Subgroups — C₂×C₂₃×#1685 #19166756

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1685	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#331	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₂₃	—	2	1	no	—
10	C₂²×SD₁₆×C₂₃	—	2	1	no	—
11	C₂²×D₈×C₂₃	—	2	1	no	—
12	C₂²×M₁₆×C₂₃	—	2	1	no	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#492	—	2	2	no	—
17	C₂₃×#492	C₂²	1	4	yes	complemented
18	C₂₃×#474	—	2	2	no	—
19	C₂₃×#474	C₂²	1	4	yes	complemented
20	C₂₃×#459	—	2	2	no	—
21	C₂₃×#459	C₂²	1	4	yes	—
22	C₂₃×#436	—	2	2	no	—
23	C₂₃×#436	C₂²	1	4	yes	complemented
24	C₂×C₂₃×#107	—	2	1	no	—
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	—	2	1	no	—
29	C₂×C₂₃×#102	—	2	1	no	—
30	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#335	C₂²	1	4	yes	complemented
33	C₂₃×#332	C₂²	1	4	yes	complemented
34	C₂₃×#331	C₂²	1	4	yes	complemented
35	C₂²×Q₈×C₂₃	—	2	1	no	—
36	C₂²×D₄×C₂₃	—	2	1	no	—
37	C₂²×D₄×C₂₃	—	2	1	no	—
38	C₂²×D₄×C₂₃	—	2	1	no	—
39	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₂₃	—	2	1	no	—
41	C₂×SD₁₆×C₂₃	—	2	4	no	—
42	C₂×SD₁₆×C₂₃	—	2	4	no	—
43	C₂×SD₁₆×C₂₃	—	4	4	no	—
44	C₂×SD₁₆×C₂₃	—	4	4	no	—
45	C₂×D₈×C₂₃	—	2	4	no	—
46	C₂×D₈×C₂₃	—	4	4	no	—
47	C₂×M₁₆×C₂₃	—	2	4	no	—
48	C₂×M₁₆×C₂₃	—	4	4	no	—
49	C₂²×C₈×C₂₃	—	2	1	no	—
50	C₂²×C₈×C₂₃	—	2	1	no	—
51	C₂²×C₈×C₂₃	—	2	1	no	—
52	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₂₃×#127	—	2	1	no	—
59	C₂₃×#127	—	2	1	no	—
60	C₂₃×#127	—	2	1	no	—
61	C₂₃×#127	C₂³	1	2	yes	—
62	C₄×D₄×C₂₃	—	2	1	no	—
63	C₄×D₄×C₂₃	—	2	1	no	—
64	C₄×D₄×C₂₃	D₄	1	2	yes	—
65	C₄×D₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×#107	—	2	4	no	—
72	C₂₃×C₄⋊C₈	—	2	1	no	—
73	C₂₃×C₄⋊C₈	—	2	1	no	—
74	C₂₃×C₄⋊C₈	—	2	1	no	—
75	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
76	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
77	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
78	C₂₃×#102	—	2	4	no	—
79	C₂₃×#102	—	2	4	no	—
80	C₂₃×C₂²⋊C₈	—	2	4	no	—
81	C₄×C₈×C₂₃	D₄	1	2	yes	—
82	C₄×C₈×C₂₃	C₂³	1	2	yes	—
83	C₂⁴×C₂₃	—	2	1	no	—
84	C₂⁴×C₂₃	—	4	1	no	—
85	C₂×Q₈×C₂₃	—	2	2	no	—
86	C₂×Q₈×C₂₃	—	4	1	no	—
87	C₂×Q₈×C₂₃	—	4	2	no	—
88	C₂×Q₈×C₂₃	—	4	2	no	—
89	C₂×D₄×C₂₃	—	2	1	no	—
90	C₂×D₄×C₂₃	—	2	2	no	—
91	C₂×D₄×C₂₃	—	2	2	no	—
92	C₂×D₄×C₂₃	—	2	2	no	—
93	C₂×D₄×C₂₃	—	2	2	no	—
94	C₂×D₄×C₂₃	—	2	2	no	—
95	C₂×D₄×C₂₃	—	4	1	no	—
96	C₂×D₄×C₂₃	—	4	1	no	—
97	C₂×D₄×C₂₃	—	4	1	no	—
98	C₂×D₄×C₂₃	—	4	2	no	—
99	C₂×D₄×C₂₃	—	4	2	no	—
100	C₂×D₄×C₂₃	—	4	2	no	—
101	C₂×D₄×C₂₃	—	4	2	no	—
102	C₂×D₄×C₂₃	—	8	2	no	—
103	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	2	no	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	4	2	no	—
108	C₂²×C₄×C₂₃	—	4	2	no	—
109	C₂²×C₄×C₂₃	—	4	1	no	—
110	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
113	SD₁₆×C₂₃	—	4	8	no	—
114	SD₁₆×C₂₃	—	4	8	no	—
115	D₈×C₂₃	—	4	8	no	—
116	M₁₆×C₂₃	—	4	8	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	—	2	2	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	4	2	no	—
122	C₂×C₈×C₂₃	—	4	2	no	—
123	C₂×C₈×C₂₃	—	4	2	no	—
124	C₂×C₈×C₂₃	D₈	1	4	yes	complemented
125	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₂₃	—	2	1	no	—
128	C₄⋊C₄×C₂₃	—	2	2	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	4	1	no	—
131	C₄⋊C₄×C₂₃	—	4	1	no	—
132	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²⋊C₄×C₂₃	—	2	2	no	—
134	C₂²⋊C₄×C₂₃	—	2	2	no	—
135	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₂₃	#41	1	1	yes	characteristic
138	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
139	C₂×#1685	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₂₃	—	2	2	no	—
141	C₂³×C₂₃	—	4	2	no	—
142	C₂³×C₂₃	—	4	2	no	—
143	C₂³×C₂₃	—	4	2	no	—
144	C₂³×C₂₃	—	4	2	no	—
145	C₂³×C₂₃	—	8	2	no	—
146	C₂³×C₂₃	—	8	2	no	—
147	C₂³×C₂₃	—	8	2	no	—
148	C₂³×C₂₃	#121	1	1	yes	characteristic
149	Q₈×C₂₃	—	4	2	no	—
150	Q₈×C₂₃	—	4	2	no	—
151	Q₈×C₂₃	—	8	2	no	—
152	D₄×C₂₃	—	2	2	no	—
153	D₄×C₂₃	—	2	2	no	—
154	D₄×C₂₃	—	4	2	no	—
155	D₄×C₂₃	—	4	2	no	—
156	D₄×C₂₃	—	4	2	no	—
157	D₄×C₂₃	—	4	2	no	—
158	D₄×C₂₃	—	4	2	no	—
159	D₄×C₂₃	—	8	2	no	—
160	D₄×C₂₃	—	8	2	no	—
161	D₄×C₂₃	—	8	2	no	—
162	D₄×C₂₃	—	8	2	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	4	2	no	—
167	C₂×C₄×C₂₃	—	4	2	no	—
168	C₂×C₄×C₂₃	—	4	4	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	—	8	2	no	—
171	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
172	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
173	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₄×C₂₃	#137	1	2	yes	—
178	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
181	C₈×C₂₃	—	2	2	no	—
182	C₈×C₂₃	—	4	2	no	—
183	C₈×C₂₃	—	4	2	no	—
184	C₈×C₂₃	—	4	2	no	—
185	C₈×C₂₃	C₂×D₈	1	4	yes	complemented
186	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
187	C₂×#474	C₂×C₂₃	1	1	yes	characteristic, complemented
188	C₂×#459	C₂×C₂₃	1	1	yes	characteristic
189	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
190	#1685	C₂×C₂₃	1	8	yes	direct product factor
191	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
192	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
193	C₂×#331	C₂×C₂₃	1	1	yes	characteristic, complemented
194	C₂²×C₂₃	—	4	2	no	—
195	C₂²×C₂₃	—	4	2	no	—
196	C₂²×C₂₃	—	4	2	no	—
197	C₂²×C₂₃	—	8	4	no	—
198	C₂²×C₂₃	—	8	2	no	—
199	C₂²×C₂₃	—	8	2	no	—
200	C₂²×C₂₃	—	8	2	no	—
201	C₂²×C₂₃	—	16	4	no	—
202	C₂²×C₂₃	#459	1	2	yes	—
203	C₂²×C₂₃	#468	1	2	yes	—
204	C₂²×C₂₃	#471	1	2	yes	—
205	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
206	C₄×C₂₃	—	2	1	no	—
207	C₄×C₂₃	—	2	1	no	—
208	C₄×C₂₃	—	4	2	no	—
209	C₄×C₂₃	—	8	2	no	—
210	C₄×C₂₃	#459	1	2	yes	—
211	C₄×C₂₃	#468	1	2	yes	—
212	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
213	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
216	C₂²×SD₁₆	—	2	1	no	—
217	C₂²×SD₁₆	—	2	1	no	—
218	C₂²×D₈	—	2	1	no	—
219	C₂²×M₁₆	—	2	1	no	—
220	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
221	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
223	#492	—	2	2	no	—
224	#492	C₂²×C₂₃	1	4	yes	complemented
225	#474	—	2	2	no	—
226	#474	C₂²×C₂₃	1	4	yes	complemented
227	#459	—	2	2	no	—
228	#459	C₂²×C₂₃	1	4	yes	—
229	#436	—	2	2	no	—
230	#436	C₂²×C₂₃	1	4	yes	complemented
231	C₂×#107	—	2	1	no	—
232	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
235	C₂×#102	—	2	1	no	—
236	C₂×#102	—	2	1	no	—
237	C₂×C₂²⋊C₈	—	2	1	no	—
238	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
239	#335	C₂²×C₂₃	1	4	yes	complemented
240	#332	C₂²×C₂₃	1	4	yes	complemented
241	#331	C₂²×C₂₃	1	4	yes	complemented
242	C₂×C₂₃	—	8	2	no	—
243	C₂×C₂₃	—	16	2	no	—
244	C₂×C₂₃	#1685	1	4	yes	direct product factor
245	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
246	C₂×C₂₃	C₂×#468	1	1	yes	characteristic
247	C₂×C₂₃	C₂×#471	1	1	yes	characteristic
248	C₂²×Q₈	—	2	1	no	—
249	C₂²×D₄	—	2	1	no	—
250	C₂²×D₄	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
253	C₂³×C₄	—	2	1	no	—
254	C₂×SD₁₆	—	2	4	no	—
255	C₂×SD₁₆	—	2	4	no	—
256	C₂×SD₁₆	—	4	4	no	—
257	C₂×SD₁₆	—	4	4	no	—
258	C₂×D₈	—	2	4	no	—
259	C₂×D₈	—	4	4	no	—
260	C₂×M₁₆	—	2	4	no	—
261	C₂×M₁₆	—	4	4	no	—
262	C₂²×C₈	—	2	1	no	—
263	C₂²×C₈	—	2	1	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
266	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
271	#127	—	2	1	no	—
272	#127	—	2	1	no	—
273	#127	—	2	1	no	—
274	#127	C₂³×C₂₃	1	2	yes	—
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	D₄×C₂₃	1	2	yes	—
278	C₄×D₄	C₂³×C₂₃	1	2	yes	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	—	2	1	no	—
283	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
284	#107	—	2	4	no	—
285	C₄⋊C₈	—	2	1	no	—
286	C₄⋊C₈	—	2	1	no	—
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
289	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
290	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
291	#102	—	2	4	no	—
292	#102	—	2	4	no	—
293	C₂²⋊C₈	—	2	4	no	—
294	C₄×C₈	D₄×C₂₃	1	2	yes	—
295	C₄×C₈	C₂³×C₂₃	1	2	yes	—
296	C₂₃	C₂×#1685	1	1	yes	characteristic, direct product factor, Sylow
297	C₂⁴	—	2	1	no	—
298	C₂⁴	—	4	1	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	4	1	no	—
301	C₂×Q₈	—	4	2	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	8	2	no	—
317	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	4	2	no	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	1	no	—
324	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
326	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
327	SD₁₆	—	4	8	no	—
328	SD₁₆	—	4	8	no	—
329	D₈	—	4	8	no	—
330	M₁₆	—	4	8	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	4	2	no	—
338	C₂×C₈	D₈×C₂₃	1	4	yes	complemented
339	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	—	4	1	no	—
346	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
347	C₂²⋊C₄	—	2	2	no	—
348	C₂²⋊C₄	—	2	2	no	—
349	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
350	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₄²	#41×C₂₃	1	1	yes	characteristic
352	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
353	C₂³	—	2	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	8	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	—	8	2	no	—
361	C₂³	C₂₃×#121	1	1	yes	characteristic
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	8	2	no	—
365	D₄	—	2	2	no	—
366	D₄	—	2	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	8	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	8	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
385	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
386	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
387	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
388	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
389	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
390	C₂×C₄	C₂₃×#137	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
394	C₈	—	2	2	no	—
395	C₈	—	4	2	no	—
396	C₈	—	4	2	no	—
397	C₈	—	4	2	no	—
398	C₈	C₂×D₈×C₂₃	1	4	yes	complemented
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	2	no	—
402	C₂²	—	8	4	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	16	4	no	—
407	C₂²	C₂₃×#459	1	2	yes	—
408	C₂²	C₂₃×#468	1	2	yes	—
409	C₂²	C₂₃×#471	1	2	yes	—
410	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	4	2	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂₃×#459	1	2	yes	—
416	C₄	C₂₃×#468	1	2	yes	—
417	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
418	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
419	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
420	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
421	C₂	—	8	2	no	—
422	C₂	—	16	2	no	—
423	C₂	C₂₃×#1685	1	4	yes	direct product factor
424	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
425	C₂	C₂×C₂₃×#468	1	1	yes	characteristic
426	C₂	C₂×C₂₃×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#492	C₂	semidirect
2. C₂×C₂₃×#474	C₂	semidirect
4. C₂×C₂₃×#436	C₂	semidirect
5. C₂₃×#1685	C₂	direct (N × H)
6. C₂×C₂₃×#335	C₂	semidirect
7. C₂×C₂₃×#332	C₂	semidirect
8. C₂×C₂₃×#331	C₂	semidirect
17. C₂₃×#492	C₂²	semidirect
19. C₂₃×#474	C₂²	semidirect
23. C₂₃×#436	C₂²	semidirect
32. C₂₃×#335	C₂²	semidirect
33. C₂₃×#332	C₂²	semidirect
34. C₂₃×#331	C₂²	semidirect
124. C₂×C₈×C₂₃	D₈	semidirect
139. C₂×#1685	C₂₃	direct (N × H)
185. C₈×C₂₃	C₂×D₈	semidirect
186. C₂×#492	C₂×C₂₃	semidirect
187. C₂×#474	C₂×C₂₃	semidirect
189. C₂×#436	C₂×C₂₃	semidirect
190. #1685	C₂×C₂₃	direct (N × H)
191. C₂×#335	C₂×C₂₃	semidirect
192. C₂×#332	C₂×C₂₃	semidirect
193. C₂×#331	C₂×C₂₃	semidirect
224. #492	C₂²×C₂₃	semidirect
226. #474	C₂²×C₂₃	semidirect
230. #436	C₂²×C₂₃	semidirect
239. #335	C₂²×C₂₃	semidirect
240. #332	C₂²×C₂₃	semidirect
241. #331	C₂²×C₂₃	semidirect
244. C₂×C₂₃	#1685	direct (N × H)
296. C₂₃	C₂×#1685	direct (N × H)
338. C₂×C₈	D₈×C₂₃	semidirect
398. C₈	C₂×D₈×C₂₃	semidirect
423. C₂	C₂₃×#1685	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×#492	2
C₂×C₂₃×#474	2
C₂×C₂₃×#459	2
C₂×C₂₃×#436	2
C₂₃×#1685	2
C₂×C₂₃×#335	2
C₂×C₂₃×#332	2
C₂×C₂₃×#331	2
C₂×#1685	23

Subgroups — C2×C23×#1685 #19166756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1685 #19166756