Subgroups — C₂₃×#34207 #19181833

Order 5888.

3254 conjugacy classes of subgroups.

The subgroup lattice has 472 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	C₂×C₂₃×#520	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2361	C₂	1	8	yes	maximal, complemented
4	C₂₃×#2354	C₂	1	4	yes	maximal, complemented
5	D₄×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2326	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2311	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#127	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#532	C₄	1	16	yes	complemented
14	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#124	C₂²	1	2	yes	—
17	C₂×C₂₃×#121	C₂²	1	4	yes	—
18	C₂₃×#520	—	2	16	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₂₃×#117	C₂²	1	4	yes	—
25	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
30	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#391	C₂²	1	4	yes	—
32	C₂₃×#390	C₂²	1	4	yes	—
33	C₂₃×#390	C₂²	1	8	yes	—
34	C₂₃×#389	C₂²	1	8	yes	—
35	C₂₃×#388	C₂²	1	16	yes	—
36	C₂₃×#386	C₂²	1	8	yes	—
37	C₂₃×#385	C₂²	1	16	yes	—
38	C₂₃×#384	C₂²	1	2	yes	—
39	C₂₃×#384	C₂²	1	2	yes	—
40	C₂₃×#384	C₂²	1	4	yes	—
41	C₂₃×#380	C₂²	1	8	yes	—
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	8	yes	—
45	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
46	C₂₃×#376	C₂²	1	2	yes	—
47	C₄³×C₂₃	C₂²	1	2	yes	—
48	C₂×#41×C₂₃	—	2	8	no	—
49	C₂×#41×C₂₃	—	2	8	no	—
50	C₂×#41×C₂₃	C₂×C₄	1	16	yes	complemented
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
55	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
56	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
58	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
59	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
61	C₂₃×#127	C₂×C₄	1	8	yes	complemented
62	C₂₃×#124	C₂×C₄	1	16	yes	complemented
63	C₂₃×#121	C₂×C₄	1	32	yes	complemented
64	C₄×Q₈×C₂₃	—	2	16	no	—
65	C₄×D₄×C₂₃	—	2	32	no	—
66	C₄×D₄×C₂₃	—	2	16	no	—
67	C₄×D₄×C₂₃	—	4	16	no	—
68	C₄×D₄×C₂₃	C₂×C₄	1	32	yes	complemented
69	C₂₃×#117	—	2	4	no	—
70	C₂₃×#117	—	2	16	no	—
71	C₂₃×#117	D₄	1	8	yes	complemented
72	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
74	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
90	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
93	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
98	C₂₃×#95	C₂³	1	4	yes	—
99	C₂₃×#95	C₂³	1	4	yes	—
100	C₂₃×#95	C₂³	1	16	yes	—
101	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
102	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
103	#41×C₂₃	—	2	64	no	—
104	C₂×Q₈×C₂₃	—	2	4	no	—
105	C₂×Q₈×C₂₃	—	2	4	no	—
106	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
107	C₂×D₄×C₂₃	—	2	8	no	—
108	C₂×D₄×C₂₃	—	2	8	no	—
109	C₂×D₄×C₂₃	—	2	4	no	—
110	C₂×D₄×C₂₃	—	2	4	no	—
111	C₂×D₄×C₂₃	—	2	8	no	—
112	C₂×D₄×C₂₃	—	2	8	no	—
113	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
114	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
115	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
116	C₂²×C₄×C₂₃	—	2	8	no	—
117	C₂²×C₄×C₂₃	—	2	8	no	—
118	C₂²×C₄×C₂₃	—	2	4	no	—
119	C₂²×C₄×C₂₃	—	2	4	no	—
120	C₂²×C₄×C₂₃	—	2	8	no	—
121	C₂²×C₄×C₂₃	—	2	16	no	—
122	C₂²×C₄×C₂₃	—	4	8	no	—
123	C₂²×C₄×C₂₃	—	4	16	no	—
124	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
125	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
126	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₂₃	#41	1	4	yes	—
129	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
138	C₄⋊C₄×C₂₃	—	2	8	no	—
139	C₄⋊C₄×C₂₃	—	2	16	no	—
140	C₄⋊C₄×C₂₃	—	2	8	no	—
141	C₄⋊C₄×C₂₃	—	2	4	no	—
142	C₄⋊C₄×C₂₃	—	2	16	no	—
143	C₄⋊C₄×C₂₃	—	4	8	no	—
144	C₄⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
145	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
146	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
147	C₂²⋊C₄×C₂₃	—	2	4	no	—
148	C₂²⋊C₄×C₂₃	—	2	16	no	—
149	C₂²⋊C₄×C₂₃	—	4	16	no	—
150	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
151	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
152	C₄²×C₂₃	—	2	16	no	—
153	C₄²×C₂₃	—	2	4	no	—
154	C₄²×C₂₃	—	2	8	no	—
155	C₄²×C₂₃	—	2	8	no	—
156	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
157	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
158	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
159	C₄²×C₂₃	C₂×D₄	1	8	yes	—
160	#34207	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	—	2	2	no	—
163	C₂³×C₂₃	—	2	4	no	—
164	C₂³×C₂₃	—	2	4	no	—
165	C₂³×C₂₃	—	2	8	no	—
166	C₂³×C₂₃	—	2	8	no	—
167	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
168	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
169	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
170	C₂³×C₂₃	C₂×#41	1	2	yes	—
171	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
172	Q₈×C₂₃	—	2	16	no	—
173	D₄×C₂₃	—	2	32	no	—
174	D₄×C₂₃	—	2	16	no	—
175	D₄×C₂₃	—	2	32	no	—
176	C₂×C₄×C₂₃	—	2	4	no	—
177	C₂×C₄×C₂₃	—	2	4	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	4	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	32	no	—
183	C₂×C₄×C₂₃	—	2	16	no	—
184	C₂×C₄×C₂₃	—	2	8	no	—
185	C₂×C₄×C₂₃	—	2	4	no	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	2	16	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	8	16	no	—
190	C₂×C₄×C₂₃	—	8	8	no	—
191	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	complemented
192	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
193	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
194	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
195	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
197	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
202	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
203	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
204	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
205	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
206	C₂×#520	C₂×C₂₃	1	2	yes	complemented
207	#2361	C₂×C₂₃	1	8	yes	complemented
208	#2354	C₂×C₂₃	1	4	yes	complemented
209	D₄×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
210	#2326	C₂×C₂₃	1	4	yes	complemented
211	#2311	C₂×C₂₃	1	2	yes	complemented
212	C₄×C₄⋊Q₈	C₂×C₂₃	1	2	yes	complemented
213	C₄×#127	C₂×C₂₃	1	2	yes	complemented
214	C₂×#384	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×C₂₃	—	2	4	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	2	no	—
219	C₂²×C₂₃	—	2	4	no	—
220	C₂²×C₂₃	—	2	8	no	—
221	C₂²×C₂₃	—	2	16	no	—
222	C₂²×C₂₃	—	4	4	no	—
223	C₂²×C₂₃	—	4	4	no	—
224	C₂²×C₂₃	—	4	8	no	—
225	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
226	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
227	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
228	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
229	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
230	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
231	C₄×C₂₃	—	2	8	no	—
232	C₄×C₂₃	—	2	4	no	—
233	C₄×C₂₃	—	2	4	no	—
234	C₄×C₂₃	—	2	4	no	—
235	C₄×C₂₃	—	4	8	no	—
236	C₄×C₂₃	—	4	4	no	—
237	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
238	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
239	C₄×C₂₃	#520	1	8	yes	complemented
240	C₂²×#41	C₂²×C₂₃	1	2	yes	—
241	#532	C₄×C₂₃	1	16	yes	complemented
242	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×#124	C₂²×C₂₃	1	2	yes	—
245	C₂×#121	C₂²×C₂₃	1	4	yes	—
246	#520	—	2	16	no	—
247	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
248	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
249	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
250	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
251	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
252	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×#117	C₂²×C₂₃	1	4	yes	—
254	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
255	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
256	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
258	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
259	#391	C₂²×C₂₃	1	4	yes	—
260	#390	C₂²×C₂₃	1	4	yes	—
261	#390	C₂²×C₂₃	1	8	yes	—
262	#389	C₂²×C₂₃	1	8	yes	—
263	#388	C₂²×C₂₃	1	16	yes	—
264	#386	C₂²×C₂₃	1	8	yes	—
265	#385	C₂²×C₂₃	1	16	yes	—
266	#384	C₂²×C₂₃	1	2	yes	—
267	#384	C₂²×C₂₃	1	2	yes	—
268	#384	C₂²×C₂₃	1	4	yes	—
269	#380	C₂²×C₂₃	1	8	yes	—
270	C₄×C₄⋊C₄	C₂²×C₂₃	1	8	yes	—
271	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
272	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
273	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
274	#376	C₂²×C₂₃	1	2	yes	—
275	C₄³	C₂²×C₂₃	1	2	yes	—
276	C₂×C₂₃	—	2	4	no	—
277	C₂×C₂₃	—	4	8	no	—
278	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
279	C₂×C₂₃	C₂×#520	1	2	yes	—
280	C₂×C₂₃	#3498	1	1	yes	characteristic
281	C₂×C₂₃	#3499	1	1	yes	characteristic
282	C₂×C₂₃	#3511	1	2	yes	—
283	C₂×#41	—	2	8	no	—
284	C₂×#41	—	2	8	no	—
285	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
286	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
287	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
288	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
289	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
290	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
291	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
292	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
293	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
294	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
295	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
296	#127	C₂×C₄×C₂₃	1	8	yes	complemented
297	#124	C₂×C₄×C₂₃	1	16	yes	complemented
298	#121	C₂×C₄×C₂₃	1	32	yes	complemented
299	C₄×Q₈	—	2	16	no	—
300	C₄×D₄	—	2	32	no	—
301	C₄×D₄	—	2	16	no	—
302	C₄×D₄	—	4	16	no	—
303	C₄×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
304	#117	—	2	4	no	—
305	#117	—	2	16	no	—
306	#117	D₄×C₂₃	1	8	yes	complemented
307	C₂×C₄⋊C₄	—	2	8	no	—
308	C₂×C₄⋊C₄	—	2	16	no	—
309	C₂×C₄⋊C₄	—	4	8	no	—
310	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
311	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
312	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
313	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
314	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
320	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
324	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
325	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
326	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
327	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
328	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
329	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
330	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
333	#95	C₂³×C₂₃	1	4	yes	—
334	#95	C₂³×C₂₃	1	4	yes	—
335	#95	C₂³×C₂₃	1	16	yes	—
336	C₂₃	#34207	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
338	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
339	#41	—	2	64	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	—	2	4	no	—
342	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	2	8	no	—
348	C₂×D₄	—	2	8	no	—
349	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
350	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
351	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	2	16	no	—
358	C₂²×C₄	—	4	8	no	—
359	C₂²×C₄	—	4	16	no	—
360	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
361	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
362	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
363	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
364	C₂²×C₄	#41×C₂₃	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
367	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
369	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	4	no	—
378	C₄⋊C₄	—	2	16	no	—
379	C₄⋊C₄	—	4	8	no	—
380	C₄⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
381	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
382	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
383	C₂²⋊C₄	—	2	4	no	—
384	C₂²⋊C₄	—	2	16	no	—
385	C₂²⋊C₄	—	4	16	no	—
386	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
387	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
388	C₄²	—	2	16	no	—
389	C₄²	—	2	4	no	—
390	C₄²	—	2	8	no	—
391	C₄²	—	2	8	no	—
392	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
393	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
394	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
395	C₄²	C₂×D₄×C₂₃	1	8	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	4	no	—
399	C₂³	—	2	4	no	—
400	C₂³	—	2	8	no	—
401	C₂³	—	2	8	no	—
402	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
403	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
404	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
405	C₂³	C₂×#41×C₂₃	1	2	yes	—
406	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
407	Q₈	—	2	16	no	—
408	D₄	—	2	32	no	—
409	D₄	—	2	16	no	—
410	D₄	—	2	32	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	32	no	—
418	C₂×C₄	—	2	16	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	16	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	8	16	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	complemented
427	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
428	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
429	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
430	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
431	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
432	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
433	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
434	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
435	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
436	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
437	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
438	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
439	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
440	C₂²	—	2	4	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	2	16	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	8	no	—
449	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
450	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
451	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
452	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
453	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
454	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
455	C₄	—	2	8	no	—
456	C₄	—	2	4	no	—
457	C₄	—	2	4	no	—
458	C₄	—	2	4	no	—
459	C₄	—	4	8	no	—
460	C₄	—	4	4	no	—
461	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
462	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
463	C₄	C₂₃×#520	1	8	yes	complemented
464	C₂	—	2	4	no	—
465	C₂	—	4	8	no	—
466	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
467	C₂	C₂×C₂₃×#520	1	2	yes	—
468	C₂	C₂₃×#3498	1	1	yes	characteristic
469	C₂	C₂₃×#3499	1	1	yes	characteristic
470	C₂	C₂₃×#3511	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. C₂×C₂₃×#520	C₂	semidirect
3. C₂₃×#2361	C₂	semidirect
4. C₂₃×#2354	C₂	semidirect
5. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#2326	C₂	semidirect
7. C₂₃×#2311	C₂	semidirect
8. C₄×C₂₃×C₄⋊Q₈	C₂	semidirect
9. C₄×C₂₃×#127	C₂	semidirect
10. C₂×C₂₃×#384	C₂	semidirect
11. C₄×C₂₃×#117	C₂	semidirect
13. C₂₃×#532	C₄	semidirect
50. C₂×#41×C₂₃	C₂×C₄	semidirect
60. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
61. C₂₃×#127	C₂×C₄	semidirect
62. C₂₃×#124	C₂×C₄	semidirect
63. C₂₃×#121	C₂×C₄	semidirect
68. C₄×D₄×C₂₃	C₂×C₄	semidirect
71. C₂₃×#117	D₄	semidirect
89. C₂×C₄²×C₂₃	C₂×C₄	semidirect
160. #34207	C₂₃	direct (N × H)
191. C₂×C₄×C₂₃	C₄×D₄	semidirect
206. C₂×#520	C₂×C₂₃	semidirect
207. #2361	C₂×C₂₃	semidirect
208. #2354	C₂×C₂₃	semidirect
209. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
210. #2326	C₂×C₂₃	semidirect
211. #2311	C₂×C₂₃	semidirect
212. C₄×C₄⋊Q₈	C₂×C₂₃	semidirect
213. C₄×#127	C₂×C₂₃	semidirect
214. C₂×#384	C₂×C₂₃	semidirect
215. C₄×#117	C₂×C₂₃	semidirect
239. C₄×C₂₃	#520	semidirect
241. #532	C₄×C₂₃	semidirect
285. C₂×#41	C₂×C₄×C₂₃	semidirect
295. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
296. #127	C₂×C₄×C₂₃	semidirect
297. #124	C₂×C₄×C₂₃	semidirect
298. #121	C₂×C₄×C₂₃	semidirect
303. C₄×D₄	C₂×C₄×C₂₃	semidirect
306. #117	D₄×C₂₃	semidirect
324. C₂×C₄²	C₂×C₄×C₂₃	semidirect
336. C₂₃	#34207	direct (N × H)
426. C₂×C₄	C₄×D₄×C₂₃	semidirect
463. C₄	C₂₃×#520	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
C₂×C₂₃×#520	2
C₂₃×#2361	2
C₂₃×#2354	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2326	2
C₂₃×#2311	2
C₄×C₂₃×C₄⋊Q₈	2
C₄×C₂₃×#127	2
C₂×C₂₃×#384	2
C₄×C₂₃×#117	2
#34207	23

Subgroups — C23×#34207 #19181833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34207 #19181833