Subgroups — C₂₃×#60618 #19208244

Order 5888.

2170 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3142	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3141	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3140	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3139	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3138	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3135	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3131	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2896	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	—	2	2	no	—
15	C₂₃×#533	—	2	2	no	—
16	C₂₃×#533	C₂²	1	2	yes	—
17	C₂₃×#533	C₂²	1	2	yes	—
18	C₂₃×#533	C₂²	1	2	yes	complemented
19	C₂₃×#533	C₂²	1	2	yes	complemented
20	C₂₃×#532	—	2	2	no	—
21	C₂₃×#532	—	2	2	no	—
22	C₂₃×#532	C₂²	1	2	yes	—
23	C₂₃×#532	C₂²	1	2	yes	—
24	C₂₃×#532	C₂²	1	2	yes	—
25	C₂₃×#532	C₂²	1	2	yes	—
26	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
28	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#125	C₂²	1	2	yes	complemented
30	C₂×C₂₃×#124	C₂²	1	2	yes	—
31	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
32	C₂×C₂₃×#121	—	2	2	no	—
33	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
34	C₂×C₄×D₄×C₂₃	—	2	2	no	—
35	C₂×C₂₃×#117	C₂²	1	2	yes	—
36	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#491	C₂²	1	8	yes	complemented
38	C₂₃×#490	C₂²	1	8	yes	complemented
39	C₂₃×#489	C₂²	1	16	yes	complemented
40	C₂₃×#488	C₂²	1	16	yes	complemented
41	C₂₃×#487	C₂²	1	8	yes	complemented
42	C₂₃×#486	C₂²	1	8	yes	complemented
43	C₂₃×#419	C₂²	1	8	yes	—
44	C₂₃×#418	C₂²	1	8	yes	—
45	C₂₃×#417	C₂²	1	16	yes	complemented
46	C₂₃×#416	C₂²	1	16	yes	complemented
47	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
48	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
49	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
50	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
51	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
52	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
53	C₂×#41×C₂₃	—	2	2	no	—
54	C₂×#41×C₂₃	—	2	2	no	—
55	C₂×#41×C₂₃	—	2	16	no	—
56	C₂×#41×C₂₃	C₂³	1	4	yes	—
57	C₂×#41×C₂₃	C₂³	1	4	yes	—
58	C₂²×Q₈×C₂₃	—	2	2	no	—
59	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₂₃	—	2	2	no	—
61	C₂²×D₄×C₂₃	—	2	2	no	—
62	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₂₃	—	2	2	no	—
64	C₂³×C₄×C₂₃	D₄	1	2	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₂₃	—	2	8	no	—
67	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
68	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
70	C₂₃×C₄⋊Q₈	—	2	2	no	—
71	C₂₃×C₄⋊Q₈	—	2	2	no	—
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
77	C₂₃×#127	—	2	2	no	—
78	C₂₃×#127	C₂³	1	2	yes	—
79	C₂₃×#127	C₂³	1	2	yes	—
80	C₂₃×#125	—	2	4	no	—
81	C₂₃×#125	C₂³	1	4	yes	—
82	C₂₃×#125	C₂³	1	4	yes	—
83	C₂₃×#124	—	2	4	no	—
84	C₂₃×#124	C₂³	1	4	yes	—
85	C₂₃×#124	C₂³	1	4	yes	—
86	C₂₃×C₂²⋊Q₈	—	2	4	no	—
87	C₂₃×C₂²⋊Q₈	—	2	4	no	—
88	C₂₃×C₂²⋊Q₈	—	2	4	no	—
89	C₂₃×C₂²⋊Q₈	—	2	4	no	—
90	C₂₃×#121	—	2	4	no	—
91	C₂₃×#121	—	2	4	no	—
92	C₂₃×#121	—	2	4	no	—
93	C₂₃×#121	—	2	4	no	—
94	C₄×Q₈×C₂₃	—	2	4	no	—
95	C₄×Q₈×C₂₃	—	2	4	no	—
96	C₄×Q₈×C₂₃	—	2	4	no	—
97	C₄×Q₈×C₂₃	—	2	4	no	—
98	C₄×D₄×C₂₃	—	2	4	no	—
99	C₄×D₄×C₂₃	—	2	4	no	—
100	C₄×D₄×C₂₃	—	2	4	no	—
101	C₄×D₄×C₂₃	—	2	4	no	—
102	C₂₃×#117	—	2	2	no	—
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	C₂³	1	4	yes	—
105	C₂₃×#117	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
112	C₂×C₄²×C₂₃	—	2	2	no	—
113	C₂×C₄²×C₂₃	D₄	1	2	yes	—
114	C₂×C₄²×C₂₃	D₄	1	2	yes	—
115	C₂×C₄²×C₂₃	D₄	1	2	yes	—
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
122	C₂₃×#103	C₂³	1	32	yes	—
123	C₂₃×#102	C₂³	1	32	yes	—
124	C₂₃×#97	C₂³	1	4	yes	—
125	C₂₃×#97	C₂³	1	4	yes	—
126	C₄×C₈×C₂₃	C₂³	1	4	yes	—
127	C₄×C₈×C₂₃	C₂³	1	4	yes	—
128	C₂⁴×C₂₃	—	2	2	no	—
129	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
130	#41×C₂₃	—	2	16	no	—
131	#41×C₂₃	—	2	16	no	—
132	#41×C₂₃	—	4	8	no	—
133	#41×C₂₃	—	4	8	no	—
134	C₂×Q₈×C₂₃	—	2	4	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	8	no	—
138	C₂×Q₈×C₂₃	—	4	8	no	—
139	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
140	C₂×D₄×C₂₃	—	2	4	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	4	no	—
144	C₂×D₄×C₂₃	—	2	2	no	—
145	C₂×D₄×C₂₃	—	2	8	no	—
146	C₂×D₄×C₂₃	—	4	8	no	—
147	C₂×D₄×C₂₃	—	4	8	no	—
148	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₂₃	—	2	4	no	—
150	C₂²×C₄×C₂₃	—	2	2	no	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	2	4	no	—
153	C₂²×C₄×C₂₃	—	4	4	no	—
154	C₂²×C₄×C₂₃	—	4	8	no	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	#41	1	8	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₂₃	—	2	8	no	—
169	M₁₆×C₂₃	—	2	8	no	—
170	C₂×C₈×C₂₃	—	2	8	no	—
171	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	8	no	—
175	C₄⋊C₄×C₂₃	—	2	8	no	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	4	no	—
179	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
180	C₂²⋊C₄×C₂₃	—	2	4	no	—
181	C₂²⋊C₄×C₂₃	—	2	4	no	—
182	C₂²⋊C₄×C₂₃	—	2	8	no	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₄²×C₂₃	—	2	4	no	—
185	C₄²×C₂₃	—	2	4	no	—
186	C₄²×C₂₃	C₂×D₄	1	2	yes	—
187	C₄²×C₂₃	C₂×D₄	1	2	yes	—
188	C₄²×C₂₃	C₂×D₄	1	2	yes	—
189	C₄²×C₂₃	C₂×D₄	1	2	yes	—
190	C₄²×C₂₃	C₂×D₄	1	2	yes	—
191	C₄²×C₂₃	C₂×D₄	1	2	yes	—
192	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
193	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
194	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
195	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
196	#60618	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₂₃	—	2	4	no	—
198	C₂³×C₂₃	—	4	2	no	—
199	C₂³×C₂₃	—	4	8	no	—
200	C₂³×C₂₃	—	4	4	no	—
201	C₂³×C₂₃	#124	1	4	yes	—
202	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
203	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
204	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₂₃	—	2	8	no	—
206	Q₈×C₂₃	—	4	8	no	—
207	Q₈×C₂₃	—	4	4	no	—
208	Q₈×C₂₃	—	4	8	no	—
209	D₄×C₂₃	—	2	8	no	—
210	D₄×C₂₃	—	4	8	no	—
211	D₄×C₂₃	—	4	8	no	—
212	D₄×C₂₃	—	4	8	no	—
213	D₄×C₂₃	—	4	4	no	—
214	D₄×C₂₃	—	4	8	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	8	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	4	2	no	—
221	C₂×C₄×C₂₃	—	4	2	no	—
222	C₂×C₄×C₂₃	—	4	2	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	—	4	4	no	—
225	C₂×C₄×C₂₃	—	4	8	no	—
226	C₂×C₄×C₂₃	—	4	8	no	—
227	C₂×C₄×C₂₃	#124	1	4	yes	—
228	C₂×C₄×C₂₃	#124	1	4	yes	—
229	C₂×C₄×C₂₃	#124	1	4	yes	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
238	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
239	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
240	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
241	C₈×C₂₃	—	2	8	no	—
242	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
244	#3142	C₂×C₂₃	1	4	yes	complemented
245	#3141	C₂×C₂₃	1	4	yes	complemented
246	#3140	C₂×C₂₃	1	4	yes	complemented
247	#3139	C₂×C₂₃	1	2	yes	complemented
248	#3138	C₂×C₂₃	1	2	yes	complemented
249	#3135	C₂×C₂₃	1	4	yes	complemented
250	#3131	C₂×C₂₃	1	4	yes	complemented
251	#2896	C₂×C₂₃	1	4	yes	complemented
252	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₂²×C₂₃	—	2	2	no	—
254	C₂²×C₂₃	—	2	2	no	—
255	C₂²×C₂₃	—	2	2	no	—
256	C₂²×C₂₃	—	4	4	no	—
257	C₂²×C₂₃	—	4	4	no	—
258	C₂²×C₂₃	—	4	8	no	—
259	C₂²×C₂₃	—	8	8	no	—
260	C₂²×C₂₃	C₂×#124	1	2	yes	—
261	C₂²×C₂₃	C₂×#124	1	2	yes	—
262	C₂²×C₂₃	C₂×#124	1	2	yes	—
263	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	2	no	—
266	C₄×C₂₃	—	2	2	no	—
267	C₄×C₂₃	—	4	4	no	—
268	C₄×C₂₃	—	4	8	no	—
269	C₄×C₂₃	C₂×#124	1	2	yes	—
270	C₄×C₂₃	C₂×#124	1	2	yes	—
271	C₄×C₂₃	C₂×#124	1	2	yes	—
272	C₄×C₂₃	#575	1	4	yes	—
273	C₄×C₂₃	C₂²×#41	1	2	yes	—
274	C₂²×#41	C₂²×C₂₃	1	2	yes	—
275	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
276	#533	—	2	2	no	—
277	#533	—	2	2	no	—
278	#533	C₂²×C₂₃	1	2	yes	—
279	#533	C₂²×C₂₃	1	2	yes	—
280	#533	C₂²×C₂₃	1	2	yes	complemented
281	#533	C₂²×C₂₃	1	2	yes	complemented
282	#532	—	2	2	no	—
283	#532	—	2	2	no	—
284	#532	C₂²×C₂₃	1	2	yes	—
285	#532	C₂²×C₂₃	1	2	yes	—
286	#532	C₂²×C₂₃	1	2	yes	—
287	#532	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
290	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
291	C₂×#125	C₂²×C₂₃	1	2	yes	complemented
292	C₂×#124	C₂²×C₂₃	1	2	yes	—
293	C₂×C₂²⋊Q₈	—	2	2	no	—
294	C₂×#121	—	2	2	no	—
295	C₂×C₄×Q₈	—	2	2	no	—
296	C₂×C₄×D₄	—	2	2	no	—
297	C₂×#117	C₂²×C₂₃	1	2	yes	—
298	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
299	#491	C₂²×C₂₃	1	8	yes	complemented
300	#490	C₂²×C₂₃	1	8	yes	complemented
301	#489	C₂²×C₂₃	1	16	yes	complemented
302	#488	C₂²×C₂₃	1	16	yes	complemented
303	#487	C₂²×C₂₃	1	8	yes	complemented
304	#486	C₂²×C₂₃	1	8	yes	complemented
305	#419	C₂²×C₂₃	1	8	yes	—
306	#418	C₂²×C₂₃	1	8	yes	—
307	#417	C₂²×C₂₃	1	16	yes	complemented
308	#416	C₂²×C₂₃	1	16	yes	complemented
309	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
310	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
311	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
312	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
313	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
314	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
315	C₂×C₂₃	—	2	2	no	—
316	C₂×C₂₃	—	2	2	no	—
317	C₂×C₂₃	—	2	2	no	—
318	C₂×C₂₃	—	8	4	no	—
319	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
320	C₂×C₂₃	C₂×#575	1	2	yes	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	2	2	no	—
323	C₂×#41	—	2	16	no	—
324	C₂×#41	C₂³×C₂₃	1	4	yes	—
325	C₂×#41	C₂³×C₂₃	1	4	yes	—
326	C₂²×Q₈	—	2	2	no	—
327	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
328	C₂²×D₄	—	2	2	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
331	C₂³×C₄	—	2	2	no	—
332	C₂³×C₄	D₄×C₂₃	1	2	yes	—
333	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
334	C₂×M₁₆	—	2	8	no	—
335	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
336	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
337	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	—	2	2	no	—
341	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
342	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
343	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
344	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
345	#127	—	2	2	no	—
346	#127	C₂³×C₂₃	1	2	yes	—
347	#127	C₂³×C₂₃	1	2	yes	—
348	#125	—	2	4	no	—
349	#125	C₂³×C₂₃	1	4	yes	—
350	#125	C₂³×C₂₃	1	4	yes	—
351	#124	—	2	4	no	—
352	#124	C₂³×C₂₃	1	4	yes	—
353	#124	C₂³×C₂₃	1	4	yes	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	#121	—	2	4	no	—
359	#121	—	2	4	no	—
360	#121	—	2	4	no	—
361	#121	—	2	4	no	—
362	C₄×Q₈	—	2	4	no	—
363	C₄×Q₈	—	2	4	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	#117	—	2	2	no	—
371	#117	—	2	2	no	—
372	#117	C₂³×C₂₃	1	4	yes	—
373	#117	C₂³×C₂₃	1	4	yes	—
374	C₂×C₄⋊C₄	—	2	2	no	—
375	C₂×C₄⋊C₄	—	2	4	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	4	no	—
380	C₂×C₄²	—	2	2	no	—
381	C₂×C₄²	D₄×C₂₃	1	2	yes	—
382	C₂×C₄²	D₄×C₂₃	1	2	yes	—
383	C₂×C₄²	D₄×C₂₃	1	2	yes	—
384	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
390	#103	C₂³×C₂₃	1	32	yes	—
391	#102	C₂³×C₂₃	1	32	yes	—
392	#97	C₂³×C₂₃	1	4	yes	—
393	#97	C₂³×C₂₃	1	4	yes	—
394	C₄×C₈	C₂³×C₂₃	1	4	yes	—
395	C₄×C₈	C₂³×C₂₃	1	4	yes	—
396	C₂₃	#60618	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	2	no	—
398	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
399	#41	—	2	16	no	—
400	#41	—	2	16	no	—
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	8	no	—
407	C₂×Q₈	—	4	8	no	—
408	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	4	8	no	—
416	C₂×D₄	—	4	8	no	—
417	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
425	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
426	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂²×C₄	#41×C₂₃	1	8	yes	—
434	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
437	M₁₆	—	2	8	no	—
438	M₁₆	—	2	8	no	—
439	C₂×C₈	—	2	8	no	—
440	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	8	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	4	no	—
455	C₄²	C₂×D₄×C₂₃	1	2	yes	—
456	C₄²	C₂×D₄×C₂₃	1	2	yes	—
457	C₄²	C₂×D₄×C₂₃	1	2	yes	—
458	C₄²	C₂×D₄×C₂₃	1	2	yes	—
459	C₄²	C₂×D₄×C₂₃	1	2	yes	—
460	C₄²	C₂×D₄×C₂₃	1	2	yes	—
461	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
462	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
463	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
464	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
465	C₂³	—	2	4	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	8	no	—
468	C₂³	—	4	4	no	—
469	C₂³	C₂₃×#124	1	4	yes	—
470	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
471	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
472	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
473	Q₈	—	2	8	no	—
474	Q₈	—	4	8	no	—
475	Q₈	—	4	8	no	—
476	Q₈	—	4	4	no	—
477	D₄	—	2	8	no	—
478	D₄	—	4	8	no	—
479	D₄	—	4	8	no	—
480	D₄	—	4	8	no	—
481	D₄	—	4	8	no	—
482	D₄	—	4	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	C₂₃×#124	1	4	yes	—
496	C₂×C₄	C₂₃×#124	1	4	yes	—
497	C₂×C₄	C₂₃×#124	1	4	yes	—
498	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
499	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
501	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
504	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
506	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
507	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
508	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
509	C₈	—	2	8	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	4	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	8	no	—
516	C₂²	—	8	8	no	—
517	C₂²	C₂×C₂₃×#124	1	2	yes	—
518	C₂²	C₂×C₂₃×#124	1	2	yes	—
519	C₂²	C₂×C₂₃×#124	1	2	yes	—
520	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	2	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	8	no	—
526	C₄	C₂×C₂₃×#124	1	2	yes	—
527	C₄	C₂×C₂₃×#124	1	2	yes	—
528	C₄	C₂×C₂₃×#124	1	2	yes	—
529	C₄	C₂₃×#575	1	4	yes	—
530	C₄	C₂²×#41×C₂₃	1	2	yes	—
531	C₂	—	2	2	no	—
532	C₂	—	2	2	no	—
533	C₂	—	2	2	no	—
534	C₂	—	8	4	no	—
535	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
536	C₂	C₂×C₂₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×D₄
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3142	C₂	semidirect
4. C₂₃×#3141	C₂	semidirect
5. C₂₃×#3140	C₂	semidirect
6. C₂₃×#3139	C₂	semidirect
7. C₂₃×#3138	C₂	semidirect
8. C₂₃×#3135	C₂	semidirect
9. C₂₃×#3131	C₂	semidirect
10. C₂₃×#2896	C₂	semidirect
11. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
18. C₂₃×#533	C₂²	semidirect
19. C₂₃×#533	C₂²	semidirect
27. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
29. C₂×C₂₃×#125	C₂²	semidirect
37. C₂₃×#491	C₂²	semidirect
38. C₂₃×#490	C₂²	semidirect
39. C₂₃×#489	C₂²	semidirect
40. C₂₃×#488	C₂²	semidirect
41. C₂₃×#487	C₂²	semidirect
42. C₂₃×#486	C₂²	semidirect
45. C₂₃×#417	C₂²	semidirect
46. C₂₃×#416	C₂²	semidirect
49. C₄×M₁₆×C₂₃	C₂²	semidirect
50. C₄×M₁₆×C₂₃	C₂²	semidirect
51. C₂×C₂₃×#97	C₂²	semidirect
52. C₂×C₄×C₈×C₂₃	C₂²	semidirect
196. #60618	C₂₃	direct (N × H)
242. C₂×#533	C₂×C₂₃	semidirect
244. #3142	C₂×C₂₃	semidirect
245. #3141	C₂×C₂₃	semidirect
246. #3140	C₂×C₂₃	semidirect
247. #3139	C₂×C₂₃	semidirect
248. #3138	C₂×C₂₃	semidirect
249. #3135	C₂×C₂₃	semidirect
250. #3131	C₂×C₂₃	semidirect
251. #2896	C₂×C₂₃	semidirect
252. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
280. #533	C₂²×C₂₃	semidirect
281. #533	C₂²×C₂₃	semidirect
289. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
291. C₂×#125	C₂²×C₂₃	semidirect
299. #491	C₂²×C₂₃	semidirect
300. #490	C₂²×C₂₃	semidirect
301. #489	C₂²×C₂₃	semidirect
302. #488	C₂²×C₂₃	semidirect
303. #487	C₂²×C₂₃	semidirect
304. #486	C₂²×C₂₃	semidirect
307. #417	C₂²×C₂₃	semidirect
308. #416	C₂²×C₂₃	semidirect
311. C₄×M₁₆	C₂²×C₂₃	semidirect
312. C₄×M₁₆	C₂²×C₂₃	semidirect
313. C₂×#97	C₂²×C₂₃	semidirect
314. C₂×C₄×C₈	C₂²×C₂₃	semidirect
396. C₂₃	#60618	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×#533	2
C₂×C₂₃×#532	2
C₂₃×#3142	2
C₂₃×#3141	2
C₂₃×#3140	2
C₂₃×#3139	2
C₂₃×#3138	2
C₂₃×#3135	2
C₂₃×#3131	2
C₂₃×#2896	2
C₂×C₄×M₁₆×C₂₃	2
#60618	23

Subgroups — C23×#60618 #19208244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60618 #19208244