Subgroups — C₂₃×#36598 #19184224

Order 5888.

1698 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2862	C₂	1	2	yes	maximal
3	C₂₃×#2582	C₂	1	2	yes	maximal
4	C₂₃×#2571	C₂	1	2	yes	maximal
5	C₂₃×#2570	C₂	1	4	yes	maximal
6	C₂₃×#2568	C₂	1	2	yes	maximal
7	C₂₃×#2511	C₂	1	4	yes	maximal
8	C₂₃×#2446	C₂	1	4	yes	maximal
9	C₂₃×#2365	C₂	1	2	yes	maximal
10	C₂₃×#2355	C₂	1	2	yes	maximal
11	C₂₃×#2355	C₂	1	2	yes	maximal
12	C₄×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal
13	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
14	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#400	C₂²	1	8	yes	—
25	C₂₃×#400	C₂²	1	8	yes	—
26	C₂₃×#398	C₂²	1	8	yes	—
27	C₂₃×#391	C₂²	1	4	yes	—
28	C₂₃×#391	C₂²	1	4	yes	—
29	C₂₃×#391	C₂²	1	4	yes	—
30	C₂₃×#391	C₂²	1	2	yes	—
31	C₂₃×#391	C₂²	1	4	yes	—
32	C₂₃×#391	C₂²	1	4	yes	—
33	C₂₃×#391	C₂²	1	4	yes	—
34	C₂₃×#389	C₂²	1	2	yes	—
35	C₂₃×#389	C₂²	1	4	yes	—
36	C₂₃×#389	C₂²	1	2	yes	—
37	C₂₃×#389	C₂²	1	4	yes	—
38	C₂₃×#389	C₂²	1	4	yes	—
39	C₂₃×#387	C₂²	1	8	yes	—
40	C₂₃×#387	C₂²	1	8	yes	—
41	C₂₃×#387	C₂²	1	8	yes	—
42	C₂₃×#384	C₂²	1	2	yes	—
43	C₂₃×#384	C₂²	1	1	yes	characteristic
44	C₂₃×#384	C₂²	1	2	yes	—
45	C₂₃×#384	C₂²	1	2	yes	—
46	C₂₃×#384	C₂²	1	2	yes	—
47	C₂₃×#382	C₂²	1	4	yes	—
48	C₂₃×#382	C₂²	1	4	yes	—
49	C₂₃×#382	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
56	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
61	C₄³×C₂₃	C₂²	1	2	yes	—
62	C₄³×C₂₃	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊Q₈	—	2	8	no	—
67	C₂₃×C₄⋊Q₈	—	2	4	no	—
68	C₂₃×C₄⋊Q₈	—	2	8	no	—
69	C₂₃×C₄⋊Q₈	—	4	4	no	—
70	C₄×Q₈×C₂₃	—	2	4	no	—
71	C₄×Q₈×C₂₃	—	2	8	no	—
72	C₄×Q₈×C₂₃	—	2	4	no	—
73	C₄×Q₈×C₂₃	—	4	8	no	—
74	C₄×Q₈×C₂₃	Q₈	1	8	yes	complemented
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
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	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
103	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
109	C₂₃×#95	C₂³	1	2	yes	—
110	C₂₃×#95	C₂³	1	2	yes	—
111	C₂₃×#95	C₂³	1	2	yes	—
112	C₂₃×#95	C₂³	1	8	yes	—
113	C₂₃×#95	C₂³	1	8	yes	—
114	C₂₃×#95	C₂³	1	4	yes	—
115	C₂₃×#95	C₂³	1	4	yes	—
116	C₂₃×#95	C₂³	1	8	yes	—
117	C₂₃×#95	C₂³	1	8	yes	—
118	C₂₃×#95	C₂³	1	8	yes	—
119	C₂₃×#95	C₂³	1	8	yes	—
120	C₂×Q₈×C₂₃	—	2	2	no	—
121	C₂×Q₈×C₂₃	—	2	2	no	—
122	C₂×Q₈×C₂₃	—	2	4	no	—
123	C₂×Q₈×C₂₃	—	2	4	no	—
124	C₂×Q₈×C₂₃	—	2	8	no	—
125	C₂×Q₈×C₂₃	—	2	8	no	—
126	C₂×Q₈×C₂₃	—	2	8	no	—
127	C₂×Q₈×C₂₃	—	2	4	no	—
128	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
129	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂⁴	1	2	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	1	yes	characteristic
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	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	4	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	8	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	8	no	—
154	C₄⋊C₄×C₂₃	—	2	4	no	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	—	4	8	no	—
158	C₄⋊C₄×C₂₃	—	4	4	no	—
159	C₄⋊C₄×C₂₃	—	4	4	no	—
160	C₄⋊C₄×C₂₃	C₂×Q₈	1	8	yes	—
161	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
162	C₄²×C₂₃	—	2	4	no	—
163	C₄²×C₂₃	—	2	4	no	—
164	C₄²×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	—	2	4	no	—
166	C₄²×C₂₃	—	2	2	no	—
167	C₄²×C₂₃	—	2	2	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	—	2	4	no	—
170	C₄²×C₂₃	—	2	4	no	—
171	C₄²×C₂₃	—	4	8	no	—
172	C₄²×C₂₃	C₂×Q₈	1	8	yes	—
173	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
174	C₄²×C₂₃	#41	1	4	yes	—
175	C₄²×C₂₃	#41	1	4	yes	—
176	C₄²×C₂₃	#41	1	4	yes	—
177	#36598	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
179	Q₈×C₂₃	—	2	8	no	—
180	Q₈×C₂₃	—	4	8	no	—
181	Q₈×C₂₃	—	4	16	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	1	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	1	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	1	no	—
191	C₂×C₄×C₂₃	—	2	1	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	4	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	8	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	8	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
208	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
209	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
210	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
211	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
212	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
213	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
214	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
215	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
216	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
217	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
218	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
219	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
220	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
221	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
222	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
223	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
224	#2868	C₂×C₂₃	1	1	yes	characteristic
225	#2862	C₂×C₂₃	1	2	yes	—
226	#2582	C₂×C₂₃	1	2	yes	—
227	#2571	C₂×C₂₃	1	2	yes	—
228	#2570	C₂×C₂₃	1	4	yes	—
229	#2568	C₂×C₂₃	1	2	yes	—
230	#2511	C₂×C₂₃	1	4	yes	—
231	#2446	C₂×C₂₃	1	4	yes	—
232	#2365	C₂×C₂₃	1	2	yes	—
233	#2355	C₂×C₂₃	1	2	yes	—
234	#2355	C₂×C₂₃	1	2	yes	—
235	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
236	C₄×C₄⋊Q₈	C₂×C₂₃	1	2	yes	—
237	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
238	C₂²×C₂₃	C₂³×Q₈	1	1	yes	characteristic
239	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
240	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
241	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
242	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
244	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄×C₂₃	—	2	4	no	—
246	C₄×C₂₃	—	2	4	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	2	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	8	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	4	2	no	—
254	C₄×C₂₃	—	4	4	no	—
255	C₄×C₂₃	#533	1	4	yes	—
256	C₄×C₂₃	#556	1	4	yes	—
257	C₄×C₂₃	#557	1	4	yes	—
258	C₄×C₂₃	#559	1	4	yes	—
259	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
261	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
264	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	—
267	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
268	#400	C₂²×C₂₃	1	8	yes	—
269	#400	C₂²×C₂₃	1	8	yes	—
270	#398	C₂²×C₂₃	1	8	yes	—
271	#391	C₂²×C₂₃	1	4	yes	—
272	#391	C₂²×C₂₃	1	2	yes	—
273	#391	C₂²×C₂₃	1	4	yes	—
274	#391	C₂²×C₂₃	1	4	yes	—
275	#391	C₂²×C₂₃	1	4	yes	—
276	#391	C₂²×C₂₃	1	4	yes	—
277	#391	C₂²×C₂₃	1	4	yes	—
278	#389	C₂²×C₂₃	1	2	yes	—
279	#389	C₂²×C₂₃	1	4	yes	—
280	#389	C₂²×C₂₃	1	4	yes	—
281	#389	C₂²×C₂₃	1	2	yes	—
282	#389	C₂²×C₂₃	1	4	yes	—
283	#387	C₂²×C₂₃	1	8	yes	—
284	#387	C₂²×C₂₃	1	8	yes	—
285	#387	C₂²×C₂₃	1	8	yes	—
286	#384	C₂²×C₂₃	1	1	yes	characteristic
287	#384	C₂²×C₂₃	1	2	yes	—
288	#384	C₂²×C₂₃	1	2	yes	—
289	#384	C₂²×C₂₃	1	2	yes	—
290	#384	C₂²×C₂₃	1	2	yes	—
291	#382	C₂²×C₂₃	1	4	yes	—
292	#382	C₂²×C₂₃	1	4	yes	—
293	#382	C₂²×C₂₃	1	4	yes	—
294	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
295	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
296	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
297	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
298	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
299	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
300	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
301	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
302	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
304	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
305	C₄³	C₂²×C₂₃	1	2	yes	—
306	C₄³	C₂²×C₂₃	1	1	yes	characteristic
307	C₂×C₂₃	C₂×#533	1	1	yes	characteristic
308	C₂×C₂₃	C₂×#556	1	1	yes	characteristic
309	C₂×C₂₃	C₂×#557	1	1	yes	characteristic
310	C₂×C₂₃	C₂×#559	1	1	yes	characteristic
311	C₂×C₂₃	#3503	1	1	yes	characteristic
312	C₂×C₂₃	#3509	1	1	yes	characteristic
313	C₂×C₂₃	#3511	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
315	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
316	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
317	C₄⋊Q₈	—	2	8	no	—
318	C₄⋊Q₈	—	2	4	no	—
319	C₄⋊Q₈	—	2	8	no	—
320	C₄⋊Q₈	—	4	4	no	—
321	C₄×Q₈	—	2	4	no	—
322	C₄×Q₈	—	2	8	no	—
323	C₄×Q₈	—	2	4	no	—
324	C₄×Q₈	—	4	8	no	—
325	C₄×Q₈	Q₈×C₂₃	1	8	yes	complemented
326	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
338	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
345	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
346	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
347	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
348	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
349	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
351	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
355	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
356	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
358	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
360	#95	C₂³×C₂₃	1	2	yes	—
361	#95	C₂³×C₂₃	1	2	yes	—
362	#95	C₂³×C₂₃	1	2	yes	—
363	#95	C₂³×C₂₃	1	8	yes	—
364	#95	C₂³×C₂₃	1	8	yes	—
365	#95	C₂³×C₂₃	1	4	yes	—
366	#95	C₂³×C₂₃	1	4	yes	—
367	#95	C₂³×C₂₃	1	8	yes	—
368	#95	C₂³×C₂₃	1	8	yes	—
369	#95	C₂³×C₂₃	1	8	yes	—
370	#95	C₂³×C₂₃	1	8	yes	—
371	C₂₃	#36598	1	1	yes	characteristic, direct product factor, Sylow
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	2	4	no	—
376	C₂×Q₈	—	2	8	no	—
377	C₂×Q₈	—	2	8	no	—
378	C₂×Q₈	—	2	8	no	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
382	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
383	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
384	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
385	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
386	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
387	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	4	8	no	—
410	C₄⋊C₄	—	4	4	no	—
411	C₄⋊C₄	—	4	4	no	—
412	C₄⋊C₄	C₂×Q₈×C₂₃	1	8	yes	—
413	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
414	C₄²	—	2	4	no	—
415	C₄²	—	2	4	no	—
416	C₄²	—	2	8	no	—
417	C₄²	—	2	4	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	4	8	no	—
424	C₄²	C₂×Q₈×C₂₃	1	8	yes	—
425	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
426	C₄²	#41×C₂₃	1	4	yes	—
427	C₄²	#41×C₂₃	1	4	yes	—
428	C₄²	#41×C₂₃	1	4	yes	—
429	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
430	Q₈	—	2	8	no	—
431	Q₈	—	4	8	no	—
432	Q₈	—	4	16	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
459	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
460	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
461	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
462	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
463	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
464	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
465	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
466	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
467	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
468	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
469	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
470	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
471	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
472	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
473	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
474	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
475	C₂²	C₂³×Q₈×C₂₃	1	1	yes	characteristic
476	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
477	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
478	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
479	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
480	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
481	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	4	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	8	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	4	no	—
492	C₄	C₂₃×#533	1	4	yes	—
493	C₄	C₂₃×#556	1	4	yes	—
494	C₄	C₂₃×#557	1	4	yes	—
495	C₄	C₂₃×#559	1	4	yes	—
496	C₂	C₂×C₂₃×#533	1	1	yes	characteristic
497	C₂	C₂×C₂₃×#556	1	1	yes	characteristic
498	C₂	C₂×C₂₃×#557	1	1	yes	characteristic
499	C₂	C₂×C₂₃×#559	1	1	yes	characteristic
500	C₂	C₂₃×#3503	1	1	yes	characteristic
501	C₂	C₂₃×#3509	1	1	yes	characteristic
502	C₂	C₂₃×#3511	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
74. C₄×Q₈×C₂₃	Q₈	semidirect
177. #36598	C₂₃	direct (N × H)
325. C₄×Q₈	Q₈×C₂₃	semidirect
371. C₂₃	#36598	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×#2868	2
C₂₃×#2862	2
C₂₃×#2582	2
C₂₃×#2571	2
C₂₃×#2570	2
C₂₃×#2568	2
C₂₃×#2511	2
C₂₃×#2446	2
C₂₃×#2365	2
C₂₃×#2355	2
C₄×C₂₃×C₄⋊Q₈	2
C₄²×Q₈×C₂₃	2
#36598	23

Subgroups — C23×#36598 #19184224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36598 #19184224