Subgroups — C₂₃×#36711 #19184337

Order 5888.

2422 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#123	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2848	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2694	C₂	1	8	yes	maximal, complemented
5	C₂₃×#2674	C₂	1	8	yes	maximal, complemented
6	C₂₃×#2626	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2583	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2482	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2482	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2435	C₂	1	2	yes	maximal
11	C₂₃×#2313	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#544	—	2	8	no	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	complemented
16	C₂×C₂₃×#125	C₂²	1	4	yes	complemented
17	C₂×C₂₃×#124	C₂²	1	4	yes	—
18	C₂×C₂₃×#123	—	2	8	no	—
19	C₂×C₂₃×#123	C₂²	1	8	yes	—
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
23	C₂×C₂₃×#117	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#399	C₂²	1	4	yes	complemented
30	C₂₃×#399	C₂²	1	4	yes	complemented
31	C₂₃×#399	C₂²	1	4	yes	complemented
32	C₂₃×#398	C₂²	1	4	yes	complemented
33	C₂₃×#397	C₂²	1	4	yes	complemented
34	C₂₃×#397	C₂²	1	4	yes	complemented
35	C₂₃×#396	C₂²	1	8	yes	complemented
36	C₂₃×#395	C₂²	1	4	yes	complemented
37	C₂₃×#394	C₂²	1	8	yes	—
38	C₂₃×#393	C₂²	1	4	yes	complemented
39	C₂₃×#390	C₂²	1	4	yes	—
40	C₂₃×#389	C₂²	1	8	yes	complemented
41	C₂₃×#388	C₂²	1	8	yes	complemented
42	C₂₃×#388	C₂²	1	8	yes	complemented
43	C₂₃×#387	C₂²	1	8	yes	complemented
44	C₂₃×#385	C₂²	1	4	yes	—
45	C₂₃×#385	C₂²	1	4	yes	—
46	C₂₃×#385	C₂²	1	8	yes	complemented
47	C₂₃×#381	C₂²	1	2	yes	—
48	C₂₃×#380	C₂²	1	2	yes	—
49	C₂₃×#380	C₂²	1	4	yes	complemented
50	C₂₃×#379	C₂²	1	4	yes	—
51	C₂₃×#379	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
53	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
54	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
57	C₂²×D₄×C₂₃	—	2	4	no	—
58	C₂²×D₄×C₂₃	—	4	4	no	—
59	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₂₃	—	2	4	no	—
61	C₂³×C₄×C₂₃	—	4	4	no	—
62	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	—	2	16	no	—
69	C₂₃×#125	—	2	16	no	—
70	C₂₃×#124	—	4	8	no	—
71	C₂₃×#123	—	2	32	no	—
72	C₂₃×C₂²⋊Q₈	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	8	no	—
74	C₂₃×C₂²⋊Q₈	—	4	8	no	—
75	C₂₃×#117	—	2	8	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
77	C₂×C₄⋊C₄×C₂₃	D₄	1	8	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
103	C₂₃×#95	C₂³	1	4	yes	—
104	C₂₃×#95	C₂³	1	4	yes	—
105	C₂₃×#95	C₂³	1	8	yes	—
106	C₂₃×#95	C₂³	1	4	yes	—
107	C₂₃×#95	C₂³	1	4	yes	—
108	C₂₃×#95	C₂³	1	8	yes	—
109	C₂⁴×C₂₃	—	2	2	no	—
110	C₂⁴×C₂₃	—	2	4	no	—
111	C₂⁴×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	—	2	4	no	—
113	C₂⁴×C₂₃	—	2	4	no	—
114	C₂⁴×C₂₃	—	4	2	no	—
115	C₂⁴×C₂₃	—	4	2	no	—
116	C₂⁴×C₂₃	—	4	2	no	—
117	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
118	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
119	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	—	4	4	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	2	8	no	—
126	C₂×D₄×C₂₃	—	4	4	no	—
127	C₂×D₄×C₂₃	—	4	16	no	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	8	no	—
132	C₂²×C₄×C₂₃	—	2	8	no	—
133	C₂²×C₄×C₂₃	—	4	2	no	—
134	C₂²×C₄×C₂₃	—	4	4	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	2	no	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	4	16	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
145	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
148	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	4	no	—
155	C₄⋊C₄×C₂₃	—	2	8	no	—
156	C₄⋊C₄×C₂₃	—	2	8	no	—
157	C₄⋊C₄×C₂₃	—	4	8	no	—
158	C₄⋊C₄×C₂₃	—	4	4	no	—
159	C₄⋊C₄×C₂₃	—	4	8	no	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	16	yes	—
161	C₂²⋊C₄×C₂₃	—	2	8	no	—
162	C₂²⋊C₄×C₂₃	—	2	16	no	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₂²⋊C₄×C₂₃	—	2	8	no	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	4	8	no	—
168	C₂²⋊C₄×C₂₃	—	4	8	no	—
169	C₂²⋊C₄×C₂₃	—	4	8	no	—
170	C₄²×C₂₃	—	2	8	no	—
171	C₄²×C₂₃	—	4	4	no	—
172	#36711	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	2	no	—
175	C₂³×C₂₃	—	2	8	no	—
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	2	4	no	—
180	C₂³×C₂₃	—	2	2	no	—
181	C₂³×C₂₃	—	2	4	no	—
182	C₂³×C₂₃	—	2	4	no	—
183	C₂³×C₂₃	—	2	4	no	—
184	C₂³×C₂₃	—	4	16	no	—
185	C₂³×C₂₃	—	4	4	no	—
186	C₂³×C₂₃	—	4	4	no	—
187	C₂³×C₂₃	—	4	4	no	—
188	C₂³×C₂₃	—	4	4	no	—
189	C₂³×C₂₃	—	4	4	no	—
190	C₂³×C₂₃	—	8	8	no	—
191	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
192	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
193	C₂³×C₂₃	2₋¹⁺⁴	1	4	yes	—
194	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
195	Q₈×C₂₃	—	8	4	no	—
196	Q₈×C₂₃	—	8	4	no	—
197	D₄×C₂₃	—	4	16	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	8	no	—
207	C₂×C₄×C₂₃	—	2	8	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	8	no	—
211	C₂×C₄×C₂₃	—	2	8	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	4	16	no	—
216	C₂×C₄×C₂₃	—	4	4	no	—
217	C₂×C₄×C₂₃	—	4	4	no	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
221	C₂×#544	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×#123	C₂×C₂₃	1	1	yes	characteristic
223	#2848	C₂×C₂₃	1	2	yes	complemented
224	#2694	C₂×C₂₃	1	8	yes	complemented
225	#2674	C₂×C₂₃	1	8	yes	complemented
226	#2626	C₂×C₂₃	1	2	yes	complemented
227	#2583	C₂×C₂₃	1	2	yes	complemented
228	#2482	C₂×C₂₃	1	2	yes	complemented
229	#2482	C₂×C₂₃	1	2	yes	complemented
230	#2435	C₂×C₂₃	1	2	yes	—
231	#2313	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	2	8	no	—
235	C₂²×C₂₃	—	2	4	no	—
236	C₂²×C₂₃	—	2	8	no	—
237	C₂²×C₂₃	—	4	16	no	—
238	C₂²×C₂₃	—	4	2	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	4	4	no	—
241	C₂²×C₂₃	—	4	2	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	4	8	no	—
245	C₂²×C₂₃	—	8	8	no	—
246	C₂²×C₂₃	#536	1	4	yes	—
247	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
248	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
249	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
250	C₂²×C₂₃	#585	1	2	yes	—
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	—	2	4	no	—
254	C₄×C₂₃	—	4	2	no	—
255	C₄×C₂₃	—	4	4	no	—
256	C₄×C₂₃	—	4	4	no	—
257	C₄×C₂₃	—	4	2	no	—
258	C₄×C₂₃	—	4	8	no	—
259	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
260	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
261	#544	—	2	8	no	—
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
263	C₂×#125	C₂²×C₂₃	1	4	yes	complemented
264	C₂×#124	C₂²×C₂₃	1	4	yes	—
265	C₂×#123	—	2	8	no	—
266	C₂×#123	C₂²×C₂₃	1	8	yes	—
267	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
268	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
269	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
270	C₂×#117	C₂²×C₂₃	1	2	yes	—
271	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
272	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
273	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
274	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
275	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
276	#399	C₂²×C₂₃	1	4	yes	complemented
277	#399	C₂²×C₂₃	1	4	yes	complemented
278	#399	C₂²×C₂₃	1	4	yes	complemented
279	#398	C₂²×C₂₃	1	4	yes	complemented
280	#397	C₂²×C₂₃	1	4	yes	complemented
281	#397	C₂²×C₂₃	1	4	yes	complemented
282	#396	C₂²×C₂₃	1	8	yes	complemented
283	#395	C₂²×C₂₃	1	4	yes	complemented
284	#394	C₂²×C₂₃	1	8	yes	—
285	#393	C₂²×C₂₃	1	4	yes	complemented
286	#390	C₂²×C₂₃	1	4	yes	—
287	#389	C₂²×C₂₃	1	8	yes	complemented
288	#388	C₂²×C₂₃	1	8	yes	complemented
289	#388	C₂²×C₂₃	1	8	yes	complemented
290	#387	C₂²×C₂₃	1	8	yes	complemented
291	#385	C₂²×C₂₃	1	4	yes	—
292	#385	C₂²×C₂₃	1	4	yes	—
293	#385	C₂²×C₂₃	1	8	yes	complemented
294	#381	C₂²×C₂₃	1	2	yes	—
295	#380	C₂²×C₂₃	1	2	yes	—
296	#380	C₂²×C₂₃	1	4	yes	complemented
297	#379	C₂²×C₂₃	1	4	yes	—
298	#379	C₂²×C₂₃	1	4	yes	—
299	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
300	C₂×C₂₃	—	2	4	no	—
301	C₂×C₂₃	—	4	4	no	—
302	C₂×C₂₃	—	8	2	no	—
303	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
304	C₂×C₂₃	#3484	1	1	yes	characteristic
305	C₂×C₂₃	#3487	1	1	yes	characteristic
306	C₂×C₂₃	#3535	1	4	yes	—
307	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
308	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
310	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
311	C₂²×D₄	—	2	4	no	—
312	C₂²×D₄	—	4	4	no	—
313	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
314	C₂³×C₄	—	2	4	no	—
315	C₂³×C₄	—	4	4	no	—
316	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
317	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
322	C₄⋊Q₈	—	2	16	no	—
323	#125	—	2	16	no	—
324	#124	—	4	8	no	—
325	#123	—	2	32	no	—
326	C₂²⋊Q₈	—	2	8	no	—
327	C₂²⋊Q₈	—	2	8	no	—
328	C₂²⋊Q₈	—	4	8	no	—
329	#117	—	2	8	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	D₄×C₂₃	1	8	yes	—
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
343	C₂×C₂²⋊C₄	—	2	4	no	—
344	C₂×C₂²⋊C₄	—	2	4	no	—
345	C₂×C₂²⋊C₄	—	2	4	no	—
346	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
355	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
356	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
357	#95	C₂³×C₂₃	1	4	yes	—
358	#95	C₂³×C₂₃	1	4	yes	—
359	#95	C₂³×C₂₃	1	8	yes	—
360	#95	C₂³×C₂₃	1	4	yes	—
361	#95	C₂³×C₂₃	1	4	yes	—
362	#95	C₂³×C₂₃	1	8	yes	—
363	C₂₃	#36711	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	2	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	—	2	2	no	—
367	C₂⁴	—	2	4	no	—
368	C₂⁴	—	2	4	no	—
369	C₂⁴	—	4	2	no	—
370	C₂⁴	—	4	2	no	—
371	C₂⁴	—	4	2	no	—
372	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
373	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
374	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	4	no	—
377	C₂×Q₈	—	4	4	no	—
378	C₂×Q₈	—	4	4	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	—	4	16	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	4	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	16	no	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₄⋊C₄	—	4	8	no	—
414	C₄⋊C₄	—	4	8	no	—
415	C₄⋊C₄	C₂×D₄×C₂₃	1	16	yes	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	16	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	4	8	no	—
423	C₂²⋊C₄	—	4	8	no	—
424	C₂²⋊C₄	—	4	8	no	—
425	C₄²	—	2	8	no	—
426	C₄²	—	4	4	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	8	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	4	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	4	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	4	16	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	4	no	—
441	C₂³	—	4	4	no	—
442	C₂³	—	4	4	no	—
443	C₂³	—	4	4	no	—
444	C₂³	—	8	8	no	—
445	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
446	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
447	C₂³	C₂₃×2₋¹⁺⁴	1	4	yes	—
448	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
449	Q₈	—	8	4	no	—
450	Q₈	—	8	4	no	—
451	D₄	—	4	16	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	8	no	—
461	C₂×C₄	—	2	8	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	16	no	—
470	C₂×C₄	—	4	4	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
474	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	8	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	8	no	—
480	C₂²	—	4	16	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	8	no	—
488	C₂²	—	8	8	no	—
489	C₂²	C₂₃×#536	1	4	yes	—
490	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
491	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
492	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
493	C₂²	C₂₃×#585	1	2	yes	—
494	C₄	—	2	4	no	—
495	C₄	—	2	4	no	—
496	C₄	—	2	4	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	8	no	—
502	C₂	—	2	4	no	—
503	C₂	—	4	4	no	—
504	C₂	—	8	2	no	—
505	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
506	C₂	C₂₃×#3484	1	1	yes	characteristic
507	C₂	C₂₃×#3487	1	1	yes	characteristic
508	C₂	C₂₃×#3535	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#544	C₂	semidirect
3. C₂₃×#2848	C₂	semidirect
4. C₂₃×#2694	C₂	semidirect
5. C₂₃×#2674	C₂	semidirect
6. C₂₃×#2626	C₂	semidirect
7. C₂₃×#2583	C₂	semidirect
8. C₂₃×#2482	C₂	semidirect
9. C₂₃×#2482	C₂	semidirect
11. C₂₃×#2313	C₂	semidirect
15. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂×C₂₃×#125	C₂²	semidirect
22. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
29. C₂₃×#399	C₂²	semidirect
30. C₂₃×#399	C₂²	semidirect
31. C₂₃×#399	C₂²	semidirect
32. C₂₃×#398	C₂²	semidirect
33. C₂₃×#397	C₂²	semidirect
34. C₂₃×#397	C₂²	semidirect
35. C₂₃×#396	C₂²	semidirect
36. C₂₃×#395	C₂²	semidirect
38. C₂₃×#393	C₂²	semidirect
40. C₂₃×#389	C₂²	semidirect
41. C₂₃×#388	C₂²	semidirect
42. C₂₃×#388	C₂²	semidirect
43. C₂₃×#387	C₂²	semidirect
46. C₂₃×#385	C₂²	semidirect
49. C₂₃×#380	C₂²	semidirect
172. #36711	C₂₃	direct (N × H)
221. C₂×#544	C₂×C₂₃	semidirect
223. #2848	C₂×C₂₃	semidirect
224. #2694	C₂×C₂₃	semidirect
225. #2674	C₂×C₂₃	semidirect
226. #2626	C₂×C₂₃	semidirect
227. #2583	C₂×C₂₃	semidirect
228. #2482	C₂×C₂₃	semidirect
229. #2482	C₂×C₂₃	semidirect
231. #2313	C₂×C₂₃	semidirect
262. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
263. C₂×#125	C₂²×C₂₃	semidirect
269. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
276. #399	C₂²×C₂₃	semidirect
277. #399	C₂²×C₂₃	semidirect
278. #399	C₂²×C₂₃	semidirect
279. #398	C₂²×C₂₃	semidirect
280. #397	C₂²×C₂₃	semidirect
281. #397	C₂²×C₂₃	semidirect
282. #396	C₂²×C₂₃	semidirect
283. #395	C₂²×C₂₃	semidirect
285. #393	C₂²×C₂₃	semidirect
287. #389	C₂²×C₂₃	semidirect
288. #388	C₂²×C₂₃	semidirect
289. #388	C₂²×C₂₃	semidirect
290. #387	C₂²×C₂₃	semidirect
293. #385	C₂²×C₂₃	semidirect
296. #380	C₂²×C₂₃	semidirect
363. C₂₃	#36711	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×#544	2
C₂²×C₂₃×#123	2
C₂₃×#2848	2
C₂₃×#2694	2
C₂₃×#2674	2
C₂₃×#2626	2
C₂₃×#2583	2
C₂₃×#2482	2
C₂₃×#2435	2
C₂₃×#2313	2
#36711	23

Subgroups — C23×#36711 #19184337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36711 #19184337