Subgroups — C₂₃×#61402 #19209028

Order 5888.

2244 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3399	C₂	1	6	yes	maximal, complemented
4	C₂₃×#3397	C₂	1	6	yes	maximal, complemented
5	C₂₃×#3392	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3391	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3151	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3038	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3036	C₂	1	3	yes	maximal, complemented
10	Q₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
12	C₂²×#41×C₂₃	—	2	1	no	—
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	3	yes	—
14	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	6	yes	—
15	C₂₃×#559	C₂²	1	3	yes	complemented
16	C₂₃×#557	C₂²	1	1	yes	characteristic, complemented
17	C₂₃×#556	C₂²	1	3	yes	complemented
18	C₂₃×#553	—	2	3	no	—
19	C₂₃×#550	—	2	1	no	—
20	C₂₃×#550	—	2	3	no	—
21	C₂₃×#550	C₂²	1	2	yes	—
22	C₂₃×#550	C₂²	1	6	yes	—
23	C₂₃×#550	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#548	—	2	1	no	—
25	C₂₃×#532	C₂²	1	3	yes	—
26	C₂×C₂₃×#127	C₂²	1	3	yes	—
27	C₂₃×#529	—	2	3	no	—
28	C₂×C₂₃×#121	—	2	3	no	—
29	C₄×#41×C₂₃	—	2	1	no	—
30	C₄×#41×C₂₃	C₂²	1	3	yes	—
31	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
33	C₂₃×#496	C₂²	1	12	yes	complemented
34	C₂₃×#493	C₂²	1	6	yes	complemented
35	C₂₃×#492	C₂²	1	6	yes	complemented
36	C₂₃×#463	C₂²	1	12	yes	complemented
37	C₂₃×#461	C₂²	1	6	yes	complemented
38	C₂₃×#460	C₂²	1	6	yes	complemented
39	C₂₃×#460	C₂²	1	6	yes	complemented
40	C₂₃×#459	C₂²	1	6	yes	complemented
41	C₂₃×#459	C₂²	1	6	yes	complemented
42	C₂₃×#459	C₂²	1	6	yes	complemented
43	C₂₃×#446	C₂²	1	6	yes	complemented
44	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
45	C₂₃×#442	C₂²	1	6	yes	complemented
46	C₂₃×#440	C₂²	1	6	yes	complemented
47	C₄×SD₁₆×C₂₃	C₂²	1	6	yes	complemented
48	C₄×D₈×C₂₃	C₂²	1	6	yes	complemented
49	C₂₃×#428	C₂²	1	1	yes	characteristic
50	C₂₃×#423	C₂²	1	3	yes	—
51	C₂₃×#418	C₂²	1	3	yes	—
52	C₂₃×#417	C₂²	1	3	yes	—
53	C₄×M₁₆×C₂₃	C₂²	1	3	yes	—
54	C₂×#41×C₂₃	—	2	1	no	—
55	C₂×#41×C₂₃	—	2	6	no	—
56	C₂×#41×C₂₃	—	2	2	no	—
57	C₂×#41×C₂₃	—	4	8	no	—
58	C₂×#41×C₂₃	C₂³	1	3	yes	—
59	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	—	2	3	no	—
61	C₂²×D₄×C₂₃	—	2	3	no	—
62	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
63	C₂³×C₄×C₂₃	—	2	3	no	—
64	C₂₃×AGL(1,ℤ/8)	—	2	12	no	—
65	C₂₃×AGL(1,ℤ/8)	—	4	12	no	—
66	C₂×SD₁₆×C₂₃	C₂³	1	6	yes	—
67	C₂×SD₁₆×C₂₃	C₂³	1	12	yes	—
68	C₂×D₈×C₂₃	C₂³	1	6	yes	—
69	C₂×D₈×C₂₃	C₂³	1	12	yes	—
70	C₂×M₁₆×C₂₃	C₂³	1	3	yes	—
71	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₂₃×#127	—	2	3	no	—
74	C₂₃×#127	—	2	3	no	—
75	C₂₃×#127	C₂³	1	6	yes	—
76	C₂₃×#127	C₂³	1	6	yes	—
77	C₂₃×#127	C₂³	1	3	yes	—
78	C₂₃×#126	—	2	6	no	—
79	C₂₃×#125	—	2	3	no	—
80	C₂₃×#124	—	2	3	no	—
81	C₂₃×#124	C₂³	1	6	yes	—
82	C₂₃×#123	—	2	6	no	—
83	C₂₃×C₂²⋊Q₈	—	2	3	no	—
84	C₂₃×#121	—	2	3	no	—
85	C₂₃×#121	—	2	3	no	—
86	C₂₃×#121	—	2	6	no	—
87	C₂₃×#121	—	2	6	no	—
88	C₂₃×#121	—	2	12	no	—
89	C₂₃×#121	—	2	3	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₂₃	D₄	1	3	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
94	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
97	C₄×D₄×C₂₃	—	2	3	no	—
98	C₄×D₄×C₂₃	—	2	3	no	—
99	C₄×D₄×C₂₃	—	2	3	no	—
100	C₄×D₄×C₂₃	—	2	3	no	—
101	C₄×D₄×C₂₃	—	2	3	no	—
102	C₄×D₄×C₂₃	—	2	3	no	—
103	C₄×D₄×C₂₃	—	2	3	no	—
104	C₄×D₄×C₂₃	C₂³	1	6	yes	—
105	C₄×D₄×C₂₃	C₂³	1	6	yes	—
106	C₄×D₄×C₂₃	C₂³	1	3	yes	—
107	C₄×D₄×C₂₃	C₂³	1	3	yes	—
108	C₂₃×#117	—	2	3	no	—
109	C₂₃×#117	—	2	3	no	—
110	C₂₃×#117	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
114	C₂×C₄²×C₂₃	—	2	3	no	—
115	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
116	C₂₃×#107	C₂³	1	2	yes	—
117	C₂₃×#106	C₂³	1	2	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
119	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
120	C₂₃×#103	C₂³	1	6	yes	—
121	C₂₃×#102	C₂³	1	6	yes	—
122	C₂₃×#102	C₂³	1	6	yes	—
123	C₂₃×#102	C₂³	1	6	yes	—
124	C₂₃×#97	C₂³	1	6	yes	—
125	C₄×C₈×C₂₃	C₂³	1	6	yes	—
126	C₂⁴×C₂₃	—	2	3	no	—
127	#41×C₂₃	—	2	6	no	—
128	#41×C₂₃	—	2	6	no	—
129	#41×C₂₃	—	4	24	no	—
130	#41×C₂₃	—	4	4	no	—
131	#41×C₂₃	—	4	8	no	—
132	C₂×Q₈×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
136	C₂×D₄×C₂₃	—	2	3	no	—
137	C₂×D₄×C₂₃	—	2	3	no	—
138	C₂×D₄×C₂₃	—	2	3	no	—
139	C₂×D₄×C₂₃	—	2	6	no	—
140	C₂×D₄×C₂₃	—	2	6	no	—
141	C₂×D₄×C₂₃	—	2	3	no	—
142	C₂×D₄×C₂₃	—	2	6	no	—
143	C₂×D₄×C₂₃	—	2	6	no	—
144	C₂×D₄×C₂₃	—	2	6	no	—
145	C₂×D₄×C₂₃	—	2	6	no	—
146	C₂×D₄×C₂₃	—	4	12	no	—
147	C₂×D₄×C₂₃	—	8	6	no	—
148	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
149	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
150	C₂²×C₄×C₂₃	—	2	1	no	—
151	C₂²×C₄×C₂₃	—	2	3	no	—
152	C₂²×C₄×C₂₃	—	2	3	no	—
153	C₂²×C₄×C₂₃	—	2	6	no	—
154	C₂²×C₄×C₂₃	—	2	6	no	—
155	C₂²×C₄×C₂₃	—	2	3	no	—
156	C₂²×C₄×C₂₃	—	4	12	no	—
157	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
160	SD₁₆×C₂₃	—	2	12	no	—
161	SD₁₆×C₂₃	—	4	12	no	—
162	D₈×C₂₃	—	2	12	no	—
163	D₈×C₂₃	—	4	12	no	—
164	M₁₆×C₂₃	—	2	6	no	—
165	M₁₆×C₂₃	#41	1	4	yes	complemented
166	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
167	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	3	no	—
170	C₄⋊C₄×C₂₃	—	2	3	no	—
171	C₄⋊C₄×C₂₃	—	2	3	no	—
172	C₄⋊C₄×C₂₃	—	2	3	no	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
175	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
176	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₂²⋊C₄×C₂₃	—	2	1	no	—
180	C₂²⋊C₄×C₂₃	—	2	6	no	—
181	C₂²⋊C₄×C₂₃	—	2	6	no	—
182	C₂²⋊C₄×C₂₃	—	2	6	no	—
183	C₂²⋊C₄×C₂₃	—	2	3	no	—
184	C₂²⋊C₄×C₂₃	—	2	3	no	—
185	C₂²⋊C₄×C₂₃	—	2	6	no	—
186	C₄²×C₂₃	—	2	1	no	—
187	C₄²×C₂₃	—	2	3	no	—
188	C₄²×C₂₃	—	2	3	no	—
189	C₄²×C₂₃	C₂×D₄	1	3	yes	—
190	C₄²×C₂₃	C₂×D₄	1	3	yes	—
191	C₄²×C₂₃	C₂⁴	1	3	yes	—
192	C₄²×C₂₃	C₂⁴	1	3	yes	—
193	#61402	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₂₃	—	2	3	no	—
195	C₂³×C₂₃	—	2	6	no	—
196	C₂³×C₂₃	—	4	6	no	—
197	C₂³×C₂₃	—	4	6	no	—
198	C₂³×C₂₃	—	4	6	no	—
199	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₂₃	—	2	6	no	—
201	Q₈×C₂₃	—	2	3	no	—
202	Q₈×C₂₃	—	2	3	no	—
203	Q₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	complemented
204	D₄×C₂₃	—	2	3	no	—
205	D₄×C₂₃	—	2	6	no	—
206	D₄×C₂₃	—	2	3	no	—
207	D₄×C₂₃	—	2	6	no	—
208	D₄×C₂₃	—	4	12	no	—
209	D₄×C₂₃	—	4	6	no	—
210	D₄×C₂₃	—	4	6	no	—
211	D₄×C₂₃	—	4	12	no	—
212	D₄×C₂₃	—	4	6	no	—
213	D₄×C₂₃	—	8	3	no	—
214	D₄×C₂₃	—	8	6	no	—
215	D₄×C₂₃	—	8	6	no	—
216	C₂×C₄×C₂₃	—	2	6	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	3	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	4	3	no	—
222	C₂×C₄×C₂₃	—	4	12	no	—
223	C₂×C₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	4	6	no	—
225	C₂×C₄×C₂₃	—	4	12	no	—
226	C₂×C₄×C₂₃	—	4	6	no	—
227	C₂×C₄×C₂₃	—	8	3	no	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
234	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
235	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
236	C₈×C₂₃	—	2	6	no	—
237	C₈×C₂₃	C₂×#41	1	4	yes	complemented
238	#3480	C₂×C₂₃	1	1	yes	characteristic, complemented
239	C₂×#550	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#3399	C₂×C₂₃	1	6	yes	complemented
241	#3397	C₂×C₂₃	1	6	yes	complemented
242	#3392	C₂×C₂₃	1	2	yes	complemented
243	#3391	C₂×C₂₃	1	2	yes	complemented
244	#3151	C₂×C₂₃	1	3	yes	complemented
245	#3038	C₂×C₂₃	1	3	yes	complemented
246	#3036	C₂×C₂₃	1	3	yes	complemented
247	Q₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₄×AGL(1,ℤ/8)	C₂×C₂₃	1	3	yes	complemented
249	C₂²×C₂₃	—	2	1	no	—
250	C₂²×C₂₃	—	2	1	no	—
251	C₂²×C₂₃	—	4	3	no	—
252	C₂²×C₂₃	—	4	6	no	—
253	C₂²×C₂₃	—	4	3	no	—
254	C₂²×C₂₃	—	4	6	no	—
255	C₂²×C₂₃	—	4	6	no	—
256	C₂²×C₂₃	—	4	3	no	—
257	C₂²×C₂₃	—	8	12	no	—
258	C₂²×C₂₃	#550	1	2	yes	—
259	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
260	C₄×C₂₃	—	2	3	no	—
261	C₄×C₂₃	—	2	3	no	—
262	C₄×C₂₃	—	2	3	no	—
263	C₄×C₂₃	—	4	1	no	—
264	C₄×C₂₃	—	4	3	no	—
265	C₄×C₂₃	—	4	1	no	—
266	C₄×C₂₃	—	4	2	no	—
267	C₄×C₂₃	#550	1	2	yes	—
268	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
269	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
270	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
271	C₂²×#41	—	2	1	no	—
272	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	3	yes	—
273	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	6	yes	—
274	#559	C₂²×C₂₃	1	3	yes	complemented
275	#557	C₂²×C₂₃	1	1	yes	characteristic, complemented
276	#556	C₂²×C₂₃	1	3	yes	complemented
277	#553	—	2	3	no	—
278	#550	—	2	1	no	—
279	#550	—	2	3	no	—
280	#550	C₂²×C₂₃	1	2	yes	—
281	#550	C₂²×C₂₃	1	6	yes	—
282	#550	C₂²×C₂₃	1	1	yes	characteristic, complemented
283	#548	—	2	1	no	—
284	#532	C₂²×C₂₃	1	3	yes	—
285	C₂×#127	C₂²×C₂₃	1	3	yes	—
286	#529	—	2	3	no	—
287	C₂×#121	—	2	3	no	—
288	C₄×#41	—	2	1	no	—
289	C₄×#41	C₂²×C₂₃	1	3	yes	—
290	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
292	#496	C₂²×C₂₃	1	12	yes	complemented
293	#493	C₂²×C₂₃	1	6	yes	complemented
294	#492	C₂²×C₂₃	1	6	yes	complemented
295	#463	C₂²×C₂₃	1	12	yes	complemented
296	#461	C₂²×C₂₃	1	6	yes	complemented
297	#460	C₂²×C₂₃	1	6	yes	complemented
298	#460	C₂²×C₂₃	1	6	yes	complemented
299	#459	C₂²×C₂₃	1	6	yes	complemented
300	#459	C₂²×C₂₃	1	6	yes	complemented
301	#459	C₂²×C₂₃	1	6	yes	complemented
302	#446	C₂²×C₂₃	1	6	yes	complemented
303	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
304	#442	C₂²×C₂₃	1	6	yes	complemented
305	#440	C₂²×C₂₃	1	6	yes	complemented
306	C₄×SD₁₆	C₂²×C₂₃	1	6	yes	complemented
307	C₄×D₈	C₂²×C₂₃	1	6	yes	complemented
308	#428	C₂²×C₂₃	1	1	yes	characteristic
309	#423	C₂²×C₂₃	1	3	yes	—
310	#418	C₂²×C₂₃	1	3	yes	—
311	#417	C₂²×C₂₃	1	3	yes	—
312	C₄×M₁₆	C₂²×C₂₃	1	3	yes	—
313	C₂×C₂₃	—	2	2	no	—
314	C₂×C₂₃	—	8	3	no	—
315	C₂×C₂₃	—	8	6	no	—
316	C₂×C₂₃	C₂×#550	1	1	yes	characteristic
317	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
318	C₂×C₂₃	#3585	1	1	yes	characteristic
319	C₂×#41	—	2	1	no	—
320	C₂×#41	—	2	6	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	—	4	8	no	—
323	C₂×#41	C₂³×C₂₃	1	3	yes	—
324	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
325	C₂²×D₄	—	2	3	no	—
326	C₂²×D₄	—	2	3	no	—
327	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
328	C₂³×C₄	—	2	3	no	—
329	AGL(1,ℤ/8)	—	2	12	no	—
330	AGL(1,ℤ/8)	—	4	12	no	—
331	C₂×SD₁₆	C₂³×C₂₃	1	6	yes	—
332	C₂×SD₁₆	C₂³×C₂₃	1	12	yes	—
333	C₂×D₈	C₂³×C₂₃	1	6	yes	—
334	C₂×D₈	C₂³×C₂₃	1	12	yes	—
335	C₂×M₁₆	C₂³×C₂₃	1	3	yes	—
336	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
338	#127	—	2	3	no	—
339	#127	—	2	3	no	—
340	#127	C₂³×C₂₃	1	6	yes	—
341	#127	C₂³×C₂₃	1	6	yes	—
342	#127	C₂³×C₂₃	1	3	yes	—
343	#126	—	2	6	no	—
344	#125	—	2	3	no	—
345	#124	—	2	3	no	—
346	#124	C₂³×C₂₃	1	6	yes	—
347	#123	—	2	6	no	—
348	C₂²⋊Q₈	—	2	3	no	—
349	#121	—	2	3	no	—
350	#121	—	2	3	no	—
351	#121	—	2	6	no	—
352	#121	—	2	6	no	—
353	#121	—	2	12	no	—
354	#121	—	2	3	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
357	C₄×Q₈	D₄×C₂₃	1	3	yes	—
358	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
359	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
360	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	—	2	3	no	—
364	C₄×D₄	—	2	3	no	—
365	C₄×D₄	—	2	3	no	—
366	C₄×D₄	—	2	3	no	—
367	C₄×D₄	—	2	3	no	—
368	C₄×D₄	—	2	3	no	—
369	C₄×D₄	C₂³×C₂₃	1	6	yes	—
370	C₄×D₄	C₂³×C₂₃	1	6	yes	—
371	C₄×D₄	C₂³×C₂₃	1	3	yes	—
372	C₄×D₄	C₂³×C₂₃	1	3	yes	—
373	#117	—	2	3	no	—
374	#117	—	2	3	no	—
375	#117	C₂³×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	—	2	3	no	—
379	C₂×C₄²	—	2	3	no	—
380	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
381	#107	C₂³×C₂₃	1	2	yes	—
382	#106	C₂³×C₂₃	1	2	yes	—
383	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
384	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
385	#103	C₂³×C₂₃	1	6	yes	—
386	#102	C₂³×C₂₃	1	6	yes	—
387	#102	C₂³×C₂₃	1	6	yes	—
388	#102	C₂³×C₂₃	1	6	yes	—
389	#97	C₂³×C₂₃	1	6	yes	—
390	C₄×C₈	C₂³×C₂₃	1	6	yes	—
391	C₂₃	#61402	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	3	no	—
393	#41	—	2	6	no	—
394	#41	—	2	6	no	—
395	#41	—	4	24	no	—
396	#41	—	4	8	no	—
397	#41	—	4	4	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
400	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	3	no	—
404	C₂×D₄	—	2	3	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	6	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	6	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	6	no	—
412	C₂×D₄	—	4	12	no	—
413	C₂×D₄	—	8	6	no	—
414	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
415	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	3	no	—
418	C₂²×C₄	—	2	3	no	—
419	C₂²×C₄	—	2	6	no	—
420	C₂²×C₄	—	2	6	no	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	4	12	no	—
423	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
424	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
425	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
426	SD₁₆	—	2	12	no	—
427	SD₁₆	—	4	12	no	—
428	D₈	—	2	12	no	—
429	D₈	—	4	12	no	—
430	M₁₆	—	2	6	no	—
431	M₁₆	#41×C₂₃	1	4	yes	complemented
432	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
433	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	3	no	—
436	C₄⋊C₄	—	2	3	no	—
437	C₄⋊C₄	—	2	3	no	—
438	C₄⋊C₄	—	2	3	no	—
439	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
440	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
441	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
442	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
443	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
444	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	2	6	no	—
447	C₂²⋊C₄	—	2	6	no	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	—	2	3	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	6	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	3	no	—
454	C₄²	—	2	3	no	—
455	C₄²	C₂×D₄×C₂₃	1	3	yes	—
456	C₄²	C₂×D₄×C₂₃	1	3	yes	—
457	C₄²	C₂⁴×C₂₃	1	3	yes	—
458	C₄²	C₂⁴×C₂₃	1	3	yes	—
459	C₂³	—	2	3	no	—
460	C₂³	—	2	6	no	—
461	C₂³	—	4	6	no	—
462	C₂³	—	4	6	no	—
463	C₂³	—	4	6	no	—
464	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
465	Q₈	—	2	6	no	—
466	Q₈	—	2	3	no	—
467	Q₈	—	2	3	no	—
468	Q₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	complemented
469	D₄	—	2	3	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	3	no	—
472	D₄	—	2	6	no	—
473	D₄	—	4	12	no	—
474	D₄	—	4	6	no	—
475	D₄	—	4	6	no	—
476	D₄	—	4	12	no	—
477	D₄	—	4	6	no	—
478	D₄	—	8	6	no	—
479	D₄	—	8	6	no	—
480	D₄	—	8	3	no	—
481	C₂×C₄	—	2	6	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	3	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	3	no	—
487	C₂×C₄	—	4	12	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	6	no	—
490	C₂×C₄	—	4	12	no	—
491	C₂×C₄	—	4	6	no	—
492	C₂×C₄	—	8	3	no	—
493	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
494	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
495	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
496	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
498	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
499	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
500	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
501	C₈	—	2	6	no	—
502	C₈	C₂×#41×C₂₃	1	4	yes	complemented
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	3	no	—
506	C₂²	—	4	6	no	—
507	C₂²	—	4	6	no	—
508	C₂²	—	4	3	no	—
509	C₂²	—	4	3	no	—
510	C₂²	—	4	6	no	—
511	C₂²	—	8	12	no	—
512	C₂²	C₂₃×#550	1	2	yes	—
513	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
514	C₄	—	2	3	no	—
515	C₄	—	2	3	no	—
516	C₄	—	2	3	no	—
517	C₄	—	4	3	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	1	no	—
521	C₄	C₂₃×#550	1	2	yes	—
522	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	6	yes	—
523	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
524	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
525	C₂	—	2	2	no	—
526	C₂	—	8	6	no	—
527	C₂	—	8	3	no	—
528	C₂	C₂×C₂₃×#550	1	1	yes	characteristic
529	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
530	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3480	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂₃×#3399	C₂	semidirect
4. C₂₃×#3397	C₂	semidirect
5. C₂₃×#3392	C₂	semidirect
6. C₂₃×#3391	C₂	semidirect
7. C₂₃×#3151	C₂	semidirect
8. C₂₃×#3038	C₂	semidirect
9. C₂₃×#3036	C₂	semidirect
10. Q₈×M₁₆×C₂₃	C₂	semidirect
11. C₄×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
15. C₂₃×#559	C₂²	semidirect
16. C₂₃×#557	C₂²	semidirect
17. C₂₃×#556	C₂²	semidirect
23. C₂₃×#550	C₂²	semidirect
33. C₂₃×#496	C₂²	semidirect
34. C₂₃×#493	C₂²	semidirect
35. C₂₃×#492	C₂²	semidirect
36. C₂₃×#463	C₂²	semidirect
37. C₂₃×#461	C₂²	semidirect
38. C₂₃×#460	C₂²	semidirect
39. C₂₃×#460	C₂²	semidirect
40. C₂₃×#459	C₂²	semidirect
41. C₂₃×#459	C₂²	semidirect
42. C₂₃×#459	C₂²	semidirect
43. C₂₃×#446	C₂²	semidirect
44. C₈×Q₈×C₂₃	C₂²	semidirect
45. C₂₃×#442	C₂²	semidirect
46. C₂₃×#440	C₂²	semidirect
47. C₄×SD₁₆×C₂₃	C₂²	semidirect
48. C₄×D₈×C₂₃	C₂²	semidirect
165. M₁₆×C₂₃	#41	semidirect
193. #61402	C₂₃	direct (N × H)
203. Q₈×C₂₃	AGL(1,ℤ/8)	semidirect
237. C₈×C₂₃	C₂×#41	semidirect
238. #3480	C₂×C₂₃	semidirect
239. C₂×#550	C₂×C₂₃	semidirect
240. #3399	C₂×C₂₃	semidirect
241. #3397	C₂×C₂₃	semidirect
242. #3392	C₂×C₂₃	semidirect
243. #3391	C₂×C₂₃	semidirect
244. #3151	C₂×C₂₃	semidirect
245. #3038	C₂×C₂₃	semidirect
246. #3036	C₂×C₂₃	semidirect
247. Q₈×M₁₆	C₂×C₂₃	semidirect
248. C₄×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
274. #559	C₂²×C₂₃	semidirect
275. #557	C₂²×C₂₃	semidirect
276. #556	C₂²×C₂₃	semidirect
282. #550	C₂²×C₂₃	semidirect
292. #496	C₂²×C₂₃	semidirect
293. #493	C₂²×C₂₃	semidirect
294. #492	C₂²×C₂₃	semidirect
295. #463	C₂²×C₂₃	semidirect
296. #461	C₂²×C₂₃	semidirect
297. #460	C₂²×C₂₃	semidirect
298. #460	C₂²×C₂₃	semidirect
299. #459	C₂²×C₂₃	semidirect
300. #459	C₂²×C₂₃	semidirect
301. #459	C₂²×C₂₃	semidirect
302. #446	C₂²×C₂₃	semidirect
303. C₈×Q₈	C₂²×C₂₃	semidirect
304. #442	C₂²×C₂₃	semidirect
305. #440	C₂²×C₂₃	semidirect
306. C₄×SD₁₆	C₂²×C₂₃	semidirect
307. C₄×D₈	C₂²×C₂₃	semidirect
391. C₂₃	#61402	direct (N × H)
431. M₁₆	#41×C₂₃	semidirect
468. Q₈	C₂₃×AGL(1,ℤ/8)	semidirect
502. C₈	C₂×#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3480	2
C₂×C₂₃×#550	2
C₂₃×#3399	2
C₂₃×#3397	2
C₂₃×#3392	2
C₂₃×#3391	2
C₂₃×#3151	2
C₂₃×#3038	2
C₂₃×#3036	2
Q₈×M₁₆×C₂₃	2
C₄×C₂₃×AGL(1,ℤ/8)	2
#61402	23

Subgroups — C23×#61402 #19209028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61402 #19209028