Subgroups — C₂₃×#10868 #19158494

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#344	C₂	1	2	yes	maximal
5	C₂×C₂₃×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	—	2	2	no	—
12	C₂×C₂₃×#108	—	2	1	no	—
13	C₂×C₂₃×#108	—	2	1	no	—
14	C₂×C₂₃×#108	C₄	1	4	yes	complemented
15	C₂₃×#427	—	2	2	no	—
16	C₂₃×#427	C₄	1	4	yes	complemented
17	C₂×C₂₃×#107	—	2	1	no	—
18	C₂×C₂₃×#106	—	2	1	no	—
19	C₂₃×#405	—	2	2	no	—
20	C₂₃×#405	—	2	2	no	—
21	C₂×C₂₃×#97	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	complemented
23	C₂₃×#344	—	2	4	no	—
24	C₂₃×H(3,ℤ/4)	—	2	4	no	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	1	no	—
27	C₂×M₁₆×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	2	2	no	—
32	C₂×M₁₆×C₂₃	—	2	2	no	—
33	C₂×M₁₆×C₂₃	—	2	2	no	—
34	C₂×M₁₆×C₂₃	—	2	2	no	—
35	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
36	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
37	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₂₃	D₄	1	2	yes	—
45	C₂²×C₈×C₂₃	D₄	1	2	yes	—
46	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
50	C₂₃×#117	—	2	1	no	—
51	C₂₃×#117	—	2	1	no	—
52	C₂₃×#117	—	2	2	no	—
53	C₂₃×#117	—	2	2	no	—
54	C₂₃×#117	—	2	2	no	—
55	C₂₃×#117	C₂×C₄	1	4	yes	—
56	C₂₃×#117	D₄	1	2	yes	—
57	C₂₃×#117	Q₈	1	2	yes	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄²×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
64	C₂₃×#108	—	2	4	no	—
65	C₂₃×#108	—	2	4	no	—
66	C₂₃×#107	—	2	4	no	—
67	C₂₃×#106	—	2	4	no	—
68	C₂₃×#97	—	2	4	no	—
69	C₄×C₈×C₂₃	—	2	4	no	—
70	C₂⁴×C₂₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
87	M₁₆×C₂₃	—	2	2	no	—
88	M₁₆×C₂₃	—	2	2	no	—
89	M₁₆×C₂₃	—	2	2	no	—
90	M₁₆×C₂₃	—	2	2	no	—
91	M₁₆×C₂₃	—	2	4	no	—
92	M₁₆×C₂₃	—	2	4	no	—
93	M₁₆×C₂₃	—	4	2	no	—
94	M₁₆×C₂₃	—	4	2	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	2	no	—
97	C₂×C₈×C₂₃	—	2	4	no	—
98	C₂×C₈×C₂₃	—	4	2	no	—
99	C₂×C₈×C₂₃	—	4	2	no	—
100	C₂×C₈×C₂₃	C₄²	1	2	yes	—
101	C₂×C₈×C₂₃	C₄²	1	2	yes	—
102	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
103	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
104	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
105	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
106	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
108	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
109	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
114	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
115	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
116	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
117	C₄⋊C₄×C₂₃	—	2	2	no	—
118	C₄⋊C₄×C₂₃	—	2	2	no	—
119	C₄⋊C₄×C₂₃	—	2	4	no	—
120	C₄⋊C₄×C₂₃	—	4	2	no	—
121	C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂²⋊C₄×C₂₃	—	2	2	no	—
123	C₂²⋊C₄×C₂₃	—	2	4	no	—
124	C₂²⋊C₄×C₂₃	—	4	2	no	—
125	C₄²×C₂₃	—	2	2	no	—
126	C₄²×C₂₃	—	2	4	no	—
127	C₄²×C₂₃	—	4	2	no	—
128	C₄²×C₂₃	C₄⋊C₄	1	4	yes	—
129	#10868	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₂₃	—	4	2	no	—
131	C₂³×C₂₃	#117	1	1	yes	characteristic
132	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
133	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
134	C₂³×C₂₃	#121	1	1	yes	characteristic
135	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂³×C₂₃	#124	1	1	yes	characteristic
137	C₂³×C₂₃	#125	1	1	yes	characteristic
138	C₂×C₄×C₂₃	—	2	4	no	—
139	C₂×C₄×C₂₃	—	4	4	no	—
140	C₂×C₄×C₂₃	—	4	4	no	—
141	C₂×C₄×C₂₃	—	8	4	no	—
142	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
152	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
159	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
160	C₂×C₄×C₂₃	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
164	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
165	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
166	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
167	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
170	C₈×C₂₃	—	2	4	no	—
171	C₈×C₂₃	—	4	4	no	—
172	C₈×C₂₃	#95	1	2	yes	—
173	C₈×C₂₃	#95	1	2	yes	—
174	C₈×C₂₃	#95	1	2	yes	—
175	C₈×C₂₃	#95	1	2	yes	—
176	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
177	C₂×#427	C₂×C₂₃	1	1	yes	characteristic
178	C₂×#405	C₂×C₂₃	1	1	yes	characteristic
179	C₂×#344	C₂×C₂₃	1	2	yes	—
180	C₂×H(3,ℤ/4)	C₂×C₂₃	1	2	yes	—
181	C₂²×C₂₃	—	2	2	no	—
182	C₂²×C₂₃	—	2	2	no	—
183	C₂²×C₂₃	—	2	2	no	—
184	C₂²×C₂₃	—	4	1	no	—
185	C₂²×C₂₃	—	4	1	no	—
186	C₂²×C₂₃	—	4	1	no	—
187	C₂²×C₂₃	—	4	1	no	—
188	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₂₃	#380	1	1	yes	characteristic
190	C₂²×C₂₃	#382	1	1	yes	characteristic
191	C₂²×C₂₃	#389	1	1	yes	characteristic
192	C₂²×C₂₃	#389	1	1	yes	characteristic
193	C₂²×C₂₃	#390	1	1	yes	characteristic
194	C₂²×C₂₃	#391	1	1	yes	characteristic
195	C₄×C₂₃	—	4	4	no	—
196	C₄×C₂₃	—	8	4	no	—
197	C₄×C₂₃	C₂×#95	1	1	yes	characteristic
198	C₄×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
199	C₄×C₂₃	#380	1	1	yes	characteristic
200	C₄×C₂₃	#384	1	1	yes	characteristic
201	C₄×C₂₃	#389	1	1	yes	characteristic
202	C₄×C₂₃	#389	1	1	yes	characteristic
203	C₄×C₂₃	#390	1	1	yes	characteristic
204	C₄×C₂₃	#391	1	1	yes	characteristic
205	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
206	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
207	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
208	C₂×#117	C₂²×C₂₃	1	2	yes	—
209	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
210	C₂×#108	—	2	2	no	—
211	C₂×#108	—	2	1	no	—
212	C₂×#108	—	2	1	no	—
213	C₂×#108	C₄×C₂₃	1	4	yes	complemented
214	#427	—	2	2	no	—
215	#427	C₄×C₂₃	1	4	yes	complemented
216	C₂×#107	—	2	1	no	—
217	C₂×#106	—	2	1	no	—
218	#405	—	2	2	no	—
219	#405	—	2	2	no	—
220	C₂×#97	C₄×C₂₃	1	2	yes	complemented
221	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	complemented
222	#344	—	2	4	no	—
223	H(3,ℤ/4)	—	2	4	no	—
224	C₂×C₂₃	—	2	1	no	—
225	C₂×C₂₃	—	2	1	no	—
226	C₂×C₂₃	—	2	1	no	—
227	C₂×C₂₃	—	2	1	no	—
228	C₂×C₂₃	—	2	1	no	—
229	C₂×C₂₃	—	2	1	no	—
230	C₂×C₂₃	C₄⋊#95	1	1	yes	characteristic
231	C₂×C₂₃	#1833	1	2	yes	—
232	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	2	1	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	2	1	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂×M₁₆	—	2	2	no	—
239	C₂×M₁₆	—	2	2	no	—
240	C₂×M₁₆	—	2	2	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
243	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
244	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
247	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
248	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
250	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
251	C₂²×C₈	D₄×C₂₃	1	2	yes	—
252	C₂²×C₈	D₄×C₂₃	1	2	yes	—
253	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
254	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
255	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
256	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
257	#117	—	2	1	no	—
258	#117	—	2	1	no	—
259	#117	—	2	2	no	—
260	#117	—	2	2	no	—
261	#117	—	2	2	no	—
262	#117	C₂×C₄×C₂₃	1	4	yes	—
263	#117	D₄×C₂₃	1	2	yes	—
264	#117	Q₈×C₂₃	1	2	yes	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	1	no	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
271	#108	—	2	4	no	—
272	#108	—	2	4	no	—
273	#107	—	2	4	no	—
274	#106	—	2	4	no	—
275	#97	—	2	4	no	—
276	C₄×C₈	—	2	4	no	—
277	C₂₃	#10868	1	1	yes	characteristic, direct product factor, Sylow
278	C₂⁴	#41×C₂₃	1	1	yes	characteristic
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
292	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
293	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
294	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	2	2	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	2	4	no	—
300	M₁₆	—	2	4	no	—
301	M₁₆	—	4	2	no	—
302	M₁₆	—	4	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	4	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	4	2	no	—
308	C₂×C₈	C₄²×C₂₃	1	2	yes	—
309	C₂×C₈	C₄²×C₂₃	1	2	yes	—
310	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
311	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
312	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
313	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
314	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
315	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
316	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
317	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
318	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
319	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
320	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
321	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
322	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
323	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
324	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	4	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₂²⋊C₄	—	4	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	4	no	—
335	C₄²	—	4	2	no	—
336	C₄²	C₄⋊C₄×C₂₃	1	4	yes	—
337	C₂³	—	4	2	no	—
338	C₂³	C₂₃×#117	1	1	yes	characteristic
339	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
340	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
341	C₂³	C₂₃×#121	1	1	yes	characteristic
342	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
343	C₂³	C₂₃×#124	1	1	yes	characteristic
344	C₂³	C₂₃×#125	1	1	yes	characteristic
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	4	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	8	4	no	—
349	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
350	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
355	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
358	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
359	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
360	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
361	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
362	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
363	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
364	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
365	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
366	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
367	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
368	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
369	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
370	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
371	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
372	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
373	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
374	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
375	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
376	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
377	C₈	—	2	4	no	—
378	C₈	—	4	4	no	—
379	C₈	C₂₃×#95	1	2	yes	—
380	C₈	C₂₃×#95	1	2	yes	—
381	C₈	C₂₃×#95	1	2	yes	—
382	C₈	C₂₃×#95	1	2	yes	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	—	4	1	no	—
389	C₂²	—	4	1	no	—
390	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
391	C₂²	C₂₃×#380	1	1	yes	characteristic
392	C₂²	C₂₃×#382	1	1	yes	characteristic
393	C₂²	C₂₃×#389	1	1	yes	characteristic
394	C₂²	C₂₃×#389	1	1	yes	characteristic
395	C₂²	C₂₃×#390	1	1	yes	characteristic
396	C₂²	C₂₃×#391	1	1	yes	characteristic
397	C₄	—	4	4	no	—
398	C₄	—	8	4	no	—
399	C₄	C₂×C₂₃×#95	1	1	yes	characteristic
400	C₄	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
401	C₄	C₂₃×#380	1	1	yes	characteristic
402	C₄	C₂₃×#384	1	1	yes	characteristic
403	C₄	C₂₃×#389	1	1	yes	characteristic
404	C₄	C₂₃×#389	1	1	yes	characteristic
405	C₄	C₂₃×#390	1	1	yes	characteristic
406	C₄	C₂₃×#391	1	1	yes	characteristic
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	—	2	1	no	—
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	C₂₃×C₄⋊#95	1	1	yes	characteristic
414	C₂	C₂₃×#1833	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#382
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂×C₂₃×#108	C₄	semidirect
16. C₂₃×#427	C₄	semidirect
21. C₂×C₂₃×#97	C₄	semidirect
22. C₂×C₄×C₈×C₂₃	C₄	semidirect
129. #10868	C₂₃	direct (N × H)
213. C₂×#108	C₄×C₂₃	semidirect
215. #427	C₄×C₂₃	semidirect
220. C₂×#97	C₄×C₂₃	semidirect
221. C₂×C₄×C₈	C₄×C₂₃	semidirect
277. C₂₃	#10868	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂×C₂₃×#427	2
C₂×C₂₃×#405	2
C₂×C₂₃×#344	2
C₂×C₂₃×H(3,ℤ/4)	2
#10868	23

Subgroups — C23×#10868 #19158494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10868 #19158494