Subgroups — C₂₃×#22455 #19170081

Order 5888.

1378 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#576	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2948	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2883	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1976	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1904	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1795	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1757	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
10	C₂₃×#576	—	2	4	no	—
11	C₂₃×#576	—	2	4	no	—
12	C₂×C₂₃×AGL(1,ℤ/8)	C₄	1	4	yes	complemented
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂×C₂₃×#135	C₄	1	2	yes	complemented
15	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₂₃	C₄	1	2	yes	complemented
17	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
19	C₂₃×#518	C₂²	1	4	yes	complemented
20	C₂₃×#442	—	2	4	no	—
21	C₄×D₈×C₂₃	—	2	4	no	—
22	C₂₃×#427	C₂²	1	2	yes	complemented
23	C₂₃×#421	C₂²	1	4	yes	complemented
24	C₂₃×#418	C₂²	1	4	yes	complemented
25	C₂₃×#413	C₂²	1	1	yes	characteristic
26	C₂₃×#410	C₂²	1	2	yes	—
27	C₂₃×#410	C₂²	1	1	yes	characteristic
28	C₂₃×#410	C₂²	1	4	yes	complemented
29	C₂×C₂₃×#99	—	2	8	no	—
30	C₂₃×#405	C₂²	1	2	yes	complemented
31	C₂₃×#344	C₂²	1	2	yes	complemented
32	C₂₃×H(3,ℤ/4)	C₂²	1	2	yes	complemented
33	C₂₃×H(5,𝔽₂)	—	2	8	no	—
34	C₂₃×H(5,𝔽₂)	—	4	4	no	—
35	C₂×#41×C₂₃	—	2	2	no	—
36	C₂×#41×C₂₃	C₂×C₄	1	4	yes	complemented
37	C₂×#41×C₂₃	C₂³	1	2	yes	—
38	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₂₃	—	2	4	no	—
40	C₂²×D₄×C₂₃	—	2	2	no	—
41	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
42	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂₃×AGL(1,ℤ/8)	—	2	8	no	—
44	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
45	C₂₃×#135	—	2	4	no	—
46	C₂₃×#135	—	2	4	no	—
47	C₂₃×#135	—	2	2	no	—
48	C₂₃×#135	—	2	2	no	—
49	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
50	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
51	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
52	C₂×D₈×C₂₃	—	2	4	no	—
53	C₂×D₈×C₂₃	—	2	4	no	—
54	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
57	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	complemented
58	C₂×D₈×C₂₃	D₄	1	4	yes	—
59	C₂×D₈×C₂₃	D₄	1	4	yes	—
60	C₂₃×#131	—	2	2	no	—
61	C₂₃×#131	—	2	2	no	—
62	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
63	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
64	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	complemented
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₂₃	—	2	8	no	—
68	C₄×D₄×C₂₃	—	4	4	no	—
69	C₂₃×#117	C₂³	1	4	yes	—
70	C₂₃×#117	C₂³	1	2	yes	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
72	C₂₃×#107	—	2	2	no	—
73	C₂₃×#106	—	2	2	no	—
74	C₂₃×#104	—	4	4	no	—
75	C₂₃×#102	—	2	8	no	—
76	C₂₃×#101	—	2	1	no	—
77	C₂₃×#100	—	2	1	no	—
78	C₂₃×#99	—	2	4	no	—
79	C₂₃×#99	—	2	8	no	—
80	C₂₃×#99	D₄	1	4	yes	—
81	C₂₃×#97	—	2	2	no	—
82	C₄×C₈×C₂₃	—	2	2	no	—
83	C₂⁴×C₂₃	—	2	2	no	—
84	C₂⁴×C₂₃	—	2	4	no	—
85	#41×C₂₃	—	2	2	no	—
86	#41×C₂₃	—	2	2	no	—
87	#41×C₂₃	—	2	8	no	—
88	#41×C₂₃	—	2	4	no	—
89	#41×C₂₃	—	2	4	no	—
90	#41×C₂₃	—	4	4	no	—
91	#41×C₂₃	—	4	8	no	—
92	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×Q₈×C₂₃	#41	1	1	yes	characteristic
94	C₂×D₄×C₂₃	—	2	8	no	—
95	C₂×D₄×C₂₃	—	2	4	no	—
96	C₂×D₄×C₂₃	—	2	2	no	—
97	C₂×D₄×C₂₃	—	2	4	no	—
98	C₂×D₄×C₂₃	—	2	2	no	—
99	C₂×D₄×C₂₃	—	2	4	no	—
100	C₂×D₄×C₂₃	—	4	4	no	—
101	C₂×D₄×C₂₃	—	4	8	no	—
102	C₂×D₄×C₂₃	—	4	16	no	—
103	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
105	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂×D₄×C₂₃	C₂×D₄	1	8	yes	—
107	C₂×D₄×C₂₃	#41	1	1	yes	characteristic
108	C₂²×C₄×C₂₃	—	2	4	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	4	8	no	—
111	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₂₃	—	2	1	no	—
114	Q₁₆×C₂₃	—	2	1	no	—
115	SD₁₆×C₂₃	—	2	8	no	—
116	SD₁₆×C₂₃	—	2	4	no	—
117	D₈×C₂₃	—	2	8	no	—
118	D₈×C₂₃	—	2	1	no	—
119	D₈×C₂₃	—	2	1	no	—
120	D₈×C₂₃	C₂²⋊C₄	1	4	yes	—
121	D₈×C₂₃	C₂²⋊C₄	1	4	yes	—
122	D₈×C₂₃	C₂²⋊C₄	1	8	yes	—
123	M₁₆×C₂₃	—	2	2	no	—
124	M₁₆×C₂₃	—	2	4	no	—
125	M₁₆×C₂₃	—	2	2	no	—
126	M₁₆×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	2	2	no	—
128	C₂×C₈×C₂₃	—	2	2	no	—
129	C₂×C₈×C₂₃	—	2	2	no	—
130	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	2	4	no	—
137	C₂²⋊C₄×C₂₃	—	2	4	no	—
138	C₂²⋊C₄×C₂₃	—	4	8	no	—
139	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
140	C₄²×C₂₃	—	2	2	no	—
141	C₄²×C₂₃	—	2	4	no	—
142	#22455	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₂₃	—	2	4	no	—
144	C₂³×C₂₃	—	2	8	no	—
145	C₂³×C₂₃	—	2	2	no	—
146	C₂³×C₂₃	—	4	8	no	—
147	C₂³×C₂₃	—	16	2	no	—
148	C₂³×C₂₃	—	16	4	no	—
149	C₂³×C₂₃	C₄×D₄	1	2	yes	—
150	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₂₃	—	2	2	no	—
152	Q₈×C₂₃	—	2	2	no	—
153	Q₈×C₂₃	—	2	2	no	—
154	D₄×C₂₃	—	2	4	no	—
155	D₄×C₂₃	—	2	2	no	—
156	D₄×C₂₃	—	2	4	no	—
157	D₄×C₂₃	—	2	2	no	—
158	D₄×C₂₃	—	2	8	no	—
159	D₄×C₂₃	—	2	2	no	—
160	D₄×C₂₃	—	4	8	no	—
161	D₄×C₂₃	—	4	8	no	—
162	D₄×C₂₃	—	4	4	no	—
163	D₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	4	no	—
168	C₂×C₄×C₂₃	—	4	8	no	—
169	C₂×C₄×C₂₃	—	4	4	no	—
170	C₂×C₄×C₂₃	—	4	4	no	—
171	C₂×C₄×C₂₃	—	16	8	no	—
172	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
173	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
177	C₈×C₂₃	—	2	2	no	—
178	C₈×C₂₃	—	4	2	no	—
179	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
180	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₈×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×#576	C₂×C₂₃	1	1	yes	characteristic
183	#2948	C₂×C₂₃	1	2	yes	complemented
184	#2883	C₂×C₂₃	1	2	yes	complemented
185	#1976	C₂×C₂₃	1	2	yes	complemented
186	#1904	C₂×C₂₃	1	4	yes	complemented
187	#1812	C₂×C₂₃	1	1	yes	characteristic, complemented
188	#1795	C₂×C₂₃	1	2	yes	complemented
189	#1757	C₂×C₂₃	1	1	yes	characteristic, complemented
190	C₂²×C₂₃	—	2	2	no	—
191	C₂²×C₂₃	—	2	8	no	—
192	C₂²×C₂₃	—	4	1	no	—
193	C₂²×C₂₃	—	4	2	no	—
194	C₂²×C₂₃	—	4	2	no	—
195	C₂²×C₂₃	—	8	8	no	—
196	C₂²×C₂₃	—	8	4	no	—
197	C₂²×C₂₃	—	8	2	no	—
198	C₂²×C₂₃	—	16	8	no	—
199	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂²×C₂₃	D₄²	1	2	yes	—
201	C₄×C₂₃	—	2	2	no	—
202	C₄×C₂₃	—	4	4	no	—
203	C₄×C₂₃	—	4	2	no	—
204	C₄×C₂₃	—	8	4	no	—
205	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
206	C₄×C₂₃	#518	1	1	yes	characteristic
207	C₄×C₂₃	#546	1	2	yes	—
208	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
209	#576	—	2	4	no	—
210	#576	—	2	4	no	—
211	C₂×AGL(1,ℤ/8)	C₄×C₂₃	1	4	yes	complemented
212	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
213	C₂×#135	C₄×C₂₃	1	2	yes	complemented
214	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
215	C₂²×D₈	C₄×C₂₃	1	2	yes	complemented
216	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
218	#518	C₂²×C₂₃	1	4	yes	complemented
219	#442	—	2	4	no	—
220	C₄×D₈	—	2	4	no	—
221	#427	C₂²×C₂₃	1	2	yes	complemented
222	#421	C₂²×C₂₃	1	4	yes	complemented
223	#418	C₂²×C₂₃	1	4	yes	complemented
224	#413	C₂²×C₂₃	1	1	yes	characteristic
225	#410	C₂²×C₂₃	1	2	yes	—
226	#410	C₂²×C₂₃	1	1	yes	characteristic
227	#410	C₂²×C₂₃	1	4	yes	complemented
228	C₂×#99	—	2	8	no	—
229	#405	C₂²×C₂₃	1	2	yes	complemented
230	#344	C₂²×C₂₃	1	2	yes	complemented
231	H(3,ℤ/4)	C₂²×C₂₃	1	2	yes	complemented
232	C₂×C₂₃	—	2	2	no	—
233	C₂×C₂₃	—	2	1	no	—
234	C₂×C₂₃	—	4	2	no	—
235	C₂×C₂₃	—	4	8	no	—
236	C₂×C₂₃	—	8	2	no	—
237	C₂×C₂₃	D₄×C₂²⋊C₄	1	1	yes	characteristic
238	H(5,𝔽₂)	—	2	8	no	—
239	H(5,𝔽₂)	—	4	4	no	—
240	C₂×#41	—	2	2	no	—
241	C₂×#41	C₂×C₄×C₂₃	1	4	yes	complemented
242	C₂×#41	C₂³×C₂₃	1	2	yes	—
243	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
244	C₂²×D₄	—	2	4	no	—
245	C₂²×D₄	—	2	2	no	—
246	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
247	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
248	AGL(1,ℤ/8)	—	2	8	no	—
249	AGL(1,ℤ/8)	—	2	16	no	—
250	#135	—	2	4	no	—
251	#135	—	2	4	no	—
252	#135	—	2	2	no	—
253	#135	—	2	2	no	—
254	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
255	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
256	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
257	C₂×D₈	—	2	4	no	—
258	C₂×D₈	—	2	4	no	—
259	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
260	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
261	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
262	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	complemented
263	C₂×D₈	D₄×C₂₃	1	4	yes	—
264	C₂×D₈	D₄×C₂₃	1	4	yes	—
265	#131	—	2	2	no	—
266	#131	—	2	2	no	—
267	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
268	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
269	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
270	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	complemented
271	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
272	C₄×D₄	—	2	8	no	—
273	C₄×D₄	—	4	4	no	—
274	#117	C₂³×C₂₃	1	4	yes	—
275	#117	C₂³×C₂₃	1	2	yes	—
276	C₂×C₂²⋊C₄	—	2	8	no	—
277	#107	—	2	2	no	—
278	#106	—	2	2	no	—
279	#104	—	4	4	no	—
280	#102	—	2	8	no	—
281	#101	—	2	1	no	—
282	#100	—	2	1	no	—
283	#99	—	2	4	no	—
284	#99	—	2	8	no	—
285	#99	D₄×C₂₃	1	4	yes	—
286	#97	—	2	2	no	—
287	C₄×C₈	—	2	2	no	—
288	C₂₃	#22455	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	—	2	2	no	—
290	C₂⁴	—	2	4	no	—
291	#41	—	2	2	no	—
292	#41	—	2	2	no	—
293	#41	—	2	8	no	—
294	#41	—	2	4	no	—
295	#41	—	2	4	no	—
296	#41	—	4	4	no	—
297	#41	—	4	8	no	—
298	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
299	C₂×Q₈	#41×C₂₃	1	1	yes	characteristic
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	4	4	no	—
307	C₂×D₄	—	4	8	no	—
308	C₂×D₄	—	4	16	no	—
309	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
310	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
311	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
312	C₂×D₄	C₂×D₄×C₂₃	1	8	yes	—
313	C₂×D₄	#41×C₂₃	1	1	yes	characteristic
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	8	no	—
317	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
319	Q₁₆	—	2	1	no	—
320	Q₁₆	—	2	1	no	—
321	SD₁₆	—	2	8	no	—
322	SD₁₆	—	2	4	no	—
323	D₈	—	2	8	no	—
324	D₈	—	2	1	no	—
325	D₈	—	2	1	no	—
326	D₈	C₂²⋊C₄×C₂₃	1	4	yes	—
327	D₈	C₂²⋊C₄×C₂₃	1	4	yes	—
328	D₈	C₂²⋊C₄×C₂₃	1	8	yes	—
329	M₁₆	—	2	2	no	—
330	M₁₆	—	2	4	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
338	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
340	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	4	8	no	—
345	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	4	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	16	2	no	—
353	C₂³	—	16	4	no	—
354	C₂³	C₄×D₄×C₂₃	1	2	yes	—
355	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
356	Q₈	—	2	2	no	—
357	Q₈	—	2	2	no	—
358	Q₈	—	2	2	no	—
359	D₄	—	2	4	no	—
360	D₄	—	2	2	no	—
361	D₄	—	2	4	no	—
362	D₄	—	2	2	no	—
363	D₄	—	2	8	no	—
364	D₄	—	2	2	no	—
365	D₄	—	4	8	no	—
366	D₄	—	4	8	no	—
367	D₄	—	4	4	no	—
368	D₄	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	4	4	no	—
376	C₂×C₄	—	16	8	no	—
377	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
378	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
380	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
385	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	4	1	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	8	8	no	—
393	C₂²	—	8	4	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	16	8	no	—
396	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
397	C₂²	D₄²×C₂₃	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	2	no	—
401	C₄	—	8	4	no	—
402	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
403	C₄	C₂₃×#518	1	1	yes	characteristic
404	C₄	C₂₃×#546	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	1	no	—
407	C₂	—	4	2	no	—
408	C₂	—	4	8	no	—
409	C₂	—	8	2	no	—
410	C₂	D₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#576	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2948	C₂	semidirect
3. C₂₃×#2883	C₂	semidirect
4. C₂₃×#1976	C₂	semidirect
5. C₂₃×#1904	C₂	semidirect
6. C₂₃×#1812	C₂	semidirect
7. C₂₃×#1795	C₂	semidirect
8. C₂₃×#1757	C₂	semidirect
12. C₂×C₂₃×AGL(1,ℤ/8)	C₄	semidirect
14. C₂×C₂₃×#135	C₄	semidirect
16. C₂²×D₈×C₂₃	C₄	semidirect
19. C₂₃×#518	C₂²	semidirect
22. C₂₃×#427	C₂²	semidirect
23. C₂₃×#421	C₂²	semidirect
24. C₂₃×#418	C₂²	semidirect
28. C₂₃×#410	C₂²	semidirect
30. C₂₃×#405	C₂²	semidirect
31. C₂₃×#344	C₂²	semidirect
32. C₂₃×H(3,ℤ/4)	C₂²	semidirect
36. C₂×#41×C₂₃	C₂×C₄	semidirect
41. C₂²×D₄×C₂₃	C₂×C₄	semidirect
49. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
50. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
51. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
56. C₂×D₈×C₂₃	C₂×C₄	semidirect
57. C₂×D₈×C₂₃	C₂×C₄	semidirect
62. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
65. C₂²×C₈×C₂₃	C₂×C₄	semidirect
142. #22455	C₂₃	direct (N × H)
183. #2948	C₂×C₂₃	semidirect
184. #2883	C₂×C₂₃	semidirect
185. #1976	C₂×C₂₃	semidirect
186. #1904	C₂×C₂₃	semidirect
187. #1812	C₂×C₂₃	semidirect
188. #1795	C₂×C₂₃	semidirect
189. #1757	C₂×C₂₃	semidirect
211. C₂×AGL(1,ℤ/8)	C₄×C₂₃	semidirect
213. C₂×#135	C₄×C₂₃	semidirect
215. C₂²×D₈	C₄×C₂₃	semidirect
218. #518	C₂²×C₂₃	semidirect
221. #427	C₂²×C₂₃	semidirect
222. #421	C₂²×C₂₃	semidirect
223. #418	C₂²×C₂₃	semidirect
227. #410	C₂²×C₂₃	semidirect
229. #405	C₂²×C₂₃	semidirect
230. #344	C₂²×C₂₃	semidirect
231. H(3,ℤ/4)	C₂²×C₂₃	semidirect
241. C₂×#41	C₂×C₄×C₂₃	semidirect
246. C₂²×D₄	C₂×C₄×C₂₃	semidirect
254. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
255. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
256. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
261. C₂×D₈	C₂×C₄×C₂₃	semidirect
262. C₂×D₈	C₂×C₄×C₂₃	semidirect
267. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
270. C₂²×C₈	C₂×C₄×C₂₃	semidirect
288. C₂₃	#22455	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×#576	2
C₂₃×#2948	2
C₂₃×#2883	2
C₂₃×#1976	2
C₂₃×#1904	2
C₂₃×#1812	2
C₂₃×#1795	2
C₂₃×#1757	2
#22455	23

Subgroups — C23×#22455 #19170081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22455 #19170081