Subgroups — C₂₃×#34127 #19181753

Order 5888.

2698 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2355	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2352	C₂	1	4	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2347	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2343	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2339	C₂	1	2	yes	maximal
8	C₂₃×#2295	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2290	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2283	C₂	1	2	yes	maximal, complemented
12	C₄²×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
13	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
16	C₄×#41×C₂₃	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂₃×#117	C₂²	1	2	yes	—
25	C₂×C₂₃×#117	C₂²	1	2	yes	—
26	C₂×C₂₃×#117	C₂²	1	2	yes	—
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
29	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
30	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#391	C₂²	1	4	yes	—
33	C₂₃×#389	C₂²	1	4	yes	—
34	C₂₃×#388	C₂²	1	8	yes	—
35	C₂₃×#387	C₂²	1	8	yes	—
36	C₂₃×#387	C₂²	1	8	yes	—
37	C₂₃×#386	C₂²	1	8	yes	—
38	C₂₃×#385	C₂²	1	8	yes	—
39	C₂₃×#384	C₂²	1	2	yes	—
40	C₂₃×#383	C₂²	1	4	yes	—
41	C₂₃×#382	C₂²	1	4	yes	—
42	C₂₃×#382	C₂²	1	2	yes	—
43	C₂₃×#381	C₂²	1	4	yes	—
44	C₂₃×#380	C₂²	1	4	yes	—
45	C₂₃×#380	C₂²	1	2	yes	—
46	C₂₃×#380	C₂²	1	2	yes	—
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	4	yes	—
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
56	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
58	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
60	C₂₃×#376	C₂²	1	2	yes	—
61	C₂₃×#376	C₂²	1	4	yes	—
62	C₂×C₂₃×#95	C₂²	1	4	yes	—
63	C₄³×C₂₃	C₂²	1	2	yes	—
64	C₂×#41×C₂₃	—	2	8	no	—
65	C₂×#41×C₂₃	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
70	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₂₃	—	2	16	no	—
79	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
83	C₂₃×#117	—	2	8	no	—
84	C₂₃×#117	—	4	4	no	—
85	C₂₃×#117	C₂×C₄	1	16	yes	complemented
86	C₂₃×#117	C₂×C₄	1	8	yes	complemented
87	C₂×C₄⋊C₄×C₂₃	—	2	16	no	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄²×C₂₃	—	4	8	no	—
105	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
106	C₂×C₄²×C₂₃	C₂×C₄	1	16	yes	complemented
107	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
108	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
109	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
112	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
113	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
116	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂₃×#95	C₂³	1	2	yes	—
121	C₂₃×#95	C₂³	1	4	yes	—
122	C₂₃×#95	C₂³	1	8	yes	—
123	C₂₃×#95	C₂³	1	2	yes	—
124	C₂₃×#95	C₂³	1	4	yes	—
125	C₂₃×#95	C₂³	1	8	yes	—
126	C₂₃×#95	C₂³	1	8	yes	—
127	C₂₃×#95	C₂³	1	4	yes	—
128	C₂₃×#95	C₂³	1	8	yes	—
129	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
130	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
131	#41×C₂₃	—	2	32	no	—
132	C₂×Q₈×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
134	C₂×D₄×C₂₃	—	2	8	no	—
135	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂×D₄×C₂₃	#41	1	8	yes	—
138	C₂²×C₄×C₂₃	—	2	8	no	—
139	C₂²×C₄×C₂₃	—	2	8	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	4	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	2	2	no	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	2	no	—
146	C₂²×C₄×C₂₃	—	2	16	no	—
147	C₂²×C₄×C₂₃	—	2	16	no	—
148	C₂²×C₄×C₂₃	—	4	8	no	—
149	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
153	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
157	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
167	C₄⋊C₄×C₂₃	—	2	16	no	—
168	C₄⋊C₄×C₂₃	—	2	8	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	8	no	—
171	C₄⋊C₄×C₂₃	—	4	4	no	—
172	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
174	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
176	C₄⋊C₄×C₂₃	#41	1	8	yes	complemented
177	C₂²⋊C₄×C₂₃	—	2	16	no	—
178	C₂²⋊C₄×C₂₃	—	2	8	no	—
179	C₂²⋊C₄×C₂₃	—	4	4	no	—
180	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
182	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
183	C₄²×C₂₃	—	2	8	no	—
184	C₄²×C₂₃	—	2	8	no	—
185	C₄²×C₂₃	—	2	4	no	—
186	C₄²×C₂₃	—	2	8	no	—
187	C₄²×C₂₃	—	2	4	no	—
188	C₄²×C₂₃	—	4	8	no	—
189	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
190	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
191	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
192	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
193	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
194	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
195	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
196	#34127	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₂₃	—	2	4	no	—
198	C₂³×C₂₃	—	2	2	no	—
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	2	2	no	—
202	C₂³×C₂₃	—	2	1	no	—
203	C₂³×C₂₃	—	2	1	no	—
204	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
205	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
206	C₂³×C₂₃	C₂×#41	1	8	yes	—
207	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	8	no	—
209	D₄×C₂₃	—	2	16	no	—
210	D₄×C₂₃	#117	1	16	yes	complemented
211	C₂×C₄×C₂₃	—	2	16	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	2	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	4	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	8	no	—
223	C₂×C₄×C₂₃	—	2	16	no	—
224	C₂×C₄×C₂₃	—	2	8	no	—
225	C₂×C₄×C₂₃	—	2	16	no	—
226	C₂×C₄×C₂₃	—	2	8	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	8	no	—
230	C₂×C₄×C₂₃	—	2	8	no	—
231	C₂×C₄×C₂₃	—	2	8	no	—
232	C₂×C₄×C₂₃	—	2	4	no	—
233	C₂×C₄×C₂₃	—	4	32	no	—
234	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
239	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
240	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
244	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
245	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
246	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
248	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic
249	#2355	C₂×C₂₃	1	2	yes	complemented
250	#2352	C₂×C₂₃	1	4	yes	complemented
251	D₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
252	#2347	C₂×C₂₃	1	2	yes	complemented
253	#2343	C₂×C₂₃	1	4	yes	complemented
254	#2339	C₂×C₂₃	1	2	yes	—
255	#2295	C₂×C₂₃	1	2	yes	complemented
256	#2290	C₂×C₂₃	1	4	yes	complemented
257	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#2283	C₂×C₂₃	1	2	yes	complemented
259	C₄²×D₄	C₂×C₂₃	1	2	yes	complemented
260	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
261	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
262	C₂²×C₂₃	—	2	4	no	—
263	C₂²×C₂₃	—	2	4	no	—
264	C₂²×C₂₃	—	2	8	no	—
265	C₂²×C₂₃	—	2	8	no	—
266	C₂²×C₂₃	—	2	2	no	—
267	C₂²×C₂₃	—	2	2	no	—
268	C₂²×C₂₃	—	2	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	C₂×#117	1	8	yes	—
272	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
273	C₂²×C₂₃	C₂²×#41	1	2	yes	—
274	C₂²×C₂₃	C₂²×#41	1	2	yes	—
275	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
277	C₄×C₂₃	—	2	2	no	—
278	C₄×C₂₃	—	2	4	no	—
279	C₄×C₂₃	—	2	2	no	—
280	C₄×C₂₃	—	2	4	no	—
281	C₄×C₂₃	—	2	4	no	—
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	8	no	—
284	C₄×C₂₃	—	4	8	no	—
285	C₄×C₂₃	—	4	4	no	—
286	C₄×C₂₃	C₂×#117	1	4	yes	—
287	C₄×C₂₃	C₄×#41	1	4	yes	—
288	C₄×C₂₃	#520	1	4	yes	—
289	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
290	C₄×#41	C₄×C₂₃	1	16	yes	complemented
291	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
292	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
294	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
297	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
298	C₂×#117	C₂²×C₂₃	1	2	yes	—
299	C₂×#117	C₂²×C₂₃	1	2	yes	—
300	C₂×#117	C₂²×C₂₃	1	2	yes	—
301	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
303	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
304	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
305	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
306	#391	C₂²×C₂₃	1	4	yes	—
307	#389	C₂²×C₂₃	1	4	yes	—
308	#388	C₂²×C₂₃	1	8	yes	—
309	#387	C₂²×C₂₃	1	8	yes	—
310	#387	C₂²×C₂₃	1	8	yes	—
311	#386	C₂²×C₂₃	1	8	yes	—
312	#385	C₂²×C₂₃	1	8	yes	—
313	#384	C₂²×C₂₃	1	2	yes	—
314	#383	C₂²×C₂₃	1	4	yes	—
315	#382	C₂²×C₂₃	1	2	yes	—
316	#382	C₂²×C₂₃	1	4	yes	—
317	#381	C₂²×C₂₃	1	4	yes	—
318	#380	C₂²×C₂₃	1	2	yes	—
319	#380	C₂²×C₂₃	1	4	yes	—
320	#380	C₂²×C₂₃	1	2	yes	—
321	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
322	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
323	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
325	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
326	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
327	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
328	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
329	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
330	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
331	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
332	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
334	#376	C₂²×C₂₃	1	2	yes	—
335	#376	C₂²×C₂₃	1	4	yes	—
336	C₂×#95	C₂²×C₂₃	1	4	yes	—
337	C₄³	C₂²×C₂₃	1	2	yes	—
338	C₂×C₂₃	—	2	8	no	—
339	C₂×C₂₃	—	4	2	no	—
340	C₂×C₂₃	C₂²×#117	1	1	yes	characteristic
341	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
342	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
343	C₂×C₂₃	#3481	1	2	yes	—
344	C₂×C₂₃	#3482	1	2	yes	—
345	C₂×#41	—	2	8	no	—
346	C₂×#41	C₂×C₄×C₂₃	1	8	yes	—
347	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
348	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
349	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
351	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
352	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
355	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
357	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
358	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
359	C₄×D₄	—	2	16	no	—
360	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
361	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
362	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
363	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
364	#117	—	2	8	no	—
365	#117	—	4	4	no	—
366	#117	C₂×C₄×C₂₃	1	16	yes	complemented
367	#117	C₂×C₄×C₂₃	1	8	yes	complemented
368	C₂×C₄⋊C₄	—	2	16	no	—
369	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
371	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
385	C₂×C₄²	—	4	8	no	—
386	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
387	C₂×C₄²	C₂×C₄×C₂₃	1	16	yes	complemented
388	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
389	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
390	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
391	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
392	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
393	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
394	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
396	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
398	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
399	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
401	#95	C₂³×C₂₃	1	2	yes	—
402	#95	C₂³×C₂₃	1	4	yes	—
403	#95	C₂³×C₂₃	1	8	yes	—
404	#95	C₂³×C₂₃	1	2	yes	—
405	#95	C₂³×C₂₃	1	4	yes	—
406	#95	C₂³×C₂₃	1	8	yes	—
407	#95	C₂³×C₂₃	1	8	yes	—
408	#95	C₂³×C₂₃	1	4	yes	—
409	#95	C₂³×C₂₃	1	8	yes	—
410	C₂₃	#34127	1	1	yes	characteristic, direct product factor, Sylow
411	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
412	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
413	#41	—	2	32	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
418	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
419	C₂×D₄	#41×C₂₃	1	8	yes	—
420	C₂²×C₄	—	2	8	no	—
421	C₂²×C₄	—	2	8	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	16	no	—
429	C₂²×C₄	—	2	16	no	—
430	C₂²×C₄	—	4	8	no	—
431	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
432	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
433	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
434	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
435	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
439	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
449	C₄⋊C₄	—	2	16	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	4	4	no	—
454	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
455	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
456	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
457	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
458	C₄⋊C₄	#41×C₂₃	1	8	yes	complemented
459	C₂²⋊C₄	—	2	16	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
463	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
464	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
465	C₄²	—	2	8	no	—
466	C₄²	—	2	8	no	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	8	no	—
469	C₄²	—	2	4	no	—
470	C₄²	—	4	8	no	—
471	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
472	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
473	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
474	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
475	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
476	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
477	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
486	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
487	C₂³	C₂×#41×C₂₃	1	8	yes	—
488	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
489	Q₈	—	2	8	no	—
490	D₄	—	2	16	no	—
491	D₄	C₂₃×#117	1	16	yes	complemented
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	16	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	16	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	16	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	4	32	no	—
515	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
518	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
520	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
521	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
523	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
524	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
525	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
526	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
527	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
528	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	8	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	C₂×C₂₃×#117	1	8	yes	—
539	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
540	C₂²	C₂²×#41×C₂₃	1	2	yes	—
541	C₂²	C₂²×#41×C₂₃	1	2	yes	—
542	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
543	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	8	no	—
551	C₄	—	4	8	no	—
552	C₄	—	4	4	no	—
553	C₄	C₂×C₂₃×#117	1	4	yes	—
554	C₄	C₄×#41×C₂₃	1	4	yes	—
555	C₄	C₂₃×#520	1	4	yes	—
556	C₂	—	2	8	no	—
557	C₂	—	4	2	no	—
558	C₂	C₂²×C₂₃×#117	1	1	yes	characteristic
559	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
560	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
561	C₂	C₂₃×#3481	1	2	yes	—
562	C₂	C₂₃×#3482	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2355	C₂	semidirect
3. C₂₃×#2352	C₂	semidirect
4. D₄×C₄⋊C₄×C₂₃	C₂	semidirect
5. C₂₃×#2347	C₂	semidirect
6. C₂₃×#2343	C₂	semidirect
8. C₂₃×#2295	C₂	semidirect
9. C₂₃×#2290	C₂	semidirect
10. C₂₃×#2285	C₂	semidirect
11. C₂₃×#2283	C₂	semidirect
12. C₄²×D₄×C₂₃	C₂	semidirect
13. C₄×C₂₃×#117	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₂₃	C₂	semidirect
16. C₄×#41×C₂₃	C₄	semidirect
76. C₄×Q₈×C₂₃	C₂×C₄	semidirect
77. C₄×Q₈×C₂₃	C₂×C₄	semidirect
81. C₄×D₄×C₂₃	C₂×C₄	semidirect
82. C₄×D₄×C₂₃	C₂×C₄	semidirect
85. C₂₃×#117	C₂×C₄	semidirect
86. C₂₃×#117	C₂×C₄	semidirect
105. C₂×C₄²×C₂₃	C₂×C₄	semidirect
106. C₂×C₄²×C₂₃	C₂×C₄	semidirect
176. C₄⋊C₄×C₂₃	#41	semidirect
196. #34127	C₂₃	direct (N × H)
210. D₄×C₂₃	#117	semidirect
249. #2355	C₂×C₂₃	semidirect
250. #2352	C₂×C₂₃	semidirect
251. D₄×C₄⋊C₄	C₂×C₂₃	semidirect
252. #2347	C₂×C₂₃	semidirect
253. #2343	C₂×C₂₃	semidirect
255. #2295	C₂×C₂₃	semidirect
256. #2290	C₂×C₂₃	semidirect
257. #2285	C₂×C₂₃	semidirect
258. #2283	C₂×C₂₃	semidirect
259. C₄²×D₄	C₂×C₂₃	semidirect
260. C₄×#117	C₂×C₂₃	semidirect
261. C₂×C₄×C₄⋊C₄	C₂×C₂₃	semidirect
290. C₄×#41	C₄×C₂₃	semidirect
357. C₄×Q₈	C₂×C₄×C₂₃	semidirect
358. C₄×Q₈	C₂×C₄×C₂₃	semidirect
362. C₄×D₄	C₂×C₄×C₂₃	semidirect
363. C₄×D₄	C₂×C₄×C₂₃	semidirect
366. #117	C₂×C₄×C₂₃	semidirect
367. #117	C₂×C₄×C₂₃	semidirect
386. C₂×C₄²	C₂×C₄×C₂₃	semidirect
387. C₂×C₄²	C₂×C₄×C₂₃	semidirect
410. C₂₃	#34127	direct (N × H)
458. C₄⋊C₄	#41×C₂₃	semidirect
491. D₄	C₂₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₂₃	2
C₂₃×#2355	2
C₂₃×#2352	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#2347	2
C₂₃×#2343	2
C₂₃×#2339	2
C₂₃×#2295	2
C₂₃×#2290	2
C₂₃×#2285	2
C₂₃×#2283	2
C₄²×D₄×C₂₃	2
C₄×C₂₃×#117	2
C₂×C₄×C₄⋊C₄×C₂₃	2
#34127	23

Subgroups — C23×#34127 #19181753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34127 #19181753