Subgroups — C₂₃×#61586 #19209212

Order 5888.

2098 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3331	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3276	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3209	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3195	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3101	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3007	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
12	C₂₃×#554	—	2	4	no	—
13	C₂₃×#554	C₂²	1	4	yes	complemented
14	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
15	C₂₃×#547	—	2	4	no	—
16	C₂₃×#547	C₂²	1	8	yes	—
17	C₂₃×#544	—	2	1	no	—
18	C₂₃×#543	—	2	2	no	—
19	C₂₃×#533	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#123	—	2	2	no	—
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
24	C₂×C₂₃×#121	—	2	1	no	—
25	C₂₃×#518	—	2	1	no	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂₃×#494	C₂²	1	4	yes	complemented
30	C₂₃×#487	C₂²	1	4	yes	complemented
31	C₂₃×#483	C₂²	1	4	yes	—
32	C₂₃×#483	C₂²	1	8	yes	complemented
33	C₂₃×#476	C₂²	1	4	yes	complemented
34	C₂₃×#470	C₂²	1	16	yes	complemented
35	C₂₃×#462	C₂²	1	8	yes	complemented
36	C₂₃×#461	C₂²	1	4	yes	complemented
37	C₂₃×#452	C₂²	1	16	yes	complemented
38	C₂₃×#451	C₂²	1	8	yes	—
39	C₂₃×#451	C₂²	1	8	yes	complemented
40	C₂₃×#450	C₂²	1	4	yes	—
41	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
42	C₂₃×#436	C₂²	1	8	yes	complemented
43	C₂₃×#431	C₂²	1	2	yes	—
44	C₂₃×#417	C₂²	1	8	yes	complemented
45	C₂₃×#407	C₂²	1	4	yes	—
46	C₂×#41×C₂₃	—	2	4	no	—
47	C₂×#41×C₂₃	—	4	8	no	—
48	C₂×#41×C₂₃	C₂³	1	8	yes	—
49	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	—	2	2	no	—
55	C₂³×C₄×C₂₃	—	2	1	no	—
56	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂₃×#137	—	4	16	no	—
58	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
59	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
60	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
61	C₂₃×C₄⋊Q₈	—	2	4	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
64	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₂₃×#125	—	2	4	no	—
67	C₂₃×#125	—	2	2	no	—
68	C₂₃×#123	—	2	16	no	—
69	C₂₃×C₂²⋊Q₈	—	2	4	no	—
70	C₂₃×C₂²⋊Q₈	—	2	4	no	—
71	C₂₃×C₂²⋊Q₈	—	2	4	no	—
72	C₂₃×C₂²⋊Q₈	—	2	2	no	—
73	C₂₃×C₂²⋊Q₈	—	2	8	no	—
74	C₂₃×C₂²⋊Q₈	—	2	8	no	—
75	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
76	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
77	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
78	C₂₃×#121	—	2	8	no	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
81	C₄×D₄×C₂₃	—	2	4	no	—
82	C₄×D₄×C₂₃	—	2	4	no	—
83	C₄×D₄×C₂₃	—	2	2	no	—
84	C₄×D₄×C₂₃	D₄	1	4	yes	—
85	C₄×D₄×C₂₃	C₂³	1	4	yes	—
86	C₄×D₄×C₂₃	C₂³	1	4	yes	—
87	C₂₃×#117	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#106	C₂³	1	8	yes	—
102	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
103	C₂₃×#103	C₂³	1	16	yes	—
104	C₂₃×#103	C₂³	1	8	yes	—
105	C₂₃×#102	C₂³	1	8	yes	—
106	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
107	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
108	C₄×C₈×C₂₃	C₂³	1	4	yes	—
109	C₂⁴×C₂₃	—	2	1	no	—
110	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
111	#41×C₂₃	—	2	16	no	—
112	#41×C₂₃	—	4	8	no	—
113	#41×C₂₃	—	4	16	no	—
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	—	4	4	no	—
118	C₂×Q₈×C₂₃	—	4	8	no	—
119	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
120	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
121	C₂×D₄×C₂₃	—	2	4	no	—
122	C₂×D₄×C₂₃	—	2	8	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	4	8	no	—
126	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
127	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	8	no	—
134	C₂²×C₄×C₂₃	—	4	4	no	—
135	C₂²×C₄×C₂₃	—	4	8	no	—
136	C₂²×C₄×C₂₃	—	4	4	no	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
143	Q₁₆×C₂₃	—	4	16	no	—
144	SD₁₆×C₂₃	—	2	16	no	—
145	M₁₆×C₂₃	—	4	8	no	—
146	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
159	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂²⋊C₄×C₂₃	—	2	4	no	—
162	C₂²⋊C₄×C₂₃	—	2	8	no	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	4	no	—
165	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
166	C₄²×C₂₃	—	2	4	no	—
167	C₄²×C₂₃	C₂×D₄	1	2	yes	—
168	C₄²×C₂₃	C₂⁴	1	2	yes	—
169	#61586	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	2	2	no	—
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	4	2	no	—
174	C₂³×C₂₃	—	4	2	no	—
175	C₂³×C₂₃	—	4	2	no	—
176	C₂³×C₂₃	—	4	2	no	—
177	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
178	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₂₃	—	2	8	no	—
180	Q₈×C₂₃	—	2	8	no	—
181	Q₈×C₂₃	—	4	4	no	—
182	Q₈×C₂₃	—	8	2	no	—
183	Q₈×C₂₃	—	8	4	no	—
184	D₄×C₂₃	—	2	4	no	—
185	D₄×C₂₃	—	2	8	no	—
186	D₄×C₂₃	—	2	2	no	—
187	D₄×C₂₃	—	4	16	no	—
188	D₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
189	C₂×C₄×C₂₃	—	2	8	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	4	2	no	—
196	C₂×C₄×C₂₃	—	4	4	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	4	16	no	—
199	C₂×C₄×C₂₃	—	4	4	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	8	no	—
202	C₂×C₄×C₂₃	—	8	4	no	—
203	C₂×C₄×C₂₃	—	8	4	no	—
204	C₂×C₄×C₂₃	#137	1	4	yes	complemented
205	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
208	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
210	C₈×C₂₃	—	2	8	no	—
211	#3501	C₂×C₂₃	1	1	yes	characteristic, complemented
212	C₂×#547	C₂×C₂₃	1	1	yes	characteristic
213	#3331	C₂×C₂₃	1	8	yes	complemented
214	#3276	C₂×C₂₃	1	8	yes	complemented
215	#3209	C₂×C₂₃	1	2	yes	complemented
216	#3195	C₂×C₂₃	1	4	yes	complemented
217	#3101	C₂×C₂₃	1	2	yes	complemented
218	#3007	C₂×C₂₃	1	4	yes	complemented
219	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	2	1	no	—
222	C₂²×C₂₃	—	2	4	no	—
223	C₂²×C₂₃	—	2	4	no	—
224	C₂²×C₂₃	—	2	4	no	—
225	C₂²×C₂₃	—	2	4	no	—
226	C₂²×C₂₃	—	4	8	no	—
227	C₂²×C₂₃	—	4	2	no	—
228	C₂²×C₂₃	—	4	2	no	—
229	C₂²×C₂₃	—	8	4	no	—
230	C₂²×C₂₃	C₂×#137	1	2	yes	—
231	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
232	C₄×C₂₃	—	2	2	no	—
233	C₄×C₂₃	—	2	4	no	—
234	C₄×C₂₃	—	2	1	no	—
235	C₄×C₂₃	—	4	4	no	—
236	C₄×C₂₃	—	4	4	no	—
237	C₄×C₂₃	—	4	4	no	—
238	C₄×C₂₃	—	4	2	no	—
239	C₄×C₂₃	C₂×#137	1	4	yes	—
240	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
241	C₂²×#41	C₂²×C₂₃	1	2	yes	—
242	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
243	#554	—	2	4	no	—
244	#554	C₂²×C₂₃	1	4	yes	complemented
245	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
246	#547	—	2	4	no	—
247	#547	C₂²×C₂₃	1	8	yes	—
248	#544	—	2	1	no	—
249	#543	—	2	2	no	—
250	#533	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#123	—	2	2	no	—
253	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
254	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
255	C₂×#121	—	2	1	no	—
256	#518	—	2	1	no	—
257	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	—	2	1	no	—
260	#494	C₂²×C₂₃	1	4	yes	complemented
261	#487	C₂²×C₂₃	1	4	yes	complemented
262	#483	C₂²×C₂₃	1	4	yes	—
263	#483	C₂²×C₂₃	1	8	yes	complemented
264	#476	C₂²×C₂₃	1	4	yes	complemented
265	#470	C₂²×C₂₃	1	16	yes	complemented
266	#462	C₂²×C₂₃	1	8	yes	complemented
267	#461	C₂²×C₂₃	1	4	yes	complemented
268	#452	C₂²×C₂₃	1	16	yes	complemented
269	#451	C₂²×C₂₃	1	8	yes	—
270	#451	C₂²×C₂₃	1	8	yes	complemented
271	#450	C₂²×C₂₃	1	4	yes	—
272	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
273	#436	C₂²×C₂₃	1	8	yes	complemented
274	#431	C₂²×C₂₃	1	2	yes	—
275	#417	C₂²×C₂₃	1	8	yes	complemented
276	#407	C₂²×C₂₃	1	4	yes	—
277	C₂×C₂₃	—	2	2	no	—
278	C₂×C₂₃	—	4	4	no	—
279	C₂×C₂₃	—	4	4	no	—
280	C₂×C₂₃	#3484	1	1	yes	characteristic
281	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
282	C₂×C₂₃	#3586	1	1	yes	characteristic
283	C₂×#41	—	2	4	no	—
284	C₂×#41	—	4	8	no	—
285	C₂×#41	C₂³×C₂₃	1	8	yes	—
286	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
287	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
288	C₂²×D₄	—	2	2	no	—
289	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
290	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	—	2	2	no	—
293	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
294	#137	—	4	16	no	—
295	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
296	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
297	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
298	C₄⋊Q₈	—	2	4	no	—
299	C₄⋊Q₈	—	2	2	no	—
300	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
301	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
302	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
303	#125	—	2	4	no	—
304	#125	—	2	2	no	—
305	#123	—	2	16	no	—
306	C₂²⋊Q₈	—	2	4	no	—
307	C₂²⋊Q₈	—	2	4	no	—
308	C₂²⋊Q₈	—	2	4	no	—
309	C₂²⋊Q₈	—	2	2	no	—
310	C₂²⋊Q₈	—	2	8	no	—
311	C₂²⋊Q₈	—	2	8	no	—
312	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
313	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
314	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
315	#121	—	2	8	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	D₄×C₂₃	1	4	yes	—
322	C₄×D₄	C₂³×C₂₃	1	4	yes	—
323	C₄×D₄	C₂³×C₂₃	1	4	yes	—
324	#117	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	8	no	—
330	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
333	C₂×C₂²⋊C₄	—	2	4	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
337	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
338	#106	C₂³×C₂₃	1	8	yes	—
339	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
340	#103	C₂³×C₂₃	1	16	yes	—
341	#103	C₂³×C₂₃	1	8	yes	—
342	#102	C₂³×C₂₃	1	8	yes	—
343	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
344	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
345	C₄×C₈	C₂³×C₂₃	1	4	yes	—
346	C₂₃	#61586	1	1	yes	characteristic, direct product factor, Sylow
347	C₂⁴	—	2	1	no	—
348	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
349	#41	—	2	16	no	—
350	#41	—	4	8	no	—
351	#41	—	4	16	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	4	4	no	—
356	C₂×Q₈	—	4	8	no	—
357	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
358	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	4	8	no	—
364	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
365	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	4	4	no	—
373	C₂²×C₄	—	4	8	no	—
374	C₂²×C₄	—	4	4	no	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
378	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
381	Q₁₆	—	4	16	no	—
382	SD₁₆	—	2	16	no	—
383	M₁₆	—	4	8	no	—
384	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	8	no	—
387	C₄⋊C₄	—	2	4	no	—
388	C₄⋊C₄	—	2	8	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
397	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
398	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
399	C₂²⋊C₄	—	2	4	no	—
400	C₂²⋊C₄	—	2	8	no	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
404	C₄²	—	2	4	no	—
405	C₄²	C₂×D₄×C₂₃	1	2	yes	—
406	C₄²	C₂⁴×C₂₃	1	2	yes	—
407	C₂³	—	2	4	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	4	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	—	4	2	no	—
414	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
415	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
416	Q₈	—	2	8	no	—
417	Q₈	—	2	8	no	—
418	Q₈	—	4	4	no	—
419	Q₈	—	8	4	no	—
420	Q₈	—	8	2	no	—
421	D₄	—	2	4	no	—
422	D₄	—	2	8	no	—
423	D₄	—	2	2	no	—
424	D₄	—	4	16	no	—
425	D₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	16	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	8	no	—
439	C₂×C₄	—	8	4	no	—
440	C₂×C₄	—	8	4	no	—
441	C₂×C₄	C₂₃×#137	1	4	yes	complemented
442	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
444	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
445	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
446	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
447	C₈	—	2	8	no	—
448	C₂²	—	2	1	no	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	4	8	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	8	4	no	—
458	C₂²	C₂×C₂₃×#137	1	2	yes	—
459	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
460	C₄	—	2	4	no	—
461	C₄	—	2	1	no	—
462	C₄	—	2	2	no	—
463	C₄	—	4	4	no	—
464	C₄	—	4	4	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	4	no	—
467	C₄	C₂×C₂₃×#137	1	4	yes	—
468	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
469	C₂	—	2	2	no	—
470	C₂	—	4	4	no	—
471	C₂	—	4	4	no	—
472	C₂	C₂₃×#3484	1	1	yes	characteristic
473	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
474	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3501	C₂	semidirect
3. C₂₃×#3331	C₂	semidirect
4. C₂₃×#3276	C₂	semidirect
5. C₂₃×#3209	C₂	semidirect
6. C₂₃×#3195	C₂	semidirect
7. C₂₃×#3101	C₂	semidirect
8. C₂₃×#3007	C₂	semidirect
9. C₂₃×#2975	C₂	semidirect
11. C₂×C₂₃×#137	C₂²	semidirect
13. C₂₃×#554	C₂²	semidirect
14. D₄×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#494	C₂²	semidirect
30. C₂₃×#487	C₂²	semidirect
32. C₂₃×#483	C₂²	semidirect
33. C₂₃×#476	C₂²	semidirect
34. C₂₃×#470	C₂²	semidirect
35. C₂₃×#462	C₂²	semidirect
36. C₂₃×#461	C₂²	semidirect
37. C₂₃×#452	C₂²	semidirect
39. C₂₃×#451	C₂²	semidirect
41. C₄×SD₁₆×C₂₃	C₂²	semidirect
42. C₂₃×#436	C₂²	semidirect
44. C₂₃×#417	C₂²	semidirect
169. #61586	C₂₃	direct (N × H)
204. C₂×C₄×C₂₃	#137	semidirect
211. #3501	C₂×C₂₃	semidirect
213. #3331	C₂×C₂₃	semidirect
214. #3276	C₂×C₂₃	semidirect
215. #3209	C₂×C₂₃	semidirect
216. #3195	C₂×C₂₃	semidirect
217. #3101	C₂×C₂₃	semidirect
218. #3007	C₂×C₂₃	semidirect
219. #2975	C₂×C₂₃	semidirect
242. C₂×#137	C₂²×C₂₃	semidirect
244. #554	C₂²×C₂₃	semidirect
245. D₄×Q₈	C₂²×C₂₃	semidirect
260. #494	C₂²×C₂₃	semidirect
261. #487	C₂²×C₂₃	semidirect
263. #483	C₂²×C₂₃	semidirect
264. #476	C₂²×C₂₃	semidirect
265. #470	C₂²×C₂₃	semidirect
266. #462	C₂²×C₂₃	semidirect
267. #461	C₂²×C₂₃	semidirect
268. #452	C₂²×C₂₃	semidirect
270. #451	C₂²×C₂₃	semidirect
272. C₄×SD₁₆	C₂²×C₂₃	semidirect
273. #436	C₂²×C₂₃	semidirect
275. #417	C₂²×C₂₃	semidirect
346. C₂₃	#61586	direct (N × H)
441. C₂×C₄	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3501	2
C₂×C₂₃×#547	2
C₂₃×#3331	2
C₂₃×#3276	2
C₂₃×#3209	2
C₂₃×#3195	2
C₂₃×#3101	2
C₂₃×#3007	2
C₂₃×#2975	2
#61586	23

Subgroups — C23×#61586 #19209212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61586 #19209212