Subgroups — C₂₃×#34493 #19182119

Order 5888.

3076 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2375	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2363	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2354	C₂	1	2	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2324	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2321	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2309	C₂	1	4	yes	maximal, complemented
10	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2293	C₂	1	2	yes	maximal, complemented
12	C₂³×D₄×C₂₃	C₂²	1	4	yes	—
13	D₄²×C₂₃	C₄	1	16	yes	complemented
14	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#121	C₂²	1	4	yes	—
16	C₂×C₂₃×#120	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
19	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
20	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
21	C₂×C₂₃×#117	C₂²	1	4	yes	complemented
22	C₂×C₂₃×#117	C₂²	1	4	yes	complemented
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
24	C₂₃×#390	C₂²	1	2	yes	—
25	C₂₃×#390	C₂²	1	2	yes	—
26	C₂₃×#390	C₂²	1	4	yes	—
27	C₂₃×#390	C₂²	1	4	yes	complemented
28	C₂₃×#390	C₂²	1	2	yes	complemented
29	C₂₃×#390	C₂²	1	8	yes	complemented
30	C₂₃×#390	C₂²	1	4	yes	complemented
31	C₂₃×#390	C₂²	1	8	yes	complemented
32	C₂₃×#389	C₂²	1	2	yes	complemented
33	C₂₃×#389	C₂²	1	2	yes	complemented
34	C₂₃×#388	C₂²	1	4	yes	complemented
35	C₂₃×#388	C₂²	1	4	yes	complemented
36	C₂₃×#388	C₂²	1	8	yes	complemented
37	C₂₃×#388	C₂²	1	8	yes	complemented
38	C₂₃×#386	C₂²	1	8	yes	complemented
39	C₂₃×#386	C₂²	1	4	yes	complemented
40	C₂₃×#386	C₂²	1	8	yes	complemented
41	C₂₃×#385	C₂²	1	4	yes	complemented
42	C₂₃×#384	C₂²	1	1	yes	characteristic, complemented
43	C₂₃×#380	C₂²	1	4	yes	complemented
44	C₂₃×#379	C₂²	1	4	yes	complemented
45	C₂₃×#379	C₂²	1	8	yes	complemented
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
50	C₄³×C₂₃	C₂²	1	1	yes	characteristic, complemented
51	C₂⁵×C₂₃	C₂³	1	4	yes	—
52	C₂²×D₄×C₂₃	—	4	8	no	—
53	C₂²×D₄×C₂₃	—	4	8	no	—
54	C₂²×D₄×C₂₃	C₂×C₄	1	32	yes	complemented
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
58	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
59	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
61	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
62	C₂₃×#127	C₂×C₄	1	8	yes	—
63	C₂₃×#121	C₂×C₄	1	32	yes	complemented
64	C₂₃×#120	C₂×C₄	1	32	yes	complemented
65	C₄×D₄×C₂₃	—	4	8	no	—
66	C₄×D₄×C₂₃	—	4	8	no	—
67	C₄×D₄×C₂₃	—	4	4	no	—
68	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
69	C₂₃×#117	—	4	8	no	—
70	C₂₃×#117	—	4	8	no	—
71	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₂₃	—	4	8	no	—
79	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂₃×#95	C₂³	1	4	yes	—
99	C₂₃×#95	C₂³	1	4	yes	—
100	C₂₃×#95	C₂³	1	8	yes	—
101	C₂₃×#95	C₂³	1	4	yes	—
102	C₂⁴×C₂₃	—	4	4	no	—
103	C₂⁴×C₂₃	—	4	8	no	—
104	C₂⁴×C₂₃	—	4	8	no	—
105	C₂⁴×C₂₃	—	4	4	no	—
106	C₂⁴×C₂₃	C₂²×C₄	1	16	yes	—
107	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
108	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
109	C₂×D₄×C₂₃	—	2	8	no	—
110	C₂×D₄×C₂₃	—	2	8	no	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	4	no	—
113	C₂×D₄×C₂₃	—	4	64	no	—
114	C₂×D₄×C₂₃	—	4	2	no	—
115	C₂×D₄×C₂₃	—	4	8	no	—
116	C₂×D₄×C₂₃	—	4	8	no	—
117	C₂×D₄×C₂₃	—	4	2	no	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
119	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
120	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₂₃	C₂²×C₄	1	32	yes	—
122	C₂²×C₄×C₂₃	—	2	8	no	—
123	C₂²×C₄×C₂₃	—	2	8	no	—
124	C₂²×C₄×C₂₃	—	2	8	no	—
125	C₂²×C₄×C₂₃	—	4	4	no	—
126	C₂²×C₄×C₂₃	—	4	8	no	—
127	C₂²×C₄×C₂₃	—	4	8	no	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
129	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
136	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
138	C₄⋊C₄×C₂₃	—	2	4	no	—
139	C₄⋊C₄×C₂₃	—	2	8	no	—
140	C₄⋊C₄×C₂₃	—	2	4	no	—
141	C₄⋊C₄×C₂₃	—	4	4	no	—
142	C₄⋊C₄×C₂₃	—	4	4	no	—
143	C₄⋊C₄×C₂₃	—	4	2	no	—
144	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
145	C₂²⋊C₄×C₂₃	—	2	8	no	—
146	C₂²⋊C₄×C₂₃	—	4	8	no	—
147	C₂²⋊C₄×C₂₃	—	4	4	no	—
148	C₂²⋊C₄×C₂₃	—	4	4	no	—
149	C₂²⋊C₄×C₂₃	—	4	4	no	—
150	C₂²⋊C₄×C₂₃	—	4	4	no	—
151	C₂²⋊C₄×C₂₃	—	4	8	no	—
152	C₂²⋊C₄×C₂₃	—	4	8	no	—
153	C₂²⋊C₄×C₂₃	—	4	4	no	—
154	C₂²⋊C₄×C₂₃	—	4	8	no	—
155	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
157	C₄²×C₂₃	—	2	4	no	—
158	C₄²×C₂₃	—	2	8	no	—
159	C₄²×C₂₃	—	2	2	no	—
160	C₄²×C₂₃	—	2	4	no	—
161	C₄²×C₂₃	—	4	1	no	—
162	C₄²×C₂₃	—	4	8	no	—
163	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
164	#34493	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₂₃	—	2	4	no	—
166	C₂³×C₂₃	—	2	4	no	—
167	C₂³×C₂₃	—	2	4	no	—
168	C₂³×C₂₃	—	2	4	no	—
169	C₂³×C₂₃	—	2	8	no	—
170	C₂³×C₂₃	—	2	8	no	—
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	—	2	4	no	—
174	C₂³×C₂₃	—	2	4	no	—
175	C₂³×C₂₃	—	4	32	no	—
176	C₂³×C₂₃	—	4	16	no	—
177	C₂³×C₂₃	—	8	8	no	—
178	C₂³×C₂₃	—	8	16	no	—
179	C₂³×C₂₃	—	8	8	no	—
180	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
181	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
182	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
183	D₄×C₂₃	—	2	16	no	—
184	D₄×C₂₃	—	2	32	no	—
185	D₄×C₂₃	—	4	32	no	—
186	D₄×C₂₃	—	4	8	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	1	no	—
192	C₂×C₄×C₂₃	—	2	32	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	1	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	8	8	no	—
207	C₂×C₄×C₂₃	—	8	8	no	—
208	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
210	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
211	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
213	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
214	C₂×D₄²	C₂×C₂₃	1	1	yes	characteristic
215	#2375	C₂×C₂₃	1	4	yes	complemented
216	#2366	C₂×C₂₃	1	1	yes	characteristic, complemented
217	#2363	C₂×C₂₃	1	4	yes	complemented
218	#2354	C₂×C₂₃	1	2	yes	complemented
219	D₄×C₂²⋊C₄	C₂×C₂₃	1	4	yes	complemented
220	#2324	C₂×C₂₃	1	4	yes	complemented
221	#2321	C₂×C₂₃	1	4	yes	complemented
222	#2309	C₂×C₂₃	1	4	yes	complemented
223	C₄×#127	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#2293	C₂×C₂₃	1	2	yes	complemented
225	C₂²×C₂₃	—	2	8	no	—
226	C₂²×C₂₃	—	2	8	no	—
227	C₂²×C₂₃	—	2	16	no	—
228	C₂²×C₂₃	—	2	8	no	—
229	C₂²×C₂₃	—	2	8	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	4	4	no	—
232	C₂²×C₂₃	—	4	8	no	—
233	C₂²×C₂₃	—	4	4	no	—
234	C₂²×C₂₃	—	4	4	no	—
235	C₂²×C₂₃	—	4	4	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	8	32	no	—
238	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
239	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
240	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
241	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
242	C₂²×C₂₃	#585	1	1	yes	characteristic
243	C₄×C₂₃	—	2	8	no	—
244	C₄×C₂₃	—	2	2	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	4	4	no	—
249	C₄×C₂₃	—	4	1	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	#518	1	8	yes	—
252	C₂³×D₄	C₂²×C₂₃	1	4	yes	—
253	D₄²	C₄×C₂₃	1	16	yes	complemented
254	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×#121	C₂²×C₂₃	1	4	yes	—
256	C₂×#120	C₂²×C₂₃	1	4	yes	—
257	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
258	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
259	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
260	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
261	C₂×#117	C₂²×C₂₃	1	4	yes	complemented
262	C₂×#117	C₂²×C₂₃	1	4	yes	complemented
263	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
264	#390	C₂²×C₂₃	1	2	yes	—
265	#390	C₂²×C₂₃	1	4	yes	—
266	#390	C₂²×C₂₃	1	2	yes	—
267	#390	C₂²×C₂₃	1	4	yes	complemented
268	#390	C₂²×C₂₃	1	2	yes	complemented
269	#390	C₂²×C₂₃	1	8	yes	complemented
270	#390	C₂²×C₂₃	1	4	yes	complemented
271	#390	C₂²×C₂₃	1	8	yes	complemented
272	#389	C₂²×C₂₃	1	2	yes	complemented
273	#389	C₂²×C₂₃	1	2	yes	complemented
274	#388	C₂²×C₂₃	1	4	yes	complemented
275	#388	C₂²×C₂₃	1	8	yes	complemented
276	#388	C₂²×C₂₃	1	8	yes	complemented
277	#388	C₂²×C₂₃	1	4	yes	complemented
278	#386	C₂²×C₂₃	1	8	yes	complemented
279	#386	C₂²×C₂₃	1	4	yes	complemented
280	#386	C₂²×C₂₃	1	8	yes	complemented
281	#385	C₂²×C₂₃	1	4	yes	complemented
282	#384	C₂²×C₂₃	1	1	yes	characteristic, complemented
283	#380	C₂²×C₂₃	1	4	yes	complemented
284	#379	C₂²×C₂₃	1	4	yes	complemented
285	#379	C₂²×C₂₃	1	8	yes	complemented
286	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
287	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
288	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
289	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
290	C₄³	C₂²×C₂₃	1	1	yes	characteristic, complemented
291	C₂×C₂₃	—	4	8	no	—
292	C₂×C₂₃	—	4	8	no	—
293	C₂×C₂₃	C₂×#518	1	2	yes	—
294	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
295	C₂×C₂₃	#3543	1	1	yes	characteristic
296	C₂×C₂₃	#3546	1	2	yes	—
297	C₂×C₂₃	#3549	1	1	yes	characteristic
298	C₂⁵	C₂³×C₂₃	1	4	yes	—
299	C₂²×D₄	—	4	8	no	—
300	C₂²×D₄	—	4	8	no	—
301	C₂²×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
302	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
303	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
304	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
305	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
306	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
307	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
308	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
309	#127	C₂×C₄×C₂₃	1	8	yes	—
310	#121	C₂×C₄×C₂₃	1	32	yes	complemented
311	#120	C₂×C₄×C₂₃	1	32	yes	complemented
312	C₄×D₄	—	4	8	no	—
313	C₄×D₄	—	4	8	no	—
314	C₄×D₄	—	4	4	no	—
315	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
316	#117	—	4	8	no	—
317	#117	—	4	8	no	—
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
320	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
323	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
325	C₂×C₂²⋊C₄	—	4	8	no	—
326	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
338	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
339	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
340	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
341	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
343	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	#95	C₂³×C₂₃	1	4	yes	—
346	#95	C₂³×C₂₃	1	4	yes	—
347	#95	C₂³×C₂₃	1	8	yes	—
348	#95	C₂³×C₂₃	1	4	yes	—
349	C₂₃	#34493	1	1	yes	characteristic, direct product factor, Sylow
350	C₂⁴	—	4	4	no	—
351	C₂⁴	—	4	8	no	—
352	C₂⁴	—	4	8	no	—
353	C₂⁴	—	4	4	no	—
354	C₂⁴	C₂²×C₄×C₂₃	1	16	yes	—
355	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
356	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
357	C₂×D₄	—	2	8	no	—
358	C₂×D₄	—	2	8	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	4	64	no	—
362	C₂×D₄	—	4	2	no	—
363	C₂×D₄	—	4	8	no	—
364	C₂×D₄	—	4	8	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
367	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
368	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
369	C₂×D₄	C₂²×C₄×C₂₃	1	32	yes	—
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	8	no	—
373	C₂²×C₄	—	4	4	no	—
374	C₂²×C₄	—	4	8	no	—
375	C₂²×C₄	—	4	8	no	—
376	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
377	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
380	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
382	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	8	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₄⋊C₄	—	4	4	no	—
391	C₄⋊C₄	—	4	2	no	—
392	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	4	8	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₂²⋊C₄	—	4	8	no	—
398	C₂²⋊C₄	—	4	8	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	4	8	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
404	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
405	C₄²	—	2	4	no	—
406	C₄²	—	2	8	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	4	no	—
409	C₄²	—	4	8	no	—
410	C₄²	—	4	1	no	—
411	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	4	no	—
416	C₂³	—	2	8	no	—
417	C₂³	—	2	8	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	4	32	no	—
423	C₂³	—	4	16	no	—
424	C₂³	—	8	8	no	—
425	C₂³	—	8	16	no	—
426	C₂³	—	8	8	no	—
427	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
428	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
429	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
430	D₄	—	2	16	no	—
431	D₄	—	2	32	no	—
432	D₄	—	4	32	no	—
433	D₄	—	4	8	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	32	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	8	8	no	—
454	C₂×C₄	—	8	8	no	—
455	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
457	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
458	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
459	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
460	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
461	C₂²	—	2	8	no	—
462	C₂²	—	2	8	no	—
463	C₂²	—	2	16	no	—
464	C₂²	—	2	8	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	8	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	32	no	—
474	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
475	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
476	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
477	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂²	C₂₃×#585	1	1	yes	characteristic
479	C₄	—	2	8	no	—
480	C₄	—	2	2	no	—
481	C₄	—	2	2	no	—
482	C₄	—	4	4	no	—
483	C₄	—	4	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	1	no	—
487	C₄	C₂₃×#518	1	8	yes	—
488	C₂	—	4	8	no	—
489	C₂	—	4	8	no	—
490	C₂	C₂×C₂₃×#518	1	2	yes	—
491	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
492	C₂	C₂₃×#3543	1	1	yes	characteristic
493	C₂	C₂₃×#3546	1	2	yes	—
494	C₂	C₂₃×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×D₄²×C₂₃	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×#2375	C₂	semidirect
3. C₂₃×#2366	C₂	semidirect
4. C₂₃×#2363	C₂	semidirect
5. C₂₃×#2354	C₂	semidirect
6. D₄×C₂²⋊C₄×C₂₃	C₂	semidirect
7. C₂₃×#2324	C₂	semidirect
8. C₂₃×#2321	C₂	semidirect
9. C₂₃×#2309	C₂	semidirect
10. C₄×C₂₃×#127	C₂	semidirect
11. C₂₃×#2293	C₂	semidirect
13. D₄²×C₂₃	C₄	semidirect
18. C₂×C₄×D₄×C₂₃	C₂²	semidirect
19. C₂×C₄×D₄×C₂₃	C₂²	semidirect
20. C₂×C₄×D₄×C₂₃	C₂²	semidirect
21. C₂×C₂₃×#117	C₂²	semidirect
22. C₂×C₂₃×#117	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
27. C₂₃×#390	C₂²	semidirect
28. C₂₃×#390	C₂²	semidirect
29. C₂₃×#390	C₂²	semidirect
30. C₂₃×#390	C₂²	semidirect
31. C₂₃×#390	C₂²	semidirect
32. C₂₃×#389	C₂²	semidirect
33. C₂₃×#389	C₂²	semidirect
34. C₂₃×#388	C₂²	semidirect
35. C₂₃×#388	C₂²	semidirect
36. C₂₃×#388	C₂²	semidirect
37. C₂₃×#388	C₂²	semidirect
38. C₂₃×#386	C₂²	semidirect
39. C₂₃×#386	C₂²	semidirect
40. C₂₃×#386	C₂²	semidirect
41. C₂₃×#385	C₂²	semidirect
42. C₂₃×#384	C₂²	semidirect
43. C₂₃×#380	C₂²	semidirect
44. C₂₃×#379	C₂²	semidirect
45. C₂₃×#379	C₂²	semidirect
46. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
47. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
50. C₄³×C₂₃	C₂²	semidirect
54. C₂²×D₄×C₂₃	C₂×C₄	semidirect
63. C₂₃×#121	C₂×C₄	semidirect
64. C₂₃×#120	C₂×C₄	semidirect
68. C₄×D₄×C₂₃	C₂×C₄	semidirect
164. #34493	C₂₃	direct (N × H)
215. #2375	C₂×C₂₃	semidirect
216. #2366	C₂×C₂₃	semidirect
217. #2363	C₂×C₂₃	semidirect
218. #2354	C₂×C₂₃	semidirect
219. D₄×C₂²⋊C₄	C₂×C₂₃	semidirect
220. #2324	C₂×C₂₃	semidirect
221. #2321	C₂×C₂₃	semidirect
222. #2309	C₂×C₂₃	semidirect
223. C₄×#127	C₂×C₂₃	semidirect
224. #2293	C₂×C₂₃	semidirect
253. D₄²	C₄×C₂₃	semidirect
258. C₂×C₄×D₄	C₂²×C₂₃	semidirect
259. C₂×C₄×D₄	C₂²×C₂₃	semidirect
260. C₂×C₄×D₄	C₂²×C₂₃	semidirect
261. C₂×#117	C₂²×C₂₃	semidirect
262. C₂×#117	C₂²×C₂₃	semidirect
263. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
267. #390	C₂²×C₂₃	semidirect
268. #390	C₂²×C₂₃	semidirect
269. #390	C₂²×C₂₃	semidirect
270. #390	C₂²×C₂₃	semidirect
271. #390	C₂²×C₂₃	semidirect
272. #389	C₂²×C₂₃	semidirect
273. #389	C₂²×C₂₃	semidirect
274. #388	C₂²×C₂₃	semidirect
275. #388	C₂²×C₂₃	semidirect
276. #388	C₂²×C₂₃	semidirect
277. #388	C₂²×C₂₃	semidirect
278. #386	C₂²×C₂₃	semidirect
279. #386	C₂²×C₂₃	semidirect
280. #386	C₂²×C₂₃	semidirect
281. #385	C₂²×C₂₃	semidirect
282. #384	C₂²×C₂₃	semidirect
283. #380	C₂²×C₂₃	semidirect
284. #379	C₂²×C₂₃	semidirect
285. #379	C₂²×C₂₃	semidirect
286. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
287. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
288. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
289. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
290. C₄³	C₂²×C₂₃	semidirect
301. C₂²×D₄	C₂×C₄×C₂₃	semidirect
310. #121	C₂×C₄×C₂₃	semidirect
311. #120	C₂×C₄×C₂₃	semidirect
315. C₄×D₄	C₂×C₄×C₂₃	semidirect
349. C₂₃	#34493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₄²×C₂₃	2
C₂₃×#2375	2
C₂₃×#2366	2
C₂₃×#2363	2
C₂₃×#2354	2
D₄×C₂²⋊C₄×C₂₃	2
C₂₃×#2324	2
C₂₃×#2321	2
C₂₃×#2309	2
C₄×C₂₃×#127	2
C₂₃×#2293	2
#34493	23

Subgroups — C23×#34493 #19182119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34493 #19182119