Subgroups — C₂₃×#34402 #19182028

Order 5888.

2410 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2377	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2367	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2353	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2341	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2339	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2333	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2328	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2327	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2318	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#532	C₄	1	16	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#124	C₂²	1	2	yes	—
18	C₂×C₂₃×#121	C₂²	1	2	yes	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#391	C₂²	1	4	yes	—
26	C₂₃×#390	C₂²	1	4	yes	—
27	C₂₃×#390	C₂²	1	8	yes	—
28	C₂₃×#388	C₂²	1	8	yes	—
29	C₂₃×#388	C₂²	1	8	yes	—
30	C₂₃×#387	C₂²	1	8	yes	—
31	C₂₃×#387	C₂²	1	8	yes	—
32	C₂₃×#386	C₂²	1	8	yes	—
33	C₂₃×#386	C₂²	1	8	yes	—
34	C₂₃×#386	C₂²	1	8	yes	—
35	C₂₃×#385	C₂²	1	8	yes	—
36	C₂₃×#383	C₂²	1	2	yes	—
37	C₂₃×#383	C₂²	1	2	yes	—
38	C₂₃×#382	C₂²	1	2	yes	—
39	C₂₃×#382	C₂²	1	2	yes	—
40	C₂₃×#381	C₂²	1	2	yes	—
41	C₂₃×#381	C₂²	1	2	yes	—
42	C₂₃×#381	C₂²	1	4	yes	—
43	C₂₃×#380	C₂²	1	2	yes	—
44	C₂₃×#380	C₂²	1	2	yes	—
45	C₂₃×#380	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
53	C₂₃×#376	C₂²	1	2	yes	—
54	C₂₃×#376	C₂²	1	2	yes	—
55	C₂×C₂₃×#95	C₂²	1	4	yes	—
56	C₂×#41×C₂₃	—	4	8	no	—
57	C₂×#41×C₂₃	C₂×C₄	1	16	yes	complemented
58	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
65	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
69	C₂₃×#127	C₂×C₄	1	8	yes	—
70	C₂₃×#124	C₂×C₄	1	16	yes	complemented
71	C₂₃×#121	C₂×C₄	1	16	yes	complemented
72	C₂₃×#121	C₂×C₄	1	16	yes	complemented
73	C₄×Q₈×C₂₃	—	4	8	no	—
74	C₄×D₄×C₂₃	—	4	8	no	—
75	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
76	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
91	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#95	C₂³	1	2	yes	—
100	C₂₃×#95	C₂³	1	4	yes	—
101	C₂₃×#95	C₂³	1	8	yes	—
102	C₂₃×#95	C₂³	1	4	yes	—
103	C₂₃×#95	C₂³	1	4	yes	—
104	C₂₃×#95	C₂³	1	4	yes	—
105	C₂₃×#95	C₂³	1	2	yes	—
106	C₂₃×#95	C₂³	1	4	yes	—
107	C₂₃×#95	C₂³	1	8	yes	—
108	C₂₃×#95	C₂³	1	4	yes	—
109	C₂₃×#95	C₂³	1	4	yes	—
110	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
111	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
112	#41×C₂₃	—	4	32	no	—
113	C₂×Q₈×C₂₃	—	2	8	no	—
114	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
115	C₂×D₄×C₂₃	—	2	8	no	—
116	C₂×D₄×C₂₃	—	4	4	no	—
117	C₂×D₄×C₂₃	—	4	4	no	—
118	C₂×D₄×C₂₃	—	4	8	no	—
119	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
120	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
121	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	4	2	no	—
125	C₂²×C₄×C₂₃	—	4	2	no	—
126	C₂²×C₄×C₂₃	—	4	8	no	—
127	C₂²×C₄×C₂₃	—	4	8	no	—
128	C₂²×C₄×C₂₃	—	4	8	no	—
129	C₂²×C₄×C₂₃	—	4	8	no	—
130	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
131	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
132	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
136	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	4	4	no	—
146	C₄⋊C₄×C₂₃	—	4	8	no	—
147	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
148	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
149	C₂²⋊C₄×C₂₃	—	4	8	no	—
150	C₂²⋊C₄×C₂₃	—	4	2	no	—
151	C₂²⋊C₄×C₂₃	—	4	8	no	—
152	C₂²⋊C₄×C₂₃	—	4	4	no	—
153	C₂²⋊C₄×C₂₃	—	4	2	no	—
154	C₂²⋊C₄×C₂₃	—	4	8	no	—
155	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
157	C₄²×C₂₃	—	2	4	no	—
158	C₄²×C₂₃	—	2	4	no	—
159	C₄²×C₂₃	—	4	4	no	—
160	C₄²×C₂₃	—	4	8	no	—
161	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
162	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
163	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
164	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
165	#34402	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₂₃	—	2	8	no	—
167	C₂³×C₂₃	—	2	2	no	—
168	C₂³×C₂₃	—	2	2	no	—
169	C₂³×C₂₃	—	2	2	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	2	8	no	—
172	C₂³×C₂₃	—	2	4	no	—
173	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
174	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
175	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
176	Q₈×C₂₃	—	2	16	no	—
177	D₄×C₂₃	—	2	16	no	—
178	D₄×C₂₃	—	4	8	no	—
179	D₄×C₂₃	—	4	8	no	—
180	D₄×C₂₃	—	4	16	no	—
181	C₂×C₄×C₂₃	—	2	8	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	8	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	8	no	—
191	C₂×C₄×C₂₃	—	2	8	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	8	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	4	4	no	—
198	C₂×C₄×C₂₃	—	4	4	no	—
199	C₂×C₄×C₂₃	—	4	16	no	—
200	C₂×C₄×C₂₃	—	8	16	no	—
201	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
202	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
207	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
208	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
209	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
210	#2377	C₂×C₂₃	1	4	yes	complemented
211	#2367	C₂×C₂₃	1	4	yes	complemented
212	#2353	C₂×C₂₃	1	4	yes	complemented
213	#2341	C₂×C₂₃	1	2	yes	complemented
214	#2339	C₂×C₂₃	1	2	yes	complemented
215	#2333	C₂×C₂₃	1	4	yes	complemented
216	#2328	C₂×C₂₃	1	2	yes	complemented
217	#2327	C₂×C₂₃	1	2	yes	complemented
218	#2318	C₂×C₂₃	1	4	yes	complemented
219	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
221	C₂²×C₂₃	—	2	2	no	—
222	C₂²×C₂₃	—	2	8	no	—
223	C₂²×C₂₃	—	2	2	no	—
224	C₂²×C₂₃	—	2	2	no	—
225	C₂²×C₂₃	—	2	8	no	—
226	C₂²×C₂₃	—	2	8	no	—
227	C₂²×C₂₃	—	4	8	no	—
228	C₂²×C₂₃	—	4	8	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
231	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
232	C₂²×C₂₃	#585	1	2	yes	—
233	C₂²×C₂₃	#585	1	2	yes	—
234	C₄×C₂₃	—	2	2	no	—
235	C₄×C₂₃	—	2	8	no	—
236	C₄×C₂₃	—	2	2	no	—
237	C₄×C₂₃	—	2	2	no	—
238	C₄×C₂₃	—	2	2	no	—
239	C₄×C₂₃	—	2	2	no	—
240	C₄×C₂₃	—	4	4	no	—
241	C₄×C₂₃	—	4	8	no	—
242	C₄×C₂₃	—	4	4	no	—
243	C₄×C₂₃	#518	1	4	yes	—
244	C₂²×#41	C₂²×C₂₃	1	2	yes	—
245	#532	C₄×C₂₃	1	16	yes	complemented
246	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#124	C₂²×C₂₃	1	2	yes	—
249	C₂×#121	C₂²×C₂₃	1	2	yes	—
250	C₂×#121	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
252	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
253	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
254	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
255	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
256	#391	C₂²×C₂₃	1	4	yes	—
257	#390	C₂²×C₂₃	1	4	yes	—
258	#390	C₂²×C₂₃	1	8	yes	—
259	#388	C₂²×C₂₃	1	8	yes	—
260	#388	C₂²×C₂₃	1	8	yes	—
261	#387	C₂²×C₂₃	1	8	yes	—
262	#387	C₂²×C₂₃	1	8	yes	—
263	#386	C₂²×C₂₃	1	8	yes	—
264	#386	C₂²×C₂₃	1	8	yes	—
265	#386	C₂²×C₂₃	1	8	yes	—
266	#385	C₂²×C₂₃	1	8	yes	—
267	#383	C₂²×C₂₃	1	2	yes	—
268	#383	C₂²×C₂₃	1	2	yes	—
269	#382	C₂²×C₂₃	1	2	yes	—
270	#382	C₂²×C₂₃	1	2	yes	—
271	#381	C₂²×C₂₃	1	4	yes	—
272	#381	C₂²×C₂₃	1	2	yes	—
273	#381	C₂²×C₂₃	1	2	yes	—
274	#380	C₂²×C₂₃	1	4	yes	—
275	#380	C₂²×C₂₃	1	2	yes	—
276	#380	C₂²×C₂₃	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
279	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
280	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
282	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
283	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
284	#376	C₂²×C₂₃	1	2	yes	—
285	#376	C₂²×C₂₃	1	2	yes	—
286	C₂×#95	C₂²×C₂₃	1	4	yes	—
287	C₂×C₂₃	—	4	2	no	—
288	C₂×C₂₃	—	4	8	no	—
289	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
290	C₂×C₂₃	#3428	1	1	yes	characteristic
291	C₂×C₂₃	#3428	1	1	yes	characteristic
292	C₂×C₂₃	#3562	1	4	yes	—
293	C₂×#41	—	4	8	no	—
294	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
295	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
296	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
297	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
298	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
299	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
300	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
301	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
302	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
303	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
306	#127	C₂×C₄×C₂₃	1	8	yes	—
307	#124	C₂×C₄×C₂₃	1	16	yes	complemented
308	#121	C₂×C₄×C₂₃	1	16	yes	complemented
309	#121	C₂×C₄×C₂₃	1	16	yes	complemented
310	C₄×Q₈	—	4	8	no	—
311	C₄×D₄	—	4	8	no	—
312	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
313	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
314	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
319	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
320	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
327	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
328	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
329	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
330	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
331	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
332	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
336	#95	C₂³×C₂₃	1	2	yes	—
337	#95	C₂³×C₂₃	1	4	yes	—
338	#95	C₂³×C₂₃	1	8	yes	—
339	#95	C₂³×C₂₃	1	4	yes	—
340	#95	C₂³×C₂₃	1	4	yes	—
341	#95	C₂³×C₂₃	1	4	yes	—
342	#95	C₂³×C₂₃	1	2	yes	—
343	#95	C₂³×C₂₃	1	4	yes	—
344	#95	C₂³×C₂₃	1	8	yes	—
345	#95	C₂³×C₂₃	1	4	yes	—
346	#95	C₂³×C₂₃	1	4	yes	—
347	C₂₃	#34402	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
349	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
350	#41	—	4	32	no	—
351	C₂×Q₈	—	2	8	no	—
352	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
353	C₂×D₄	—	2	8	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	4	no	—
357	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
358	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
359	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
360	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	4	8	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	8	no	—
366	C₂²×C₄	—	4	8	no	—
367	C₂²×C₄	—	4	8	no	—
368	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
369	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
370	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
371	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
372	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
373	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
374	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	4	8	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
386	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
387	C₂²⋊C₄	—	4	8	no	—
388	C₂²⋊C₄	—	4	8	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	8	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	2	no	—
393	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
394	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
395	C₄²	—	2	4	no	—
396	C₄²	—	2	4	no	—
397	C₄²	—	4	8	no	—
398	C₄²	—	4	4	no	—
399	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
400	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
401	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
402	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
403	C₂³	—	2	8	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	4	no	—
408	C₂³	—	2	8	no	—
409	C₂³	—	2	4	no	—
410	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
411	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
412	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
413	Q₈	—	2	16	no	—
414	D₄	—	2	16	no	—
415	D₄	—	4	16	no	—
416	D₄	—	4	8	no	—
417	D₄	—	4	8	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	8	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	8	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	4	16	no	—
435	C₂×C₄	—	4	4	no	—
436	C₂×C₄	—	4	4	no	—
437	C₂×C₄	—	8	16	no	—
438	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
440	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
442	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
443	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
444	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
445	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	8	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	8	no	—
451	C₂²	—	2	8	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	8	no	—
454	C₂²	—	4	4	no	—
455	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
456	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
457	C₂²	C₂₃×#585	1	2	yes	—
458	C₂²	C₂₃×#585	1	2	yes	—
459	C₄	—	2	2	no	—
460	C₄	—	2	8	no	—
461	C₄	—	2	2	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	8	no	—
467	C₄	—	4	4	no	—
468	C₄	C₂₃×#518	1	4	yes	—
469	C₂	—	4	2	no	—
470	C₂	—	4	8	no	—
471	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
472	C₂	C₂₃×#3428	1	1	yes	characteristic
473	C₂	C₂₃×#3428	1	1	yes	characteristic
474	C₂	C₂₃×#3562	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#532	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. C₂₃×#2377	C₂	semidirect
3. C₂₃×#2367	C₂	semidirect
4. C₂₃×#2353	C₂	semidirect
5. C₂₃×#2341	C₂	semidirect
6. C₂₃×#2339	C₂	semidirect
7. C₂₃×#2333	C₂	semidirect
8. C₂₃×#2328	C₂	semidirect
9. C₂₃×#2327	C₂	semidirect
10. C₂₃×#2318	C₂	semidirect
11. C₂₃×#2283	C₂	semidirect
12. C₂₃×#2283	C₂	semidirect
14. C₂₃×#532	C₄	semidirect
57. C₂×#41×C₂₃	C₂×C₄	semidirect
68. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
70. C₂₃×#124	C₂×C₄	semidirect
71. C₂₃×#121	C₂×C₄	semidirect
72. C₂₃×#121	C₂×C₄	semidirect
75. C₄×D₄×C₂₃	C₂×C₄	semidirect
76. C₄×D₄×C₂₃	C₂×C₄	semidirect
90. C₂×C₄²×C₂₃	C₂×C₄	semidirect
165. #34402	C₂₃	direct (N × H)
210. #2377	C₂×C₂₃	semidirect
211. #2367	C₂×C₂₃	semidirect
212. #2353	C₂×C₂₃	semidirect
213. #2341	C₂×C₂₃	semidirect
214. #2339	C₂×C₂₃	semidirect
215. #2333	C₂×C₂₃	semidirect
216. #2328	C₂×C₂₃	semidirect
217. #2327	C₂×C₂₃	semidirect
218. #2318	C₂×C₂₃	semidirect
219. #2283	C₂×C₂₃	semidirect
220. #2283	C₂×C₂₃	semidirect
245. #532	C₄×C₂₃	semidirect
294. C₂×#41	C₂×C₄×C₂₃	semidirect
305. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
307. #124	C₂×C₄×C₂₃	semidirect
308. #121	C₂×C₄×C₂₃	semidirect
309. #121	C₂×C₄×C₂₃	semidirect
312. C₄×D₄	C₂×C₄×C₂₃	semidirect
313. C₄×D₄	C₂×C₄×C₂₃	semidirect
327. C₂×C₄²	C₂×C₄×C₂₃	semidirect
347. C₂₃	#34402	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×#532	2
C₂₃×#2377	2
C₂₃×#2367	2
C₂₃×#2353	2
C₂₃×#2341	2
C₂₃×#2339	2
C₂₃×#2333	2
C₂₃×#2328	2
C₂₃×#2327	2
C₂₃×#2318	2
C₂₃×#2283	2
#34402	23

Subgroups — C23×#34402 #19182028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34402 #19182028