Subgroups — C₂₃×#23047 #19170673

Order 5888.

840 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#473	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2947	C₂	1	2	yes	maximal
4	C₂₃×#2912	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1951	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1930	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1882	C₂	1	2	yes	maximal
9	C₂₃×#1857	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1765	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
15	C₂₃×#520	C₂²	1	2	yes	—
16	C₂₃×#519	C₂²	1	2	yes	—
17	C₂₃×#473	C₄	1	8	yes	complemented
18	C₂₃×#441	—	2	1	no	—
19	C₂₃×#441	—	2	1	no	—
20	C₂₃×#441	—	2	2	no	—
21	C₂₃×#440	—	2	2	no	—
22	C₄×Q₁₆×C₂₃	—	2	2	no	—
23	C₄×Q₁₆×C₂₃	—	2	2	no	—
24	C₄×SD₁₆×C₂₃	—	2	2	no	—
25	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
26	C₂₃×#428	—	2	2	no	—
27	C₂₃×#428	C₂²	1	2	yes	—
28	C₂₃×#428	C₂²	1	2	yes	—
29	C₂₃×#427	—	2	1	no	—
30	C₂₃×#427	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#106	—	2	1	no	—
32	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#104	C₂²	1	1	yes	characteristic
34	C₂₃×#419	C₂²	1	2	yes	—
35	C₂₃×#416	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#103	C₂²	1	2	yes	—
37	C₂×C₂₃×#101	C₂²	1	2	yes	—
38	C₂₃×#405	C₂²	1	2	yes	—
39	C₂₃×#405	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	2	yes	—
42	C₂₃×#343	C₂²	1	2	yes	—
43	C₂₃×#341	C₂²	1	1	yes	characteristic
44	C₂₃×#339	C₂²	1	1	yes	characteristic
45	C₂₃×#339	C₂²	1	2	yes	—
46	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂₃×#137	—	2	4	no	—
49	C₂₃×#137	C₂×C₄	1	8	yes	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×Q₁₆×C₂₃	—	2	1	no	—
52	C₂×Q₁₆×C₂₃	—	2	1	no	—
53	C₂×Q₁₆×C₂₃	—	2	2	no	—
54	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂₃×#131	—	2	2	no	—
57	C₂₃×#131	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₂₃	—	2	1	no	—
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₂₃	—	2	2	no	—
66	C₄×Q₈×C₂₃	—	2	2	no	—
67	C₄×Q₈×C₂₃	—	2	2	no	—
68	C₄×Q₈×C₂₃	—	2	2	no	—
69	C₄×Q₈×C₂₃	—	2	2	no	—
70	C₄×D₄×C₂₃	—	2	2	no	—
71	C₄×D₄×C₂₃	—	4	2	no	—
72	C₂₃×#117	—	2	2	no	—
73	C₂₃×#117	—	4	2	no	—
74	C₂₃×#117	C₂³	1	1	yes	characteristic
75	C₂₃×#117	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#108	C₂×C₄	1	4	yes	complemented
80	C₂₃×#107	—	2	2	no	—
81	C₂₃×#107	—	2	2	no	—
82	C₂₃×#107	D₄	1	2	yes	—
83	C₂₃×#106	—	2	1	no	—
84	C₂₃×#106	—	2	1	no	—
85	C₂₃×#106	—	2	2	no	—
86	C₂₃×#106	—	2	2	no	—
87	C₂₃×#106	D₄	1	1	yes	characteristic
88	C₂₃×#106	D₄	1	1	yes	characteristic
89	C₂₃×#104	—	2	2	no	—
90	C₂₃×#103	—	2	2	no	—
91	C₂₃×#103	—	2	2	no	—
92	C₂₃×#103	—	2	2	no	—
93	C₂₃×#103	—	2	2	no	—
94	C₂₃×#103	D₄	1	2	yes	—
95	C₂₃×#102	D₄	1	2	yes	—
96	C₂₃×#101	C₂×C₄	1	8	yes	complemented
97	C₂₃×#97	—	2	2	no	—
98	C₂₃×#97	—	2	1	no	—
99	C₂₃×#97	—	2	2	no	—
100	C₄×C₈×C₂₃	—	2	2	no	—
101	C₄×C₈×C₂₃	—	2	1	no	—
102	C₂₃×#95	—	4	2	no	—
103	#41×C₂₃	—	2	1	no	—
104	#41×C₂₃	—	2	1	no	—
105	#41×C₂₃	C₂²×C₄	1	2	yes	—
106	#41×C₂₃	#41	1	2	yes	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
113	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×D₄×C₂₃	—	2	1	no	—
115	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	2	1	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	4	2	no	—
121	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
122	Q₁₆×C₂₃	—	2	8	no	—
123	Q₁₆×C₂₃	—	2	4	no	—
124	Q₁₆×C₂₃	—	2	4	no	—
125	SD₁₆×C₂₃	—	2	8	no	—
126	M₁₆×C₂₃	—	2	1	no	—
127	M₁₆×C₂₃	—	2	2	no	—
128	M₁₆×C₂₃	—	2	4	no	—
129	M₁₆×C₂₃	—	2	1	no	—
130	M₁₆×C₂₃	—	2	1	no	—
131	M₁₆×C₂₃	C₂²×C₄	1	2	yes	—
132	M₁₆×C₂₃	C₂²×C₄	1	4	yes	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	1	no	—
136	C₂×C₈×C₂₃	—	2	1	no	—
137	C₂×C₈×C₂₃	—	2	4	no	—
138	C₂×C₈×C₂₃	—	2	1	no	—
139	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₂₃	#41	1	2	yes	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	4	2	no	—
147	C₄⋊C₄×C₂₃	—	4	2	no	—
148	C₄⋊C₄×C₂₃	—	4	2	no	—
149	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₂₃	—	2	1	no	—
153	C₂²⋊C₄×C₂₃	—	2	2	no	—
154	C₂²⋊C₄×C₂₃	—	4	2	no	—
155	C₄²×C₂₃	—	2	2	no	—
156	C₄²×C₂₃	—	2	1	no	—
157	C₄²×C₂₃	—	2	1	no	—
158	C₄²×C₂₃	—	2	1	no	—
159	C₄²×C₂₃	—	4	2	no	—
160	C₄²×C₂₃	—	4	2	no	—
161	C₄²×C₂₃	—	4	2	no	—
162	#23047	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₂₃	—	2	1	no	—
164	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Q₈×C₂₃	—	2	1	no	—
166	Q₈×C₂₃	—	2	4	no	—
167	Q₈×C₂₃	—	2	4	no	—
168	Q₈×C₂₃	—	4	2	no	—
169	Q₈×C₂₃	—	4	2	no	—
170	Q₈×C₂₃	C₄×D₄	1	2	yes	—
171	D₄×C₂₃	—	2	2	no	—
172	D₄×C₂₃	—	2	2	no	—
173	D₄×C₂₃	—	2	1	no	—
174	D₄×C₂₃	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	8	2	no	—
188	C₂×C₄×C₂₃	—	16	2	no	—
189	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₂₃	—	2	2	no	—
196	C₈×C₂₃	—	2	4	no	—
197	C₈×C₂₃	—	2	2	no	—
198	C₈×C₂₃	—	4	1	no	—
199	C₈×C₂₃	C₄×D₄	1	2	yes	—
200	C₈×C₂₃	C₄×D₄	1	2	yes	—
201	C₂×#473	C₂×C₂₃	1	1	yes	characteristic
202	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#2947	C₂×C₂₃	1	2	yes	—
204	#2912	C₂×C₂₃	1	1	yes	characteristic
205	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#1951	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1930	C₂×C₂₃	1	2	yes	complemented
208	#1882	C₂×C₂₃	1	2	yes	—
209	#1857	C₂×C₂₃	1	2	yes	complemented
210	#1773	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#1765	C₂×C₂₃	1	1	yes	characteristic
212	C₂²×C₂₃	—	2	2	no	—
213	C₂²×C₂₃	—	2	2	no	—
214	C₂²×C₂₃	—	4	2	no	—
215	C₂²×C₂₃	#520	1	1	yes	characteristic
216	C₂²×C₂₃	D₄²	1	1	yes	characteristic
217	C₂²×C₂₃	#553	1	1	yes	characteristic
218	C₄×C₂₃	—	2	2	no	—
219	C₄×C₂₃	—	4	4	no	—
220	C₄×C₂₃	—	4	2	no	—
221	C₄×C₂₃	—	4	2	no	—
222	C₄×C₂₃	—	4	2	no	—
223	C₄×C₂₃	—	8	1	no	—
224	C₄×C₂₃	—	8	2	no	—
225	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₂₃	#547	1	1	yes	characteristic
228	C₄×C₂₃	#548	1	1	yes	characteristic
229	C₂×#137	C₂²×C₂₃	1	2	yes	—
230	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
231	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
232	#520	C₂²×C₂₃	1	2	yes	—
233	#519	C₂²×C₂₃	1	2	yes	—
234	#473	C₄×C₂₃	1	8	yes	complemented
235	#441	—	2	1	no	—
236	#441	—	2	1	no	—
237	#441	—	2	2	no	—
238	#440	—	2	2	no	—
239	C₄×Q₁₆	—	2	2	no	—
240	C₄×Q₁₆	—	2	2	no	—
241	C₄×SD₁₆	—	2	2	no	—
242	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
243	#428	—	2	2	no	—
244	#428	C₂²×C₂₃	1	2	yes	—
245	#428	C₂²×C₂₃	1	2	yes	—
246	#427	—	2	1	no	—
247	#427	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#106	—	2	1	no	—
249	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×#104	C₂²×C₂₃	1	1	yes	characteristic
251	#419	C₂²×C₂₃	1	2	yes	—
252	#416	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#103	C₂²×C₂₃	1	2	yes	—
254	C₂×#101	C₂²×C₂₃	1	2	yes	—
255	#405	C₂²×C₂₃	1	2	yes	—
256	#405	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
258	#391	C₂²×C₂₃	1	2	yes	—
259	#343	C₂²×C₂₃	1	2	yes	—
260	#341	C₂²×C₂₃	1	1	yes	characteristic
261	#339	C₂²×C₂₃	1	1	yes	characteristic
262	#339	C₂²×C₂₃	1	2	yes	—
263	C₂×C₂₃	—	2	1	no	—
264	C₂×C₂₃	—	2	1	no	—
265	C₂×C₂₃	—	4	2	no	—
266	C₂×C₂₃	#2361	1	1	yes	characteristic
267	C₂×C₂₃	#3289	1	2	yes	—
268	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
269	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
270	#137	—	2	4	no	—
271	#137	C₂×C₄×C₂₃	1	8	yes	—
272	C₂×Q₁₆	—	2	2	no	—
273	C₂×Q₁₆	—	2	1	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	2	2	no	—
276	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
277	C₂×SD₁₆	—	2	2	no	—
278	#131	—	2	2	no	—
279	#131	C₂×C₄×C₂₃	1	4	yes	—
280	C₂×M₁₆	—	2	1	no	—
281	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
282	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
283	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
284	C₂²×C₈	—	2	1	no	—
285	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
287	C₄×Q₈	—	2	2	no	—
288	C₄×Q₈	—	2	2	no	—
289	C₄×Q₈	—	2	2	no	—
290	C₄×Q₈	—	2	2	no	—
291	C₄×Q₈	—	2	2	no	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	—	4	2	no	—
294	#117	—	2	2	no	—
295	#117	—	4	2	no	—
296	#117	C₂³×C₂₃	1	1	yes	characteristic
297	#117	C₂³×C₂₃	1	2	yes	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
300	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
301	#108	C₂×C₄×C₂₃	1	4	yes	complemented
302	#107	—	2	2	no	—
303	#107	—	2	2	no	—
304	#107	D₄×C₂₃	1	2	yes	—
305	#106	—	2	1	no	—
306	#106	—	2	1	no	—
307	#106	—	2	2	no	—
308	#106	—	2	2	no	—
309	#106	D₄×C₂₃	1	1	yes	characteristic
310	#106	D₄×C₂₃	1	1	yes	characteristic
311	#104	—	2	2	no	—
312	#103	—	2	2	no	—
313	#103	—	2	2	no	—
314	#103	—	2	2	no	—
315	#103	—	2	2	no	—
316	#103	D₄×C₂₃	1	2	yes	—
317	#102	D₄×C₂₃	1	2	yes	—
318	#101	C₂×C₄×C₂₃	1	8	yes	complemented
319	#97	—	2	2	no	—
320	#97	—	2	1	no	—
321	#97	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	2	1	no	—
324	#95	—	4	2	no	—
325	C₂₃	#23047	1	1	yes	characteristic, direct product factor, Sylow
326	#41	—	2	1	no	—
327	#41	—	2	1	no	—
328	#41	C₂²×C₄×C₂₃	1	2	yes	—
329	#41	#41×C₂₃	1	2	yes	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
336	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
345	Q₁₆	—	2	8	no	—
346	Q₁₆	—	2	4	no	—
347	Q₁₆	—	2	4	no	—
348	SD₁₆	—	2	8	no	—
349	M₁₆	—	2	1	no	—
350	M₁₆	—	2	2	no	—
351	M₁₆	—	2	4	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	2	1	no	—
354	M₁₆	C₂²×C₄×C₂₃	1	2	yes	—
355	M₁₆	C₂²×C₄×C₂₃	1	4	yes	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	4	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
364	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
365	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂×C₈	#41×C₂₃	1	2	yes	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
373	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
374	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂²⋊C₄	—	2	1	no	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	2	no	—
383	C₄²	—	4	2	no	—
384	C₄²	—	4	2	no	—
385	C₂³	—	2	1	no	—
386	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
387	Q₈	—	2	1	no	—
388	Q₈	—	2	4	no	—
389	Q₈	—	2	4	no	—
390	Q₈	—	4	2	no	—
391	Q₈	—	4	2	no	—
392	Q₈	C₄×D₄×C₂₃	1	2	yes	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	—	2	1	no	—
396	D₄	C₄×D₄×C₂₃	1	2	yes	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	8	2	no	—
410	C₂×C₄	—	16	2	no	—
411	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
416	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
417	C₈	—	2	2	no	—
418	C₈	—	2	2	no	—
419	C₈	—	2	4	no	—
420	C₈	—	4	1	no	—
421	C₈	C₄×D₄×C₂₃	1	2	yes	—
422	C₈	C₄×D₄×C₂₃	1	2	yes	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	4	2	no	—
426	C₂²	C₂₃×#520	1	1	yes	characteristic
427	C₂²	D₄²×C₂₃	1	1	yes	characteristic
428	C₂²	C₂₃×#553	1	1	yes	characteristic
429	C₄	—	2	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	4	no	—
434	C₄	—	8	1	no	—
435	C₄	—	8	2	no	—
436	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
437	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
438	C₄	C₂₃×#547	1	1	yes	characteristic
439	C₄	C₂₃×#548	1	1	yes	characteristic
440	C₂	—	2	1	no	—
441	C₂	—	2	1	no	—
442	C₂	—	4	2	no	—
443	C₂	C₂₃×#2361	1	1	yes	characteristic
444	C₂	C₂₃×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2950	C₂	semidirect
5. C₂₃×#1971	C₂	semidirect
6. C₂₃×#1951	C₂	semidirect
7. C₂₃×#1930	C₂	semidirect
9. C₂₃×#1857	C₂	semidirect
10. C₂₃×#1773	C₂	semidirect
17. C₂₃×#473	C₄	semidirect
54. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
79. C₂₃×#108	C₂×C₄	semidirect
96. C₂₃×#101	C₂×C₄	semidirect
162. #23047	C₂₃	direct (N × H)
202. #2950	C₂×C₂₃	semidirect
205. #1971	C₂×C₂₃	semidirect
206. #1951	C₂×C₂₃	semidirect
207. #1930	C₂×C₂₃	semidirect
209. #1857	C₂×C₂₃	semidirect
210. #1773	C₂×C₂₃	semidirect
234. #473	C₄×C₂₃	semidirect
276. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
301. #108	C₂×C₄×C₂₃	semidirect
318. #101	C₂×C₄×C₂₃	semidirect
325. C₂₃	#23047	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×#473	2
C₂₃×#2950	2
C₂₃×#2947	2
C₂₃×#2912	2
C₂₃×#1971	2
C₂₃×#1951	2
C₂₃×#1930	2
C₂₃×#1882	2
C₂₃×#1857	2
C₂₃×#1773	2
C₂₃×#1765	2
#23047	23

Subgroups — C23×#23047 #19170673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23047 #19170673