Subgroups — C₂₃×#21304 #19168930

Order 5888.

1162 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1985	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1984	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#533	—	2	4	no	—
15	C₂₃×#533	—	2	4	no	—
16	C₂×C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
17	C₂×C₂₃×#125	C₄	1	2	yes	—
18	C₂×C₂₃×C₂²⋊Q₈	C₄	1	4	yes	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#424	—	2	4	no	—
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
25	C₂₃×#405	—	2	4	no	—
26	C₂×C₂₃×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂₃×#336	C₂²	1	2	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂₃×#336	C₂²	1	2	yes	—
32	C₂₃×#336	C₂²	1	4	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₂₃	—	2	2	no	—
38	C₂×M₁₆×C₂₃	—	2	2	no	—
39	C₂×M₁₆×C₂₃	—	2	2	no	—
40	C₂×M₁₆×C₂₃	—	2	2	no	—
41	C₂×M₁₆×C₂₃	—	2	2	no	—
42	C₂×M₁₆×C₂₃	Q₈	1	4	yes	complemented
43	C₂²×C₈×C₂₃	—	2	2	no	—
44	C₂²×C₈×C₂₃	—	2	2	no	—
45	C₂²×C₈×C₂₃	—	2	2	no	—
46	C₂²×C₈×C₂₃	—	2	2	no	—
47	C₂²×C₈×C₂₃	—	2	2	no	—
48	C₂²×C₈×C₂₃	Q₈	1	4	yes	complemented
49	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂₃×C₄⋊Q₈	—	2	8	no	—
55	C₂₃×#125	—	2	8	no	—
56	C₂₃×C₂²⋊Q₈	—	2	16	no	—
57	C₄×Q₈×C₂₃	—	2	8	no	—
58	C₄×Q₈×C₂₃	—	2	8	no	—
59	C₂₃×#117	—	2	2	no	—
60	C₂₃×#117	—	2	2	no	—
61	C₂₃×#117	D₄	1	4	yes	—
62	C₂₃×#117	D₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₂₃	—	2	2	no	—
70	C₂×C₄²×C₂₃	—	2	2	no	—
71	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
76	C₂₃×C₄⋊C₈	—	2	8	no	—
77	C₂₃×#97	—	2	4	no	—
78	C₄×C₈×C₂₃	—	2	4	no	—
79	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×Q₈×C₂₃	—	2	8	no	—
81	C₂×Q₈×C₂₃	—	2	8	no	—
82	C₂×Q₈×C₂₃	—	4	4	no	—
83	C₂²×C₄×C₂₃	—	2	4	no	—
84	C₂²×C₄×C₂₃	—	2	4	no	—
85	C₂²×C₄×C₂₃	—	4	2	no	—
86	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
95	C₂²×C₄×C₂₃	#41	1	2	yes	—
96	C₂²×C₄×C₂₃	#41	1	2	yes	—
97	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
98	M₁₆×C₂₃	—	2	8	no	—
99	M₁₆×C₂₃	—	4	4	no	—
100	C₂×C₈×C₂₃	—	2	8	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	—	2	2	no	—
104	C₂×C₈×C₂₃	—	2	1	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	—	2	2	no	—
107	C₂×C₈×C₂₃	—	2	2	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	2	no	—
113	C₂×C₈×C₂₃	—	4	4	no	—
114	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
116	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
117	C₄⋊C₄×C₂₃	—	2	4	no	—
118	C₄⋊C₄×C₂₃	—	2	16	no	—
119	C₄⋊C₄×C₂₃	—	2	8	no	—
120	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
121	C₂²⋊C₄×C₂₃	—	2	4	no	—
122	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
123	C₄²×C₂₃	—	2	2	no	—
124	C₄²×C₂₃	—	2	2	no	—
125	C₄²×C₂₃	—	2	2	no	—
126	C₄²×C₂₃	—	2	2	no	—
127	C₄²×C₂₃	—	2	2	no	—
128	C₄²×C₂₃	—	2	2	no	—
129	C₄²×C₂₃	C₂×D₄	1	4	yes	—
130	C₄²×C₂₃	C₂×D₄	1	4	yes	—
131	#21304	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂³×C₂₃	—	2	2	no	—
133	C₂³×C₂₃	—	2	2	no	—
134	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
136	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
137	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
138	Q₈×C₂₃	—	4	16	no	—
139	C₂×C₄×C₂₃	—	2	2	no	—
140	C₂×C₄×C₂₃	—	2	4	no	—
141	C₂×C₄×C₂₃	—	2	2	no	—
142	C₂×C₄×C₂₃	—	2	4	no	—
143	C₂×C₄×C₂₃	—	2	2	no	—
144	C₂×C₄×C₂₃	—	2	4	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	4	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	4	no	—
149	C₂×C₄×C₂₃	—	4	8	no	—
150	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
151	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
153	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
154	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
155	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
156	C₂×C₄×C₂₃	#124	1	4	yes	—
157	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
158	C₂×C₄×C₂₃	#131	1	4	yes	—
159	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₂₃	—	2	2	no	—
166	C₈×C₂₃	—	2	2	no	—
167	C₈×C₂₃	—	2	4	no	—
168	C₈×C₂₃	—	4	2	no	—
169	C₈×C₂₃	—	4	2	no	—
170	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
171	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
172	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
173	#1985	C₂×C₂₃	1	4	yes	complemented
174	#1984	C₂×C₂₃	1	4	yes	complemented
175	#1729	C₂×C₂₃	1	1	yes	characteristic
176	#1729	C₂×C₂₃	1	1	yes	characteristic
177	#1728	C₂×C₂₃	1	1	yes	characteristic
178	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
179	C₂²×C₂₃	—	2	4	no	—
180	C₂²×C₂₃	—	2	2	no	—
181	C₂²×C₂₃	—	2	2	no	—
182	C₂²×C₂₃	—	2	4	no	—
183	C₂²×C₂₃	#391	1	4	yes	—
184	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₂²×C₂₃	C₂×#131	1	2	yes	—
188	C₂²×C₂₃	#568	1	2	yes	—
189	C₄×C₂₃	—	2	4	no	—
190	C₄×C₂₃	—	2	8	no	—
191	C₄×C₂₃	—	4	2	no	—
192	C₄×C₂₃	—	4	4	no	—
193	C₄×C₂₃	#391	1	4	yes	—
194	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
195	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
196	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
197	C₄×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
198	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
199	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
200	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
201	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
202	#533	—	2	4	no	—
203	#533	—	2	4	no	—
204	C₂×C₄⋊Q₈	C₄×C₂₃	1	2	yes	—
205	C₂×#125	C₄×C₂₃	1	2	yes	—
206	C₂×C₂²⋊Q₈	C₄×C₂₃	1	4	yes	—
207	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
208	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
209	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
210	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
211	#424	—	2	4	no	—
212	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
213	#405	—	2	4	no	—
214	C₂×#97	C₂²×C₂₃	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
216	#336	C₂²×C₂₃	1	2	yes	—
217	#336	C₂²×C₂₃	1	2	yes	—
218	#336	C₂²×C₂₃	1	4	yes	—
219	#336	C₂²×C₂₃	1	2	yes	—
220	#336	C₂²×C₂₃	1	4	yes	—
221	#336	C₂²×C₂₃	1	2	yes	—
222	C₂×C₂₃	—	2	4	no	—
223	C₂×C₂₃	C₂×#391	1	1	yes	characteristic
224	C₂×C₂₃	#2880	1	2	yes	—
225	C₂×C₂₃	#2960	1	1	yes	characteristic
226	C₂×C₂₃	#2961	1	1	yes	characteristic
227	C₂×C₂₃	#2962	1	2	yes	—
228	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
229	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
230	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	2	2	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	—	2	2	no	—
236	C₂×M₁₆	Q₈×C₂₃	1	4	yes	complemented
237	C₂²×C₈	—	2	2	no	—
238	C₂²×C₈	—	2	2	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	2	2	no	—
241	C₂²×C₈	—	2	2	no	—
242	C₂²×C₈	Q₈×C₂₃	1	4	yes	complemented
243	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
244	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
245	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
248	C₄⋊Q₈	—	2	8	no	—
249	#125	—	2	8	no	—
250	C₂²⋊Q₈	—	2	16	no	—
251	C₄×Q₈	—	2	8	no	—
252	C₄×Q₈	—	2	8	no	—
253	#117	—	2	2	no	—
254	#117	—	2	2	no	—
255	#117	D₄×C₂₃	1	4	yes	—
256	#117	D₄×C₂₃	1	4	yes	—
257	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
260	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
261	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
263	C₂×C₄²	—	2	2	no	—
264	C₂×C₄²	—	2	2	no	—
265	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
268	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
269	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
270	C₄⋊C₈	—	2	8	no	—
271	#97	—	2	4	no	—
272	C₄×C₈	—	2	4	no	—
273	C₂₃	#21304	1	1	yes	characteristic, direct product factor, Sylow
274	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
275	C₂×Q₈	—	2	8	no	—
276	C₂×Q₈	—	2	8	no	—
277	C₂×Q₈	—	4	4	no	—
278	C₂²×C₄	—	2	4	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
282	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
283	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
284	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
287	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
288	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
289	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
290	C₂²×C₄	#41×C₂₃	1	2	yes	—
291	C₂²×C₄	#41×C₂₃	1	2	yes	—
292	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
293	M₁₆	—	2	8	no	—
294	M₁₆	—	4	4	no	—
295	C₂×C₈	—	2	8	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	4	4	no	—
309	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
310	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
311	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
312	C₄⋊C₄	—	2	4	no	—
313	C₄⋊C₄	—	2	16	no	—
314	C₄⋊C₄	—	2	8	no	—
315	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
316	C₂²⋊C₄	—	2	4	no	—
317	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
318	C₄²	—	2	2	no	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂×D₄×C₂₃	1	4	yes	—
325	C₄²	C₂×D₄×C₂₃	1	4	yes	—
326	C₂³	—	2	2	no	—
327	C₂³	—	2	2	no	—
328	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
329	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
330	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
331	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
332	Q₈	—	4	16	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	4	8	no	—
344	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
345	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
346	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
347	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
348	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
349	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
350	C₂×C₄	C₂₃×#124	1	4	yes	—
351	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
352	C₂×C₄	C₂₃×#131	1	4	yes	—
353	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	2	4	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	4	no	—
367	C₂²	—	2	4	no	—
368	C₂²	C₂₃×#391	1	4	yes	—
369	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
370	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
371	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
372	C₂²	C₂×C₂₃×#131	1	2	yes	—
373	C₂²	C₂₃×#568	1	2	yes	—
374	C₄	—	2	4	no	—
375	C₄	—	2	8	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	4	no	—
378	C₄	C₂₃×#391	1	4	yes	—
379	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
380	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
381	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
382	C₄	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
383	C₂	—	2	4	no	—
384	C₂	C₂×C₂₃×#391	1	1	yes	characteristic
385	C₂	C₂₃×#2880	1	2	yes	—
386	C₂	C₂₃×#2960	1	1	yes	characteristic
387	C₂	C₂₃×#2961	1	1	yes	characteristic
388	C₂	C₂₃×#2962	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1985	C₂	semidirect
5. C₂₃×#1984	C₂	semidirect
42. C₂×M₁₆×C₂₃	Q₈	semidirect
48. C₂²×C₈×C₂₃	Q₈	semidirect
131. #21304	C₂₃	direct (N × H)
173. #1985	C₂×C₂₃	semidirect
174. #1984	C₂×C₂₃	semidirect
236. C₂×M₁₆	Q₈×C₂₃	semidirect
242. C₂²×C₈	Q₈×C₂₃	semidirect
273. C₂₃	#21304	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×C₂₃×#533	2
C₂×C₂₃×#424	2
C₂×C₂₃×#405	2
C₂₃×#1985	2
C₂₃×#1984	2
C₂₃×#1729	2
C₂₃×#1728	2
C₂×C₂₃×#336	2
#21304	23

Subgroups — C23×#21304 #19168930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21304 #19168930