Subgroups — C₂₃×#34236 #19181862

Order 5888.

2750 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2369	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2366	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2361	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2357	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2349	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2339	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2337	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2319	C₂	1	2	yes	maximal, complemented
10	C₄×C₂₃×#121	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2275	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#532	C₄	1	16	yes	complemented
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#124	C₂²	1	2	yes	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₂₃×#121	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#391	C₂²	1	8	yes	—
30	C₂₃×#390	C₂²	1	4	yes	—
31	C₂₃×#390	C₂²	1	4	yes	—
32	C₂₃×#389	C₂²	1	4	yes	—
33	C₂₃×#389	C₂²	1	4	yes	—
34	C₂₃×#388	C₂²	1	8	yes	—
35	C₂₃×#388	C₂²	1	8	yes	—
36	C₂₃×#386	C₂²	1	8	yes	—
37	C₂₃×#386	C₂²	1	8	yes	—
38	C₂₃×#385	C₂²	1	8	yes	—
39	C₂₃×#383	C₂²	1	2	yes	—
40	C₂₃×#382	C₂²	1	1	yes	characteristic
41	C₂₃×#382	C₂²	1	1	yes	characteristic
42	C₂₃×#381	C₂²	1	2	yes	—
43	C₂₃×#381	C₂²	1	4	yes	—
44	C₂₃×#380	C₂²	1	2	yes	—
45	C₂₃×#380	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
54	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
55	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
56	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
57	C₂₃×#376	C₂²	1	2	yes	—
58	C₂×C₂₃×#95	C₂²	1	2	yes	—
59	C₂×C₂₃×#95	C₂²	1	4	yes	—
60	C₂×#41×C₂₃	—	4	8	no	—
61	C₂×#41×C₂₃	C₂×C₄	1	16	yes	complemented
62	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
72	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
73	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
74	C₂₃×#127	C₂×C₄	1	8	yes	complemented
75	C₂₃×#124	C₂×C₄	1	16	yes	complemented
76	C₂₃×#121	C₂×C₄	1	16	yes	complemented
77	C₂₃×#121	C₂×C₄	1	16	yes	complemented
78	C₄×D₄×C₂₃	—	2	16	no	—
79	C₄×D₄×C₂₃	—	4	8	no	—
80	C₄×D₄×C₂₃	—	4	16	no	—
81	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
83	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
98	C₂×C₄²×C₂₃	—	4	4	no	—
99	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
100	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
103	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#95	C₂³	1	2	yes	—
110	C₂₃×#95	C₂³	1	2	yes	—
111	C₂₃×#95	C₂³	1	8	yes	—
112	C₂₃×#95	C₂³	1	2	yes	—
113	C₂₃×#95	C₂³	1	4	yes	—
114	C₂₃×#95	C₂³	1	4	yes	—
115	C₂₃×#95	C₂³	1	2	yes	—
116	C₂₃×#95	C₂³	1	2	yes	—
117	C₂₃×#95	C₂³	1	2	yes	—
118	C₂₃×#95	C₂³	1	4	yes	—
119	C₂₃×#95	C₂³	1	8	yes	—
120	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
121	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
122	#41×C₂₃	—	4	32	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
125	C₂×D₄×C₂₃	—	2	16	no	—
126	C₂×D₄×C₂₃	—	2	8	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	4	no	—
129	C₂×D₄×C₂₃	—	4	4	no	—
130	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
131	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
132	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
133	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	8	no	—
139	C₂²×C₄×C₂₃	—	2	8	no	—
140	C₂²×C₄×C₂₃	—	2	16	no	—
141	C₂²×C₄×C₂₃	—	4	4	no	—
142	C₂²×C₄×C₂₃	—	4	8	no	—
143	C₂²×C₄×C₂₃	—	4	8	no	—
144	C₂²×C₄×C₂₃	—	4	8	no	—
145	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	4	8	no	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
167	C₄⋊C₄×C₂₃	#41	1	8	yes	complemented
168	C₂²⋊C₄×C₂₃	—	2	8	no	—
169	C₂²⋊C₄×C₂₃	—	2	16	no	—
170	C₂²⋊C₄×C₂₃	—	4	8	no	—
171	C₂²⋊C₄×C₂₃	—	4	8	no	—
172	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
173	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
174	C₄²×C₂₃	—	2	8	no	—
175	C₄²×C₂₃	—	2	8	no	—
176	C₄²×C₂₃	—	4	2	no	—
177	C₄²×C₂₃	—	4	2	no	—
178	C₄²×C₂₃	—	4	8	no	—
179	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
180	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
181	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
182	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
183	#34236	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
184	C₂³×C₂₃	—	2	2	no	—
185	C₂³×C₂₃	—	2	1	no	—
186	C₂³×C₂₃	—	2	1	no	—
187	C₂³×C₂₃	—	2	8	no	—
188	C₂³×C₂₃	—	2	4	no	—
189	C₂³×C₂₃	—	2	8	no	—
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
192	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
193	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
194	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
195	Q₈×C₂₃	—	4	8	no	—
196	D₄×C₂₃	—	2	32	no	—
197	D₄×C₂₃	—	2	16	no	—
198	D₄×C₂₃	—	2	16	no	—
199	D₄×C₂₃	—	4	8	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	8	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	8	no	—
210	C₂×C₄×C₂₃	—	2	8	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	4	no	—
220	C₂×C₄×C₂₃	—	4	16	no	—
221	C₂×C₄×C₂₃	—	4	16	no	—
222	C₂×C₄×C₂₃	—	8	16	no	—
223	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
224	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
226	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
227	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
228	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
229	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
230	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
231	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
232	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
233	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
234	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
235	#2369	C₂×C₂₃	1	2	yes	complemented
236	#2366	C₂×C₂₃	1	2	yes	complemented
237	#2361	C₂×C₂₃	1	4	yes	complemented
238	#2357	C₂×C₂₃	1	2	yes	complemented
239	#2349	C₂×C₂₃	1	2	yes	complemented
240	#2339	C₂×C₂₃	1	4	yes	complemented
241	#2337	C₂×C₂₃	1	4	yes	complemented
242	#2319	C₂×C₂₃	1	2	yes	complemented
243	C₄×#121	C₂×C₂₃	1	4	yes	complemented
244	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#2275	C₂×C₂₃	1	2	yes	complemented
246	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂²×C₂₃	—	2	2	no	—
248	C₂²×C₂₃	—	2	2	no	—
249	C₂²×C₂₃	—	2	4	no	—
250	C₂²×C₂₃	—	2	4	no	—
251	C₂²×C₂₃	—	2	8	no	—
252	C₂²×C₂₃	—	2	8	no	—
253	C₂²×C₂₃	—	2	8	no	—
254	C₂²×C₂₃	—	4	8	no	—
255	C₂²×C₂₃	—	4	8	no	—
256	C₂²×C₂₃	#518	1	4	yes	—
257	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
258	C₂²×C₂₃	C₂²×#41	1	2	yes	—
259	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
260	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	2	2	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	8	no	—
266	C₄×C₂₃	—	2	2	no	—
267	C₄×C₂₃	—	2	8	no	—
268	C₄×C₂₃	—	4	4	no	—
269	C₄×C₂₃	—	4	4	no	—
270	C₄×C₂₃	—	4	4	no	—
271	C₄×C₂₃	C₄×#41	1	4	yes	—
272	C₄×C₂₃	#520	1	4	yes	—
273	C₂²×#41	C₂²×C₂₃	1	2	yes	—
274	#532	C₄×C₂₃	1	16	yes	complemented
275	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×#124	C₂²×C₂₃	1	2	yes	—
278	C₂×#121	C₂²×C₂₃	1	2	yes	—
279	C₂×#121	C₂²×C₂₃	1	2	yes	—
280	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
281	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
282	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
283	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
284	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
285	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
286	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
287	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
288	#391	C₂²×C₂₃	1	8	yes	—
289	#390	C₂²×C₂₃	1	4	yes	—
290	#390	C₂²×C₂₃	1	4	yes	—
291	#389	C₂²×C₂₃	1	4	yes	—
292	#389	C₂²×C₂₃	1	4	yes	—
293	#388	C₂²×C₂₃	1	8	yes	—
294	#388	C₂²×C₂₃	1	8	yes	—
295	#386	C₂²×C₂₃	1	8	yes	—
296	#386	C₂²×C₂₃	1	8	yes	—
297	#385	C₂²×C₂₃	1	8	yes	—
298	#383	C₂²×C₂₃	1	2	yes	—
299	#382	C₂²×C₂₃	1	1	yes	characteristic
300	#382	C₂²×C₂₃	1	1	yes	characteristic
301	#381	C₂²×C₂₃	1	4	yes	—
302	#381	C₂²×C₂₃	1	2	yes	—
303	#380	C₂²×C₂₃	1	4	yes	—
304	#380	C₂²×C₂₃	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
306	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
308	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
309	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
310	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
312	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
314	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
315	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
316	#376	C₂²×C₂₃	1	2	yes	—
317	C₂×#95	C₂²×C₂₃	1	2	yes	—
318	C₂×#95	C₂²×C₂₃	1	4	yes	—
319	C₂×C₂₃	—	2	4	no	—
320	C₂×C₂₃	—	4	8	no	—
321	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
322	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
323	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
324	C₂×C₂₃	#3519	1	2	yes	—
325	C₂×C₂₃	#3522	1	2	yes	—
326	C₂×#41	—	4	8	no	—
327	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
328	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
329	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
330	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
331	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
332	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
333	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
334	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
335	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
336	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
338	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
339	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
340	#127	C₂×C₄×C₂₃	1	8	yes	complemented
341	#124	C₂×C₄×C₂₃	1	16	yes	complemented
342	#121	C₂×C₄×C₂₃	1	16	yes	complemented
343	#121	C₂×C₄×C₂₃	1	16	yes	complemented
344	C₄×D₄	—	2	16	no	—
345	C₄×D₄	—	4	8	no	—
346	C₄×D₄	—	4	16	no	—
347	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
348	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
349	C₂×C₄⋊C₄	—	2	8	no	—
350	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
351	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
352	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
364	C₂×C₄²	—	4	4	no	—
365	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
366	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
367	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
368	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
369	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
370	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
371	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
372	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
375	#95	C₂³×C₂₃	1	2	yes	—
376	#95	C₂³×C₂₃	1	2	yes	—
377	#95	C₂³×C₂₃	1	8	yes	—
378	#95	C₂³×C₂₃	1	2	yes	—
379	#95	C₂³×C₂₃	1	4	yes	—
380	#95	C₂³×C₂₃	1	4	yes	—
381	#95	C₂³×C₂₃	1	2	yes	—
382	#95	C₂³×C₂₃	1	2	yes	—
383	#95	C₂³×C₂₃	1	2	yes	—
384	#95	C₂³×C₂₃	1	4	yes	—
385	#95	C₂³×C₂₃	1	8	yes	—
386	C₂₃	#34236	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
388	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
389	#41	—	4	32	no	—
390	C₂×Q₈	—	4	4	no	—
391	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
392	C₂×D₄	—	2	16	no	—
393	C₂×D₄	—	2	8	no	—
394	C₂×D₄	—	2	4	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	4	4	no	—
397	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
398	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
399	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
400	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	8	no	—
406	C₂²×C₄	—	2	8	no	—
407	C₂²×C₄	—	2	16	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	—	4	8	no	—
410	C₂²×C₄	—	4	8	no	—
411	C₂²×C₄	—	4	8	no	—
412	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
413	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
414	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
415	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
416	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
417	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
418	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
419	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
420	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
422	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
425	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	4	8	no	—
432	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
433	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
434	C₄⋊C₄	#41×C₂₃	1	8	yes	complemented
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	16	no	—
437	C₂²⋊C₄	—	4	8	no	—
438	C₂²⋊C₄	—	4	8	no	—
439	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
440	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
441	C₄²	—	2	8	no	—
442	C₄²	—	2	8	no	—
443	C₄²	—	4	2	no	—
444	C₄²	—	4	2	no	—
445	C₄²	—	4	8	no	—
446	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
447	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
448	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
449	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	1	no	—
452	C₂³	—	2	1	no	—
453	C₂³	—	2	8	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	8	no	—
456	C₂³	—	2	4	no	—
457	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
458	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
459	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
460	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
461	Q₈	—	4	8	no	—
462	D₄	—	2	32	no	—
463	D₄	—	2	16	no	—
464	D₄	—	2	16	no	—
465	D₄	—	4	8	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	8	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	4	16	no	—
487	C₂×C₄	—	4	16	no	—
488	C₂×C₄	—	8	16	no	—
489	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
490	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
491	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
492	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
493	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
494	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
495	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
496	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
497	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
498	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
499	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
500	C₂²	—	2	2	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	2	4	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	8	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	4	8	no	—
508	C₂²	—	4	8	no	—
509	C₂²	C₂₃×#518	1	4	yes	—
510	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
511	C₂²	C₂²×#41×C₂₃	1	2	yes	—
512	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
513	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
514	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	8	no	—
519	C₄	—	2	2	no	—
520	C₄	—	2	8	no	—
521	C₄	—	4	4	no	—
522	C₄	—	4	4	no	—
523	C₄	—	4	4	no	—
524	C₄	C₄×#41×C₂₃	1	4	yes	—
525	C₄	C₂₃×#520	1	4	yes	—
526	C₂	—	2	4	no	—
527	C₂	—	4	8	no	—
528	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
529	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
530	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
531	C₂	C₂₃×#3519	1	2	yes	—
532	C₂	C₂₃×#3522	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2369	C₂	semidirect
3. C₂₃×#2366	C₂	semidirect
4. C₂₃×#2361	C₂	semidirect
5. C₂₃×#2357	C₂	semidirect
6. C₂₃×#2349	C₂	semidirect
7. C₂₃×#2339	C₂	semidirect
8. C₂₃×#2337	C₂	semidirect
9. C₂₃×#2319	C₂	semidirect
10. C₄×C₂₃×#121	C₂	semidirect
11. C₂₃×#2283	C₂	semidirect
12. C₂₃×#2275	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
15. C₂₃×#532	C₄	semidirect
61. C₂×#41×C₂₃	C₂×C₄	semidirect
73. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
74. C₂₃×#127	C₂×C₄	semidirect
75. C₂₃×#124	C₂×C₄	semidirect
76. C₂₃×#121	C₂×C₄	semidirect
77. C₂₃×#121	C₂×C₄	semidirect
81. C₄×D₄×C₂₃	C₂×C₄	semidirect
82. C₄×D₄×C₂₃	C₂×C₄	semidirect
99. C₂×C₄²×C₂₃	C₂×C₄	semidirect
167. C₄⋊C₄×C₂₃	#41	semidirect
183. #34236	C₂₃	direct (N × H)
235. #2369	C₂×C₂₃	semidirect
236. #2366	C₂×C₂₃	semidirect
237. #2361	C₂×C₂₃	semidirect
238. #2357	C₂×C₂₃	semidirect
239. #2349	C₂×C₂₃	semidirect
240. #2339	C₂×C₂₃	semidirect
241. #2337	C₂×C₂₃	semidirect
242. #2319	C₂×C₂₃	semidirect
243. C₄×#121	C₂×C₂₃	semidirect
244. #2283	C₂×C₂₃	semidirect
245. #2275	C₂×C₂₃	semidirect
246. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
274. #532	C₄×C₂₃	semidirect
327. C₂×#41	C₂×C₄×C₂₃	semidirect
339. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
340. #127	C₂×C₄×C₂₃	semidirect
341. #124	C₂×C₄×C₂₃	semidirect
342. #121	C₂×C₄×C₂₃	semidirect
343. #121	C₂×C₄×C₂₃	semidirect
347. C₄×D₄	C₂×C₄×C₂₃	semidirect
348. C₄×D₄	C₂×C₄×C₂₃	semidirect
365. C₂×C₄²	C₂×C₄×C₂₃	semidirect
386. C₂₃	#34236	direct (N × H)
434. C₄⋊C₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#532	2
C₂₃×#2369	2
C₂₃×#2366	2
C₂₃×#2361	2
C₂₃×#2357	2
C₂₃×#2349	2
C₂₃×#2339	2
C₂₃×#2337	2
C₂₃×#2319	2
C₄×C₂₃×#121	2
C₂₃×#2283	2
C₂₃×#2275	2
C₂×C₄×C₄⋊C₄×C₂₃	2
#34236	23

Subgroups — C23×#34236 #19181862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34236 #19181862