Subgroups — C₂₃×#22686 #19170312

Order 5888.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#444	C₂	1	2	yes	maximal
2	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2906	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1946	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1866	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1806	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	C₂²	1	4	yes	—
11	C₄×#41×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#444	—	2	8	no	—
13	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
14	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
15	C₄×SD₁₆×C₂₃	C₄	1	2	yes	complemented
16	C₄×SD₁₆×C₂₃	C₄	1	2	yes	complemented
17	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
18	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₂₃×#108	C₂²	1	2	yes	—
20	C₂×C₂₃×#108	C₂²	1	2	yes	—
21	C₂₃×#427	C₂²	1	1	yes	characteristic
22	C₂₃×#424	—	2	4	no	—
23	C₂₃×#424	C₂²	1	4	yes	—
24	C₂₃×#423	—	2	2	no	—
25	C₂₃×#423	C₂²	1	2	yes	—
26	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
27	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#104	C₂²	1	4	yes	—
29	C₂₃×#416	C₂²	1	2	yes	—
30	C₂×C₂₃×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#339	C₂²	1	4	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂×#41×C₂₃	C₂³	1	2	yes	—
36	C₂₃×#135	D₄	1	4	yes	—
37	C₂₃×#135	Q₈	1	4	yes	—
38	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
40	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂₃×#131	—	2	8	no	—
42	C₂₃×#131	—	4	4	no	—
43	C₂×M₁₆×C₂₃	—	2	4	no	—
44	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
45	C₂²×C₈×C₂₃	—	2	4	no	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
51	C₄×D₄×C₂₃	—	2	2	no	—
52	C₄×D₄×C₂₃	—	2	2	no	—
53	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
55	C₂₃×#117	—	2	2	no	—
56	C₂₃×#117	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₄²×C₂₃	—	2	2	no	—
59	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
61	C₂₃×#108	—	2	4	no	—
62	C₂₃×#108	—	4	2	no	—
63	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
64	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
65	C₂₃×#106	C₂×C₄	1	2	yes	—
66	C₂₃×C₄⋊C₈	—	2	4	no	—
67	C₂₃×C₄⋊C₈	—	2	4	no	—
68	C₂₃×C₄⋊C₈	—	2	2	no	—
69	C₂₃×C₄⋊C₈	—	2	2	no	—
70	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
71	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
72	C₂₃×#104	—	2	8	no	—
73	C₂₃×#103	C₂×C₄	1	2	yes	—
74	C₂₃×#103	C₂×C₄	1	2	yes	—
75	C₂₃×#102	C₂×C₄	1	2	yes	—
76	C₂₃×#102	C₂×C₄	1	2	yes	—
77	C₂₃×#97	—	2	4	no	—
78	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
81	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#95	—	2	1	no	—
83	#41×C₂₃	—	2	2	no	—
84	#41×C₂₃	—	2	2	no	—
85	#41×C₂₃	C₂×D₄	1	4	yes	—
86	#41×C₂₃	C₂×Q₈	1	4	yes	—
87	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₂₃	—	2	2	no	—
89	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₂₃	—	2	2	no	—
91	C₂²×C₄×C₂₃	—	2	2	no	—
92	C₂²×C₄×C₂₃	—	2	2	no	—
93	C₂²×C₄×C₂₃	—	2	2	no	—
94	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
95	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	Q₁₆×C₂₃	C₄⋊C₄	1	4	yes	complemented
98	SD₁₆×C₂₃	C₄⋊C₄	1	8	yes	complemented
99	D₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
100	M₁₆×C₂₃	—	2	4	no	—
101	M₁₆×C₂₃	—	2	4	no	—
102	M₁₆×C₂₃	—	4	8	no	—
103	C₂×C₈×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	—	2	2	no	—
106	C₂×C₈×C₂₃	—	2	1	no	—
107	C₂×C₈×C₂₃	—	2	4	no	—
108	C₂×C₈×C₂₃	—	4	8	no	—
109	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
114	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	4	2	no	—
119	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
121	C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂²⋊C₄×C₂₃	—	2	2	no	—
123	C₂²⋊C₄×C₂₃	—	2	2	no	—
124	C₄²×C₂₃	—	2	2	no	—
125	C₄²×C₂₃	—	2	2	no	—
126	C₄²×C₂₃	—	2	2	no	—
127	C₄²×C₂₃	—	2	2	no	—
128	C₄²×C₂₃	—	2	2	no	—
129	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
133	#22686	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₂₃	—	2	2	no	—
135	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
136	Q₈×C₂₃	—	2	2	no	—
137	Q₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
138	D₄×C₂₃	—	2	4	no	—
139	D₄×C₂₃	—	2	4	no	—
140	D₄×C₂₃	—	2	2	no	—
141	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
142	C₂×C₄×C₂₃	—	2	2	no	—
143	C₂×C₄×C₂₃	—	2	4	no	—
144	C₂×C₄×C₂₃	—	2	4	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	4	1	no	—
149	C₂×C₄×C₂₃	—	4	4	no	—
150	C₂×C₄×C₂₃	—	4	2	no	—
151	C₂×C₄×C₂₃	—	4	2	no	—
152	C₂×C₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	8	2	no	—
154	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
164	C₈×C₂₃	—	2	1	no	—
165	C₈×C₂₃	—	2	1	no	—
166	C₈×C₂₃	—	2	4	no	—
167	C₈×C₂₃	—	4	4	no	—
168	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
169	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
170	C₂×#444	C₂×C₂₃	1	2	yes	—
171	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
172	#2906	C₂×C₂₃	1	2	yes	complemented
173	#1946	C₂×C₂₃	1	2	yes	complemented
174	#1866	C₂×C₂₃	1	4	yes	complemented
175	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
176	#1806	C₂×C₂₃	1	2	yes	complemented
177	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
178	C₂²×C₂₃	—	2	1	no	—
179	C₂²×C₂₃	—	2	1	no	—
180	C₂²×C₂₃	—	2	4	no	—
181	C₂²×C₂₃	—	4	2	no	—
182	C₂²×C₂₃	—	4	2	no	—
183	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₂²×C₂₃	#547	1	1	yes	characteristic
185	C₂²×C₂₃	#554	1	1	yes	characteristic
186	C₄×C₂₃	—	2	4	no	—
187	C₄×C₂₃	—	2	1	no	—
188	C₄×C₂₃	—	2	1	no	—
189	C₄×C₂₃	—	2	1	no	—
190	C₄×C₂₃	—	4	2	no	—
191	C₄×C₂₃	—	4	2	no	—
192	C₄×C₂₃	—	8	2	no	—
193	C₄×C₂₃	#444	1	2	yes	complemented
194	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
195	C₄×C₂₃	#520	1	1	yes	characteristic
196	C₄×C₂₃	D₄²	1	1	yes	characteristic
197	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
198	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
199	C₂×#131	C₂²×C₂₃	1	4	yes	—
200	C₄×#41	C₂²×C₂₃	1	2	yes	—
201	#444	—	2	8	no	—
202	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
203	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
204	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
205	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
206	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
207	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
208	C₂×#108	C₂²×C₂₃	1	2	yes	—
209	C₂×#108	C₂²×C₂₃	1	2	yes	—
210	#427	C₂²×C₂₃	1	1	yes	characteristic
211	#424	—	2	4	no	—
212	#424	C₂²×C₂₃	1	4	yes	—
213	#423	—	2	2	no	—
214	#423	C₂²×C₂₃	1	2	yes	—
215	C₂×C₄⋊C₈	—	2	2	no	—
216	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×#104	C₂²×C₂₃	1	4	yes	—
218	#416	C₂²×C₂₃	1	2	yes	—
219	C₂×#97	C₂²×C₂₃	1	2	yes	—
220	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
221	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
222	#339	C₂²×C₂₃	1	4	yes	—
223	#336	C₂²×C₂₃	1	2	yes	—
224	C₂×C₂₃	—	2	1	no	—
225	C₂×C₂₃	—	2	1	no	—
226	C₂×C₂₃	—	4	4	no	—
227	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
229	C₂×C₂₃	#2957	1	1	yes	characteristic
230	C₂×#41	C₂³×C₂₃	1	2	yes	—
231	#135	D₄×C₂₃	1	4	yes	—
232	#135	Q₈×C₂₃	1	4	yes	—
233	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
234	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
235	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
236	#131	—	2	8	no	—
237	#131	—	4	4	no	—
238	C₂×M₁₆	—	2	4	no	—
239	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
240	C₂²×C₈	—	2	4	no	—
241	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
242	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
243	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
244	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
245	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
246	C₄×D₄	—	2	2	no	—
247	C₄×D₄	—	2	2	no	—
248	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
249	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
250	#117	—	2	2	no	—
251	#117	C₂³×C₂₃	1	2	yes	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
255	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
256	#108	—	2	4	no	—
257	#108	—	4	2	no	—
258	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
259	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
260	#106	C₂×C₄×C₂₃	1	2	yes	—
261	C₄⋊C₈	—	2	4	no	—
262	C₄⋊C₈	—	2	4	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
266	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
267	#104	—	2	8	no	—
268	#103	C₂×C₄×C₂₃	1	2	yes	—
269	#103	C₂×C₄×C₂₃	1	2	yes	—
270	#102	C₂×C₄×C₂₃	1	2	yes	—
271	#102	C₂×C₄×C₂₃	1	2	yes	—
272	#97	—	2	4	no	—
273	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
274	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
275	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
276	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
277	#95	—	2	1	no	—
278	C₂₃	#22686	1	1	yes	characteristic, direct product factor, Sylow
279	#41	—	2	2	no	—
280	#41	—	2	2	no	—
281	#41	C₂×D₄×C₂₃	1	4	yes	—
282	#41	C₂×Q₈×C₂₃	1	4	yes	—
283	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
291	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
292	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
293	Q₁₆	C₄⋊C₄×C₂₃	1	4	yes	complemented
294	SD₁₆	C₄⋊C₄×C₂₃	1	8	yes	complemented
295	D₈	C₄⋊C₄×C₂₃	1	4	yes	complemented
296	M₁₆	—	2	4	no	—
297	M₁₆	—	2	4	no	—
298	M₁₆	—	4	8	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	4	8	no	—
305	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
306	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
307	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
308	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
310	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
311	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
316	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
326	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
327	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
329	C₂³	—	2	2	no	—
330	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
331	Q₈	—	2	2	no	—
332	Q₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
333	D₄	—	2	4	no	—
334	D₄	—	2	4	no	—
335	D₄	—	2	2	no	—
336	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	8	2	no	—
349	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
358	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	4	no	—
362	C₈	—	4	4	no	—
363	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
364	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	4	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
371	C₂²	C₂₃×#547	1	1	yes	characteristic
372	C₂²	C₂₃×#554	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	8	2	no	—
380	C₄	C₂₃×#444	1	2	yes	complemented
381	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
382	C₄	C₂₃×#520	1	1	yes	characteristic
383	C₄	D₄²×C₂₃	1	1	yes	characteristic
384	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	4	4	no	—
388	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
389	C₂	C₂×C₂₃×#444	1	1	yes	characteristic
390	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2906	C₂	semidirect
4. C₂₃×#1946	C₂	semidirect
5. C₂₃×#1866	C₂	semidirect
6. C₂₃×#1848	C₂	semidirect
7. C₂₃×#1806	C₂	semidirect
8. C₂₃×#1770	C₂	semidirect
13. C₄×Q₁₆×C₂₃	C₄	semidirect
14. C₄×Q₁₆×C₂₃	C₄	semidirect
15. C₄×SD₁₆×C₂₃	C₄	semidirect
16. C₄×SD₁₆×C₂₃	C₄	semidirect
17. C₄×D₈×C₂₃	C₄	semidirect
18. C₄×D₈×C₂₃	C₄	semidirect
97. Q₁₆×C₂₃	C₄⋊C₄	semidirect
98. SD₁₆×C₂₃	C₄⋊C₄	semidirect
99. D₈×C₂₃	C₄⋊C₄	semidirect
133. #22686	C₂₃	direct (N × H)
172. #2906	C₂×C₂₃	semidirect
173. #1946	C₂×C₂₃	semidirect
174. #1866	C₂×C₂₃	semidirect
175. #1848	C₂×C₂₃	semidirect
176. #1806	C₂×C₂₃	semidirect
177. #1770	C₂×C₂₃	semidirect
193. C₄×C₂₃	#444	semidirect
202. C₄×Q₁₆	C₄×C₂₃	semidirect
203. C₄×Q₁₆	C₄×C₂₃	semidirect
204. C₄×SD₁₆	C₄×C₂₃	semidirect
205. C₄×SD₁₆	C₄×C₂₃	semidirect
206. C₄×D₈	C₄×C₂₃	semidirect
207. C₄×D₈	C₄×C₂₃	semidirect
278. C₂₃	#22686	direct (N × H)
293. Q₁₆	C₄⋊C₄×C₂₃	semidirect
294. SD₁₆	C₄⋊C₄×C₂₃	semidirect
295. D₈	C₄⋊C₄×C₂₃	semidirect
380. C₄	C₂₃×#444	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#444	2
C₄×C₂₃×#135	2
C₂₃×#2906	2
C₂₃×#1946	2
C₂₃×#1866	2
C₂₃×#1848	2
C₂₃×#1806	2
C₂₃×#1770	2
#22686	23

Subgroups — C23×#22686 #19170312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22686 #19170312