Subgroups — C₂₃×#22285 #19169911

Order 5888.

772 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3156	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2957	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2950	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1979	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1976	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1912	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1909	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1905	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1902	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1833	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#137	C₂²	1	2	yes	—
13	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#568	C₂²	1	2	yes	—
16	C₂₃×#536	C₂²	1	2	yes	—
17	C₂₃×#519	C₂²	1	2	yes	—
18	C₂₃×#497	C₄	1	4	yes	complemented
19	C₂₃×#495	C₄	1	2	yes	complemented
20	C₂₃×#494	C₄	1	2	yes	complemented
21	C₂₃×#444	—	2	2	no	—
22	C₂₃×#443	—	2	2	no	—
23	C₂₃×#441	—	2	2	no	—
24	C₄×Q₁₆×C₂₃	—	2	2	no	—
25	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
26	C₂₃×#427	C₂²	1	1	yes	characteristic
27	C₂₃×#421	C₂²	1	2	yes	—
28	C₂₃×#421	C₂²	1	2	yes	—
29	C₂₃×#419	C₂²	1	2	yes	—
30	C₂₃×#419	C₂²	1	2	yes	—
31	C₂×C₂₃×#101	C₂²	1	2	yes	—
32	C₂₃×#410	C₂²	1	2	yes	—
33	C₂₃×#405	C₂²	1	1	yes	characteristic
34	C₂₃×#405	C₂²	1	1	yes	characteristic
35	C₂₃×#405	C₂²	1	1	yes	characteristic
36	C₂₃×#344	C₂²	1	2	yes	—
37	C₂₃×#338	C₂²	1	1	yes	characteristic
38	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
40	C₂₃×H(3,ℤ/4)	C₂²	1	2	yes	—
41	C₂₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
42	C₂×#41×C₂₃	C₂³	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
44	C₂₃×#137	—	4	2	no	—
45	C₂₃×#137	—	4	2	no	—
46	C₂₃×#135	—	2	1	no	—
47	C₂₃×#135	—	2	1	no	—
48	C₂₃×#135	—	2	1	no	—
49	C₂₃×#135	—	2	1	no	—
50	C₂×Q₁₆×C₂₃	—	2	1	no	—
51	C₂×Q₁₆×C₂₃	—	2	1	no	—
52	C₂×Q₁₆×C₂₃	—	2	1	no	—
53	C₂×Q₁₆×C₂₃	—	2	1	no	—
54	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
58	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
60	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂₃×#131	—	2	4	no	—
62	C₂₃×#131	—	4	2	no	—
63	C₂×M₁₆×C₂₃	—	4	2	no	—
64	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
70	C₂₃×#124	C₂×C₄	1	4	yes	—
71	C₂₃×#123	—	4	2	no	—
72	C₂₃×C₂²⋊Q₈	—	4	2	no	—
73	C₄×Q₈×C₂₃	—	2	4	no	—
74	C₄×Q₈×C₂₃	—	4	2	no	—
75	C₂₃×#117	—	4	2	no	—
76	C₂₃×#117	C₂³	1	1	yes	characteristic
77	C₂₃×#117	C₂³	1	2	yes	—
78	C₂₃×#117	C₂³	1	1	yes	characteristic
79	C₂₃×#117	C₂³	1	1	yes	characteristic
80	C₂₃×#108	—	2	1	no	—
81	C₂₃×#108	—	2	1	no	—
82	C₂₃×#107	—	2	1	no	—
83	C₂₃×#106	—	2	1	no	—
84	C₂₃×#104	—	2	4	no	—
85	C₂₃×#104	—	4	2	no	—
86	C₂₃×#103	—	2	4	no	—
87	C₂₃×#103	—	4	2	no	—
88	C₂₃×#101	—	4	2	no	—
89	C₂₃×#99	—	4	2	no	—
90	C₂₃×#97	—	2	1	no	—
91	C₂₃×#97	C₂×C₄	1	2	yes	—
92	C₂₃×#97	D₄	1	2	yes	—
93	C₄×C₈×C₂₃	—	2	1	no	—
94	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
95	C₄×C₈×C₂₃	D₄	1	2	yes	—
96	#41×C₂₃	—	2	2	no	—
97	#41×C₂₃	—	2	2	no	—
98	#41×C₂₃	—	2	2	no	—
99	#41×C₂₃	—	2	2	no	—
100	C₂×Q₈×C₂₃	—	2	2	no	—
101	C₂×Q₈×C₂₃	—	2	2	no	—
102	C₂×Q₈×C₂₃	—	2	2	no	—
103	C₂×Q₈×C₂₃	—	2	2	no	—
104	C₂×Q₈×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
106	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
107	C₂×D₄×C₂₃	—	2	2	no	—
108	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	2	no	—
111	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
112	Q₁₆×C₂₃	—	2	2	no	—
113	Q₁₆×C₂₃	—	2	2	no	—
114	Q₁₆×C₂₃	—	2	2	no	—
115	Q₁₆×C₂₃	—	2	2	no	—
116	Q₁₆×C₂₃	—	2	1	no	—
117	Q₁₆×C₂₃	—	2	1	no	—
118	Q₁₆×C₂₃	—	4	4	no	—
119	SD₁₆×C₂₃	—	2	2	no	—
120	SD₁₆×C₂₃	—	2	2	no	—
121	SD₁₆×C₂₃	—	4	4	no	—
122	D₈×C₂₃	—	2	1	no	—
123	D₈×C₂₃	—	2	1	no	—
124	M₁₆×C₂₃	—	2	1	no	—
125	M₁₆×C₂₃	—	2	1	no	—
126	M₁₆×C₂₃	—	2	1	no	—
127	M₁₆×C₂₃	—	2	1	no	—
128	M₁₆×C₂₃	—	2	2	no	—
129	M₁₆×C₂₃	—	2	1	no	—
130	M₁₆×C₂₃	—	2	2	no	—
131	M₁₆×C₂₃	—	2	1	no	—
132	M₁₆×C₂₃	—	4	4	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	4	4	no	—
136	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
140	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
143	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	4	4	no	—
149	C₄⋊C₄×C₂₃	—	4	4	no	—
150	C₂²⋊C₄×C₂₃	—	2	1	no	—
151	C₂²⋊C₄×C₂₃	—	2	1	no	—
152	C₂²⋊C₄×C₂₃	—	2	1	no	—
153	C₂²⋊C₄×C₂₃	—	2	2	no	—
154	C₂²⋊C₄×C₂₃	—	4	4	no	—
155	C₄²×C₂₃	—	2	1	no	—
156	C₄²×C₂₃	—	2	2	no	—
157	C₄²×C₂₃	—	4	4	no	—
158	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
159	C₄²×C₂₃	C₂×D₄	1	2	yes	—
160	#22285	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
163	Q₈×C₂₃	—	2	2	no	—
164	Q₈×C₂₃	—	2	4	no	—
165	Q₈×C₂₃	—	2	4	no	—
166	Q₈×C₂₃	—	2	2	no	—
167	Q₈×C₂₃	—	4	2	no	—
168	Q₈×C₂₃	—	4	2	no	—
169	D₄×C₂₃	—	2	2	no	—
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	4	2	no	—
172	D₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	2	4	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	4	2	no	—
178	C₂×C₄×C₂₃	—	4	2	no	—
179	C₂×C₄×C₂₃	—	4	2	no	—
180	C₂×C₄×C₂₃	—	4	2	no	—
181	C₂×C₄×C₂₃	—	4	2	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
188	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
189	C₈×C₂₃	—	2	2	no	—
190	C₈×C₂₃	—	2	2	no	—
191	C₈×C₂₃	—	4	2	no	—
192	C₈×C₂₃	—	4	1	no	—
193	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
194	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
195	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
196	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
197	#3156	C₂×C₂₃	1	1	yes	characteristic
198	#2957	C₂×C₂₃	1	1	yes	characteristic
199	#2950	C₂×C₂₃	1	1	yes	characteristic, complemented
200	#1979	C₂×C₂₃	1	1	yes	characteristic, complemented
201	#1976	C₂×C₂₃	1	1	yes	characteristic
202	#1912	C₂×C₂₃	1	2	yes	complemented
203	#1909	C₂×C₂₃	1	2	yes	complemented
204	#1905	C₂×C₂₃	1	2	yes	complemented
205	#1902	C₂×C₂₃	1	2	yes	complemented
206	#1833	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1747	C₂×C₂₃	1	1	yes	characteristic, complemented
208	C₂²×C₂₃	—	4	1	no	—
209	C₂²×C₂₃	—	4	2	no	—
210	C₂²×C₂₃	—	8	2	no	—
211	C₂²×C₂₃	#518	1	1	yes	characteristic
212	C₂²×C₂₃	#535	1	1	yes	characteristic
213	C₂²×C₂₃	#540	1	1	yes	characteristic
214	C₄×C₂₃	—	4	2	no	—
215	C₄×C₂₃	—	4	2	no	—
216	C₄×C₂₃	—	4	2	no	—
217	C₄×C₂₃	—	4	4	no	—
218	C₄×C₂₃	—	8	2	no	—
219	C₄×C₂₃	—	8	1	no	—
220	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₂₃	#518	1	1	yes	characteristic
222	C₄×C₂₃	#535	1	1	yes	characteristic
223	C₄×C₂₃	#540	1	1	yes	characteristic
224	C₂×#137	C₂²×C₂₃	1	2	yes	—
225	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
226	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
227	#568	C₂²×C₂₃	1	2	yes	—
228	#536	C₂²×C₂₃	1	2	yes	—
229	#519	C₂²×C₂₃	1	2	yes	—
230	#497	C₄×C₂₃	1	4	yes	complemented
231	#495	C₄×C₂₃	1	2	yes	complemented
232	#494	C₄×C₂₃	1	2	yes	complemented
233	#444	—	2	2	no	—
234	#443	—	2	2	no	—
235	#441	—	2	2	no	—
236	C₄×Q₁₆	—	2	2	no	—
237	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
238	#427	C₂²×C₂₃	1	1	yes	characteristic
239	#421	C₂²×C₂₃	1	2	yes	—
240	#421	C₂²×C₂₃	1	2	yes	—
241	#419	C₂²×C₂₃	1	2	yes	—
242	#419	C₂²×C₂₃	1	2	yes	—
243	C₂×#101	C₂²×C₂₃	1	2	yes	—
244	#410	C₂²×C₂₃	1	2	yes	—
245	#405	C₂²×C₂₃	1	1	yes	characteristic
246	#405	C₂²×C₂₃	1	1	yes	characteristic
247	#405	C₂²×C₂₃	1	1	yes	characteristic
248	#344	C₂²×C₂₃	1	2	yes	—
249	#338	C₂²×C₂₃	1	1	yes	characteristic
250	H(3,ℤ/4)	C₂²×C₂₃	1	2	yes	—
251	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
252	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
253	H(3,ℤ/4)	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×C₂₃	—	2	1	no	—
255	C₂×C₂₃	—	2	1	no	—
256	C₂×C₂₃	—	2	1	no	—
257	C₂×C₂₃	—	8	2	no	—
258	C₂×C₂₃	#2328	1	1	yes	characteristic
259	C₂×#41	C₂³×C₂₃	1	2	yes	—
260	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
261	#137	—	4	2	no	—
262	#137	—	4	2	no	—
263	#135	—	2	1	no	—
264	#135	—	2	1	no	—
265	#135	—	2	1	no	—
266	#135	—	2	1	no	—
267	C₂×Q₁₆	—	2	1	no	—
268	C₂×Q₁₆	—	2	1	no	—
269	C₂×Q₁₆	—	2	1	no	—
270	C₂×Q₁₆	—	2	1	no	—
271	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
272	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
273	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
274	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
275	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
276	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
277	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
278	#131	—	2	4	no	—
279	#131	—	4	2	no	—
280	C₂×M₁₆	—	4	2	no	—
281	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
282	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
283	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
284	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
286	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
287	#124	C₂×C₄×C₂₃	1	4	yes	—
288	#123	—	4	2	no	—
289	C₂²⋊Q₈	—	4	2	no	—
290	C₄×Q₈	—	2	4	no	—
291	C₄×Q₈	—	4	2	no	—
292	#117	—	4	2	no	—
293	#117	C₂³×C₂₃	1	2	yes	—
294	#117	C₂³×C₂₃	1	1	yes	characteristic
295	#117	C₂³×C₂₃	1	1	yes	characteristic
296	#117	C₂³×C₂₃	1	1	yes	characteristic
297	#108	—	2	1	no	—
298	#108	—	2	1	no	—
299	#107	—	2	1	no	—
300	#106	—	2	1	no	—
301	#104	—	2	4	no	—
302	#104	—	4	2	no	—
303	#103	—	2	4	no	—
304	#103	—	4	2	no	—
305	#101	—	4	2	no	—
306	#99	—	4	2	no	—
307	#97	—	2	1	no	—
308	#97	C₂×C₄×C₂₃	1	2	yes	—
309	#97	D₄×C₂₃	1	2	yes	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
312	C₄×C₈	D₄×C₂₃	1	2	yes	—
313	C₂₃	#22285	1	1	yes	characteristic, direct product factor, Sylow
314	#41	—	2	2	no	—
315	#41	—	2	2	no	—
316	#41	—	2	2	no	—
317	#41	—	2	2	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
324	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
330	Q₁₆	—	2	2	no	—
331	Q₁₆	—	2	2	no	—
332	Q₁₆	—	2	2	no	—
333	Q₁₆	—	2	2	no	—
334	Q₁₆	—	2	1	no	—
335	Q₁₆	—	2	1	no	—
336	Q₁₆	—	4	4	no	—
337	SD₁₆	—	2	2	no	—
338	SD₁₆	—	2	2	no	—
339	SD₁₆	—	4	4	no	—
340	D₈	—	2	1	no	—
341	D₈	—	2	1	no	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	2	1	no	—
344	M₁₆	—	2	1	no	—
345	M₁₆	—	2	1	no	—
346	M₁₆	—	2	2	no	—
347	M₁₆	—	2	1	no	—
348	M₁₆	—	2	2	no	—
349	M₁₆	—	2	1	no	—
350	M₁₆	—	4	4	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	4	4	no	—
354	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
355	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
356	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
357	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
358	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
360	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
361	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	2	no	—
372	C₂²⋊C₄	—	4	4	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	4	4	no	—
376	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
377	C₄²	C₂×D₄×C₂₃	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	2	4	no	—
382	Q₈	—	2	4	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	4	2	no	—
386	D₄	—	2	2	no	—
387	D₄	—	2	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
403	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
404	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
405	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
406	C₈	—	2	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	2	no	—
409	C₈	—	4	1	no	—
410	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
411	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
412	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
413	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
414	C₂²	—	4	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	C₂₃×#518	1	1	yes	characteristic
418	C₂²	C₂₃×#535	1	1	yes	characteristic
419	C₂²	C₂₃×#540	1	1	yes	characteristic
420	C₄	—	4	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	4	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	1	no	—
426	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
427	C₄	C₂₃×#518	1	1	yes	characteristic
428	C₄	C₂₃×#535	1	1	yes	characteristic
429	C₄	C₂₃×#540	1	1	yes	characteristic
430	C₂	—	2	1	no	—
431	C₂	—	2	1	no	—
432	C₂	—	2	1	no	—
433	C₂	—	8	2	no	—
434	C₂	C₂₃×#2328	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#2328
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2950	C₂	semidirect
4. C₂₃×#1979	C₂	semidirect
6. C₂₃×#1912	C₂	semidirect
7. C₂₃×#1909	C₂	semidirect
8. C₂₃×#1905	C₂	semidirect
9. C₂₃×#1902	C₂	semidirect
10. C₂₃×#1833	C₂	semidirect
11. C₂₃×#1747	C₂	semidirect
18. C₂₃×#497	C₄	semidirect
19. C₂₃×#495	C₄	semidirect
20. C₂₃×#494	C₄	semidirect
160. #22285	C₂₃	direct (N × H)
199. #2950	C₂×C₂₃	semidirect
200. #1979	C₂×C₂₃	semidirect
202. #1912	C₂×C₂₃	semidirect
203. #1909	C₂×C₂₃	semidirect
204. #1905	C₂×C₂₃	semidirect
205. #1902	C₂×C₂₃	semidirect
206. #1833	C₂×C₂₃	semidirect
207. #1747	C₂×C₂₃	semidirect
230. #497	C₄×C₂₃	semidirect
231. #495	C₄×C₂₃	semidirect
232. #494	C₄×C₂₃	semidirect
313. C₂₃	#22285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3156	2
C₂₃×#2957	2
C₂₃×#2950	2
C₂₃×#1979	2
C₂₃×#1976	2
C₂₃×#1912	2
C₂₃×#1909	2
C₂₃×#1905	2
C₂₃×#1902	2
C₂₃×#1833	2
C₂₃×#1747	2
#22285	23

Subgroups — C23×#22285 #19169911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22285 #19169911