Subgroups — C₂₃×#60800 #19208426

Order 5888.

2134 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3218	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3209	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3206	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3152	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3139	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3039	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3004	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×#137	C₂²	1	8	yes	—
13	C₂₃×#533	C₂²	1	2	yes	—
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₂₃×#532	C₂²	1	1	yes	characteristic
16	C₂₃×#532	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
18	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
19	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	4	yes	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×#529	—	2	2	no	—
25	C₂₃×#529	—	2	2	no	—
26	C₂₃×#529	C₂²	1	4	yes	—
27	C₂₃×#525	—	2	2	no	—
28	C₂₃×#525	C₂²	1	2	yes	—
29	C₄×#41×C₂₃	—	2	2	no	—
30	C₄×#41×C₂₃	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#497	C₂²	1	8	yes	complemented
34	C₂₃×#494	C₂²	1	4	yes	complemented
35	C₂₃×#492	C₂²	1	4	yes	complemented
36	C₂₃×#491	C₂²	1	8	yes	complemented
37	C₂₃×#487	C₂²	1	8	yes	complemented
38	C₂₃×#462	C₂²	1	16	yes	complemented
39	C₂₃×#461	C₂²	1	16	yes	complemented
40	C₂₃×#451	C₂²	1	16	yes	—
41	C₂₃×#448	C₂²	1	16	yes	—
42	C₂₃×#432	C₂²	1	4	yes	—
43	C₂₃×#431	C₂²	1	4	yes	—
44	C₂₃×#423	C₂²	1	2	yes	—
45	C₂₃×#419	C₂²	1	8	yes	—
46	C₂₃×#407	C₂²	1	2	yes	—
47	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
48	C₂×#41×C₂₃	—	2	2	no	—
49	C₂×#41×C₂₃	C₂³	1	2	yes	—
50	C₂²×Q₈×C₂₃	—	2	4	no	—
51	C₂²×Q₈×C₂₃	—	2	2	no	—
52	C₂²×Q₈×C₂₃	—	4	8	no	—
53	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
54	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
55	C₂³×C₄×C₂₃	—	2	2	no	—
56	C₂³×C₄×C₂₃	D₄	1	2	yes	—
57	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#137	—	4	16	no	—
59	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
60	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
61	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	4	no	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	—	2	4	no	—
66	C₂₃×C₄⋊Q₈	—	2	4	no	—
67	C₂₃×C₄⋊Q₈	—	2	4	no	—
68	C₂₃×C₄⋊Q₈	—	2	4	no	—
69	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
70	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
71	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
74	C₂₃×#127	C₂³	1	1	yes	characteristic
75	C₂₃×#126	—	2	4	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#125	C₂³	1	2	yes	—
78	C₂₃×#124	—	2	2	no	—
79	C₂₃×#124	C₂³	1	2	yes	—
80	C₂₃×#123	—	2	4	no	—
81	C₂₃×#123	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
85	C₂₃×#121	—	2	2	no	—
86	C₂₃×#121	—	2	2	no	—
87	C₂₃×#121	C₂³	1	4	yes	—
88	C₂₃×#120	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
92	C₄×D₄×C₂₃	—	2	4	no	—
93	C₄×D₄×C₂₃	—	2	4	no	—
94	C₄×D₄×C₂₃	—	2	4	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	C₂³	1	4	yes	—
98	C₂₃×#117	—	2	2	no	—
99	C₂₃×#117	—	2	4	no	—
100	C₂₃×#117	—	2	2	no	—
101	C₂₃×#117	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
108	C₂×C₄²×C₂₃	—	2	2	no	—
109	C₂×C₄²×C₂₃	D₄	1	2	yes	—
110	C₂×C₄²×C₂₃	D₄	1	2	yes	—
111	C₂×C₄²×C₂₃	D₄	1	2	yes	—
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
117	C₂₃×#103	C₂³	1	32	yes	—
118	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
119	C₂₃×#97	C₂³	1	4	yes	—
120	C₄×C₈×C₂₃	C₂³	1	4	yes	—
121	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
122	#41×C₂₃	—	2	8	no	—
123	#41×C₂₃	—	4	8	no	—
124	C₂×Q₈×C₂₃	—	2	2	no	—
125	C₂×Q₈×C₂₃	—	2	8	no	—
126	C₂×Q₈×C₂₃	—	2	4	no	—
127	C₂×Q₈×C₂₃	—	2	16	no	—
128	C₂×Q₈×C₂₃	—	4	16	no	—
129	C₂×Q₈×C₂₃	—	4	8	no	—
130	C₂×Q₈×C₂₃	—	4	16	no	—
131	C₂×Q₈×C₂₃	C₂⁴	1	8	yes	—
132	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
133	C₂×D₄×C₂₃	—	2	2	no	—
134	C₂×D₄×C₂₃	—	2	4	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₂₃	—	2	8	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	8	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	4	8	no	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₂₃	—	4	16	no	—
155	SD₁₆×C₂₃	—	4	16	no	—
156	M₁₆×C₂₃	—	2	8	no	—
157	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
158	C₄⋊C₄×C₂₃	—	2	8	no	—
159	C₄⋊C₄×C₂₃	—	2	8	no	—
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	8	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	4	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	2	no	—
170	C₂²⋊C₄×C₂₃	—	2	4	no	—
171	C₂²⋊C₄×C₂₃	—	2	4	no	—
172	C₂²⋊C₄×C₂₃	—	2	4	no	—
173	C₄²×C₂₃	—	2	4	no	—
174	C₄²×C₂₃	—	2	2	no	—
175	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₂₃	C₂×D₄	1	2	yes	—
177	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂×D₄	1	2	yes	—
179	C₄²×C₂₃	C₂×D₄	1	2	yes	—
180	C₄²×C₂₃	C₂×D₄	1	2	yes	—
181	C₄²×C₂₃	C₂×D₄	1	2	yes	—
182	C₄²×C₂₃	C₂⁴	1	2	yes	—
183	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
185	#60800	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₂₃	—	2	2	no	—
187	C₂³×C₂₃	—	2	4	no	—
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
191	Q₈×C₂₃	—	2	16	no	—
192	Q₈×C₂₃	—	2	4	no	—
193	Q₈×C₂₃	—	4	16	no	—
194	Q₈×C₂₃	—	4	8	no	—
195	Q₈×C₂₃	—	4	4	no	—
196	D₄×C₂₃	—	2	4	no	—
197	D₄×C₂₃	—	4	4	no	—
198	D₄×C₂₃	—	4	8	no	—
199	D₄×C₂₃	—	4	4	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	2	no	—
209	C₂×C₄×C₂₃	—	2	4	no	—
210	C₂×C₄×C₂₃	—	2	8	no	—
211	C₂×C₄×C₂₃	—	4	16	no	—
212	C₂×C₄×C₂₃	—	4	8	no	—
213	C₂×C₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	—	4	8	no	—
215	C₂×C₄×C₂₃	#137	1	8	yes	complemented
216	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
223	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
224	C₈×C₂₃	—	2	8	no	—
225	#3444	C₂×C₂₃	1	1	yes	characteristic
226	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#3218	C₂×C₂₃	1	8	yes	complemented
228	#3209	C₂×C₂₃	1	4	yes	complemented
229	#3206	C₂×C₂₃	1	4	yes	complemented
230	#3152	C₂×C₂₃	1	2	yes	complemented
231	#3139	C₂×C₂₃	1	2	yes	complemented
232	#3039	C₂×C₂₃	1	4	yes	complemented
233	#3004	C₂×C₂₃	1	4	yes	complemented
234	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	8	no	—
236	C₂²×C₂₃	—	2	2	no	—
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	4	4	no	—
241	C₂²×C₂₃	#535	1	2	yes	—
242	C₂²×C₂₃	C₂×#137	1	4	yes	—
243	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
244	C₄×C₂₃	—	2	2	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	4	8	no	—
248	C₄×C₂₃	—	4	2	no	—
249	C₄×C₂₃	—	4	4	no	—
250	C₄×C₂₃	#535	1	2	yes	—
251	C₄×C₂₃	C₂×#137	1	4	yes	—
252	C₄×C₂₃	C₂×#137	1	4	yes	—
253	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
254	C₂³×Q₈	C₂²×C₂₃	1	2	yes	—
255	C₂×#137	C₂²×C₂₃	1	8	yes	—
256	#533	C₂²×C₂₃	1	2	yes	—
257	#533	C₂²×C₂₃	1	1	yes	characteristic
258	#532	C₂²×C₂₃	1	1	yes	characteristic
259	#532	C₂²×C₂₃	1	2	yes	—
260	C₂×C₄⋊Q₈	—	2	2	no	—
261	C₂×C₄⋊Q₈	—	2	2	no	—
262	C₂×C₄⋊Q₈	—	2	2	no	—
263	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
264	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
265	C₂×C₄⋊Q₈	C₂²×C₂₃	1	4	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
267	#529	—	2	2	no	—
268	#529	—	2	2	no	—
269	#529	C₂²×C₂₃	1	4	yes	—
270	#525	—	2	2	no	—
271	#525	C₂²×C₂₃	1	2	yes	—
272	C₄×#41	—	2	2	no	—
273	C₄×#41	C₂²×C₂₃	1	2	yes	—
274	C₂²×C₄⋊C₄	—	2	2	no	—
275	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
276	#497	C₂²×C₂₃	1	8	yes	complemented
277	#494	C₂²×C₂₃	1	4	yes	complemented
278	#492	C₂²×C₂₃	1	4	yes	complemented
279	#491	C₂²×C₂₃	1	8	yes	complemented
280	#487	C₂²×C₂₃	1	8	yes	complemented
281	#462	C₂²×C₂₃	1	16	yes	complemented
282	#461	C₂²×C₂₃	1	16	yes	complemented
283	#451	C₂²×C₂₃	1	16	yes	—
284	#448	C₂²×C₂₃	1	16	yes	—
285	#432	C₂²×C₂₃	1	4	yes	—
286	#431	C₂²×C₂₃	1	4	yes	—
287	#423	C₂²×C₂₃	1	2	yes	—
288	#419	C₂²×C₂₃	1	8	yes	—
289	#407	C₂²×C₂₃	1	2	yes	—
290	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
291	C₂×C₂₃	—	2	2	no	—
292	C₂×C₂₃	—	2	4	no	—
293	C₂×C₂₃	—	8	2	no	—
294	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
295	C₂×C₂₃	C₂²×#137	1	2	yes	—
296	C₂×#41	—	2	2	no	—
297	C₂×#41	C₂³×C₂₃	1	2	yes	—
298	C₂²×Q₈	—	2	4	no	—
299	C₂²×Q₈	—	2	2	no	—
300	C₂²×Q₈	—	4	8	no	—
301	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
302	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
303	C₂³×C₄	—	2	2	no	—
304	C₂³×C₄	D₄×C₂₃	1	2	yes	—
305	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
306	#137	—	4	16	no	—
307	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
308	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
309	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	4	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	4	no	—
314	C₄⋊Q₈	—	2	4	no	—
315	C₄⋊Q₈	—	2	4	no	—
316	C₄⋊Q₈	—	2	4	no	—
317	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
318	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
319	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
320	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
322	#127	C₂³×C₂₃	1	1	yes	characteristic
323	#126	—	2	4	no	—
324	#125	—	2	2	no	—
325	#125	C₂³×C₂₃	1	2	yes	—
326	#124	—	2	2	no	—
327	#124	C₂³×C₂₃	1	2	yes	—
328	#123	—	2	4	no	—
329	#123	—	2	2	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
333	#121	—	2	2	no	—
334	#121	—	2	2	no	—
335	#121	C₂³×C₂₃	1	4	yes	—
336	#120	—	2	2	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
340	C₄×D₄	—	2	4	no	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	C₂³×C₂₃	1	4	yes	—
346	#117	—	2	2	no	—
347	#117	—	2	4	no	—
348	#117	—	2	2	no	—
349	#117	C₂³×C₂₃	1	2	yes	—
350	C₂×C₄⋊C₄	—	2	4	no	—
351	C₂×C₄⋊C₄	—	2	4	no	—
352	C₂×C₄⋊C₄	—	2	4	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	4	no	—
355	C₂×C₄⋊C₄	—	2	4	no	—
356	C₂×C₄²	—	2	2	no	—
357	C₂×C₄²	D₄×C₂₃	1	2	yes	—
358	C₂×C₄²	D₄×C₂₃	1	2	yes	—
359	C₂×C₄²	D₄×C₂₃	1	2	yes	—
360	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
362	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
364	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
365	#103	C₂³×C₂₃	1	32	yes	—
366	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
367	#97	C₂³×C₂₃	1	4	yes	—
368	C₄×C₈	C₂³×C₂₃	1	4	yes	—
369	C₂₃	#60800	1	1	yes	characteristic, direct product factor, Sylow
370	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
371	#41	—	2	8	no	—
372	#41	—	4	8	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	8	no	—
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	16	no	—
377	C₂×Q₈	—	4	16	no	—
378	C₂×Q₈	—	4	8	no	—
379	C₂×Q₈	—	4	16	no	—
380	C₂×Q₈	C₂⁴×C₂₃	1	8	yes	—
381	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	4	8	no	—
393	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
397	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
402	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
403	Q₁₆	—	4	16	no	—
404	SD₁₆	—	4	16	no	—
405	M₁₆	—	2	8	no	—
406	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	2	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	2	4	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	2	no	—
424	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₂₃	1	2	yes	—
426	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
427	C₄²	C₂×D₄×C₂₃	1	2	yes	—
428	C₄²	C₂×D₄×C₂₃	1	2	yes	—
429	C₄²	C₂×D₄×C₂₃	1	2	yes	—
430	C₄²	C₂×D₄×C₂₃	1	2	yes	—
431	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
432	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
433	C₄²	C₂⁴×C₂₃	1	2	yes	—
434	C₂³	—	2	4	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
438	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
439	Q₈	—	2	16	no	—
440	Q₈	—	2	4	no	—
441	Q₈	—	4	16	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	8	no	—
444	D₄	—	2	4	no	—
445	D₄	—	4	8	no	—
446	D₄	—	4	4	no	—
447	D₄	—	4	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	4	16	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	C₂₃×#137	1	8	yes	complemented
464	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
470	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
471	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
472	C₈	—	2	8	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	4	no	—
479	C₂²	C₂₃×#535	1	2	yes	—
480	C₂²	C₂×C₂₃×#137	1	4	yes	—
481	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	4	8	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	4	no	—
488	C₄	C₂₃×#535	1	2	yes	—
489	C₄	C₂×C₂₃×#137	1	4	yes	—
490	C₄	C₂×C₂₃×#137	1	4	yes	—
491	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
492	C₂	—	2	4	no	—
493	C₂	—	2	2	no	—
494	C₂	—	8	2	no	—
495	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
496	C₂	C₂²×C₂₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#3218	C₂	semidirect
4. C₂₃×#3209	C₂	semidirect
5. C₂₃×#3206	C₂	semidirect
6. C₂₃×#3152	C₂	semidirect
7. C₂₃×#3139	C₂	semidirect
8. C₂₃×#3039	C₂	semidirect
9. C₂₃×#3004	C₂	semidirect
10. C₂₃×#2920	C₂	semidirect
33. C₂₃×#497	C₂²	semidirect
34. C₂₃×#494	C₂²	semidirect
35. C₂₃×#492	C₂²	semidirect
36. C₂₃×#491	C₂²	semidirect
37. C₂₃×#487	C₂²	semidirect
38. C₂₃×#462	C₂²	semidirect
39. C₂₃×#461	C₂²	semidirect
185. #60800	C₂₃	direct (N × H)
215. C₂×C₄×C₂₃	#137	semidirect
226. C₂²×C₄⋊Q₈	C₂×C₂₃	semidirect
227. #3218	C₂×C₂₃	semidirect
228. #3209	C₂×C₂₃	semidirect
229. #3206	C₂×C₂₃	semidirect
230. #3152	C₂×C₂₃	semidirect
231. #3139	C₂×C₂₃	semidirect
232. #3039	C₂×C₂₃	semidirect
233. #3004	C₂×C₂₃	semidirect
234. #2920	C₂×C₂₃	semidirect
276. #497	C₂²×C₂₃	semidirect
277. #494	C₂²×C₂₃	semidirect
278. #492	C₂²×C₂₃	semidirect
279. #491	C₂²×C₂₃	semidirect
280. #487	C₂²×C₂₃	semidirect
281. #462	C₂²×C₂₃	semidirect
282. #461	C₂²×C₂₃	semidirect
369. C₂₃	#60800	direct (N × H)
463. 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×#3444	2
C₂²×C₂₃×C₄⋊Q₈	2
C₂₃×#3218	2
C₂₃×#3209	2
C₂₃×#3206	2
C₂₃×#3152	2
C₂₃×#3139	2
C₂₃×#3039	2
C₂₃×#3004	2
C₂₃×#2920	2
#60800	23

Subgroups — C23×#60800 #19208426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60800 #19208426