Subgroups — C₂₃×#24659 #19172285

Order 5888.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2036	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2000	C₂	1	2	yes	maximal
7	C₂₃×#1981	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1973	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1935	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₂₃	—	2	1	no	—
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#461	—	2	4	no	—
20	C₂₃×#432	—	2	4	no	—
21	C₂×C₂₃×#107	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#103	C₂²	1	2	yes	—
25	C₂×C₂₃×#103	C₂²	1	2	yes	—
26	C₂×C₂₃×#103	C₂²	1	2	yes	—
27	C₂×C₂₃×#102	C₂²	1	2	yes	—
28	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₂₃×#97	C₂²	1	2	yes	—
30	C₂₃×#391	C₂²	1	2	yes	—
31	C₂₃×#390	C₂²	1	1	yes	characteristic
32	C₂₃×#389	—	2	1	no	—
33	C₂₃×#389	—	2	1	no	—
34	C₂₃×#389	C₂²	1	1	yes	characteristic
35	C₂₃×#388	—	2	1	no	—
36	C₂₃×#386	—	2	1	no	—
37	C₂₃×#384	C₂²	1	1	yes	characteristic
38	C₂₃×#380	—	2	1	no	—
39	C₂₃×#380	—	2	1	no	—
40	C₂₃×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
43	C₂₃×#340	C₂²	1	2	yes	—
44	C₂₃×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×D₄×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	—	2	1	no	—
49	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×SD₁₆×C₂₃	—	2	4	no	—
53	C₂×SD₁₆×C₂₃	—	2	4	no	—
54	C₂×SD₁₆×C₂₃	—	4	2	no	—
55	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₂²⋊Q₈	—	4	2	no	—
60	C₂₃×C₂²⋊Q₈	—	4	2	no	—
61	C₂₃×#121	—	2	2	no	—
62	C₂₃×#121	—	2	2	no	—
63	C₄×D₄×C₂₃	—	2	2	no	—
64	C₄×D₄×C₂₃	—	2	2	no	—
65	C₄×D₄×C₂₃	—	4	4	no	—
66	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄²×C₂₃	—	2	2	no	—
83	C₂×C₄²×C₂₃	—	2	2	no	—
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
89	C₂₃×#107	—	4	2	no	—
90	C₂₃×C₄⋊C₈	—	2	2	no	—
91	C₂₃×C₄⋊C₈	D₄	1	4	yes	complemented
92	C₂₃×#103	—	2	4	no	—
93	C₂₃×#103	—	2	4	no	—
94	C₂₃×#103	—	4	2	no	—
95	C₂₃×#102	—	4	2	no	—
96	C₂₃×C₂²⋊C₈	—	2	4	no	—
97	C₂₃×#97	—	2	4	no	—
98	C₂₃×#95	—	2	1	no	—
99	C₂₃×#95	—	2	1	no	—
100	C₂₃×#95	—	2	1	no	—
101	C₂₃×#95	—	2	2	no	—
102	C₂₃×#95	—	2	2	no	—
103	C₂⁴×C₂₃	—	2	1	no	—
104	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Q₈×C₂₃	—	2	2	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	1	no	—
113	C₂×D₄×C₂₃	—	2	1	no	—
114	C₂×D₄×C₂₃	—	2	1	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	1	no	—
117	C₂×D₄×C₂₃	—	4	2	no	—
118	C₂×D₄×C₂₃	—	4	2	no	—
119	C₂×D₄×C₂₃	#41	1	2	yes	—
120	C₂²×C₄×C₂₃	—	2	2	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	4	1	no	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	4	2	no	—
139	C₂²×C₄×C₂₃	—	4	2	no	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	SD₁₆×C₂₃	—	4	8	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	2	2	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	4	2	no	—
161	C₄⋊C₄×C₂₃	—	4	2	no	—
162	C₄⋊C₄×C₂₃	—	4	4	no	—
163	C₄⋊C₄×C₂₃	—	4	2	no	—
164	C₄⋊C₄×C₂₃	—	4	2	no	—
165	C₄⋊C₄×C₂₃	—	4	2	no	—
166	C₄⋊C₄×C₂₃	#41	1	2	yes	—
167	C₂²⋊C₄×C₂₃	—	2	4	no	—
168	C₂²⋊C₄×C₂₃	—	4	2	no	—
169	C₂²⋊C₄×C₂₃	—	4	2	no	—
170	C₂²⋊C₄×C₂₃	—	4	2	no	—
171	C₂²⋊C₄×C₂₃	—	4	4	no	—
172	C₂²⋊C₄×C₂₃	—	4	2	no	—
173	C₄²×C₂₃	—	2	2	no	—
174	C₄²×C₂₃	—	2	2	no	—
175	C₄²×C₂₃	—	2	1	no	—
176	C₄²×C₂₃	—	2	1	no	—
177	C₄²×C₂₃	—	4	2	no	—
178	C₄²×C₂₃	C₂×D₄	1	2	yes	—
179	C₄²×C₂₃	#41	1	2	yes	—
180	#24659	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂³×C₂₃	—	2	1	no	—
182	C₂³×C₂₃	—	2	1	no	—
183	C₂³×C₂₃	—	2	1	no	—
184	C₂³×C₂₃	—	2	1	no	—
185	C₂³×C₂₃	—	2	1	no	—
186	C₂³×C₂₃	—	2	1	no	—
187	C₂³×C₂₃	—	2	2	no	—
188	C₂³×C₂₃	—	4	1	no	—
189	C₂³×C₂₃	—	4	1	no	—
190	C₂³×C₂₃	—	4	1	no	—
191	C₂³×C₂₃	—	4	1	no	—
192	C₂³×C₂₃	—	4	1	no	—
193	C₂³×C₂₃	—	4	1	no	—
194	C₂³×C₂₃	#137	1	2	yes	—
195	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₂₃	—	4	2	no	—
197	Q₈×C₂₃	—	4	2	no	—
198	Q₈×C₂₃	—	4	2	no	—
199	Q₈×C₂₃	—	4	2	no	—
200	D₄×C₂₃	—	2	2	no	—
201	D₄×C₂₃	—	2	2	no	—
202	D₄×C₂₃	—	4	8	no	—
203	D₄×C₂₃	—	4	1	no	—
204	D₄×C₂₃	—	4	1	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	1	no	—
207	C₂×C₄×C₂₃	—	2	1	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	1	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	4	4	no	—
218	C₂×C₄×C₂₃	—	4	1	no	—
219	C₂×C₄×C₂₃	—	4	1	no	—
220	C₂×C₄×C₂₃	—	4	1	no	—
221	C₂×C₄×C₂₃	—	4	1	no	—
222	C₂×C₄×C₂₃	—	4	1	no	—
223	C₂×C₄×C₂₃	—	4	1	no	—
224	C₂×C₄×C₂₃	—	4	4	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	2	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	2	no	—
229	C₂×C₄×C₂₃	—	4	2	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	8	4	no	—
238	C₂×C₄×C₂₃	#121	1	2	yes	—
239	C₂×C₄×C₂₃	#121	1	2	yes	—
240	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
241	C₂×C₄×C₂₃	#137	1	2	yes	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
248	C₈×C₂₃	—	2	4	no	—
249	C₈×C₂₃	—	4	2	no	—
250	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
251	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
252	#2857	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₄×#121	C₂×C₂₃	1	1	yes	characteristic
254	#2036	C₂×C₂₃	1	2	yes	complemented
255	#2000	C₂×C₂₃	1	2	yes	—
256	#1981	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#1973	C₂×C₂₃	1	2	yes	complemented
258	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#1935	C₂×C₂₃	1	2	yes	complemented
260	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂²×C₂₃	—	2	2	no	—
262	C₂²×C₂₃	—	2	2	no	—
263	C₂²×C₂₃	—	2	2	no	—
264	C₂²×C₂₃	—	4	1	no	—
265	C₂²×C₂₃	—	4	1	no	—
266	C₂²×C₂₃	—	4	1	no	—
267	C₂²×C₂₃	—	4	1	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	8	1	no	—
272	C₂²×C₂₃	—	8	1	no	—
273	C₂²×C₂₃	—	8	1	no	—
274	C₂²×C₂₃	—	8	1	no	—
275	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
276	C₂²×C₂₃	#525	1	1	yes	characteristic
277	C₂²×C₂₃	#535	1	1	yes	characteristic
278	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
279	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
280	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
281	C₂²×C₂₃	#575	1	1	yes	characteristic
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	8	2	no	—
288	C₄×C₂₃	—	8	2	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	—	8	1	no	—
291	C₄×C₂₃	#468	1	4	yes	—
292	C₄×C₂₃	#546	1	1	yes	characteristic
293	C₄×C₂₃	#550	1	1	yes	characteristic
294	C₄×C₂₃	#553	1	1	yes	characteristic
295	C₄×C₂₃	#555	1	1	yes	characteristic
296	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
297	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
298	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
299	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×C₄×D₄	—	2	1	no	—
301	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
302	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
303	#461	—	2	4	no	—
304	#432	—	2	4	no	—
305	C₂×#107	C₂²×C₂₃	1	2	yes	—
306	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#103	C₂²×C₂₃	1	2	yes	—
309	C₂×#103	C₂²×C₂₃	1	2	yes	—
310	C₂×#103	C₂²×C₂₃	1	2	yes	—
311	C₂×#102	C₂²×C₂₃	1	2	yes	—
312	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
313	C₂×#97	C₂²×C₂₃	1	2	yes	—
314	#391	C₂²×C₂₃	1	2	yes	—
315	#390	C₂²×C₂₃	1	1	yes	characteristic
316	#389	—	2	1	no	—
317	#389	—	2	1	no	—
318	#389	C₂²×C₂₃	1	1	yes	characteristic
319	#388	—	2	1	no	—
320	#386	—	2	1	no	—
321	#384	C₂²×C₂₃	1	1	yes	characteristic
322	#380	—	2	1	no	—
323	#380	—	2	1	no	—
324	#380	C₂²×C₂₃	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
326	C₄×C₂²⋊C₄	—	2	1	no	—
327	#340	C₂²×C₂₃	1	2	yes	—
328	#340	C₂²×C₂₃	1	2	yes	—
329	C₂×C₂₃	—	4	2	no	—
330	C₂×C₂₃	—	8	2	no	—
331	C₂×C₂₃	#2414	1	1	yes	characteristic
332	C₂×C₂₃	C₂×#468	1	1	yes	characteristic
333	C₂×C₂₃	#3056	1	1	yes	characteristic
334	C₂×C₂₃	#3110	1	1	yes	characteristic
335	C₂×C₂₃	#3116	1	1	yes	characteristic
336	C₂×C₂₃	#3214	1	1	yes	characteristic
337	C₂×C₂₃	#3218	1	1	yes	characteristic
338	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
341	C₂³×C₄	—	2	1	no	—
342	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
343	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₂×SD₁₆	—	2	4	no	—
346	C₂×SD₁₆	—	2	4	no	—
347	C₂×SD₁₆	—	4	2	no	—
348	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
349	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
350	C₄⋊Q₈	—	2	2	no	—
351	C₄⋊Q₈	—	2	2	no	—
352	C₂²⋊Q₈	—	4	2	no	—
353	C₂²⋊Q₈	—	4	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	4	4	no	—
359	C₂×C₄⋊C₄	—	2	1	no	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	1	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
366	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
369	C₂×C₂²⋊C₄	—	2	1	no	—
370	C₂×C₂²⋊C₄	—	2	1	no	—
371	C₂×C₂²⋊C₄	—	2	2	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	—	2	2	no	—
377	C₂×C₄²	—	2	1	no	—
378	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
379	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
381	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
382	#107	—	4	2	no	—
383	C₄⋊C₈	—	2	2	no	—
384	C₄⋊C₈	D₄×C₂₃	1	4	yes	complemented
385	#103	—	2	4	no	—
386	#103	—	2	4	no	—
387	#103	—	4	2	no	—
388	#102	—	4	2	no	—
389	C₂²⋊C₈	—	2	4	no	—
390	#97	—	2	4	no	—
391	#95	—	2	1	no	—
392	#95	—	2	1	no	—
393	#95	—	2	1	no	—
394	#95	—	2	2	no	—
395	#95	—	2	2	no	—
396	C₂₃	#24659	1	1	yes	characteristic, direct product factor, Sylow
397	C₂⁴	—	2	1	no	—
398	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	2	2	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	4	2	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	#41×C₂₃	1	2	yes	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	4	1	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	SD₁₆	—	4	8	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	2	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	4	2	no	—
455	C₄⋊C₄	—	4	4	no	—
456	C₄⋊C₄	—	4	2	no	—
457	C₄⋊C₄	—	4	2	no	—
458	C₄⋊C₄	—	4	2	no	—
459	C₄⋊C₄	—	4	2	no	—
460	C₄⋊C₄	#41×C₂₃	1	2	yes	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	4	2	no	—
463	C₂²⋊C₄	—	4	2	no	—
464	C₂²⋊C₄	—	4	2	no	—
465	C₂²⋊C₄	—	4	4	no	—
466	C₂²⋊C₄	—	4	2	no	—
467	C₄²	—	2	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	4	2	no	—
472	C₄²	C₂×D₄×C₂₃	1	2	yes	—
473	C₄²	#41×C₂₃	1	2	yes	—
474	C₂³	—	2	1	no	—
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	—	2	1	no	—
478	C₂³	—	2	1	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	1	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	4	1	no	—
486	C₂³	—	4	1	no	—
487	C₂³	C₂₃×#137	1	2	yes	—
488	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
489	Q₈	—	4	2	no	—
490	Q₈	—	4	2	no	—
491	Q₈	—	4	2	no	—
492	Q₈	—	4	2	no	—
493	D₄	—	2	2	no	—
494	D₄	—	2	2	no	—
495	D₄	—	4	8	no	—
496	D₄	—	4	1	no	—
497	D₄	—	4	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	1	no	—
514	C₂×C₄	—	4	1	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	8	4	no	—
531	C₂×C₄	C₂₃×#121	1	2	yes	—
532	C₂×C₄	C₂₃×#121	1	2	yes	—
533	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
534	C₂×C₄	C₂₃×#137	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
536	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
537	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
538	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
539	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
540	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
541	C₈	—	2	4	no	—
542	C₈	—	4	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
558	C₂²	C₂₃×#525	1	1	yes	characteristic
559	C₂²	C₂₃×#535	1	1	yes	characteristic
560	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
561	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
562	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
563	C₂²	C₂₃×#575	1	1	yes	characteristic
564	C₄	—	2	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	2	no	—
569	C₄	—	8	2	no	—
570	C₄	—	8	1	no	—
571	C₄	—	8	2	no	—
572	C₄	—	8	1	no	—
573	C₄	C₂₃×#468	1	4	yes	—
574	C₄	C₂₃×#546	1	1	yes	characteristic
575	C₄	C₂₃×#550	1	1	yes	characteristic
576	C₄	C₂₃×#553	1	1	yes	characteristic
577	C₄	C₂₃×#555	1	1	yes	characteristic
578	C₂	—	4	2	no	—
579	C₂	—	8	2	no	—
580	C₂	C₂₃×#2414	1	1	yes	characteristic
581	C₂	C₂×C₂₃×#468	1	1	yes	characteristic
582	C₂	C₂₃×#3056	1	1	yes	characteristic
583	C₂	C₂₃×#3110	1	1	yes	characteristic
584	C₂	C₂₃×#3116	1	1	yes	characteristic
585	C₂	C₂₃×#3214	1	1	yes	characteristic
586	C₂	C₂₃×#3218	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#461	C₂	semidirect
2. C₂×C₂₃×#432	C₂	semidirect
3. C₂₃×#2857	C₂	semidirect
5. C₂₃×#2036	C₂	semidirect
7. C₂₃×#1981	C₂	semidirect
8. C₂₃×#1973	C₂	semidirect
9. C₂₃×#1947	C₂	semidirect
10. C₂₃×#1935	C₂	semidirect
11. C₂₃×#1802	C₂	semidirect
91. C₂₃×C₄⋊C₈	D₄	semidirect
180. #24659	C₂₃	direct (N × H)
250. C₂×#461	C₂×C₂₃	semidirect
251. C₂×#432	C₂×C₂₃	semidirect
252. #2857	C₂×C₂₃	semidirect
254. #2036	C₂×C₂₃	semidirect
256. #1981	C₂×C₂₃	semidirect
257. #1973	C₂×C₂₃	semidirect
258. #1947	C₂×C₂₃	semidirect
259. #1935	C₂×C₂₃	semidirect
260. #1802	C₂×C₂₃	semidirect
384. C₄⋊C₈	D₄×C₂₃	semidirect
396. C₂₃	#24659	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×#461	2
C₂×C₂₃×#432	2
C₂₃×#2857	2
C₄×C₂₃×#121	2
C₂₃×#2036	2
C₂₃×#2000	2
C₂₃×#1981	2
C₂₃×#1973	2
C₂₃×#1947	2
C₂₃×#1935	2
C₂₃×#1802	2
#24659	23

Subgroups — C23×#24659 #19172285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24659 #19172285