Subgroups — C₂₃×#60498 #19208124

Order 5888.

2006 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3083	C₂	1	6	yes	maximal, complemented
4	C₂₃×#3082	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3079	C₂	1	3	yes	maximal, complemented
6	C₂₃×#3078	C₂	1	3	yes	maximal, complemented
7	C₂₃×#3076	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×#477	C₂	1	3	yes	maximal, complemented
9	C₂₃×#2913	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2898	C₂	1	2	yes	maximal
12	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	Q₈²×C₂₃	—	2	1	no	—
15	C₂₃×#557	—	2	1	no	—
16	C₂₃×#554	—	2	3	no	—
17	D₄×Q₈×C₂₃	—	2	3	no	—
18	C₂₃×#544	C₂²	1	6	yes	—
19	C₂₃×#533	C₂²	1	3	yes	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
21	C₂₃×#520	—	2	3	no	—
22	C₂₃×#520	C₂²	1	3	yes	—
23	C₂₃×#520	C₂²	1	1	yes	characteristic
24	C₂₃×#520	C₂²	1	2	yes	—
25	C₂₃×#519	—	2	1	no	—
26	C₂₃×#519	C₂²	1	3	yes	—
27	C₄×#41×C₂₃	—	2	3	no	—
28	C₄×#41×C₂₃	C₂²	1	3	yes	—
29	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
31	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
32	C₂₃×#479	C₂²	1	6	yes	—
33	C₂₃×#479	C₂²	1	12	yes	—
34	C₂₃×#478	C₂²	1	6	yes	—
35	C₂₃×#477	C₂²	1	3	yes	—
36	C₂₃×#477	C₂²	1	6	yes	—
37	C₂₃×#477	C₂²	1	12	yes	—
38	C₂₃×#476	C₂²	1	3	yes	—
39	C₂₃×#475	C₂²	1	3	yes	—
40	C₂₃×#475	C₂²	1	6	yes	—
41	C₂₃×#474	C₂²	1	1	yes	characteristic
42	C₂₃×#474	C₂²	1	6	yes	—
43	C₂₃×#428	C₂²	1	6	yes	—
44	C₂₃×#428	C₂²	1	6	yes	—
45	C₂₃×#427	C₂²	1	6	yes	—
46	C₂×C₂₃×#107	C₂²	1	6	yes	—
47	C₂₃×#424	C₂²	1	6	yes	—
48	C₂₃×#423	C₂²	1	3	yes	—
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
50	C₂₃×#419	C₂²	1	6	yes	—
51	C₂₃×#417	C₂²	1	6	yes	—
52	C₂₃×#416	C₂²	1	6	yes	—
53	C₂×C₂₃×#103	C₂²	1	6	yes	—
54	C₂₃×2₋¹⁺⁴	—	2	4	no	—
55	C₂₃×2₋¹⁺⁴	Q₈	1	8	yes	complemented
56	C₂×#41×C₂₃	—	2	3	no	—
57	C₂×#41×C₂₃	C₂³	1	3	yes	—
58	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	—	2	3	no	—
61	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
62	C₂₃×#131	—	2	8	no	—
63	C₂×M₁₆×C₂₃	C₂³	1	6	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
65	C₂₃×C₄⋊Q₈	—	2	6	no	—
66	C₂₃×C₄⋊Q₈	—	2	3	no	—
67	C₂₃×C₄⋊Q₈	—	2	3	no	—
68	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
70	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
71	C₂₃×#125	—	2	6	no	—
72	C₂₃×#125	C₂³	1	6	yes	—
73	C₂₃×C₂²⋊Q₈	—	2	3	no	—
74	C₂₃×C₂²⋊Q₈	—	2	3	no	—
75	C₂₃×C₂²⋊Q₈	—	2	12	no	—
76	C₄×Q₈×C₂₃	—	2	2	no	—
77	C₄×Q₈×C₂₃	—	2	1	no	—
78	C₄×Q₈×C₂₃	—	2	3	no	—
79	C₄×Q₈×C₂₃	—	2	3	no	—
80	C₄×Q₈×C₂₃	—	2	3	no	—
81	C₄×Q₈×C₂₃	—	2	3	no	—
82	C₄×Q₈×C₂₃	—	2	3	no	—
83	C₄×Q₈×C₂₃	—	2	3	no	—
84	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
86	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
87	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
88	C₄×Q₈×C₂₃	C₂³	1	6	yes	—
89	C₄×D₄×C₂₃	—	2	6	no	—
90	C₄×D₄×C₂₃	—	2	3	no	—
91	C₄×D₄×C₂₃	—	2	3	no	—
92	C₄×D₄×C₂₃	—	2	3	no	—
93	C₄×D₄×C₂₃	—	2	6	no	—
94	C₄×D₄×C₂₃	D₄	1	3	yes	—
95	C₄×D₄×C₂₃	D₄	1	3	yes	—
96	C₄×D₄×C₂₃	C₂³	1	3	yes	—
97	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×#117	—	2	3	no	—
99	C₂₃×#117	—	2	6	no	—
100	C₂₃×#117	—	2	3	no	—
101	C₂₃×#117	C₂³	1	6	yes	—
102	C₂₃×#117	C₂³	1	3	yes	—
103	C₂₃×#117	C₂³	1	3	yes	—
104	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
109	C₂×C₄²×C₂₃	—	2	3	no	—
110	C₂×C₄²×C₂₃	—	2	3	no	—
111	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
112	C₂₃×#107	C₂³	1	6	yes	—
113	C₂₃×#107	C₂³	1	12	yes	—
114	C₂₃×#107	C₂³	1	2	yes	—
115	C₂₃×#106	C₂³	1	6	yes	—
116	C₂₃×#106	C₂³	1	6	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
120	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
121	C₂₃×#103	C₂³	1	6	yes	—
122	C₂₃×#103	C₂³	1	12	yes	—
123	C₂₃×#103	C₂³	1	6	yes	—
124	C₂₃×#102	C₂³	1	2	yes	—
125	C₂₃×#102	C₂³	1	6	yes	—
126	#41×C₂₃	—	2	24	no	—
127	#41×C₂₃	—	2	6	no	—
128	#41×C₂₃	C₂×Q₈	1	4	yes	—
129	#41×C₂₃	C₂×Q₈	1	12	yes	complemented
130	#41×C₂₃	#41	1	4	yes	—
131	C₂×Q₈×C₂₃	—	2	8	no	—
132	C₂×Q₈×C₂₃	—	2	3	no	—
133	C₂×Q₈×C₂₃	—	2	1	no	—
134	C₂×Q₈×C₂₃	—	2	3	no	—
135	C₂×Q₈×C₂₃	—	2	6	no	—
136	C₂×Q₈×C₂₃	—	2	6	no	—
137	C₂×Q₈×C₂₃	—	8	6	no	—
138	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₂₃	C₂×Q₈	1	12	yes	complemented
140	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
141	C₂×D₄×C₂₃	—	2	3	no	—
142	C₂×D₄×C₂₃	C₂×D₄	1	3	yes	—
143	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	—	2	3	no	—
145	C₂²×C₄×C₂₃	—	2	3	no	—
146	C₂²×C₄×C₂₃	—	2	3	no	—
147	C₂²×C₄×C₂₃	—	2	6	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	3	no	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
153	M₁₆×C₂₃	—	2	12	no	—
154	C₂×C₈×C₂₃	—	2	12	no	—
155	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
156	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
157	C₄⋊C₄×C₂₃	—	2	6	no	—
158	C₄⋊C₄×C₂₃	—	2	3	no	—
159	C₄⋊C₄×C₂₃	—	2	3	no	—
160	C₄⋊C₄×C₂₃	—	2	3	no	—
161	C₄⋊C₄×C₂₃	—	2	6	no	—
162	C₄⋊C₄×C₂₃	—	2	6	no	—
163	C₄⋊C₄×C₂₃	—	2	3	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	6	no	—
166	C₄⋊C₄×C₂₃	—	2	6	no	—
167	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₂₃	C₂⁴	1	6	yes	—
171	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
173	C₂²⋊C₄×C₂₃	—	2	3	no	—
174	C₂²⋊C₄×C₂₃	—	2	6	no	—
175	C₂²⋊C₄×C₂₃	—	2	1	no	—
176	C₂²⋊C₄×C₂₃	—	2	3	no	—
177	C₂²⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
178	C₄²×C₂₃	—	2	3	no	—
179	C₄²×C₂₃	—	2	3	no	—
180	C₄²×C₂₃	—	2	6	no	—
181	C₄²×C₂₃	—	2	3	no	—
182	C₄²×C₂₃	—	2	6	no	—
183	C₄²×C₂₃	C₂×D₄	1	3	yes	—
184	C₄²×C₂₃	C₂×D₄	1	3	yes	—
185	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
186	C₄²×C₂₃	C₂⁴	1	3	yes	—
187	#60498	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	1	no	—
189	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
190	Q₈×C₂₃	—	2	12	no	—
191	Q₈×C₂₃	—	2	3	no	—
192	Q₈×C₂₃	—	4	6	no	—
193	Q₈×C₂₃	—	4	2	no	—
194	Q₈×C₂₃	—	8	6	no	—
195	Q₈×C₂₃	C₂²⋊Q₈	1	4	yes	—
196	Q₈×C₂₃	C₂²×Q₈	1	6	yes	—
197	D₄×C₂₃	—	2	12	no	—
198	D₄×C₂₃	—	2	1	no	—
199	D₄×C₂₃	C₂²⋊Q₈	1	12	yes	—
200	D₄×C₂₃	C₂²×Q₈	1	2	yes	—
201	C₂×C₄×C₂₃	—	2	12	no	—
202	C₂×C₄×C₂₃	—	2	3	no	—
203	C₂×C₄×C₂₃	—	2	12	no	—
204	C₂×C₄×C₂₃	—	2	6	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	3	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	6	no	—
209	C₂×C₄×C₂₃	—	4	3	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	8	3	no	—
212	C₂×C₄×C₂₃	—	8	3	no	—
213	C₂×C₄×C₂₃	—	8	6	no	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂²×Q₈	1	6	yes	—
220	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₂₃	—	2	2	no	—
223	C₈×C₂₃	—	2	6	no	—
224	#3503	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₄×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic
226	#3083	C₂×C₂₃	1	6	yes	complemented
227	#3082	C₂×C₂₃	1	6	yes	complemented
228	#3079	C₂×C₂₃	1	3	yes	complemented
229	#3078	C₂×C₂₃	1	3	yes	complemented
230	#3076	C₂×C₂₃	1	3	yes	complemented
231	C₂×#477	C₂×C₂₃	1	3	yes	complemented
232	#2913	C₂×C₂₃	1	2	yes	complemented
233	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#2898	C₂×C₂₃	1	2	yes	—
235	C₂²×C₂₃	—	2	3	no	—
236	C₂²×C₂₃	—	2	3	no	—
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	4	1	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	6	yes	—
241	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₂₃	—	2	1	no	—
243	C₄×C₂₃	—	2	3	no	—
244	C₄×C₂₃	—	2	6	no	—
245	C₄×C₂₃	—	2	3	no	—
246	C₄×C₂₃	—	4	3	no	—
247	C₄×C₂₃	—	4	6	no	—
248	C₄×C₂₃	—	4	1	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	C₂×C₂²⋊Q₈	1	6	yes	—
251	C₄×C₂₃	#578	1	2	yes	—
252	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
253	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
254	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×#131	C₂²×C₂₃	1	2	yes	—
256	Q₈²	—	2	1	no	—
257	#557	—	2	1	no	—
258	#554	—	2	3	no	—
259	D₄×Q₈	—	2	3	no	—
260	#544	C₂²×C₂₃	1	6	yes	—
261	#533	C₂²×C₂₃	1	3	yes	—
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
263	#520	—	2	3	no	—
264	#520	C₂²×C₂₃	1	3	yes	—
265	#520	C₂²×C₂₃	1	2	yes	—
266	#520	C₂²×C₂₃	1	1	yes	characteristic
267	#519	—	2	1	no	—
268	#519	C₂²×C₂₃	1	3	yes	—
269	C₄×#41	—	2	3	no	—
270	C₄×#41	C₂²×C₂₃	1	3	yes	—
271	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₄×Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
274	#479	C₂²×C₂₃	1	6	yes	—
275	#479	C₂²×C₂₃	1	12	yes	—
276	#478	C₂²×C₂₃	1	6	yes	—
277	#477	C₂²×C₂₃	1	3	yes	—
278	#477	C₂²×C₂₃	1	6	yes	—
279	#477	C₂²×C₂₃	1	12	yes	—
280	#476	C₂²×C₂₃	1	3	yes	—
281	#475	C₂²×C₂₃	1	3	yes	—
282	#475	C₂²×C₂₃	1	6	yes	—
283	#474	C₂²×C₂₃	1	1	yes	characteristic
284	#474	C₂²×C₂₃	1	6	yes	—
285	#428	C₂²×C₂₃	1	6	yes	—
286	#428	C₂²×C₂₃	1	6	yes	—
287	#427	C₂²×C₂₃	1	6	yes	—
288	C₂×#107	C₂²×C₂₃	1	6	yes	—
289	#424	C₂²×C₂₃	1	6	yes	—
290	#423	C₂²×C₂₃	1	3	yes	—
291	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
292	#419	C₂²×C₂₃	1	6	yes	—
293	#417	C₂²×C₂₃	1	6	yes	—
294	#416	C₂²×C₂₃	1	6	yes	—
295	C₂×#103	C₂²×C₂₃	1	6	yes	—
296	C₂×C₂₃	—	2	6	no	—
297	C₂×C₂₃	—	4	2	no	—
298	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
300	C₂×C₂₃	#3585	1	1	yes	characteristic
301	2₋¹⁺⁴	—	2	4	no	—
302	2₋¹⁺⁴	Q₈×C₂₃	1	8	yes	complemented
303	C₂×#41	—	2	3	no	—
304	C₂×#41	C₂³×C₂₃	1	3	yes	—
305	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	—	2	3	no	—
308	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
309	#131	—	2	8	no	—
310	C₂×M₁₆	C₂³×C₂₃	1	6	yes	—
311	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
312	C₄⋊Q₈	—	2	6	no	—
313	C₄⋊Q₈	—	2	3	no	—
314	C₄⋊Q₈	—	2	3	no	—
315	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
316	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
317	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
318	#125	—	2	6	no	—
319	#125	C₂³×C₂₃	1	6	yes	—
320	C₂²⋊Q₈	—	2	3	no	—
321	C₂²⋊Q₈	—	2	3	no	—
322	C₂²⋊Q₈	—	2	12	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×Q₈	—	2	3	no	—
326	C₄×Q₈	—	2	3	no	—
327	C₄×Q₈	—	2	3	no	—
328	C₄×Q₈	—	2	3	no	—
329	C₄×Q₈	—	2	3	no	—
330	C₄×Q₈	—	2	3	no	—
331	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
332	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
333	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
334	C₄×Q₈	C₂³×C₂₃	1	6	yes	—
335	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
336	C₄×D₄	—	2	3	no	—
337	C₄×D₄	—	2	6	no	—
338	C₄×D₄	—	2	3	no	—
339	C₄×D₄	—	2	3	no	—
340	C₄×D₄	—	2	6	no	—
341	C₄×D₄	D₄×C₂₃	1	3	yes	—
342	C₄×D₄	D₄×C₂₃	1	3	yes	—
343	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
344	C₄×D₄	C₂³×C₂₃	1	3	yes	—
345	#117	—	2	3	no	—
346	#117	—	2	6	no	—
347	#117	—	2	3	no	—
348	#117	C₂³×C₂₃	1	3	yes	—
349	#117	C₂³×C₂₃	1	6	yes	—
350	#117	C₂³×C₂₃	1	3	yes	—
351	C₂×C₄⋊C₄	—	2	3	no	—
352	C₂×C₄⋊C₄	—	2	3	no	—
353	C₂×C₄⋊C₄	—	2	3	no	—
354	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
356	C₂×C₄²	—	2	3	no	—
357	C₂×C₄²	—	2	3	no	—
358	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
359	#107	C₂³×C₂₃	1	6	yes	—
360	#107	C₂³×C₂₃	1	12	yes	—
361	#107	C₂³×C₂₃	1	2	yes	—
362	#106	C₂³×C₂₃	1	6	yes	—
363	#106	C₂³×C₂₃	1	6	yes	—
364	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
366	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
367	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
368	#103	C₂³×C₂₃	1	6	yes	—
369	#103	C₂³×C₂₃	1	12	yes	—
370	#103	C₂³×C₂₃	1	6	yes	—
371	#102	C₂³×C₂₃	1	2	yes	—
372	#102	C₂³×C₂₃	1	6	yes	—
373	C₂₃	#60498	1	1	yes	characteristic, direct product factor, Sylow
374	#41	—	2	24	no	—
375	#41	—	2	6	no	—
376	#41	C₂×Q₈×C₂₃	1	4	yes	—
377	#41	C₂×Q₈×C₂₃	1	12	yes	complemented
378	#41	#41×C₂₃	1	4	yes	—
379	C₂×Q₈	—	2	8	no	—
380	C₂×Q₈	—	2	3	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	3	no	—
383	C₂×Q₈	—	2	6	no	—
384	C₂×Q₈	—	2	6	no	—
385	C₂×Q₈	—	8	6	no	—
386	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
387	C₂×Q₈	C₂×Q₈×C₂₃	1	12	yes	complemented
388	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
389	C₂×D₄	—	2	3	no	—
390	C₂×D₄	C₂×D₄×C₂₃	1	3	yes	—
391	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	—	2	3	no	—
394	C₂²×C₄	—	2	3	no	—
395	C₂²×C₄	—	2	6	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	3	no	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
399	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
400	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
401	M₁₆	—	2	12	no	—
402	C₂×C₈	—	2	12	no	—
403	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
404	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
405	C₄⋊C₄	—	2	6	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	6	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	6	no	—
414	C₄⋊C₄	—	2	6	no	—
415	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
416	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
417	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
418	C₄⋊C₄	C₂⁴×C₂₃	1	6	yes	—
419	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
420	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
421	C₂²⋊C₄	—	2	3	no	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
426	C₄²	—	2	3	no	—
427	C₄²	—	2	3	no	—
428	C₄²	—	2	6	no	—
429	C₄²	—	2	3	no	—
430	C₄²	—	2	6	no	—
431	C₄²	C₂×D₄×C₂₃	1	3	yes	—
432	C₄²	C₂×D₄×C₂₃	1	3	yes	—
433	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
434	C₄²	C₂⁴×C₂₃	1	3	yes	—
435	C₂³	—	2	1	no	—
436	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
437	Q₈	—	2	3	no	—
438	Q₈	—	2	12	no	—
439	Q₈	—	4	6	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	8	6	no	—
442	Q₈	C₂₃×C₂²⋊Q₈	1	4	yes	—
443	Q₈	C₂²×Q₈×C₂₃	1	6	yes	—
444	D₄	—	2	12	no	—
445	D₄	—	2	1	no	—
446	D₄	C₂₃×C₂²⋊Q₈	1	12	yes	—
447	D₄	C₂²×Q₈×C₂₃	1	2	yes	—
448	C₂×C₄	—	2	3	no	—
449	C₂×C₄	—	2	12	no	—
450	C₂×C₄	—	2	6	no	—
451	C₂×C₄	—	2	12	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	4	3	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	3	no	—
459	C₂×C₄	—	8	6	no	—
460	C₂×C₄	—	8	3	no	—
461	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
462	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
466	C₂×C₄	C₂²×Q₈×C₂₃	1	6	yes	—
467	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
468	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	2	6	no	—
471	C₂²	—	2	3	no	—
472	C₂²	—	2	3	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	6	yes	—
477	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
478	C₄	—	2	6	no	—
479	C₄	—	2	3	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	3	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	3	no	—
485	C₄	—	4	6	no	—
486	C₄	C₂×C₂₃×C₂²⋊Q₈	1	6	yes	—
487	C₄	C₂₃×#578	1	2	yes	—
488	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
489	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
490	C₂	—	2	6	no	—
491	C₂	—	4	2	no	—
492	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
493	C₂	C₂×C₂₃×#578	1	1	yes	characteristic
494	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3503	C₂	semidirect
3. C₂₃×#3083	C₂	semidirect
4. C₂₃×#3082	C₂	semidirect
5. C₂₃×#3079	C₂	semidirect
6. C₂₃×#3078	C₂	semidirect
7. C₂₃×#3076	C₂	semidirect
8. C₂×C₂₃×#477	C₂	semidirect
9. C₂₃×#2913	C₂	semidirect
10. C₂₃×#2906	C₂	semidirect
55. C₂₃×2₋¹⁺⁴	Q₈	semidirect
129. #41×C₂₃	C₂×Q₈	semidirect
139. C₂×Q₈×C₂₃	C₂×Q₈	semidirect
187. #60498	C₂₃	direct (N × H)
224. #3503	C₂×C₂₃	semidirect
226. #3083	C₂×C₂₃	semidirect
227. #3082	C₂×C₂₃	semidirect
228. #3079	C₂×C₂₃	semidirect
229. #3078	C₂×C₂₃	semidirect
230. #3076	C₂×C₂₃	semidirect
231. C₂×#477	C₂×C₂₃	semidirect
232. #2913	C₂×C₂₃	semidirect
233. #2906	C₂×C₂₃	semidirect
302. 2₋¹⁺⁴	Q₈×C₂₃	semidirect
373. C₂₃	#60498	direct (N × H)
377. #41	C₂×Q₈×C₂₃	semidirect
387. C₂×Q₈	C₂×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₄×C₂₃×2₋¹⁺⁴	2
C₂₃×#3083	2
C₂₃×#3082	2
C₂₃×#3079	2
C₂₃×#3078	2
C₂₃×#3076	2
C₂×C₂₃×#477	2
C₂₃×#2913	2
C₂₃×#2906	2
C₂₃×#2898	2
#60498	23

Subgroups — C23×#60498 #19208124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60498 #19208124