Subgroups — C₂₃×#34086 #19181712

Order 5888.

2638 conjugacy classes of subgroups.

The subgroup lattice has 492 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×#2337	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2333	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2331	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2328	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2324	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2297	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2293	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2290	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₂₃	C₂	1	2	yes	maximal, complemented
12	C₄×C₂₃×#117	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×D₄×C₂₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂₃×#117	C₂²	1	2	yes	—
24	C₂×C₂₃×#117	C₂²	1	4	yes	—
25	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
26	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#391	C₂²	1	8	yes	—
28	C₂₃×#390	C₂²	1	4	yes	—
29	C₂₃×#390	C₂²	1	8	yes	—
30	C₂₃×#389	C₂²	1	4	yes	—
31	C₂₃×#388	C₂²	1	8	yes	—
32	C₂₃×#388	C₂²	1	8	yes	—
33	C₂₃×#387	C₂²	1	8	yes	—
34	C₂₃×#386	C₂²	1	8	yes	—
35	C₂₃×#386	C₂²	1	8	yes	—
36	C₂₃×#386	C₂²	1	8	yes	—
37	C₂₃×#385	C₂²	1	8	yes	—
38	C₂₃×#383	C₂²	1	2	yes	—
39	C₂₃×#382	C₂²	1	2	yes	—
40	C₂₃×#381	C₂²	1	2	yes	—
41	C₂₃×#380	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
46	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
47	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₂₃×#376	C₂²	1	2	yes	—
53	C₂₃×#376	C₂²	1	2	yes	—
54	C₄³×C₂₃	C₂²	1	1	yes	characteristic
55	C₄³×C₂₃	C₂²	1	1	yes	characteristic
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	2	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
62	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
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×D₄×C₂₃	—	2	16	no	—
74	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
75	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
76	C₂₃×#117	—	2	8	no	—
77	C₂₃×#117	—	4	8	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	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	—	4	8	no	—
91	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
92	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
104	C₂₃×#95	C₂³	1	4	yes	—
105	C₂₃×#95	C₂³	1	4	yes	—
106	C₂₃×#95	C₂³	1	4	yes	—
107	C₂₃×#95	C₂³	1	4	yes	—
108	C₂₃×#95	C₂³	1	4	yes	—
109	C₂₃×#95	C₂³	1	8	yes	—
110	C₂₃×#95	C₂³	1	4	yes	—
111	C₂₃×#95	C₂³	1	8	yes	—
112	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
113	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
114	#41×C₂₃	—	4	32	no	—
115	C₂×Q₈×C₂₃	—	4	4	no	—
116	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
117	C₂×D₄×C₂₃	—	2	8	no	—
118	C₂×D₄×C₂₃	—	4	4	no	—
119	C₂×D₄×C₂₃	—	4	4	no	—
120	C₂×D₄×C₂₃	—	4	4	no	—
121	C₂×D₄×C₂₃	—	4	4	no	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
123	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
125	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	2	8	no	—
128	C₂²×C₄×C₂₃	—	2	8	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	16	no	—
131	C₂²×C₄×C₂₃	—	4	8	no	—
132	C₂²×C₄×C₂₃	—	4	2	no	—
133	C₂²×C₄×C₂₃	—	4	8	no	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
137	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	8	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	4	yes	—
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	—	4	8	no	—
151	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
152	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	16	no	—
155	C₂²⋊C₄×C₂₃	—	4	8	no	—
156	C₂²⋊C₄×C₂₃	—	4	4	no	—
157	C₂²⋊C₄×C₂₃	—	4	8	no	—
158	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
159	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
160	C₄²×C₂₃	—	2	4	no	—
161	C₄²×C₂₃	—	2	4	no	—
162	C₄²×C₂₃	—	2	8	no	—
163	C₄²×C₂₃	—	2	4	no	—
164	C₄²×C₂₃	—	4	8	no	—
165	C₄²×C₂₃	—	4	2	no	—
166	C₄²×C₂₃	—	4	2	no	—
167	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
168	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
169	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
170	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
171	C₄²×C₂₃	#41	1	8	yes	complemented
172	#34086	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	8	no	—
175	C₂³×C₂₃	—	2	4	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	8	no	—
180	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
181	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
182	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
183	Q₈×C₂₃	—	4	8	no	—
184	D₄×C₂₃	—	2	16	no	—
185	D₄×C₂₃	—	4	8	no	—
186	D₄×C₂₃	—	4	8	no	—
187	D₄×C₂₃	—	4	16	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	8	no	—
197	C₂×C₄×C₂₃	—	2	32	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	8	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	8	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	—	4	32	no	—
207	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂×#41	1	8	yes	—
215	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
216	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
217	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
218	#2337	C₂×C₂₃	1	4	yes	complemented
219	#2333	C₂×C₂₃	1	4	yes	complemented
220	#2331	C₂×C₂₃	1	2	yes	complemented
221	#2328	C₂×C₂₃	1	2	yes	complemented
222	#2324	C₂×C₂₃	1	4	yes	complemented
223	#2297	C₂×C₂₃	1	4	yes	complemented
224	#2293	C₂×C₂₃	1	2	yes	complemented
225	#2290	C₂×C₂₃	1	4	yes	complemented
226	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₄²×D₄	C₂×C₂₃	1	2	yes	complemented
228	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂²×C₂₃	—	2	2	no	—
230	C₂²×C₂₃	—	2	8	no	—
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	2	8	no	—
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	2	8	no	—
235	C₂²×C₂₃	—	4	8	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	8	no	—
238	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
239	C₂²×C₂₃	C₂²×#41	1	2	yes	—
240	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
241	C₂²×C₂₃	#585	1	2	yes	—
242	C₄×C₂₃	—	2	2	no	—
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	2	8	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	8	no	—
248	C₄×C₂₃	—	4	4	no	—
249	C₄×C₂₃	—	4	8	no	—
250	C₄×C₂₃	C₄×#41	1	4	yes	—
251	C₄×C₂₃	#518	1	4	yes	—
252	C₂²×#41	C₂²×C₂₃	1	2	yes	—
253	#532	C₄×C₂₃	1	16	yes	complemented
254	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×#124	C₂²×C₂₃	1	2	yes	—
257	C₂×#121	C₂²×C₂₃	1	2	yes	—
258	C₂×#121	C₂²×C₂₃	1	2	yes	—
259	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
260	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
261	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
262	C₂×#117	C₂²×C₂₃	1	2	yes	—
263	C₂×#117	C₂²×C₂₃	1	4	yes	—
264	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
265	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
266	#391	C₂²×C₂₃	1	8	yes	—
267	#390	C₂²×C₂₃	1	4	yes	—
268	#390	C₂²×C₂₃	1	8	yes	—
269	#389	C₂²×C₂₃	1	4	yes	—
270	#388	C₂²×C₂₃	1	8	yes	—
271	#388	C₂²×C₂₃	1	8	yes	—
272	#387	C₂²×C₂₃	1	8	yes	—
273	#386	C₂²×C₂₃	1	8	yes	—
274	#386	C₂²×C₂₃	1	8	yes	—
275	#386	C₂²×C₂₃	1	8	yes	—
276	#385	C₂²×C₂₃	1	8	yes	—
277	#383	C₂²×C₂₃	1	2	yes	—
278	#382	C₂²×C₂₃	1	2	yes	—
279	#381	C₂²×C₂₃	1	2	yes	—
280	#380	C₂²×C₂₃	1	2	yes	—
281	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
282	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
283	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
284	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
285	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
286	C₄×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
287	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
288	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
289	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
290	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
291	#376	C₂²×C₂₃	1	2	yes	—
292	#376	C₂²×C₂₃	1	2	yes	—
293	C₄³	C₂²×C₂₃	1	1	yes	characteristic
294	C₄³	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₂₃	—	4	2	no	—
296	C₂×C₂₃	—	4	8	no	—
297	C₂×C₂₃	C₂×C₄×#41	1	1	yes	characteristic
298	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
299	C₂×C₂₃	#3428	1	1	yes	characteristic
300	C₂×C₂₃	#3460	1	4	yes	—
301	C₂×#41	—	4	8	no	—
302	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
303	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
304	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
305	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
306	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
307	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
308	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
309	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
310	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
312	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
314	#127	C₂×C₄×C₂₃	1	8	yes	—
315	#124	C₂×C₄×C₂₃	1	16	yes	complemented
316	#121	C₂×C₄×C₂₃	1	16	yes	complemented
317	#121	C₂×C₄×C₂₃	1	16	yes	complemented
318	C₄×D₄	—	2	16	no	—
319	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
320	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
321	#117	—	2	8	no	—
322	#117	—	4	8	no	—
323	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
326	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂×C₄²	—	4	8	no	—
336	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
337	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
338	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
339	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
340	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
341	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
342	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
349	#95	C₂³×C₂₃	1	4	yes	—
350	#95	C₂³×C₂₃	1	4	yes	—
351	#95	C₂³×C₂₃	1	4	yes	—
352	#95	C₂³×C₂₃	1	4	yes	—
353	#95	C₂³×C₂₃	1	4	yes	—
354	#95	C₂³×C₂₃	1	8	yes	—
355	#95	C₂³×C₂₃	1	4	yes	—
356	#95	C₂³×C₂₃	1	8	yes	—
357	C₂₃	#34086	1	1	yes	characteristic, direct product factor, Sylow
358	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
359	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
360	#41	—	4	32	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	—	4	4	no	—
367	C₂×D₄	—	4	4	no	—
368	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
369	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
370	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
371	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	2	8	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	16	no	—
377	C₂²×C₄	—	4	8	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	8	no	—
380	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
382	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
383	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
385	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
387	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₄⋊C₄	—	2	8	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	4	8	no	—
397	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
398	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	16	no	—
401	C₂²⋊C₄	—	4	8	no	—
402	C₂²⋊C₄	—	4	8	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
405	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
406	C₄²	—	2	4	no	—
407	C₄²	—	2	4	no	—
408	C₄²	—	2	8	no	—
409	C₄²	—	2	4	no	—
410	C₄²	—	4	8	no	—
411	C₄²	—	4	2	no	—
412	C₄²	—	4	2	no	—
413	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
414	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
415	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
416	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
417	C₄²	#41×C₂₃	1	8	yes	complemented
418	C₂³	—	2	2	no	—
419	C₂³	—	2	8	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	8	no	—
425	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
426	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
427	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
428	Q₈	—	4	8	no	—
429	D₄	—	2	16	no	—
430	D₄	—	4	8	no	—
431	D₄	—	4	16	no	—
432	D₄	—	4	8	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	32	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	—	4	32	no	—
452	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
453	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
455	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
456	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
457	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
458	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
459	C₂×C₄	C₂×#41×C₂₃	1	8	yes	—
460	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
461	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	8	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	8	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	2	8	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	8	no	—
471	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
472	C₂²	C₂²×#41×C₂₃	1	2	yes	—
473	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
474	C₂²	C₂₃×#585	1	2	yes	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	8	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	8	no	—
481	C₄	—	4	8	no	—
482	C₄	—	4	4	no	—
483	C₄	C₄×#41×C₂₃	1	4	yes	—
484	C₄	C₂₃×#518	1	4	yes	—
485	C₂	—	4	2	no	—
486	C₂	—	4	8	no	—
487	C₂	C₂×C₄×#41×C₂₃	1	1	yes	characteristic
488	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
489	C₂	C₂₃×#3428	1	1	yes	characteristic
490	C₂	C₂₃×#3460	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2337	C₂	semidirect
3. C₂₃×#2333	C₂	semidirect
4. C₂₃×#2331	C₂	semidirect
5. C₂₃×#2328	C₂	semidirect
6. C₂₃×#2324	C₂	semidirect
7. C₂₃×#2297	C₂	semidirect
8. C₂₃×#2293	C₂	semidirect
9. C₂₃×#2290	C₂	semidirect
10. C₂₃×#2283	C₂	semidirect
11. C₄²×D₄×C₂₃	C₂	semidirect
12. C₄×C₂₃×#117	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
74. C₄×D₄×C₂₃	C₂×C₄	semidirect
75. C₄×D₄×C₂₃	C₂×C₄	semidirect
91. C₂×C₄²×C₂₃	C₂×C₄	semidirect
171. C₄²×C₂₃	#41	semidirect
172. #34086	C₂₃	direct (N × H)
218. #2337	C₂×C₂₃	semidirect
219. #2333	C₂×C₂₃	semidirect
220. #2331	C₂×C₂₃	semidirect
221. #2328	C₂×C₂₃	semidirect
222. #2324	C₂×C₂₃	semidirect
223. #2297	C₂×C₂₃	semidirect
224. #2293	C₂×C₂₃	semidirect
225. #2290	C₂×C₂₃	semidirect
226. #2283	C₂×C₂₃	semidirect
227. C₄²×D₄	C₂×C₂₃	semidirect
228. C₄×#117	C₂×C₂₃	semidirect
253. #532	C₄×C₂₃	semidirect
302. C₂×#41	C₂×C₄×C₂₃	semidirect
313. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
315. #124	C₂×C₄×C₂₃	semidirect
316. #121	C₂×C₄×C₂₃	semidirect
317. #121	C₂×C₄×C₂₃	semidirect
319. C₄×D₄	C₂×C₄×C₂₃	semidirect
320. C₄×D₄	C₂×C₄×C₂₃	semidirect
336. C₂×C₄²	C₂×C₄×C₂₃	semidirect
357. C₂₃	#34086	direct (N × H)
417. C₄²	#41×C₂₃	semidirect

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×#2337	2
C₂₃×#2333	2
C₂₃×#2331	2
C₂₃×#2328	2
C₂₃×#2324	2
C₂₃×#2297	2
C₂₃×#2293	2
C₂₃×#2290	2
C₂₃×#2283	2
C₄²×D₄×C₂₃	2
C₄×C₂₃×#117	2
#34086	23

Subgroups — C23×#34086 #19181712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34086 #19181712