Subgroups — C₂₃×#34210 #19181836

Order 5888.

4006 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#518	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2366	C₂	1	8	yes	maximal, complemented
4	D₄×C₂²⋊C₄×C₂₃	C₂	1	8	yes	maximal, complemented
5	C₂₃×#2328	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2309	C₂	1	2	yes	maximal, complemented
7	C₄×C₂₃×#127	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂³×D₄×C₂₃	C₂²	1	4	yes	—
12	C₂×C₂₃×#127	C₄	1	16	yes	complemented
13	C₂×C₂₃×#127	C₂²	1	2	yes	—
14	C₂×C₂₃×#127	C₂²	1	2	yes	—
15	C₂×C₂₃×#127	C₂²	1	4	yes	—
16	C₂₃×#518	—	2	16	no	—
17	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
18	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
19	C₂×C₄×D₄×C₂₃	C₂²	1	16	yes	complemented
20	C₂×C₂₃×#117	C₂²	1	2	yes	—
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#390	C₂²	1	8	yes	complemented
26	C₂₃×#390	C₂²	1	16	yes	complemented
27	C₂₃×#390	C₂²	1	8	yes	complemented
28	C₂₃×#386	C₂²	1	32	yes	complemented
29	C₂₃×#384	C₂²	1	2	yes	complemented
30	C₂₃×#382	C₂²	1	2	yes	complemented
31	C₂₃×#380	C₂²	1	4	yes	—
32	C₂₃×#379	C₂²	1	8	yes	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
36	C₂₃×#376	C₂²	1	2	yes	complemented
37	C₄³×C₂₃	C₂²	1	2	yes	complemented
38	C₂⁵×C₂₃	C₂³	1	4	yes	—
39	C₂²×D₄×C₂₃	—	2	8	no	—
40	C₂²×D₄×C₂₃	—	2	8	no	—
41	C₂²×D₄×C₂₃	—	4	16	no	—
42	C₂²×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
43	C₂²×D₄×C₂₃	C₂×C₄	1	32	yes	complemented
44	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
45	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
46	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
47	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
48	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₂₃	C₂³	1	16	yes	—
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#127	C₂×C₄	1	16	yes	complemented
52	C₂₃×#127	C₂×C₄	1	32	yes	complemented
53	C₂₃×#127	C₂×C₄	1	16	yes	complemented
54	C₄×D₄×C₂₃	—	2	32	no	—
55	C₄×D₄×C₂₃	—	2	32	no	—
56	C₄×D₄×C₂₃	—	4	32	no	—
57	C₂₃×#117	—	2	8	no	—
58	C₂₃×#117	—	2	4	no	—
59	C₂₃×#117	D₄	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₂₃	—	2	32	no	—
65	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
67	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
68	C₂×C₂²⋊C₄×C₂₃	C₂³	1	32	yes	—
69	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
71	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
73	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
74	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#95	C₂³	1	8	yes	—
79	C₂₃×#95	C₂³	1	8	yes	—
80	C₂⁴×C₂₃	—	2	8	no	—
81	C₂⁴×C₂₃	—	2	8	no	—
82	C₂⁴×C₂₃	—	4	16	no	—
83	C₂⁴×C₂₃	C₂²×C₄	1	16	yes	—
84	C₂⁴×C₂₃	C₂⁴	1	8	yes	—
85	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₂×D₄×C₂₃	—	2	64	no	—
87	C₂×D₄×C₂₃	—	2	8	no	—
88	C₂×D₄×C₂₃	—	2	8	no	—
89	C₂×D₄×C₂₃	—	2	8	no	—
90	C₂×D₄×C₂₃	—	2	8	no	—
91	C₂×D₄×C₂₃	—	2	32	no	—
92	C₂×D₄×C₂₃	—	4	64	no	—
93	C₂×D₄×C₂₃	—	4	16	no	—
94	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
95	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
96	C₂×D₄×C₂₃	C₂²×C₄	1	32	yes	—
97	C₂×D₄×C₂₃	C₂²×C₄	1	32	yes	—
98	C₂²×C₄×C₂₃	—	2	8	no	—
99	C₂²×C₄×C₂₃	—	2	32	no	—
100	C₂²×C₄×C₂₃	—	4	32	no	—
101	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
102	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
104	C₂²×C₄×C₂₃	#41	1	4	yes	—
105	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
109	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
112	C₄⋊C₄×C₂₃	—	2	8	no	—
113	C₄⋊C₄×C₂₃	—	2	4	no	—
114	C₄⋊C₄×C₂₃	—	2	16	no	—
115	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	complemented
116	C₂²⋊C₄×C₂₃	—	2	8	no	—
117	C₂²⋊C₄×C₂₃	—	2	4	no	—
118	C₂²⋊C₄×C₂₃	—	2	32	no	—
119	C₂²⋊C₄×C₂₃	—	4	32	no	—
120	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
121	C₄²×C₂₃	—	2	8	no	—
122	C₄²×C₂₃	—	2	4	no	—
123	C₄²×C₂₃	—	2	16	no	—
124	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
125	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
126	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
127	C₄²×C₂₃	C₂×D₄	1	8	yes	complemented
128	#34210	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	2	4	no	—
130	C₂³×C₂₃	—	2	8	no	—
131	C₂³×C₂₃	—	2	8	no	—
132	C₂³×C₂₃	—	2	32	no	—
133	C₂³×C₂₃	—	2	32	no	—
134	C₂³×C₂₃	—	4	32	no	—
135	C₂³×C₂₃	—	8	16	no	—
136	C₂³×C₂₃	—	8	16	no	—
137	C₂³×C₂₃	C₂³×C₄	1	16	yes	—
138	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
139	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
140	C₂³×C₂₃	C₂×#41	1	2	yes	—
141	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
142	D₄×C₂₃	—	2	32	no	—
143	D₄×C₂₃	—	2	32	no	—
144	D₄×C₂₃	—	2	64	no	—
145	D₄×C₂₃	—	4	32	no	—
146	C₂×C₄×C₂₃	—	2	8	no	—
147	C₂×C₄×C₂₃	—	2	4	no	—
148	C₂×C₄×C₂₃	—	2	16	no	—
149	C₂×C₄×C₂₃	—	2	8	no	—
150	C₂×C₄×C₂₃	—	2	4	no	—
151	C₂×C₄×C₂₃	—	2	8	no	—
152	C₂×C₄×C₂₃	—	2	16	no	—
153	C₂×C₄×C₂₃	—	8	32	no	—
154	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	complemented
155	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
156	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
157	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
158	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
159	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
160	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
161	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
163	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
164	C₂×C₄×C₂₃	H(5,𝔽₂)	1	8	yes	—
165	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic
166	C₂×#518	C₂×C₂₃	1	2	yes	complemented
167	#2366	C₂×C₂₃	1	8	yes	complemented
168	D₄×C₂²⋊C₄	C₂×C₂₃	1	8	yes	complemented
169	#2328	C₂×C₂₃	1	4	yes	complemented
170	#2309	C₂×C₂₃	1	2	yes	complemented
171	C₄×#127	C₂×C₂₃	1	4	yes	complemented
172	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
173	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
174	C₂²×C₂₃	—	2	4	no	—
175	C₂²×C₂₃	—	2	4	no	—
176	C₂²×C₂₃	—	2	4	no	—
177	C₂²×C₂₃	—	2	16	no	—
178	C₂²×C₂₃	—	2	32	no	—
179	C₂²×C₂₃	—	4	16	no	—
180	C₂²×C₂₃	—	4	16	no	—
181	C₂²×C₂₃	—	8	32	no	—
182	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
183	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
184	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
185	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
186	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	4	yes	—
187	C₄×C₂₃	—	2	4	no	—
188	C₄×C₂₃	—	2	8	no	—
189	C₄×C₂₃	—	4	8	no	—
190	C₄×C₂₃	—	4	4	no	—
191	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
192	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
193	C₄×C₂₃	#518	1	8	yes	complemented
194	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
195	C₂³×D₄	C₂²×C₂₃	1	4	yes	—
196	C₂×#127	C₄×C₂₃	1	16	yes	complemented
197	C₂×#127	C₂²×C₂₃	1	2	yes	—
198	C₂×#127	C₂²×C₂₃	1	2	yes	—
199	C₂×#127	C₂²×C₂₃	1	4	yes	—
200	#518	—	2	16	no	—
201	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	complemented
202	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	complemented
203	C₂×C₄×D₄	C₂²×C₂₃	1	16	yes	complemented
204	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
205	C₂×#117	C₂²×C₂₃	1	2	yes	—
206	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
207	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
208	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
209	#390	C₂²×C₂₃	1	8	yes	complemented
210	#390	C₂²×C₂₃	1	16	yes	complemented
211	#390	C₂²×C₂₃	1	8	yes	complemented
212	#386	C₂²×C₂₃	1	32	yes	complemented
213	#384	C₂²×C₂₃	1	2	yes	complemented
214	#382	C₂²×C₂₃	1	2	yes	complemented
215	#380	C₂²×C₂₃	1	4	yes	—
216	#379	C₂²×C₂₃	1	8	yes	—
217	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
218	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
219	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
220	#376	C₂²×C₂₃	1	2	yes	complemented
221	C₄³	C₂²×C₂₃	1	2	yes	complemented
222	C₂×C₂₃	—	2	4	no	—
223	C₂×C₂₃	—	4	16	no	—
224	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
225	C₂×C₂₃	C₂×#518	1	2	yes	—
226	C₂×C₂₃	#3498	1	2	yes	—
227	C₂×C₂₃	#3508	1	2	yes	—
228	C₂⁵	C₂³×C₂₃	1	4	yes	—
229	C₂²×D₄	—	2	8	no	—
230	C₂²×D₄	—	2	8	no	—
231	C₂²×D₄	—	4	16	no	—
232	C₂²×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
233	C₂²×D₄	C₂×C₄×C₂₃	1	32	yes	complemented
234	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
235	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
236	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
237	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
238	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
239	C₂³×C₄	C₂³×C₂₃	1	16	yes	—
240	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
241	#127	C₂×C₄×C₂₃	1	16	yes	complemented
242	#127	C₂×C₄×C₂₃	1	32	yes	complemented
243	#127	C₂×C₄×C₂₃	1	16	yes	complemented
244	C₄×D₄	—	2	32	no	—
245	C₄×D₄	—	2	32	no	—
246	C₄×D₄	—	4	32	no	—
247	#117	—	2	8	no	—
248	#117	—	2	4	no	—
249	#117	D₄×C₂₃	1	8	yes	complemented
250	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
251	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
252	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
253	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
254	C₂×C₂²⋊C₄	—	2	32	no	—
255	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
256	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
257	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
258	C₂×C₂²⋊C₄	C₂³×C₂₃	1	32	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
260	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
261	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
262	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
263	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
264	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
267	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
268	#95	C₂³×C₂₃	1	8	yes	—
269	#95	C₂³×C₂₃	1	8	yes	—
270	C₂₃	#34210	1	1	yes	characteristic, direct product factor, Sylow
271	C₂⁴	—	2	8	no	—
272	C₂⁴	—	2	8	no	—
273	C₂⁴	—	4	16	no	—
274	C₂⁴	C₂²×C₄×C₂₃	1	16	yes	—
275	C₂⁴	C₂⁴×C₂₃	1	8	yes	—
276	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₂×D₄	—	2	64	no	—
278	C₂×D₄	—	2	8	no	—
279	C₂×D₄	—	2	8	no	—
280	C₂×D₄	—	2	8	no	—
281	C₂×D₄	—	2	8	no	—
282	C₂×D₄	—	2	32	no	—
283	C₂×D₄	—	4	64	no	—
284	C₂×D₄	—	4	16	no	—
285	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
286	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
287	C₂×D₄	C₂²×C₄×C₂₃	1	32	yes	—
288	C₂×D₄	C₂²×C₄×C₂₃	1	32	yes	—
289	C₂²×C₄	—	2	8	no	—
290	C₂²×C₄	—	2	32	no	—
291	C₂²×C₄	—	4	32	no	—
292	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
293	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
294	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
295	C₂²×C₄	#41×C₂₃	1	4	yes	—
296	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
297	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
298	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
299	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
300	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
301	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
302	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
303	C₄⋊C₄	—	2	8	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	16	no	—
306	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	complemented
307	C₂²⋊C₄	—	2	8	no	—
308	C₂²⋊C₄	—	2	4	no	—
309	C₂²⋊C₄	—	2	32	no	—
310	C₂²⋊C₄	—	4	32	no	—
311	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
312	C₄²	—	2	8	no	—
313	C₄²	—	2	4	no	—
314	C₄²	—	2	16	no	—
315	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
316	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
317	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
318	C₄²	C₂×D₄×C₂₃	1	8	yes	complemented
319	C₂³	—	2	4	no	—
320	C₂³	—	2	8	no	—
321	C₂³	—	2	8	no	—
322	C₂³	—	2	32	no	—
323	C₂³	—	2	32	no	—
324	C₂³	—	4	32	no	—
325	C₂³	—	8	16	no	—
326	C₂³	—	8	16	no	—
327	C₂³	C₂³×C₄×C₂₃	1	16	yes	—
328	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
329	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
330	C₂³	C₂×#41×C₂₃	1	2	yes	—
331	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
332	D₄	—	2	32	no	—
333	D₄	—	2	32	no	—
334	D₄	—	2	64	no	—
335	D₄	—	4	32	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	4	no	—
338	C₂×C₄	—	2	16	no	—
339	C₂×C₄	—	2	8	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	8	no	—
342	C₂×C₄	—	2	16	no	—
343	C₂×C₄	—	8	32	no	—
344	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	complemented
345	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
346	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
347	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
348	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
349	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
350	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
351	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
352	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
353	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
354	C₂×C₄	C₂₃×H(5,𝔽₂)	1	8	yes	—
355	C₂²	—	2	4	no	—
356	C₂²	—	2	4	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	16	no	—
359	C₂²	—	2	32	no	—
360	C₂²	—	4	16	no	—
361	C₂²	—	4	16	no	—
362	C₂²	—	8	32	no	—
363	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
364	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
365	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
366	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
367	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	4	yes	—
368	C₄	—	2	4	no	—
369	C₄	—	2	8	no	—
370	C₄	—	4	8	no	—
371	C₄	—	4	4	no	—
372	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
373	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
374	C₄	C₂₃×#518	1	8	yes	complemented
375	C₂	—	2	4	no	—
376	C₂	—	4	16	no	—
377	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
378	C₂	C₂×C₂₃×#518	1	2	yes	—
379	C₂	C₂₃×#3498	1	2	yes	—
380	C₂	C₂₃×#3508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#127	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×#518	C₂	semidirect
3. C₂₃×#2366	C₂	semidirect
4. D₄×C₂²⋊C₄×C₂₃	C₂	semidirect
5. C₂₃×#2328	C₂	semidirect
6. C₂₃×#2309	C₂	semidirect
7. C₄×C₂₃×#127	C₂	semidirect
8. C₂₃×#2285	C₂	semidirect
9. C₄×C₂₃×#117	C₂	semidirect
12. C₂×C₂₃×#127	C₄	semidirect
17. C₂×C₄×D₄×C₂₃	C₂²	semidirect
18. C₂×C₄×D₄×C₂₃	C₂²	semidirect
19. C₂×C₄×D₄×C₂₃	C₂²	semidirect
25. C₂₃×#390	C₂²	semidirect
26. C₂₃×#390	C₂²	semidirect
27. C₂₃×#390	C₂²	semidirect
28. C₂₃×#386	C₂²	semidirect
29. C₂₃×#384	C₂²	semidirect
30. C₂₃×#382	C₂²	semidirect
33. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
34. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
36. C₂₃×#376	C₂²	semidirect
37. C₄³×C₂₃	C₂²	semidirect
42. C₂²×D₄×C₂₃	C₂×C₄	semidirect
43. C₂²×D₄×C₂₃	C₂×C₄	semidirect
51. C₂₃×#127	C₂×C₄	semidirect
52. C₂₃×#127	C₂×C₄	semidirect
53. C₂₃×#127	C₂×C₄	semidirect
59. C₂₃×#117	D₄	semidirect
70. C₂×C₄²×C₂₃	C₂×C₄	semidirect
115. C₄⋊C₄×C₂₃	C₂×D₄	semidirect
127. C₄²×C₂₃	C₂×D₄	semidirect
128. #34210	C₂₃	direct (N × H)
154. C₂×C₄×C₂₃	C₄×D₄	semidirect
166. C₂×#518	C₂×C₂₃	semidirect
167. #2366	C₂×C₂₃	semidirect
168. D₄×C₂²⋊C₄	C₂×C₂₃	semidirect
169. #2328	C₂×C₂₃	semidirect
170. #2309	C₂×C₂₃	semidirect
171. C₄×#127	C₂×C₂₃	semidirect
172. #2285	C₂×C₂₃	semidirect
173. C₄×#117	C₂×C₂₃	semidirect
193. C₄×C₂₃	#518	semidirect
196. C₂×#127	C₄×C₂₃	semidirect
201. C₂×C₄×D₄	C₂²×C₂₃	semidirect
202. C₂×C₄×D₄	C₂²×C₂₃	semidirect
203. C₂×C₄×D₄	C₂²×C₂₃	semidirect
209. #390	C₂²×C₂₃	semidirect
210. #390	C₂²×C₂₃	semidirect
211. #390	C₂²×C₂₃	semidirect
212. #386	C₂²×C₂₃	semidirect
213. #384	C₂²×C₂₃	semidirect
214. #382	C₂²×C₂₃	semidirect
217. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
218. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
220. #376	C₂²×C₂₃	semidirect
221. C₄³	C₂²×C₂₃	semidirect
232. C₂²×D₄	C₂×C₄×C₂₃	semidirect
233. C₂²×D₄	C₂×C₄×C₂₃	semidirect
241. #127	C₂×C₄×C₂₃	semidirect
242. #127	C₂×C₄×C₂₃	semidirect
243. #127	C₂×C₄×C₂₃	semidirect
249. #117	D₄×C₂₃	semidirect
260. C₂×C₄²	C₂×C₄×C₂₃	semidirect
270. C₂₃	#34210	direct (N × H)
306. C₄⋊C₄	C₂×D₄×C₂₃	semidirect
318. C₄²	C₂×D₄×C₂₃	semidirect
344. C₂×C₄	C₄×D₄×C₂₃	semidirect
374. C₄	C₂₃×#518	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#127	2
C₂×C₂₃×#518	2
C₂₃×#2366	2
D₄×C₂²⋊C₄×C₂₃	2
C₂₃×#2328	2
C₂₃×#2309	2
C₄×C₂₃×#127	2
C₂₃×#2285	2
C₄×C₂₃×#117	2
#34210	23

Subgroups — C23×#34210 #19181836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34210 #19181836