Subgroups — C₂₃×#35019 #19182645

Order 5888.

2994 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2850	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2639	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2591	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2561	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2504	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2503	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2436	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2435	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2308	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#124	C₂²	1	4	yes	complemented
18	C₂×C₂₃×C₂²⋊Q₈	—	2	8	no	—
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
23	C₂×C₂₃×#121	—	2	8	no	—
24	C₂×C₂₃×#121	C₂²	1	2	yes	—
25	C₂×C₂₃×#121	C₂²	1	2	yes	—
26	C₂×C₂₃×#121	C₂²	1	2	yes	—
27	C₂×C₂₃×#121	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
31	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₂²×C₄²×C₂₃	C₂²	1	2	yes	complemented
35	C₂₃×#400	C₂²	1	8	yes	complemented
36	C₂₃×#397	C₂²	1	8	yes	complemented
37	C₂₃×#394	C₂²	1	8	yes	complemented
38	C₂₃×#393	C₂²	1	8	yes	complemented
39	C₂₃×#391	C₂²	1	4	yes	complemented
40	C₂₃×#390	C₂²	1	4	yes	complemented
41	C₂₃×#388	C₂²	1	16	yes	complemented
42	C₂₃×#386	C₂²	1	8	yes	complemented
43	C₂₃×#386	C₂²	1	8	yes	complemented
44	C₂₃×#386	C₂²	1	16	yes	complemented
45	C₂₃×#382	C₂²	1	4	yes	complemented
46	C₂₃×#381	C₂²	1	8	yes	—
47	C₂₃×#379	C₂²	1	8	yes	—
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	complemented
52	C₂×C₂₃×#95	C₂²	1	2	yes	complemented
53	C₂⁵×C₂₃	C₂³	1	2	yes	—
54	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	4	no	—
56	C₂²×Q₈×C₂₃	—	4	4	no	—
57	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
58	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₂₃	—	2	8	no	—
60	C₂²×D₄×C₂₃	—	2	16	no	—
61	C₂²×D₄×C₂₃	—	2	4	no	—
62	C₂²×D₄×C₂₃	—	4	4	no	—
63	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
67	C₂³×C₄×C₂₃	—	4	4	no	—
68	C₂³×C₄×C₂₃	D₄	1	8	yes	—
69	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
75	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
76	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#124	—	4	8	no	—
80	C₂₃×C₂²⋊Q₈	—	2	8	no	—
81	C₂₃×C₂²⋊Q₈	—	2	8	no	—
82	C₂₃×C₂²⋊Q₈	—	2	8	no	—
83	C₂₃×C₂²⋊Q₈	—	2	8	no	—
84	C₂₃×#121	—	2	8	no	—
85	C₂₃×#121	—	2	8	no	—
86	C₂₃×#121	—	2	8	no	—
87	C₂₃×#121	—	2	8	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
100	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
107	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
108	C₂×C₄²×C₂₃	—	4	4	no	—
109	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
111	C₂₃×#95	C₂³	1	4	yes	—
112	C₂₃×#95	C₂³	1	4	yes	—
113	C₂₃×#95	C₂³	1	16	yes	—
114	C₂₃×#95	C₂³	1	16	yes	—
115	C₂⁴×C₂₃	—	2	4	no	—
116	C₂⁴×C₂₃	—	2	8	no	—
117	C₂⁴×C₂₃	—	4	2	no	—
118	C₂⁴×C₂₃	—	4	2	no	—
119	C₂⁴×C₂₃	—	4	4	no	—
120	C₂⁴×C₂₃	—	4	4	no	—
121	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
122	C₂⁴×C₂₃	#41	1	8	yes	—
123	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
124	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
125	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
126	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
127	C₂×Q₈×C₂₃	—	2	4	no	—
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	—	2	8	no	—
130	C₂×Q₈×C₂₃	—	4	4	no	—
131	C₂×Q₈×C₂₃	—	4	16	no	—
132	C₂×D₄×C₂₃	—	2	8	no	—
133	C₂×D₄×C₂₃	—	2	8	no	—
134	C₂×D₄×C₂₃	—	2	16	no	—
135	C₂×D₄×C₂₃	—	2	16	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	4	no	—
138	C₂×D₄×C₂₃	—	2	8	no	—
139	C₂×D₄×C₂₃	—	4	4	no	—
140	C₂×D₄×C₂₃	—	4	16	no	—
141	C₂×D₄×C₂₃	—	4	32	no	—
142	C₂²×C₄×C₂₃	—	2	4	no	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	4	no	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	—	2	8	no	—
149	C₂²×C₄×C₂₃	—	2	8	no	—
150	C₂²×C₄×C₂₃	—	4	16	no	—
151	C₂²×C₄×C₂₃	—	4	4	no	—
152	C₂²×C₄×C₂₃	—	4	2	no	—
153	C₂²×C₄×C₂₃	—	4	2	no	—
154	C₂²×C₄×C₂₃	—	4	2	no	—
155	C₂²×C₄×C₂₃	—	4	4	no	—
156	C₂²×C₄×C₂₃	—	4	4	no	—
157	C₂²×C₄×C₂₃	—	4	4	no	—
158	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
165	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	4	no	—
175	C₄⋊C₄×C₂₃	—	2	4	no	—
176	C₄⋊C₄×C₂₃	—	2	8	no	—
177	C₄⋊C₄×C₂₃	—	2	8	no	—
178	C₂²⋊C₄×C₂₃	—	2	8	no	—
179	C₂²⋊C₄×C₂₃	—	2	8	no	—
180	C₂²⋊C₄×C₂₃	—	2	8	no	—
181	C₂²⋊C₄×C₂₃	—	2	8	no	—
182	C₂²⋊C₄×C₂₃	—	2	8	no	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₂²⋊C₄×C₂₃	—	4	16	no	—
185	C₂²⋊C₄×C₂₃	—	4	16	no	—
186	C₄²×C₂₃	—	2	8	no	—
187	C₄²×C₂₃	—	4	4	no	—
188	#35019	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	—	2	2	no	—
191	C₂³×C₂₃	—	2	2	no	—
192	C₂³×C₂₃	—	2	2	no	—
193	C₂³×C₂₃	—	2	2	no	—
194	C₂³×C₂₃	—	2	2	no	—
195	C₂³×C₂₃	—	2	4	no	—
196	C₂³×C₂₃	—	2	8	no	—
197	C₂³×C₂₃	—	2	8	no	—
198	C₂³×C₂₃	—	2	8	no	—
199	C₂³×C₂₃	—	4	4	no	—
200	C₂³×C₂₃	—	4	4	no	—
201	C₂³×C₂₃	—	4	16	no	—
202	C₂³×C₂₃	—	4	8	no	—
203	C₂³×C₂₃	—	4	8	no	—
204	C₂³×C₂₃	—	4	8	no	—
205	C₂³×C₂₃	—	8	16	no	—
206	C₂³×C₂₃	#124	1	16	yes	complemented
207	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
208	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
209	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
210	C₂³×C₂₃	C₂×#41	1	4	yes	—
211	C₂³×C₂₃	C₂×#41	1	4	yes	—
212	C₂³×C₂₃	C₂×#41	1	4	yes	—
213	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
214	Q₈×C₂₃	—	4	8	no	—
215	Q₈×C₂₃	—	4	8	no	—
216	D₄×C₂₃	—	4	8	no	—
217	D₄×C₂₃	—	4	16	no	—
218	D₄×C₂₃	—	4	16	no	—
219	D₄×C₂₃	—	4	8	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	8	no	—
224	C₂×C₄×C₂₃	—	2	8	no	—
225	C₂×C₄×C₂₃	—	2	8	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	4	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	8	no	—
230	C₂×C₄×C₂₃	—	2	8	no	—
231	C₂×C₄×C₂₃	—	2	8	no	—
232	C₂×C₄×C₂₃	—	2	8	no	—
233	C₂×C₄×C₂₃	—	2	8	no	—
234	C₂×C₄×C₂₃	—	2	8	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	4	no	—
237	C₂×C₄×C₂₃	—	4	4	no	—
238	C₂×C₄×C₂₃	—	4	8	no	—
239	C₂×C₄×C₂₃	—	4	16	no	—
240	C₂×C₄×C₂₃	—	4	8	no	—
241	C₂×C₄×C₂₃	—	4	8	no	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
247	C₂²×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
248	C₂²×#121	C₂×C₂₃	1	1	yes	characteristic
249	#2850	C₂×C₂₃	1	4	yes	complemented
250	#2639	C₂×C₂₃	1	4	yes	complemented
251	#2591	C₂×C₂₃	1	4	yes	complemented
252	#2561	C₂×C₂₃	1	4	yes	complemented
253	#2504	C₂×C₂₃	1	2	yes	complemented
254	#2503	C₂×C₂₃	1	2	yes	complemented
255	#2436	C₂×C₂₃	1	2	yes	complemented
256	#2435	C₂×C₂₃	1	2	yes	complemented
257	#2308	C₂×C₂₃	1	4	yes	complemented
258	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂²×C₂₃	—	2	2	no	—
260	C₂²×C₂₃	—	2	2	no	—
261	C₂²×C₂₃	—	2	2	no	—
262	C₂²×C₂₃	—	2	2	no	—
263	C₂²×C₂₃	—	2	2	no	—
264	C₂²×C₂₃	—	2	2	no	—
265	C₂²×C₂₃	—	2	8	no	—
266	C₂²×C₂₃	—	2	8	no	—
267	C₂²×C₂₃	—	2	8	no	—
268	C₂²×C₂₃	—	4	16	no	—
269	C₂²×C₂₃	—	4	4	no	—
270	C₂²×C₂₃	—	4	4	no	—
271	C₂²×C₂₃	—	4	4	no	—
272	C₂²×C₂₃	—	4	16	no	—
273	C₂²×C₂₃	—	8	16	no	—
274	C₂²×C₂₃	C₂×#124	1	4	yes	—
275	C₂²×C₂₃	C₂×#124	1	4	yes	—
276	C₂²×C₂₃	C₂×#124	1	4	yes	—
277	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
278	C₂²×C₂₃	C₂²×#41	1	2	yes	—
279	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₂₃	#585	1	2	yes	—
281	C₄×C₂₃	—	2	2	no	—
282	C₄×C₂₃	—	2	2	no	—
283	C₄×C₂₃	—	2	2	no	—
284	C₄×C₂₃	—	4	4	no	—
285	C₄×C₂₃	—	4	4	no	—
286	C₄×C₂₃	—	4	4	no	—
287	C₄×C₂₃	—	4	8	no	—
288	C₄×C₂₃	#534	1	4	yes	—
289	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
290	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
291	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×#124	C₂²×C₂₃	1	4	yes	complemented
294	C₂×C₂²⋊Q₈	—	2	8	no	—
295	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
296	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
297	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
298	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
299	C₂×#121	—	2	8	no	—
300	C₂×#121	C₂²×C₂₃	1	2	yes	—
301	C₂×#121	C₂²×C₂₃	1	2	yes	—
302	C₂×#121	C₂²×C₂₃	1	2	yes	—
303	C₂×#121	C₂²×C₂₃	1	2	yes	—
304	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
306	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
307	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
308	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
309	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
310	C₂²×C₄²	C₂²×C₂₃	1	2	yes	complemented
311	#400	C₂²×C₂₃	1	8	yes	complemented
312	#397	C₂²×C₂₃	1	8	yes	complemented
313	#394	C₂²×C₂₃	1	8	yes	complemented
314	#393	C₂²×C₂₃	1	8	yes	complemented
315	#391	C₂²×C₂₃	1	4	yes	complemented
316	#390	C₂²×C₂₃	1	4	yes	complemented
317	#388	C₂²×C₂₃	1	16	yes	complemented
318	#386	C₂²×C₂₃	1	8	yes	complemented
319	#386	C₂²×C₂₃	1	8	yes	complemented
320	#386	C₂²×C₂₃	1	16	yes	complemented
321	#382	C₂²×C₂₃	1	4	yes	complemented
322	#381	C₂²×C₂₃	1	8	yes	—
323	#379	C₂²×C₂₃	1	8	yes	—
324	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
325	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
326	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
327	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	complemented
328	C₂×#95	C₂²×C₂₃	1	2	yes	complemented
329	C₂×C₂₃	—	2	4	no	—
330	C₂×C₂₃	—	2	4	no	—
331	C₂×C₂₃	—	2	4	no	—
332	C₂×C₂₃	—	8	4	no	—
333	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
334	C₂×C₂₃	C₂×#534	1	1	yes	characteristic
335	C₂×C₂₃	#3449	1	1	yes	characteristic
336	C₂×C₂₃	#3505	1	4	yes	—
337	C₂⁵	C₂³×C₂₃	1	2	yes	—
338	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
339	C₂²×Q₈	—	2	4	no	—
340	C₂²×Q₈	—	4	4	no	—
341	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
342	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
343	C₂²×D₄	—	2	8	no	—
344	C₂²×D₄	—	2	16	no	—
345	C₂²×D₄	—	2	4	no	—
346	C₂²×D₄	—	4	4	no	—
347	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
348	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
349	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
350	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
351	C₂³×C₄	—	4	4	no	—
352	C₂³×C₄	D₄×C₂₃	1	8	yes	—
353	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
354	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
358	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
359	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
360	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
362	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
363	#124	—	4	8	no	—
364	C₂²⋊Q₈	—	2	8	no	—
365	C₂²⋊Q₈	—	2	8	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	8	no	—
368	#121	—	2	8	no	—
369	#121	—	2	8	no	—
370	#121	—	2	8	no	—
371	#121	—	2	8	no	—
372	C₂×C₄⋊C₄	—	2	4	no	—
373	C₂×C₄⋊C₄	—	2	4	no	—
374	C₂×C₄⋊C₄	—	2	8	no	—
375	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
381	C₂×C₂²⋊C₄	—	2	8	no	—
382	C₂×C₂²⋊C₄	—	2	8	no	—
383	C₂×C₂²⋊C₄	—	2	8	no	—
384	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
392	C₂×C₄²	—	4	4	no	—
393	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
394	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
395	#95	C₂³×C₂₃	1	4	yes	—
396	#95	C₂³×C₂₃	1	4	yes	—
397	#95	C₂³×C₂₃	1	16	yes	—
398	#95	C₂³×C₂₃	1	16	yes	—
399	C₂₃	#35019	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	8	no	—
402	C₂⁴	—	4	2	no	—
403	C₂⁴	—	4	2	no	—
404	C₂⁴	—	4	4	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
407	C₂⁴	#41×C₂₃	1	8	yes	—
408	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
409	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
410	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
411	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	—	2	8	no	—
415	C₂×Q₈	—	4	4	no	—
416	C₂×Q₈	—	4	16	no	—
417	C₂×D₄	—	2	8	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	16	no	—
420	C₂×D₄	—	2	16	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	16	no	—
426	C₂×D₄	—	4	32	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	8	no	—
434	C₂²×C₄	—	2	8	no	—
435	C₂²×C₄	—	4	16	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	4	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
444	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
445	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
448	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
451	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	4	16	no	—
470	C₂²⋊C₄	—	4	16	no	—
471	C₄²	—	2	8	no	—
472	C₄²	—	4	4	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	8	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	4	4	no	—
485	C₂³	—	4	16	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	8	16	no	—
490	C₂³	C₂₃×#124	1	16	yes	complemented
491	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
492	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
493	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂³	C₂×#41×C₂₃	1	4	yes	—
495	C₂³	C₂×#41×C₂₃	1	4	yes	—
496	C₂³	C₂×#41×C₂₃	1	4	yes	—
497	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
498	Q₈	—	4	8	no	—
499	Q₈	—	4	8	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	16	no	—
502	D₄	—	4	16	no	—
503	D₄	—	4	8	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	16	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
528	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
530	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	8	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	4	16	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	16	no	—
545	C₂²	—	8	16	no	—
546	C₂²	C₂×C₂₃×#124	1	4	yes	—
547	C₂²	C₂×C₂₃×#124	1	4	yes	—
548	C₂²	C₂×C₂₃×#124	1	4	yes	—
549	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
550	C₂²	C₂²×#41×C₂₃	1	2	yes	—
551	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
552	C₂²	C₂₃×#585	1	2	yes	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	4	4	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	4	no	—
559	C₄	—	4	8	no	—
560	C₄	C₂₃×#534	1	4	yes	—
561	C₂	—	2	4	no	—
562	C₂	—	2	4	no	—
563	C₂	—	2	4	no	—
564	C₂	—	8	4	no	—
565	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
566	C₂	C₂×C₂₃×#534	1	1	yes	characteristic
567	C₂	C₂₃×#3449	1	1	yes	characteristic
568	C₂	C₂₃×#3505	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#121	C₂
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
1. C₂²×C₂₃×C₂²⋊Q₈	C₂	semidirect
3. C₂₃×#2850	C₂	semidirect
4. C₂₃×#2639	C₂	semidirect
5. C₂₃×#2591	C₂	semidirect
6. C₂₃×#2561	C₂	semidirect
7. C₂₃×#2504	C₂	semidirect
8. C₂₃×#2503	C₂	semidirect
9. C₂₃×#2436	C₂	semidirect
10. C₂₃×#2435	C₂	semidirect
11. C₂₃×#2308	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
13. C₂³×Q₈×C₂₃	C₂²	semidirect
17. C₂×C₂₃×#124	C₂²	semidirect
21. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
22. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
29. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
34. C₂²×C₄²×C₂₃	C₂²	semidirect
35. C₂₃×#400	C₂²	semidirect
36. C₂₃×#397	C₂²	semidirect
37. C₂₃×#394	C₂²	semidirect
38. C₂₃×#393	C₂²	semidirect
39. C₂₃×#391	C₂²	semidirect
40. C₂₃×#390	C₂²	semidirect
41. C₂₃×#388	C₂²	semidirect
42. C₂₃×#386	C₂²	semidirect
43. C₂₃×#386	C₂²	semidirect
44. C₂₃×#386	C₂²	semidirect
45. C₂₃×#382	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
51. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
52. C₂×C₂₃×#95	C₂²	semidirect
188. #35019	C₂₃	direct (N × H)
206. C₂³×C₂₃	#124	semidirect
247. C₂²×C₂²⋊Q₈	C₂×C₂₃	semidirect
249. #2850	C₂×C₂₃	semidirect
250. #2639	C₂×C₂₃	semidirect
251. #2591	C₂×C₂₃	semidirect
252. #2561	C₂×C₂₃	semidirect
253. #2504	C₂×C₂₃	semidirect
254. #2503	C₂×C₂₃	semidirect
255. #2436	C₂×C₂₃	semidirect
256. #2435	C₂×C₂₃	semidirect
257. #2308	C₂×C₂₃	semidirect
258. C₂×C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
289. C₂³×Q₈	C₂²×C₂₃	semidirect
293. C₂×#124	C₂²×C₂₃	semidirect
297. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
298. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
305. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
306. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
310. C₂²×C₄²	C₂²×C₂₃	semidirect
311. #400	C₂²×C₂₃	semidirect
312. #397	C₂²×C₂₃	semidirect
313. #394	C₂²×C₂₃	semidirect
314. #393	C₂²×C₂₃	semidirect
315. #391	C₂²×C₂₃	semidirect
316. #390	C₂²×C₂₃	semidirect
317. #388	C₂²×C₂₃	semidirect
318. #386	C₂²×C₂₃	semidirect
319. #386	C₂²×C₂₃	semidirect
320. #386	C₂²×C₂₃	semidirect
321. #382	C₂²×C₂₃	semidirect
326. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
327. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
328. C₂×#95	C₂²×C₂₃	semidirect
399. C₂₃	#35019	direct (N × H)
490. C₂³	C₂₃×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₂²⋊Q₈	2
C₂²×C₂₃×#121	2
C₂₃×#2850	2
C₂₃×#2639	2
C₂₃×#2591	2
C₂₃×#2561	2
C₂₃×#2504	2
C₂₃×#2503	2
C₂₃×#2436	2
C₂₃×#2435	2
C₂₃×#2308	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
#35019	23

Subgroups — C23×#35019 #19182645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#35019 #19182645