Subgroups — C₂₃×#61888 #19209514

Order 5888.

1970 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3397	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3367	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3213	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3206	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3112	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3100	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
11	C₂₃×#556	—	2	4	no	—
12	C₂₃×#556	C₂²	1	8	yes	—
13	C₂₃×#550	C₂²	1	4	yes	complemented
14	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
15	C₂₃×#548	—	2	4	no	—
16	C₂₃×#537	—	2	1	no	—
17	C₂₃×#537	—	2	2	no	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#532	C₂²	1	4	yes	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#126	—	2	2	no	—
22	C₂×C₂₃×#124	C₂²	1	2	yes	—
23	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#117	—	2	1	no	—
28	C₂₃×#496	C₂²	1	8	yes	—
29	C₂₃×#492	C₂²	1	4	yes	complemented
30	C₂₃×#489	C₂²	1	8	yes	—
31	C₂₃×#487	C₂²	1	4	yes	complemented
32	C₂₃×#480	C₂²	1	4	yes	—
33	C₂₃×#477	C₂²	1	8	yes	complemented
34	C₂₃×#474	C₂²	1	4	yes	—
35	C₂₃×#463	C₂²	1	16	yes	—
36	C₂₃×#461	C₂²	1	8	yes	complemented
37	C₂₃×#460	C₂²	1	8	yes	complemented
38	C₂₃×#459	C₂²	1	4	yes	—
39	C₂₃×#448	C₂²	1	8	yes	—
40	C₂₃×#447	C₂²	1	4	yes	—
41	C₂₃×#446	C₂²	1	8	yes	complemented
42	C₂₃×#440	C₂²	1	16	yes	complemented
43	C₄×D₈×C₂₃	C₂²	1	8	yes	—
44	C₂₃×#432	C₂²	1	4	yes	—
45	C₂₃×#431	C₂²	1	2	yes	—
46	C₂₃×2₋¹⁺⁴	—	4	8	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	2	no	—
50	C₂²×Q₈×C₂₃	—	2	4	no	—
51	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₂₃	—	2	1	no	—
55	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
56	C₂×D₈×C₂₃	C₂³	1	8	yes	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
59	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
61	C₂₃×#127	C₂³	1	2	yes	—
62	C₂₃×#127	C₂³	1	4	yes	—
63	C₂₃×#126	—	2	16	no	—
64	C₂₃×#124	—	2	2	no	—
65	C₂₃×#124	—	2	4	no	—
66	C₂₃×#124	—	2	4	no	—
67	C₂₃×#124	C₂³	1	8	yes	—
68	C₂₃×C₂²⋊Q₈	—	2	8	no	—
69	C₂₃×C₂²⋊Q₈	—	2	4	no	—
70	C₂₃×C₂²⋊Q₈	—	2	8	no	—
71	C₂₃×#121	—	2	4	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#121	—	2	4	no	—
74	C₂₃×#121	—	2	8	no	—
75	C₂₃×#121	C₂³	1	4	yes	—
76	C₄×Q₈×C₂₃	—	2	4	no	—
77	C₄×Q₈×C₂₃	D₄	1	4	yes	—
78	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
79	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	4	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	C₂³	1	4	yes	—
84	C₄×D₄×C₂₃	C₂³	1	8	yes	—
85	C₄×D₄×C₂₃	C₂³	1	4	yes	—
86	C₂₃×#117	—	2	8	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₄²×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
99	C₂₃×#107	C₂³	1	8	yes	—
100	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
101	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
102	C₂₃×#103	C₂³	1	16	yes	—
103	C₂₃×#102	C₂³	1	8	yes	—
104	C₂₃×#102	C₂³	1	8	yes	—
105	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
106	C₂₃×#97	C₂³	1	8	yes	—
107	C₄×C₈×C₂₃	C₂³	1	4	yes	—
108	C₂⁴×C₂₃	—	2	1	no	—
109	#41×C₂₃	—	4	16	no	—
110	#41×C₂₃	—	4	16	no	—
111	#41×C₂₃	—	4	8	no	—
112	C₂×Q₈×C₂₃	—	2	4	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	4	4	no	—
117	C₂×Q₈×C₂₃	—	4	16	no	—
118	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
119	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
120	C₂×D₄×C₂₃	—	2	4	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	2	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	2	no	—
126	C₂×D₄×C₂₃	—	4	2	no	—
127	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
128	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	8	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	4	4	no	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₂₃	—	4	16	no	—
141	D₈×C₂₃	—	4	8	no	—
142	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
143	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	1	no	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
151	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
152	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	4	no	—
155	C₂²⋊C₄×C₂₃	—	2	8	no	—
156	C₂²⋊C₄×C₂₃	—	2	4	no	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	4	no	—
159	C₄²×C₂₃	—	2	8	no	—
160	C₄²×C₂₃	C₂×D₄	1	2	yes	—
161	C₄²×C₂₃	C₂×D₄	1	4	yes	—
162	C₄²×C₂₃	C₂⁴	1	2	yes	—
163	C₄²×C₂₃	C₂⁴	1	4	yes	—
164	#61888	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₂₃	—	2	2	no	—
166	C₂³×C₂₃	—	2	4	no	—
167	C₂³×C₂₃	—	4	2	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	—	4	2	no	—
170	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₂₃	—	2	8	no	—
172	Q₈×C₂₃	—	2	8	no	—
173	Q₈×C₂₃	—	4	4	no	—
174	Q₈×C₂₃	—	4	8	no	—
175	Q₈×C₂₃	—	4	4	no	—
176	D₄×C₂₃	—	2	4	no	—
177	D₄×C₂₃	—	2	8	no	—
178	D₄×C₂₃	—	4	2	no	—
179	D₄×C₂₃	—	4	8	no	—
180	D₄×C₂₃	—	4	8	no	—
181	D₄×C₂₃	—	4	4	no	—
182	D₄×C₂₃	—	8	4	no	—
183	D₄×C₂₃	—	8	2	no	—
184	C₂×C₄×C₂₃	—	2	4	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	16	no	—
192	C₂×C₄×C₂₃	—	4	4	no	—
193	C₂×C₄×C₂₃	—	4	2	no	—
194	C₂×C₄×C₂₃	—	4	8	no	—
195	C₂×C₄×C₂₃	—	4	8	no	—
196	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
201	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
203	C₈×C₂₃	—	2	8	no	—
204	#3497	C₂×C₂₃	1	1	yes	characteristic, complemented
205	C₂×#556	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#3397	C₂×C₂₃	1	8	yes	complemented
207	#3367	C₂×C₂₃	1	8	yes	complemented
208	#3213	C₂×C₂₃	1	4	yes	complemented
209	#3206	C₂×C₂₃	1	2	yes	complemented
210	#3112	C₂×C₂₃	1	4	yes	complemented
211	#3100	C₂×C₂₃	1	2	yes	complemented
212	#2983	C₂×C₂₃	1	1	yes	characteristic, complemented
213	C₂²×C₂₃	—	2	1	no	—
214	C₂²×C₂₃	—	2	1	no	—
215	C₂²×C₂₃	—	4	4	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	—	4	2	no	—
218	C₂²×C₂₃	—	4	4	no	—
219	C₂²×C₂₃	—	4	4	no	—
220	C₂²×C₂₃	—	4	2	no	—
221	C₂²×C₂₃	—	8	4	no	—
222	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
223	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
224	C₄×C₂₃	—	2	4	no	—
225	C₄×C₂₃	—	2	2	no	—
226	C₄×C₂₃	—	2	1	no	—
227	C₄×C₂₃	—	2	4	no	—
228	C₄×C₂₃	—	4	4	no	—
229	C₄×C₂₃	—	4	2	no	—
230	C₄×C₂₃	—	4	4	no	—
231	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
232	C₄×C₂₃	#585	1	2	yes	—
233	C₂×2₋¹⁺⁴	—	2	1	no	—
234	#556	—	2	4	no	—
235	#556	C₂²×C₂₃	1	8	yes	—
236	#550	C₂²×C₂₃	1	4	yes	complemented
237	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
238	#548	—	2	4	no	—
239	#537	—	2	1	no	—
240	#537	—	2	2	no	—
241	#532	C₂²×C₂₃	1	2	yes	—
242	#532	C₂²×C₂₃	1	4	yes	—
243	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×#126	—	2	2	no	—
245	C₂×#124	C₂²×C₂₃	1	2	yes	—
246	C₂×C₂²⋊Q₈	—	2	1	no	—
247	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
248	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×#117	—	2	1	no	—
251	#496	C₂²×C₂₃	1	8	yes	—
252	#492	C₂²×C₂₃	1	4	yes	complemented
253	#489	C₂²×C₂₃	1	8	yes	—
254	#487	C₂²×C₂₃	1	4	yes	complemented
255	#480	C₂²×C₂₃	1	4	yes	—
256	#477	C₂²×C₂₃	1	8	yes	complemented
257	#474	C₂²×C₂₃	1	4	yes	—
258	#463	C₂²×C₂₃	1	16	yes	—
259	#461	C₂²×C₂₃	1	8	yes	complemented
260	#460	C₂²×C₂₃	1	8	yes	complemented
261	#459	C₂²×C₂₃	1	4	yes	—
262	#448	C₂²×C₂₃	1	8	yes	—
263	#447	C₂²×C₂₃	1	4	yes	—
264	#446	C₂²×C₂₃	1	8	yes	complemented
265	#440	C₂²×C₂₃	1	16	yes	complemented
266	C₄×D₈	C₂²×C₂₃	1	8	yes	—
267	#432	C₂²×C₂₃	1	4	yes	—
268	#431	C₂²×C₂₃	1	2	yes	—
269	C₂×C₂₃	—	2	2	no	—
270	C₂×C₂₃	—	8	4	no	—
271	C₂×C₂₃	—	8	2	no	—
272	C₂×C₂₃	#3487	1	1	yes	characteristic
273	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₂₃	#3586	1	1	yes	characteristic
275	2₋¹⁺⁴	—	4	8	no	—
276	C₂×#41	—	2	4	no	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	—	2	2	no	—
279	C₂²×Q₈	—	2	4	no	—
280	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
281	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
282	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
283	C₂³×C₄	—	2	1	no	—
284	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
285	C₂×D₈	C₂³×C₂₃	1	8	yes	—
286	C₄⋊Q₈	—	2	2	no	—
287	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
288	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
289	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
290	#127	C₂³×C₂₃	1	2	yes	—
291	#127	C₂³×C₂₃	1	4	yes	—
292	#126	—	2	16	no	—
293	#124	—	2	4	no	—
294	#124	—	2	2	no	—
295	#124	—	2	4	no	—
296	#124	C₂³×C₂₃	1	8	yes	—
297	C₂²⋊Q₈	—	2	8	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	—	2	8	no	—
300	#121	—	2	4	no	—
301	#121	—	2	2	no	—
302	#121	—	2	4	no	—
303	#121	—	2	8	no	—
304	#121	C₂³×C₂₃	1	4	yes	—
305	C₄×Q₈	—	2	4	no	—
306	C₄×Q₈	D₄×C₂₃	1	4	yes	—
307	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
308	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	C₄×D₄	—	2	4	no	—
312	C₄×D₄	C₂³×C₂₃	1	4	yes	—
313	C₄×D₄	C₂³×C₂₃	1	4	yes	—
314	C₄×D₄	C₂³×C₂₃	1	8	yes	—
315	#117	—	2	8	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₄²	—	2	2	no	—
326	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
328	#107	C₂³×C₂₃	1	8	yes	—
329	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
330	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
331	#103	C₂³×C₂₃	1	16	yes	—
332	#102	C₂³×C₂₃	1	8	yes	—
333	#102	C₂³×C₂₃	1	8	yes	—
334	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
335	#97	C₂³×C₂₃	1	8	yes	—
336	C₄×C₈	C₂³×C₂₃	1	4	yes	—
337	C₂₃	#61888	1	1	yes	characteristic, direct product factor, Sylow
338	C₂⁴	—	2	1	no	—
339	#41	—	4	16	no	—
340	#41	—	4	16	no	—
341	#41	—	4	8	no	—
342	C₂×Q₈	—	2	4	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	4	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×Q₈	—	4	16	no	—
348	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
349	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
358	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	8	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
370	SD₁₆	—	4	16	no	—
371	D₈	—	4	8	no	—
372	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
380	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
381	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
382	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
383	C₂²⋊C₄	—	2	8	no	—
384	C₂²⋊C₄	—	2	4	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	—	2	4	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₄²	—	2	8	no	—
390	C₄²	C₂×D₄×C₂₃	1	2	yes	—
391	C₄²	C₂×D₄×C₂₃	1	4	yes	—
392	C₄²	C₂⁴×C₂₃	1	2	yes	—
393	C₄²	C₂⁴×C₂₃	1	4	yes	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	4	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
400	Q₈	—	2	8	no	—
401	Q₈	—	2	8	no	—
402	Q₈	—	4	8	no	—
403	Q₈	—	4	4	no	—
404	Q₈	—	4	4	no	—
405	D₄	—	2	4	no	—
406	D₄	—	2	8	no	—
407	D₄	—	4	2	no	—
408	D₄	—	4	8	no	—
409	D₄	—	4	8	no	—
410	D₄	—	4	4	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	2	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	16	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
427	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
430	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
431	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
432	C₈	—	2	8	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	2	1	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	8	4	no	—
442	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
443	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
444	C₄	—	2	1	no	—
445	C₄	—	2	4	no	—
446	C₄	—	2	4	no	—
447	C₄	—	2	2	no	—
448	C₄	—	4	4	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	2	no	—
451	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
452	C₄	C₂₃×#585	1	2	yes	—
453	C₂	—	2	2	no	—
454	C₂	—	8	2	no	—
455	C₂	—	8	4	no	—
456	C₂	C₂₃×#3487	1	1	yes	characteristic
457	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
458	C₂	C₂₃×#3586	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×#3497	C₂	semidirect
2. C₂×C₂₃×#556	C₂	semidirect
3. C₂₃×#3397	C₂	semidirect
4. C₂₃×#3367	C₂	semidirect
5. C₂₃×#3213	C₂	semidirect
6. C₂₃×#3206	C₂	semidirect
7. C₂₃×#3112	C₂	semidirect
8. C₂₃×#3100	C₂	semidirect
9. C₂₃×#2983	C₂	semidirect
13. C₂₃×#550	C₂²	semidirect
14. D₄×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#492	C₂²	semidirect
31. C₂₃×#487	C₂²	semidirect
33. C₂₃×#477	C₂²	semidirect
36. C₂₃×#461	C₂²	semidirect
37. C₂₃×#460	C₂²	semidirect
41. C₂₃×#446	C₂²	semidirect
42. C₂₃×#440	C₂²	semidirect
164. #61888	C₂₃	direct (N × H)
204. #3497	C₂×C₂₃	semidirect
205. C₂×#556	C₂×C₂₃	semidirect
206. #3397	C₂×C₂₃	semidirect
207. #3367	C₂×C₂₃	semidirect
208. #3213	C₂×C₂₃	semidirect
209. #3206	C₂×C₂₃	semidirect
210. #3112	C₂×C₂₃	semidirect
211. #3100	C₂×C₂₃	semidirect
212. #2983	C₂×C₂₃	semidirect
236. #550	C₂²×C₂₃	semidirect
237. D₄×Q₈	C₂²×C₂₃	semidirect
252. #492	C₂²×C₂₃	semidirect
254. #487	C₂²×C₂₃	semidirect
256. #477	C₂²×C₂₃	semidirect
259. #461	C₂²×C₂₃	semidirect
260. #460	C₂²×C₂₃	semidirect
264. #446	C₂²×C₂₃	semidirect
265. #440	C₂²×C₂₃	semidirect
337. C₂₃	#61888	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3497	2
C₂×C₂₃×#556	2
C₂₃×#3397	2
C₂₃×#3367	2
C₂₃×#3213	2
C₂₃×#3206	2
C₂₃×#3112	2
C₂₃×#3100	2
C₂₃×#2983	2
#61888	23

Subgroups — C23×#61888 #19209514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61888 #19209514