Subgroups — C₂₃×#61408 #19209034

Order 5888.

1858 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3507	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3402	C₂	1	4	yes	maximal, complemented
3	C₂₃×#3400	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3396	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3395	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3393	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3156	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#3046	C₂	1	4	yes	maximal, complemented
10	C₂₃×#3037	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2951	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₂₃×#137	C₂²	1	4	yes	—
15	C₂×C₂₃×#135	C₂²	1	2	yes	—
16	C₂×C₂₃×#135	C₂²	1	2	yes	—
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#559	C₂²	1	4	yes	—
19	C₂₃×#557	C₂²	1	4	yes	—
20	C₂₃×#556	—	2	2	no	—
21	C₂₃×#556	C₂²	1	8	yes	—
22	C₂₃×#554	—	2	2	no	—
23	C₂₃×#552	—	2	4	no	—
24	C₂₃×#539	—	2	4	no	—
25	C₂₃×#536	C₂²	1	4	yes	—
26	C₂₃×#533	C₂²	1	1	yes	characteristic
27	C₂₃×#532	C₂²	1	2	yes	—
28	C₂₃×#529	—	2	4	no	—
29	C₂₃×#520	C₂²	1	4	yes	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
31	C₂₃×#497	C₂²	1	8	yes	—
32	C₂₃×#495	C₂²	1	2	yes	—
33	C₂₃×#495	C₂²	1	4	yes	—
34	C₂₃×#495	C₂²	1	2	yes	—
35	C₂₃×#494	C₂²	1	4	yes	—
36	C₂₃×#494	C₂²	1	1	yes	characteristic
37	C₂₃×#493	C₂²	1	1	yes	characteristic
38	C₂₃×#492	C₂²	1	2	yes	—
39	C₂₃×#464	C₂²	1	8	yes	—
40	C₂₃×#464	C₂²	1	8	yes	—
41	C₂₃×#463	C₂²	1	8	yes	—
42	C₂₃×#462	C₂²	1	4	yes	—
43	C₂₃×#462	C₂²	1	8	yes	—
44	C₂₃×#461	C₂²	1	8	yes	—
45	C₂₃×#460	C₂²	1	4	yes	—
46	C₂₃×#446	C₂²	1	2	yes	—
47	C₂₃×#446	C₂²	1	2	yes	—
48	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
49	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
50	C₂₃×#440	C₂²	1	8	yes	—
51	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
52	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
53	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
54	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
55	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
56	C₄×D₈×C₂₃	C₂²	1	4	yes	—
57	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
58	C₂₃×#424	C₂²	1	2	yes	—
59	C₂₃×#423	C₂²	1	1	yes	characteristic
60	C₂₃×#417	C₂²	1	4	yes	—
61	C₂×C₂₃×#103	C₂²	1	2	yes	—
62	C₂₃×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
64	C₂×#41×C₂₃	C₂³	1	4	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
66	C₂₃×#137	—	4	8	no	—
67	C₂₃×#135	—	2	8	no	—
68	C₂₃×#135	—	4	4	no	—
69	C₂×Q₁₆×C₂₃	—	2	4	no	—
70	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
74	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
77	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
78	C₂×D₈×C₂₃	C₂³	1	2	yes	—
79	C₂×D₈×C₂₃	C₂³	1	2	yes	—
80	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
81	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
85	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
86	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
87	C₂₃×#127	C₂³	1	2	yes	—
88	C₂₃×#126	—	2	8	no	—
89	C₂₃×#126	—	2	8	no	—
90	C₂₃×#125	—	2	8	no	—
91	C₂₃×#125	—	2	4	no	—
92	C₂₃×#125	D₄	1	2	yes	—
93	C₂₃×#124	—	2	4	no	—
94	C₂₃×#124	C₂³	1	4	yes	—
95	C₂₃×#124	C₂³	1	8	yes	—
96	C₂₃×#123	—	2	8	no	—
97	C₂₃×#123	—	2	8	no	—
98	C₂₃×C₂²⋊Q₈	—	2	8	no	—
99	C₂₃×C₂²⋊Q₈	—	2	4	no	—
100	C₂₃×C₂²⋊Q₈	—	2	4	no	—
101	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
102	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
103	C₂₃×#121	—	2	2	no	—
104	C₂₃×#121	—	2	2	no	—
105	C₄×Q₈×C₂₃	—	2	4	no	—
106	C₄×Q₈×C₂₃	—	2	2	no	—
107	C₄×Q₈×C₂₃	—	2	2	no	—
108	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
109	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
110	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
111	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
112	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
113	C₄×D₄×C₂₃	—	2	4	no	—
114	C₄×D₄×C₂₃	—	2	8	no	—
115	C₄×D₄×C₂₃	—	2	2	no	—
116	C₄×D₄×C₂₃	—	2	2	no	—
117	C₄×D₄×C₂₃	C₂³	1	8	yes	—
118	C₂₃×#117	—	2	4	no	—
119	C₂₃×#117	—	2	8	no	—
120	C₂₃×#117	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
122	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
123	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
124	C₂×C₄²×C₂₃	—	2	2	no	—
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂₃×#107	C₂³	1	2	yes	—
127	C₂₃×#107	C₂³	1	2	yes	—
128	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
129	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
130	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
131	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
132	C₂₃×#103	C₂³	1	4	yes	—
133	C₂₃×#103	C₂³	1	4	yes	—
134	C₂₃×#103	C₂³	1	8	yes	—
135	C₂₃×#102	C₂³	1	8	yes	—
136	C₂₃×#97	C₂³	1	4	yes	—
137	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₂₃	C₂³	1	4	yes	—
142	#41×C₂₃	—	2	8	no	—
143	#41×C₂₃	—	2	8	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	2	4	no	—
146	C₂×Q₈×C₂₃	—	2	4	no	—
147	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
150	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
151	C₂×D₄×C₂₃	—	2	4	no	—
152	C₂×D₄×C₂₃	—	2	2	no	—
153	C₂×D₄×C₂₃	—	2	2	no	—
154	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	2	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	8	no	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₂₃	—	2	4	no	—
163	Q₁₆×C₂₃	—	2	4	no	—
164	Q₁₆×C₂₃	—	2	4	no	—
165	Q₁₆×C₂₃	—	4	2	no	—
166	Q₁₆×C₂₃	—	4	8	no	—
167	SD₁₆×C₂₃	—	2	8	no	—
168	SD₁₆×C₂₃	—	4	4	no	—
169	SD₁₆×C₂₃	—	4	8	no	—
170	D₈×C₂₃	—	2	4	no	—
171	D₈×C₂₃	—	4	2	no	—
172	M₁₆×C₂₃	—	2	4	no	—
173	C₂×C₈×C₂₃	—	2	2	no	—
174	C₂×C₈×C₂₃	#41	1	4	yes	—
175	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
176	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
179	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	8	no	—
182	C₄⋊C₄×C₂₃	—	2	4	no	—
183	C₄⋊C₄×C₂₃	—	2	4	no	—
184	C₄⋊C₄×C₂₃	—	2	8	no	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	8	no	—
188	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
191	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
195	C₂²⋊C₄×C₂₃	—	2	4	no	—
196	C₂²⋊C₄×C₂₃	—	2	8	no	—
197	C₂²⋊C₄×C₂₃	—	2	8	no	—
198	C₂²⋊C₄×C₂₃	—	2	4	no	—
199	C₂²⋊C₄×C₂₃	—	2	8	no	—
200	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
201	C₄²×C₂₃	—	2	4	no	—
202	C₄²×C₂₃	—	2	4	no	—
203	C₄²×C₂₃	—	2	8	no	—
204	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
207	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
208	C₄²×C₂₃	C₂⁴	1	4	yes	—
209	#61408	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	4	no	—
211	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₂₃	—	2	4	no	—
213	Q₈×C₂₃	—	2	4	no	—
214	Q₈×C₂₃	—	2	4	no	—
215	Q₈×C₂₃	—	2	4	no	—
216	Q₈×C₂₃	—	2	4	no	—
217	Q₈×C₂₃	—	4	2	no	—
218	Q₈×C₂₃	—	4	4	no	—
219	D₄×C₂₃	—	2	4	no	—
220	D₄×C₂₃	—	2	4	no	—
221	D₄×C₂₃	—	4	8	no	—
222	D₄×C₂₃	—	4	4	no	—
223	D₄×C₂₃	—	8	2	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	4	no	—
230	C₂×C₄×C₂₃	—	4	8	no	—
231	C₂×C₄×C₂₃	—	8	8	no	—
232	C₂×C₄×C₂₃	—	8	2	no	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
241	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
242	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
243	C₈×C₂₃	—	2	1	no	—
244	C₈×C₂₃	—	2	1	no	—
245	C₈×C₂₃	—	2	4	no	—
246	C₈×C₂₃	C₂×#41	1	2	yes	—
247	C₈×C₂₃	C₂×#41	1	2	yes	—
248	#3507	C₂×C₂₃	1	2	yes	complemented
249	#3402	C₂×C₂₃	1	4	yes	complemented
250	#3400	C₂×C₂₃	1	4	yes	complemented
251	#3396	C₂×C₂₃	1	2	yes	complemented
252	#3395	C₂×C₂₃	1	2	yes	complemented
253	#3393	C₂×C₂₃	1	4	yes	complemented
254	#3156	C₂×C₂₃	1	2	yes	complemented
255	#3147	C₂×C₂₃	1	1	yes	characteristic
256	#3046	C₂×C₂₃	1	4	yes	complemented
257	#3037	C₂×C₂₃	1	2	yes	complemented
258	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#2951	C₂×C₂₃	1	2	yes	complemented
260	C₂×C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂²×C₂₃	—	2	1	no	—
262	C₂²×C₂₃	—	2	1	no	—
263	C₂²×C₂₃	—	4	4	no	—
264	C₂²×C₂₃	—	4	4	no	—
265	C₂²×C₂₃	—	4	4	no	—
266	C₂²×C₂₃	#550	1	2	yes	—
267	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
268	C₄×C₂₃	—	2	1	no	—
269	C₄×C₂₃	—	2	1	no	—
270	C₄×C₂₃	—	2	4	no	—
271	C₄×C₂₃	—	2	1	no	—
272	C₄×C₂₃	—	2	4	no	—
273	C₄×C₂₃	—	4	4	no	—
274	C₄×C₂₃	—	4	4	no	—
275	C₄×C₂₃	—	4	4	no	—
276	C₄×C₂₃	#550	1	2	yes	—
277	C₄×C₂₃	#578	1	2	yes	—
278	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
279	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂×#137	C₂²×C₂₃	1	4	yes	—
281	C₂×#135	C₂²×C₂₃	1	2	yes	—
282	C₂×#135	C₂²×C₂₃	1	2	yes	—
283	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
284	#559	C₂²×C₂₃	1	4	yes	—
285	#557	C₂²×C₂₃	1	4	yes	—
286	#556	—	2	2	no	—
287	#556	C₂²×C₂₃	1	8	yes	—
288	#554	—	2	2	no	—
289	#552	—	2	4	no	—
290	#539	—	2	4	no	—
291	#536	C₂²×C₂₃	1	4	yes	—
292	#533	C₂²×C₂₃	1	1	yes	characteristic
293	#532	C₂²×C₂₃	1	2	yes	—
294	#529	—	2	4	no	—
295	#520	C₂²×C₂₃	1	4	yes	—
296	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
297	#497	C₂²×C₂₃	1	8	yes	—
298	#495	C₂²×C₂₃	1	2	yes	—
299	#495	C₂²×C₂₃	1	4	yes	—
300	#495	C₂²×C₂₃	1	2	yes	—
301	#494	C₂²×C₂₃	1	4	yes	—
302	#494	C₂²×C₂₃	1	1	yes	characteristic
303	#493	C₂²×C₂₃	1	1	yes	characteristic
304	#492	C₂²×C₂₃	1	2	yes	—
305	#464	C₂²×C₂₃	1	8	yes	—
306	#464	C₂²×C₂₃	1	8	yes	—
307	#463	C₂²×C₂₃	1	8	yes	—
308	#462	C₂²×C₂₃	1	4	yes	—
309	#462	C₂²×C₂₃	1	8	yes	—
310	#461	C₂²×C₂₃	1	8	yes	—
311	#460	C₂²×C₂₃	1	4	yes	—
312	#446	C₂²×C₂₃	1	2	yes	—
313	#446	C₂²×C₂₃	1	2	yes	—
314	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
315	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
316	#440	C₂²×C₂₃	1	8	yes	—
317	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
318	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
319	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
320	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
321	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
322	C₄×D₈	C₂²×C₂₃	1	4	yes	—
323	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
324	#424	C₂²×C₂₃	1	2	yes	—
325	#423	C₂²×C₂₃	1	1	yes	characteristic
326	#417	C₂²×C₂₃	1	4	yes	—
327	C₂×#103	C₂²×C₂₃	1	2	yes	—
328	#405	C₂²×C₂₃	1	2	yes	—
329	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×C₂₃	—	2	2	no	—
331	C₂×C₂₃	—	8	4	no	—
332	C₂×C₂₃	C₂×#550	1	1	yes	characteristic
333	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
334	C₂×C₂₃	#3584	1	1	yes	characteristic
335	C₂×#41	C₂³×C₂₃	1	4	yes	—
336	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
337	#137	—	4	8	no	—
338	#135	—	2	8	no	—
339	#135	—	4	4	no	—
340	C₂×Q₁₆	—	2	4	no	—
341	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
342	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
343	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
344	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
345	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
346	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
347	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
348	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
349	C₂×D₈	C₂³×C₂₃	1	2	yes	—
350	C₂×D₈	C₂³×C₂₃	1	2	yes	—
351	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
352	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
354	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
355	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
357	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
358	#127	C₂³×C₂₃	1	2	yes	—
359	#126	—	2	8	no	—
360	#126	—	2	8	no	—
361	#125	—	2	4	no	—
362	#125	—	2	8	no	—
363	#125	D₄×C₂₃	1	2	yes	—
364	#124	—	2	4	no	—
365	#124	C₂³×C₂₃	1	4	yes	—
366	#124	C₂³×C₂₃	1	8	yes	—
367	#123	—	2	8	no	—
368	#123	—	2	8	no	—
369	C₂²⋊Q₈	—	2	8	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	—	2	4	no	—
372	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
373	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
374	#121	—	2	2	no	—
375	#121	—	2	2	no	—
376	C₄×Q₈	—	2	4	no	—
377	C₄×Q₈	—	2	2	no	—
378	C₄×Q₈	—	2	2	no	—
379	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
380	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
381	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
382	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
383	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	8	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	C₂³×C₂₃	1	8	yes	—
389	#117	—	2	4	no	—
390	#117	—	2	8	no	—
391	#117	C₂³×C₂₃	1	2	yes	—
392	C₂×C₄⋊C₄	—	2	2	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
397	#107	C₂³×C₂₃	1	2	yes	—
398	#107	C₂³×C₂₃	1	2	yes	—
399	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
400	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
401	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
402	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
403	#103	C₂³×C₂₃	1	4	yes	—
404	#103	C₂³×C₂₃	1	4	yes	—
405	#103	C₂³×C₂₃	1	8	yes	—
406	#102	C₂³×C₂₃	1	8	yes	—
407	#97	C₂³×C₂₃	1	4	yes	—
408	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
411	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
412	C₄×C₈	C₂³×C₂₃	1	4	yes	—
413	C₂₃	#61408	1	1	yes	characteristic, direct product factor, Sylow
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	4	no	—
419	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
420	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
421	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
422	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
434	Q₁₆	—	2	4	no	—
435	Q₁₆	—	2	4	no	—
436	Q₁₆	—	2	4	no	—
437	Q₁₆	—	4	2	no	—
438	Q₁₆	—	4	8	no	—
439	SD₁₆	—	2	8	no	—
440	SD₁₆	—	4	4	no	—
441	SD₁₆	—	4	8	no	—
442	D₈	—	2	4	no	—
443	D₈	—	4	2	no	—
444	M₁₆	—	2	4	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	#41×C₂₃	1	4	yes	—
447	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
448	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
449	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
450	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
451	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
461	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
462	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
463	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
465	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
466	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	4	no	—
475	C₄²	—	2	8	no	—
476	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
479	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
480	C₄²	C₂⁴×C₂₃	1	4	yes	—
481	C₂³	—	2	4	no	—
482	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
483	Q₈	—	2	4	no	—
484	Q₈	—	2	4	no	—
485	Q₈	—	2	4	no	—
486	Q₈	—	2	4	no	—
487	Q₈	—	2	4	no	—
488	Q₈	—	4	2	no	—
489	Q₈	—	4	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	2	4	no	—
492	D₄	—	4	8	no	—
493	D₄	—	4	4	no	—
494	D₄	—	8	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	8	8	no	—
503	C₂×C₄	—	8	2	no	—
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
506	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
512	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
513	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
514	C₈	—	2	1	no	—
515	C₈	—	2	1	no	—
516	C₈	—	2	4	no	—
517	C₈	C₂×#41×C₂₃	1	2	yes	—
518	C₈	C₂×#41×C₂₃	1	2	yes	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	4	no	—
524	C₂²	C₂₃×#550	1	2	yes	—
525	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	1	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	4	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	C₂₃×#550	1	2	yes	—
535	C₄	C₂₃×#578	1	2	yes	—
536	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
537	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
538	C₂	—	2	2	no	—
539	C₂	—	8	4	no	—
540	C₂	C₂×C₂₃×#550	1	1	yes	characteristic
541	C₂	C₂×C₂₃×#578	1	1	yes	characteristic
542	C₂	C₂₃×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3507	C₂	semidirect
2. C₂₃×#3402	C₂	semidirect
3. C₂₃×#3400	C₂	semidirect
4. C₂₃×#3396	C₂	semidirect
5. C₂₃×#3395	C₂	semidirect
6. C₂₃×#3393	C₂	semidirect
7. C₂₃×#3156	C₂	semidirect
9. C₂₃×#3046	C₂	semidirect
10. C₂₃×#3037	C₂	semidirect
11. C₂₃×#2960	C₂	semidirect
12. C₂₃×#2951	C₂	semidirect
13. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
209. #61408	C₂₃	direct (N × H)
248. #3507	C₂×C₂₃	semidirect
249. #3402	C₂×C₂₃	semidirect
250. #3400	C₂×C₂₃	semidirect
251. #3396	C₂×C₂₃	semidirect
252. #3395	C₂×C₂₃	semidirect
253. #3393	C₂×C₂₃	semidirect
254. #3156	C₂×C₂₃	semidirect
256. #3046	C₂×C₂₃	semidirect
257. #3037	C₂×C₂₃	semidirect
258. #2960	C₂×C₂₃	semidirect
259. #2951	C₂×C₂₃	semidirect
260. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
413. C₂₃	#61408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3507	2
C₂₃×#3402	2
C₂₃×#3400	2
C₂₃×#3396	2
C₂₃×#3395	2
C₂₃×#3393	2
C₂₃×#3156	2
C₂₃×#3147	2
C₂₃×#3046	2
C₂₃×#3037	2
C₂₃×#2960	2
C₂₃×#2951	2
C₂×C₄×Q₁₆×C₂₃	2
#61408	23

Subgroups — C23×#61408 #19209034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61408 #19209034