Subgroups — C₂₃×#22688 #19170314

Order 5888.

870 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#443	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×#1945	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	C₈×C₄⋊C₄×C₂₃	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×#443	—	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×C₄×C₈×C₂₃	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	1	yes	characteristic
71	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₂₃×#104	—	2	8	no	—
74	C₂₃×#103	C₂×C₄	1	2	yes	—
75	C₂₃×#103	C₂×C₄	1	2	yes	—
76	C₂₃×#102	C₂×C₄	1	2	yes	—
77	C₂₃×#102	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₂₃	—	2	4	no	—
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	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
83	C₂₃×#95	—	2	1	no	—
84	#41×C₂₃	—	2	2	no	—
85	#41×C₂₃	—	2	2	no	—
86	#41×C₂₃	C₂×D₄	1	4	yes	—
87	#41×C₂₃	C₂×Q₈	1	4	yes	—
88	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₂₃	—	2	2	no	—
90	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
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	—	2	2	no	—
95	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
97	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
98	Q₁₆×C₂₃	C₄⋊C₄	1	4	yes	complemented
99	SD₁₆×C₂₃	C₄⋊C₄	1	8	yes	complemented
100	D₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
101	M₁₆×C₂₃	—	2	4	no	—
102	M₁₆×C₂₃	—	2	4	no	—
103	M₁₆×C₂₃	—	4	8	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	—	2	2	no	—
107	C₂×C₈×C₂₃	—	2	1	no	—
108	C₂×C₈×C₂₃	—	2	4	no	—
109	C₂×C₈×C₂₃	—	4	8	no	—
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	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
118	C₄⋊C₄×C₂₃	—	2	4	no	—
119	C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
122	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂²⋊C₄×C₂₃	—	2	2	no	—
124	C₂²⋊C₄×C₂₃	—	2	2	no	—
125	C₂²⋊C₄×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	—	2	2	no	—
130	C₄²×C₂₃	—	2	2	no	—
131	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
135	#22688	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂³×C₂₃	—	2	2	no	—
137	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
138	Q₈×C₂₃	—	2	2	no	—
139	Q₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
140	D₄×C₂₃	—	2	4	no	—
141	D₄×C₂₃	—	2	4	no	—
142	D₄×C₂₃	—	2	2	no	—
143	D₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×C₄×C₂₃	—	2	2	no	—
145	C₂×C₄×C₂₃	—	2	4	no	—
146	C₂×C₄×C₂₃	—	2	4	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	4	no	—
149	C₂×C₄×C₂₃	—	2	2	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	4	1	no	—
152	C₂×C₄×C₂₃	—	4	2	no	—
153	C₂×C₄×C₂₃	—	4	2	no	—
154	C₂×C₄×C₂₃	—	4	4	no	—
155	C₂×C₄×C₂₃	—	8	2	no	—
156	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
166	C₈×C₂₃	—	2	1	no	—
167	C₈×C₂₃	—	2	1	no	—
168	C₈×C₂₃	—	2	2	no	—
169	C₈×C₂₃	—	2	2	no	—
170	C₈×C₂₃	—	4	4	no	—
171	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
172	C₈×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×#443	C₂×C₂₃	1	2	yes	—
174	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
175	#2906	C₂×C₂₃	1	2	yes	complemented
176	#1945	C₂×C₂₃	1	2	yes	complemented
177	#1866	C₂×C₂₃	1	4	yes	complemented
178	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
179	#1806	C₂×C₂₃	1	2	yes	complemented
180	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
181	C₂²×C₂₃	—	2	1	no	—
182	C₂²×C₂₃	—	2	1	no	—
183	C₂²×C₂₃	—	2	4	no	—
184	C₂²×C₂₃	—	4	2	no	—
185	C₂²×C₂₃	—	4	2	no	—
186	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₂₃	#547	1	1	yes	characteristic
188	C₂²×C₂₃	#554	1	1	yes	characteristic
189	C₄×C₂₃	—	2	4	no	—
190	C₄×C₂₃	—	2	1	no	—
191	C₄×C₂₃	—	2	1	no	—
192	C₄×C₂₃	—	2	1	no	—
193	C₄×C₂₃	—	4	4	no	—
194	C₄×C₂₃	—	4	2	no	—
195	C₄×C₂₃	—	4	2	no	—
196	C₄×C₂₃	#443	1	2	yes	complemented
197	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
198	C₄×C₂₃	#520	1	1	yes	characteristic
199	C₄×C₂₃	D₄²	1	1	yes	characteristic
200	C₄×C₂₃	D₄×Q₈	1	1	yes	characteristic
201	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
202	C₂×#131	C₂²×C₂₃	1	4	yes	—
203	C₄×#41	C₂²×C₂₃	1	2	yes	—
204	#443	—	2	8	no	—
205	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
206	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
207	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
208	C₄×SD₁₆	C₄×C₂₃	1	2	yes	complemented
209	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
210	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
211	C₂×#108	C₂²×C₂₃	1	2	yes	—
212	C₂×#108	C₂²×C₂₃	1	2	yes	—
213	#427	C₂²×C₂₃	1	1	yes	characteristic
214	#424	—	2	4	no	—
215	#424	C₂²×C₂₃	1	4	yes	—
216	#423	—	2	2	no	—
217	#423	C₂²×C₂₃	1	2	yes	—
218	C₂×C₄⋊C₈	—	2	2	no	—
219	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×#104	C₂²×C₂₃	1	4	yes	—
221	#416	C₂²×C₂₃	1	2	yes	—
222	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
223	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
224	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
225	#339	C₂²×C₂₃	1	4	yes	—
226	#336	C₂²×C₂₃	1	2	yes	—
227	C₂×C₂₃	—	2	1	no	—
228	C₂×C₂₃	—	2	1	no	—
229	C₂×C₂₃	—	4	4	no	—
230	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
232	C₂×C₂₃	#2957	1	1	yes	characteristic
233	C₂×#41	C₂³×C₂₃	1	2	yes	—
234	#135	D₄×C₂₃	1	4	yes	—
235	#135	Q₈×C₂₃	1	4	yes	—
236	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
237	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
238	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
239	#131	—	2	8	no	—
240	#131	—	4	4	no	—
241	C₂×M₁₆	—	2	4	no	—
242	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
243	C₂²×C₈	—	2	4	no	—
244	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
246	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
247	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
248	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
252	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
253	#117	—	2	2	no	—
254	#117	C₂³×C₂₃	1	2	yes	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
258	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
259	#108	—	2	4	no	—
260	#108	—	4	2	no	—
261	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
262	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
263	#106	C₂×C₄×C₂₃	1	2	yes	—
264	C₄⋊C₈	—	2	4	no	—
265	C₄⋊C₈	—	2	4	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
269	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
270	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
271	#104	—	2	8	no	—
272	#103	C₂×C₄×C₂₃	1	2	yes	—
273	#103	C₂×C₄×C₂₃	1	2	yes	—
274	#102	C₂×C₄×C₂₃	1	2	yes	—
275	#102	C₂×C₄×C₂₃	1	2	yes	—
276	C₄×C₈	—	2	4	no	—
277	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
278	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
279	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
280	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
281	#95	—	2	1	no	—
282	C₂₃	#22688	1	1	yes	characteristic, direct product factor, Sylow
283	#41	—	2	2	no	—
284	#41	—	2	2	no	—
285	#41	C₂×D₄×C₂₃	1	4	yes	—
286	#41	C₂×Q₈×C₂₃	1	4	yes	—
287	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	2	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
295	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
297	Q₁₆	C₄⋊C₄×C₂₃	1	4	yes	complemented
298	SD₁₆	C₄⋊C₄×C₂₃	1	8	yes	complemented
299	D₈	C₄⋊C₄×C₂₃	1	4	yes	complemented
300	M₁₆	—	2	4	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	4	8	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	4	8	no	—
309	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
310	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
311	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
313	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
314	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
315	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
316	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
321	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
331	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
333	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
336	Q₈	—	2	2	no	—
337	Q₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
338	D₄	—	2	4	no	—
339	D₄	—	2	4	no	—
340	D₄	—	2	2	no	—
341	D₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
355	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
358	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
359	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
361	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
362	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
363	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	2	2	no	—
367	C₈	—	2	2	no	—
368	C₈	—	4	4	no	—
369	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
370	C₈	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
377	C₂²	C₂₃×#547	1	1	yes	characteristic
378	C₂²	C₂₃×#554	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	2	4	no	—
383	C₄	—	4	4	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	C₂₃×#443	1	2	yes	complemented
387	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
388	C₄	C₂₃×#520	1	1	yes	characteristic
389	C₄	D₄²×C₂₃	1	1	yes	characteristic
390	C₄	D₄×Q₈×C₂₃	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	4	4	no	—
394	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
395	C₂	C₂×C₂₃×#443	1	1	yes	characteristic
396	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×#1945	C₂	semidirect
5. C₂₃×#1866	C₂	semidirect
6. C₂₃×#1848	C₂	semidirect
7. C₂₃×#1806	C₂	semidirect
8. C₈×C₄⋊C₄×C₂₃	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
98. Q₁₆×C₂₃	C₄⋊C₄	semidirect
99. SD₁₆×C₂₃	C₄⋊C₄	semidirect
100. D₈×C₂₃	C₄⋊C₄	semidirect
135. #22688	C₂₃	direct (N × H)
175. #2906	C₂×C₂₃	semidirect
176. #1945	C₂×C₂₃	semidirect
177. #1866	C₂×C₂₃	semidirect
178. #1848	C₂×C₂₃	semidirect
179. #1806	C₂×C₂₃	semidirect
180. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
196. C₄×C₂₃	#443	semidirect
205. C₄×Q₁₆	C₄×C₂₃	semidirect
206. C₄×Q₁₆	C₄×C₂₃	semidirect
207. C₄×SD₁₆	C₄×C₂₃	semidirect
208. C₄×SD₁₆	C₄×C₂₃	semidirect
209. C₄×D₈	C₄×C₂₃	semidirect
210. C₄×D₈	C₄×C₂₃	semidirect
282. C₂₃	#22688	direct (N × H)
297. Q₁₆	C₄⋊C₄×C₂₃	semidirect
298. SD₁₆	C₄⋊C₄×C₂₃	semidirect
299. D₈	C₄⋊C₄×C₂₃	semidirect
386. C₄	C₂₃×#443	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#443	2
C₄×C₂₃×#135	2
C₂₃×#2906	2
C₂₃×#1945	2
C₂₃×#1866	2
C₂₃×#1848	2
C₂₃×#1806	2
C₈×C₄⋊C₄×C₂₃	2
#22688	23

Subgroups — C23×#22688 #19170314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22688 #19170314