	no	—
381	C₂⁴	—	4	1	no	—
382	#41	—	2	1	no	—
383	#41	—	2	1	no	—
384	#41	—	2	8	no	—
385	#41	—	2	2	no	—
386	#41	—	2	2	no	—
387	#41	—	4	4	no	—
388	#41	—	4	4	no	—
389	#41	—	4	4	no	—
390	#41	—	4	1	no	—
391	#41	—	4	1	no	—
392	#41	D₈×C₂₃	1	2	yes	—
393	#41	C₂×D₄×C₂₃	1	2	yes	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	D₈×C₂₃	1	1	yes	characteristic
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	8	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	4	4	no	—
405	C₂×D₄	—	4	8	no	—
406	C₂×D₄	—	4	8	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	4	2	no	—
409	C₂×D₄	—	4	1	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	1	no	—
412	C₂×D₄	—	4	1	no	—
413	C₂×D₄	—	4	1	no	—
414	C₂×D₄	—	4	1	no	—
415	C₂×D₄	—	4	1	no	—
416	C₂×D₄	—	4	1	no	—
417	C₂×D₄	—	4	1	no	—
418	C₂×D₄	—	4	1	no	—
419	C₂×D₄	D₈×C₂₃	1	1	yes	characteristic
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	1	no	—
423	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
424	Q₁₆	—	2	1	no	—
425	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
426	SD₁₆	—	2	8	no	—
427	SD₁₆	—	2	2	no	—
428	SD₁₆	—	2	2	no	—
429	SD₁₆	—	4	4	no	—
430	D₈	—	2	8	no	—
431	D₈	—	2	1	no	—
432	D₈	—	2	2	no	—
433	D₈	—	2	2	no	—
434	D₈	—	2	2	no	—
435	D₈	—	2	2	no	—
436	D₈	—	2	2	no	—
437	D₈	—	4	4	no	—
438	D₈	—	4	1	no	—
439	D₈	—	4	1	no	—
440	D₈	C₂×D₄×C₂₃	1	2	yes	—
441	D₈	C₂×D₄×C₂₃	1	4	yes	—
442	M₁₆	—	2	1	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	—	2	2	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	4	1	no	—
447	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	4	1	no	—
452	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
458	C₁₆	—	2	2	no	—
459	C₁₆	—	2	2	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	4	8	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	4	2	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	8	1	no	—
468	C₂³	—	8	1	no	—
469	C₂³	—	16	2	no	—
470	C₂³	—	16	2	no	—
471	C₂³	—	16	4	no	—
472	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
473	Q₈	—	2	1	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	4	1	no	—
476	Q₈	C₂×D₈×C₂₃	1	2	yes	—
477	D₄	—	2	2	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	1	no	—
480	D₄	—	2	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	4	8	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	2	no	—
487	D₄	—	4	2	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	1	no	—
493	D₄	—	8	1	no	—
494	D₄	C₂×D₈×C₂₃	1	2	yes	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	4	8	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
503	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
504	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	4	2	no	—
510	C₈	C₂₃×#120	1	1	yes	characteristic
511	C₈	C₂₃×#120	1	1	yes	characteristic
512	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
513	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	8	4	no	—
520	C₂²	—	8	2	no	—
521	C₂²	—	8	2	no	—
522	C₂²	—	16	2	no	—
523	C₂²	—	16	2	no	—
524	C₂²	—	16	2	no	—
525	C₂²	—	16	2	no	—
526	C₂²	—	16	8	no	—
527	C₂²	—	16	2	no	—
528	C₂²	C₂₃×#447	1	1	yes	characteristic
529	C₂²	C₂₃×#447	1	1	yes	characteristic
530	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
531	C₄	—	2	2	no	—
532	C₄	—	4	2	no	—
533	C₄	C₂₃×#447	1	1	yes	characteristic
534	C₄	C₂₃×#447	1	1	yes	characteristic
535	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
536	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
537	C₂	—	2	1	no	—
538	C₂	—	2	1	no	—
539	C₂	—	4	2	no	—
540	C₂	—	8	4	no	—
541	C₂	—	8	2	no	—
542	C₂	—	16	2	no	—
543	C₂	C₂₃×#2192	1	2	yes	direct product factor
544	C₂	C₂×C₂₃×#447	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#447
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#576	C₂	semidirect
2. C₂×C₂₃×#509	C₂	semidirect
3. C₂²×C₂₃×D₁₆	C₂	semidirect
4. C₂×C₂₃×#472	C₂	semidirect
5. C₂₃×#2192	C₂	direct (N × H)
6. C₂×C₂₃×#361	C₂	semidirect
7. C₂×C₂₃×#359	C₂	semidirect
8. C₂×C₂₃×#350	C₂	semidirect
13. C₂₃×#576	C₂²	semidirect
21. C₂²×D₈×C₂₃	C₂²	semidirect
24. C₂₃×#509	C₂²	semidirect
29. C₂×C₂₃×D₁₆	C₂²	semidirect
30. C₂×C₂₃×M₃₂	C₂²	semidirect
31. C₂²×C₁₆×C₂₃	C₂²	semidirect
33. C₂₃×#472	C₂²	semidirect
34. C₂×C₂₃×#108	C₂²	semidirect
36. C₂₃×#361	C₂²	semidirect
37. C₂₃×#359	C₂²	semidirect
38. C₂₃×#350	C₂²	semidirect
84. C₂×D₈×C₂₃	C₂³	semidirect
98. C₂₃×M₃₂	C₂³	semidirect
102. C₂×C₁₆×C₂₃	C₂³	semidirect
104. C₂₃×#108	C₂³	semidirect
187. C₂×#2192	C₂₃	direct (N × H)
242. C₂×#576	C₂×C₂₃	semidirect
243. C₂×#509	C₂×C₂₃	semidirect
244. C₂²×D₁₆	C₂×C₂₃	semidirect
245. C₂×#472	C₂×C₂₃	semidirect
246. #2192	C₂×C₂₃	direct (N × H)
247. C₂×#361	C₂×C₂₃	semidirect
248. C₂×#359	C₂×C₂₃	semidirect
249. C₂×#350	C₂×C₂₃	semidirect
277. #576	C₂²×C₂₃	semidirect
285. C₂²×D₈	C₂²×C₂₃	semidirect
288. #509	C₂²×C₂₃	semidirect
293. C₂×D₁₆	C₂²×C₂₃	semidirect
294. C₂×M₃₂	C₂²×C₂₃	semidirect
295. C₂²×C₁₆	C₂²×C₂₃	semidirect
297. #472	C₂²×C₂₃	semidirect
298. C₂×#108	C₂²×C₂₃	semidirect
300. #361	C₂²×C₂₃	semidirect
301. #359	C₂²×C₂₃	semidirect
302. #350	C₂²×C₂₃	semidirect
309. C₂×C₂₃	#2192	direct (N × H)
356. C₂×D₈	C₂³×C₂₃	semidirect
370. M₃₂	C₂³×C₂₃	semidirect
374. C₂×C₁₆	C₂³×C₂₃	semidirect
376. #108	C₂³×C₂₃	semidirect
378. C₂₃	C₂×#2192	direct (N × H)
543. C₂	C₂₃×#2192	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#576	2
C₂×C₂₃×#509	2
C₂²×C₂₃×D₁₆	2
C₂×C₂₃×#472	2
C₂₃×#2192	2
C₂×C₂₃×#361	2
C₂×C₂₃×#359	2
C₂×C₂₃×#350	2
C₂×#2192	23

Subgroups — C2×C23×#2192 #19181160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2192 #19181160