Subgroups — C₂₃×#60205 #19207831

Order 5888.

1950 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2967	C₂	1	8	yes	maximal, complemented
3	C₂₃×#2965	C₂	1	8	yes	maximal, complemented
4	C₂₃×#2961	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2932	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2930	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2922	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂₃×#131	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#131	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#533	C₄	1	2	yes	—
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₂₃×#532	C₄	1	2	yes	—
16	C₂₃×#532	C₂²	1	1	yes	characteristic
17	C₂₃×#529	C₄	1	4	yes	—
18	C₂₃×#529	C₄	1	4	yes	—
19	C₂₃×#529	C₂²	1	4	yes	—
20	C₂₃×#525	C₄	1	4	yes	—
21	C₂₃×#525	C₂²	1	2	yes	—
22	C₄×#41×C₂₃	C₂²	1	4	yes	—
23	C₄×#41×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#446	C₂²	1	16	yes	complemented
26	C₂₃×#436	C₂²	1	16	yes	—
27	C₂₃×#435	C₂²	1	16	yes	—
28	C₂₃×#435	C₂²	1	16	yes	complemented
29	C₂₃×#433	C₂²	1	8	yes	complemented
30	C₂₃×#432	C₂²	1	4	yes	—
31	C₂₃×#431	C₂²	1	4	yes	—
32	C₂₃×#431	C₂²	1	8	yes	complemented
33	C₂₃×#424	C₂²	1	4	yes	—
34	C₂₃×#423	C₂²	1	2	yes	—
35	C₂₃×#408	C₂²	1	4	yes	—
36	C₂₃×#407	C₂²	1	2	yes	—
37	C₂₃×#405	C₂²	1	4	yes	—
38	C₂₃×#405	C₂²	1	8	yes	complemented
39	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
40	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
41	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
42	C₄×M₁₆×C₂₃	C₂²	1	4	yes	complemented
43	C₂×C₂₃×#97	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	complemented
47	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
48	C₂×#41×C₂₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#131	—	2	16	no	—
52	C₂×M₁₆×C₂₃	—	2	8	no	—
53	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
54	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
55	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
58	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
61	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
62	C₂₃×#126	C₂×C₄	1	8	yes	—
63	C₂₃×#125	C₂×C₄	1	2	yes	—
64	C₂₃×#125	C₂×C₄	1	4	yes	—
65	C₂₃×#124	C₂×C₄	1	2	yes	—
66	C₂₃×#124	C₂×C₄	1	4	yes	—
67	C₂₃×#123	C₂×C₄	1	8	yes	—
68	C₂₃×#123	C₂×C₄	1	4	yes	—
69	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
70	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
71	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
72	C₂₃×#121	C₂×C₄	1	4	yes	—
73	C₂₃×#121	C₂×C₄	1	4	yes	—
74	C₂₃×#121	C₂×C₄	1	4	yes	—
75	C₂₃×#120	C₂×C₄	1	4	yes	—
76	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
77	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
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	4	yes	—
82	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
83	C₄×D₄×C₂₃	C₂³	1	8	yes	—
84	C₄×D₄×C₂₃	C₂³	1	4	yes	—
85	C₂₃×#117	C₂×C₄	1	4	yes	—
86	C₂₃×#117	C₂×C₄	1	8	yes	—
87	C₂₃×#117	C₂³	1	2	yes	—
88	C₂₃×#117	C₂³	1	4	yes	—
89	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
91	C₂×C₄²×C₂₃	C₂×C₄	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	1	yes	characteristic
95	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
97	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
98	C₂₃×C₄⋊C₈	C₂³	1	16	yes	—
99	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
100	C₂₃×C₂²⋊C₈	C₂³	1	16	yes	—
101	C₂₃×#97	C₂³	1	8	yes	—
102	C₂₃×#97	C₂³	1	4	yes	—
103	C₂₃×#97	C₂³	1	8	yes	—
104	C₄×C₈×C₂₃	C₂³	1	2	yes	—
105	C₄×C₈×C₂₃	C₂³	1	2	yes	—
106	C₄×C₈×C₂₃	C₂³	1	8	yes	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₄×C₈×C₂₃	C₂³	1	2	yes	—
109	C₄×C₈×C₂₃	C₂³	1	4	yes	—
110	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	#41×C₂₃	—	2	8	no	—
112	#41×C₂₃	—	4	8	no	—
113	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
115	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
116	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
117	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
118	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
119	C₂²×C₄×C₂₃	—	2	4	no	—
120	C₂²×C₄×C₂₃	—	4	4	no	—
121	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
127	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
133	M₁₆×C₂₃	—	2	8	no	—
134	M₁₆×C₂₃	—	2	16	no	—
135	M₁₆×C₂₃	—	2	16	no	—
136	C₂×C₈×C₂₃	—	2	4	no	—
137	C₂×C₈×C₂₃	—	2	16	no	—
138	C₂×C₈×C₂₃	#41	1	8	yes	complemented
139	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
140	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
141	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
142	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
143	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
144	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
145	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
146	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
147	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
148	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
149	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
150	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
151	C₂²⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
152	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
153	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
154	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
155	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
156	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
157	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
159	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
161	C₄²×C₂₃	C₂⁴	1	2	yes	—
162	C₄²×C₂₃	C₂⁴	1	4	yes	—
163	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
164	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
165	#60205	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₂₃	—	2	2	no	—
167	C₂³×C₂₃	—	4	2	no	—
168	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
169	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
170	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
171	Q₈×C₂₃	—	2	4	no	—
172	Q₈×C₂₃	—	4	4	no	—
173	D₄×C₂₃	—	2	8	no	—
174	D₄×C₂₃	—	2	4	no	—
175	D₄×C₂₃	—	4	4	no	—
176	D₄×C₂₃	—	4	8	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	4	no	—
180	C₂×C₄×C₂₃	—	2	4	no	—
181	C₂×C₄×C₂₃	—	2	4	no	—
182	C₂×C₄×C₂₃	—	2	16	no	—
183	C₂×C₄×C₂₃	—	2	8	no	—
184	C₂×C₄×C₂₃	—	2	4	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
189	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
190	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
191	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
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	1	yes	characteristic
195	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
197	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
198	C₈×C₂₃	—	2	4	no	—
199	C₈×C₂₃	—	2	8	no	—
200	C₈×C₂₃	C₂×#41	1	8	yes	—
201	#3444	C₂×C₂₃	1	1	yes	characteristic
202	#2967	C₂×C₂₃	1	8	yes	complemented
203	#2965	C₂×C₂₃	1	8	yes	complemented
204	#2961	C₂×C₂₃	1	2	yes	complemented
205	#2932	C₂×C₂₃	1	2	yes	complemented
206	#2930	C₂×C₂₃	1	4	yes	complemented
207	#2922	C₂×C₂₃	1	2	yes	complemented
208	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₄×#131	C₂×C₂₃	1	2	yes	complemented
210	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
211	C₂²×C₂₃	—	2	2	no	—
212	C₂²×C₂₃	—	2	2	no	—
213	C₂²×C₂₃	—	2	4	no	—
214	C₂²×C₂₃	—	2	2	no	—
215	C₂²×C₂₃	—	2	2	no	—
216	C₂²×C₂₃	—	2	8	no	—
217	C₂²×C₂₃	—	4	4	no	—
218	C₂²×C₂₃	C₄×#41	1	2	yes	—
219	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
220	C₄×C₂₃	—	2	2	no	—
221	C₄×C₂₃	—	2	4	no	—
222	C₄×C₂₃	—	2	2	no	—
223	C₄×C₂₃	—	2	2	no	—
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	2	8	no	—
226	C₄×C₂₃	C₄×#41	1	2	yes	—
227	C₄×C₂₃	#568	1	4	yes	—
228	C₄×C₂₃	C₂²×#41	1	2	yes	—
229	C₂×#131	C₂²×C₂₃	1	4	yes	—
230	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
231	#533	C₄×C₂₃	1	2	yes	—
232	#533	C₂²×C₂₃	1	1	yes	characteristic
233	#532	C₄×C₂₃	1	2	yes	—
234	#532	C₂²×C₂₃	1	1	yes	characteristic
235	#529	C₄×C₂₃	1	4	yes	—
236	#529	C₄×C₂₃	1	4	yes	—
237	#529	C₂²×C₂₃	1	4	yes	—
238	#525	C₄×C₂₃	1	4	yes	—
239	#525	C₂²×C₂₃	1	2	yes	—
240	C₄×#41	C₂²×C₂₃	1	4	yes	—
241	C₄×#41	C₂²×C₂₃	1	2	yes	—
242	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
243	#446	C₂²×C₂₃	1	16	yes	complemented
244	#436	C₂²×C₂₃	1	16	yes	—
245	#435	C₂²×C₂₃	1	16	yes	—
246	#435	C₂²×C₂₃	1	16	yes	complemented
247	#433	C₂²×C₂₃	1	8	yes	complemented
248	#432	C₂²×C₂₃	1	4	yes	—
249	#431	C₂²×C₂₃	1	4	yes	—
250	#431	C₂²×C₂₃	1	8	yes	complemented
251	#424	C₂²×C₂₃	1	4	yes	—
252	#423	C₂²×C₂₃	1	2	yes	—
253	#408	C₂²×C₂₃	1	4	yes	—
254	#407	C₂²×C₂₃	1	2	yes	—
255	#405	C₂²×C₂₃	1	4	yes	—
256	#405	C₂²×C₂₃	1	8	yes	complemented
257	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
258	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
259	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
260	C₄×M₁₆	C₂²×C₂₃	1	4	yes	complemented
261	C₂×#97	C₂²×C₂₃	1	2	yes	—
262	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	complemented
265	C₂×C₂₃	—	2	2	no	—
266	C₂×C₂₃	—	4	4	no	—
267	C₂×C₂₃	—	4	2	no	—
268	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
269	C₂×C₂₃	C₂×#568	1	2	yes	—
270	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
271	C₂×#41	C₂³×C₂₃	1	2	yes	—
272	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
273	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
274	#131	—	2	16	no	—
275	C₂×M₁₆	—	2	8	no	—
276	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
277	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
278	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
279	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
280	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
281	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
282	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
283	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
284	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
285	#126	C₂×C₄×C₂₃	1	8	yes	—
286	#125	C₂×C₄×C₂₃	1	2	yes	—
287	#125	C₂×C₄×C₂₃	1	4	yes	—
288	#124	C₂×C₄×C₂₃	1	2	yes	—
289	#124	C₂×C₄×C₂₃	1	4	yes	—
290	#123	C₂×C₄×C₂₃	1	8	yes	—
291	#123	C₂×C₄×C₂₃	1	4	yes	—
292	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
293	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
294	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
295	#121	C₂×C₄×C₂₃	1	4	yes	—
296	#121	C₂×C₄×C₂₃	1	4	yes	—
297	#121	C₂×C₄×C₂₃	1	4	yes	—
298	#120	C₂×C₄×C₂₃	1	4	yes	—
299	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
300	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
301	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
302	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
303	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
304	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
305	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
306	C₄×D₄	C₂³×C₂₃	1	8	yes	—
307	C₄×D₄	C₂³×C₂₃	1	4	yes	—
308	#117	C₂×C₄×C₂₃	1	4	yes	—
309	#117	C₂×C₄×C₂₃	1	8	yes	—
310	#117	C₂³×C₂₃	1	2	yes	—
311	#117	C₂³×C₂₃	1	4	yes	—
312	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
313	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
314	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
315	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
316	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
318	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
321	C₄⋊C₈	C₂³×C₂₃	1	16	yes	—
322	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
323	C₂²⋊C₈	C₂³×C₂₃	1	16	yes	—
324	#97	C₂³×C₂₃	1	8	yes	—
325	#97	C₂³×C₂₃	1	4	yes	—
326	#97	C₂³×C₂₃	1	8	yes	—
327	C₄×C₈	C₂³×C₂₃	1	2	yes	—
328	C₄×C₈	C₂³×C₂₃	1	2	yes	—
329	C₄×C₈	C₂³×C₂₃	1	2	yes	—
330	C₄×C₈	C₂³×C₂₃	1	2	yes	—
331	C₄×C₈	C₂³×C₂₃	1	4	yes	—
332	C₄×C₈	C₂³×C₂₃	1	8	yes	—
333	C₂₃	#60205	1	1	yes	characteristic, direct product factor, Sylow
334	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
335	#41	—	2	8	no	—
336	#41	—	4	8	no	—
337	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
338	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
339	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
340	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
341	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
342	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
346	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
351	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
356	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
357	M₁₆	—	2	8	no	—
358	M₁₆	—	2	16	no	—
359	M₁₆	—	2	16	no	—
360	C₂×C₈	—	2	4	no	—
361	C₂×C₈	—	2	16	no	—
362	C₂×C₈	#41×C₂₃	1	8	yes	complemented
363	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
364	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
365	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
366	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
367	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
368	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
369	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
370	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
371	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
372	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
373	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
374	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
375	C₂²⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
376	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
377	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
378	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
379	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
380	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
381	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
382	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
383	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
384	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
385	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
386	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
387	C₄²	C₂⁴×C₂₃	1	4	yes	—
388	C₄²	C₂⁴×C₂₃	1	2	yes	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
392	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
393	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
394	Q₈	—	2	4	no	—
395	Q₈	—	4	4	no	—
396	D₄	—	2	8	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	4	no	—
399	D₄	—	4	8	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	4	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	16	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
413	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
414	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
415	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
416	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
417	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
418	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
420	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
421	C₈	—	2	4	no	—
422	C₈	—	2	8	no	—
423	C₈	C₂×#41×C₂₃	1	8	yes	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	4	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	8	no	—
430	C₂²	—	4	4	no	—
431	C₂²	C₄×#41×C₂₃	1	2	yes	—
432	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	8	no	—
439	C₄	C₄×#41×C₂₃	1	2	yes	—
440	C₄	C₂₃×#568	1	4	yes	—
441	C₄	C₂²×#41×C₂₃	1	2	yes	—
442	C₂	—	2	2	no	—
443	C₂	—	4	2	no	—
444	C₂	—	4	4	no	—
445	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
446	C₂	C₂×C₂₃×#568	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂₃×#3444	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2967	C₂	semidirect
3. C₂₃×#2965	C₂	semidirect
4. C₂₃×#2961	C₂	semidirect
5. C₂₃×#2932	C₂	semidirect
6. C₂₃×#2930	C₂	semidirect
7. C₂₃×#2922	C₂	semidirect
8. C₂₃×#2920	C₂	semidirect
9. C₄×C₂₃×#131	C₂	semidirect
10. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
25. C₂₃×#446	C₂²	semidirect
28. C₂₃×#435	C₂²	semidirect
29. C₂₃×#433	C₂²	semidirect
32. C₂₃×#431	C₂²	semidirect
38. C₂₃×#405	C₂²	semidirect
42. C₄×M₁₆×C₂₃	C₂²	semidirect
46. C₂×C₄×C₈×C₂₃	C₂²	semidirect
138. C₂×C₈×C₂₃	#41	semidirect
165. #60205	C₂₃	direct (N × H)
202. #2967	C₂×C₂₃	semidirect
203. #2965	C₂×C₂₃	semidirect
204. #2961	C₂×C₂₃	semidirect
205. #2932	C₂×C₂₃	semidirect
206. #2930	C₂×C₂₃	semidirect
207. #2922	C₂×C₂₃	semidirect
208. #2920	C₂×C₂₃	semidirect
209. C₄×#131	C₂×C₂₃	semidirect
210. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
243. #446	C₂²×C₂₃	semidirect
246. #435	C₂²×C₂₃	semidirect
247. #433	C₂²×C₂₃	semidirect
250. #431	C₂²×C₂₃	semidirect
256. #405	C₂²×C₂₃	semidirect
260. C₄×M₁₆	C₂²×C₂₃	semidirect
264. C₂×C₄×C₈	C₂²×C₂₃	semidirect
333. C₂₃	#60205	direct (N × H)
362. C₂×C₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3444	2
C₂₃×#2967	2
C₂₃×#2965	2
C₂₃×#2961	2
C₂₃×#2932	2
C₂₃×#2930	2
C₂₃×#2922	2
C₂₃×#2920	2
C₄×C₂₃×#131	2
C₂×C₄×M₁₆×C₂₃	2
#60205	23

Subgroups — C23×#60205 #19207831

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60205 #19207831