Subgroups — C₂×C₂₃×#3182 #19208315

Order 5888.

2234 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3182	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#468	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#440	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	—	2	1	no	—
12	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	—	2	4	no	—
15	C₂₃×#533	C₂²	1	8	yes	complemented
16	C₂₃×#532	—	2	4	no	—
17	C₂₃×#532	C₂²	1	8	yes	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#125	—	2	1	no	—
23	C₂×C₂₃×#124	—	2	1	no	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₂₃×#121	—	2	1	no	—
27	C₂×C₂₃×#121	C₂²	1	2	yes	—
28	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₂₃	—	2	1	no	—
31	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
32	C₂×C₂₃×#117	—	2	1	no	—
33	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#500	C₂²	1	8	yes	complemented
35	C₂₃×#492	C₂²	1	8	yes	complemented
36	C₂₃×#487	C₂²	1	8	yes	complemented
37	C₂₃×#486	C₂²	1	8	yes	complemented
38	C₂₃×#468	C₂²	1	32	yes	complemented
39	C₂₃×#440	C₂²	1	32	yes	complemented
40	C₂×C₂₃×#107	C₂²	1	4	yes	—
41	C₂×C₂₃×#103	C₂²	1	4	yes	—
42	C₂×C₂₃×#102	C₂²	1	4	yes	—
43	C₄×M₁₆×C₂₃	C₂²	1	8	yes	complemented
44	C₂×C₂₃×#97	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₂×#41×C₂₃	—	2	8	no	—
48	C₂×#41×C₂₃	—	4	8	no	—
49	C₂²×Q₈×C₂₃	—	2	1	no	—
50	C₂²×Q₈×C₂₃	—	2	1	no	—
51	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₂₃	—	2	2	no	—
57	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₂₃	—	2	16	no	—
61	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
62	C₂×M₁₆×C₂₃	—	2	8	no	—
63	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
64	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	—	2	2	no	—
68	C₂₃×C₄⋊Q₈	—	2	2	no	—
69	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
70	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
71	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₂₃×#127	—	2	2	no	—
73	C₂₃×#127	C₂³	1	4	yes	—
74	C₂₃×#125	—	2	8	no	—
75	C₂₃×#124	—	2	8	no	—
76	C₂₃×C₂²⋊Q₈	—	2	8	no	—
77	C₂₃×C₂²⋊Q₈	—	2	4	no	—
78	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
79	C₂₃×#121	—	2	8	no	—
80	C₂₃×#121	—	2	4	no	—
81	C₂₃×#121	C₂³	1	8	yes	—
82	C₄×Q₈×C₂₃	—	2	8	no	—
83	C₄×Q₈×C₂₃	—	2	4	no	—
84	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
85	C₄×D₄×C₂₃	—	2	8	no	—
86	C₄×D₄×C₂₃	—	2	4	no	—
87	C₄×D₄×C₂₃	C₂³	1	8	yes	—
88	C₂₃×#117	—	2	8	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄²×C₂₃	—	2	2	no	—
101	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	D₄	1	4	yes	—
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
107	C₂₃×#107	C₂³	1	16	yes	—
108	C₂₃×#103	C₂³	1	16	yes	—
109	C₂₃×#102	C₂³	1	16	yes	—
110	C₂₃×#97	C₂³	1	8	yes	—
111	C₄×C₈×C₂₃	C₂³	1	4	yes	—
112	C₄×C₈×C₂₃	C₂³	1	4	yes	—
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
115	#41×C₂₃	—	4	32	no	—
116	C₂×Q₈×C₂₃	—	2	2	no	—
117	C₂×Q₈×C₂₃	—	2	4	no	—
118	C₂×Q₈×C₂₃	—	2	4	no	—
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	4	4	no	—
121	C₂×Q₈×C₂₃	—	4	4	no	—
122	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
123	C₂×D₄×C₂₃	—	2	8	no	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	4	no	—
127	C₂×D₄×C₂₃	—	2	8	no	—
128	C₂×D₄×C₂₃	—	4	8	no	—
129	C₂×D₄×C₂₃	—	4	4	no	—
130	C₂×D₄×C₂₃	—	4	4	no	—
131	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	8	no	—
139	C₂²×C₄×C₂₃	—	4	8	no	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
150	SD₁₆×C₂₃	—	2	32	no	—
151	M₁₆×C₂₃	—	2	16	no	—
152	C₂×C₈×C₂₃	—	2	4	no	—
153	C₂×C₈×C₂₃	#41	1	8	yes	complemented
154	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
155	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	8	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	2	8	no	—
168	C₂²⋊C₄×C₂₃	—	2	8	no	—
169	C₄²×C₂₃	—	2	8	no	—
170	C₄²×C₂₃	C₂×D₄	1	2	yes	—
171	C₄²×C₂₃	C₂×D₄	1	4	yes	—
172	C₄²×C₂₃	C₂×D₄	1	4	yes	—
173	C₄²×C₂₃	C₂×D₄	1	2	yes	—
174	C₄²×C₂₃	C₂⁴	1	2	yes	—
175	C₄²×C₂₃	C₂⁴	1	2	yes	—
176	C₂×#3182	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	2	no	—
180	C₂³×C₂₃	—	2	4	no	—
181	C₂³×C₂₃	—	4	4	no	—
182	C₂³×C₂₃	—	4	4	no	—
183	C₂³×C₂₃	—	4	4	no	—
184	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
185	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₂₃	—	2	8	no	—
187	Q₈×C₂₃	—	4	4	no	—
188	Q₈×C₂₃	—	4	8	no	—
189	Q₈×C₂₃	—	4	8	no	—
190	D₄×C₂₃	—	2	8	no	—
191	D₄×C₂₃	—	4	16	no	—
192	D₄×C₂₃	—	4	4	no	—
193	D₄×C₂₃	—	4	8	no	—
194	D₄×C₂₃	—	4	8	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	4	16	no	—
206	C₂×C₄×C₂₃	—	4	4	no	—
207	C₂×C₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
210	C₂×C₄×C₂₃	#137	1	4	yes	—
211	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
218	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
219	C₈×C₂₃	—	2	4	no	—
220	C₈×C₂₃	C₂×#41	1	8	yes	complemented
221	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
223	#3182	C₂×C₂₃	1	16	yes	direct product factor
224	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂×#468	C₂×C₂₃	1	4	yes	complemented
229	C₂×#440	C₂×C₂₃	1	4	yes	complemented
230	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	2	no	—
234	C₂²×C₂₃	—	4	4	no	—
235	C₂²×C₂₃	—	4	4	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	4	no	—
238	C₂²×C₂₃	—	8	8	no	—
239	C₂²×C₂₃	#532	1	2	yes	—
240	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
241	C₂²×C₂₃	C₂×#137	1	2	yes	—
242	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₂₃	—	2	2	no	—
244	C₄×C₂₃	—	2	2	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	4	4	no	—
247	C₄×C₂₃	—	4	4	no	—
248	C₄×C₂₃	—	4	4	no	—
249	C₄×C₂₃	#532	1	2	yes	—
250	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
251	C₄×C₂₃	C₂×#137	1	4	yes	—
252	C₄×C₂₃	C₂²×#41	1	2	yes	—
253	C₂²×#41	—	2	1	no	—
254	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
255	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
256	#533	—	2	4	no	—
257	#533	C₂²×C₂₃	1	8	yes	complemented
258	#532	—	2	4	no	—
259	#532	C₂²×C₂₃	1	8	yes	—
260	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
263	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×#125	—	2	1	no	—
265	C₂×#124	—	2	1	no	—
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
268	C₂×#121	—	2	1	no	—
269	C₂×#121	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄×Q₈	—	2	1	no	—
271	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
272	C₂×C₄×D₄	—	2	1	no	—
273	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
274	C₂×#117	—	2	1	no	—
275	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
276	#500	C₂²×C₂₃	1	8	yes	complemented
277	#492	C₂²×C₂₃	1	8	yes	complemented
278	#487	C₂²×C₂₃	1	8	yes	complemented
279	#486	C₂²×C₂₃	1	8	yes	complemented
280	#468	C₂²×C₂₃	1	32	yes	complemented
281	#440	C₂²×C₂₃	1	32	yes	complemented
282	C₂×#107	C₂²×C₂₃	1	4	yes	—
283	C₂×#103	C₂²×C₂₃	1	4	yes	—
284	C₂×#102	C₂²×C₂₃	1	4	yes	—
285	C₄×M₁₆	C₂²×C₂₃	1	8	yes	complemented
286	C₂×#97	C₂²×C₂₃	1	2	yes	—
287	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×C₂₃	—	4	2	no	—
290	C₂×C₂₃	—	8	4	no	—
291	C₂×C₂₃	#3182	1	4	yes	direct product factor
292	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
293	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
295	C₂×#41	—	2	8	no	—
296	C₂×#41	—	4	8	no	—
297	C₂²×Q₈	—	2	1	no	—
298	C₂²×Q₈	—	2	1	no	—
299	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
304	C₂³×C₄	—	2	2	no	—
305	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
306	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
307	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
308	C₂×SD₁₆	—	2	16	no	—
309	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
310	C₂×M₁₆	—	2	8	no	—
311	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
312	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
313	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
318	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
319	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
320	#127	—	2	2	no	—
321	#127	C₂³×C₂₃	1	4	yes	—
322	#125	—	2	8	no	—
323	#124	—	2	8	no	—
324	C₂²⋊Q₈	—	2	8	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
327	#121	—	2	8	no	—
328	#121	—	2	4	no	—
329	#121	C₂³×C₂₃	1	8	yes	—
330	C₄×Q₈	—	2	8	no	—
331	C₄×Q₈	—	2	4	no	—
332	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
333	C₄×D₄	—	2	8	no	—
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	C₂³×C₂₃	1	8	yes	—
336	#117	—	2	8	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₄²	—	2	2	no	—
349	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₂₃	1	4	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	4	yes	—
355	#107	C₂³×C₂₃	1	16	yes	—
356	#103	C₂³×C₂₃	1	16	yes	—
357	#102	C₂³×C₂₃	1	16	yes	—
358	#97	C₂³×C₂₃	1	8	yes	—
359	C₄×C₈	C₂³×C₂₃	1	4	yes	—
360	C₄×C₈	C₂³×C₂₃	1	4	yes	—
361	C₂₃	C₂×#3182	1	1	yes	characteristic, direct product factor, Sylow
362	C₂⁴	—	2	2	no	—
363	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
364	#41	—	4	32	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	2	8	no	—
369	C₂×Q₈	—	4	4	no	—
370	C₂×Q₈	—	4	4	no	—
371	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	—	4	4	no	—
379	C₂×D₄	—	4	4	no	—
380	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	4	8	no	—
389	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
399	SD₁₆	—	2	32	no	—
400	M₁₆	—	2	16	no	—
401	C₂×C₈	—	2	4	no	—
402	C₂×C₈	#41×C₂₃	1	8	yes	complemented
403	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
404	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
414	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₄²	—	2	8	no	—
419	C₄²	C₂×D₄×C₂₃	1	2	yes	—
420	C₄²	C₂×D₄×C₂₃	1	4	yes	—
421	C₄²	C₂×D₄×C₂₃	1	4	yes	—
422	C₄²	C₂×D₄×C₂₃	1	2	yes	—
423	C₄²	C₂⁴×C₂₃	1	2	yes	—
424	C₄²	C₂⁴×C₂₃	1	2	yes	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	4	4	no	—
430	C₂³	—	4	4	no	—
431	C₂³	—	4	4	no	—
432	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
433	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
434	Q₈	—	2	8	no	—
435	Q₈	—	4	8	no	—
436	Q₈	—	4	8	no	—
437	Q₈	—	4	4	no	—
438	D₄	—	2	8	no	—
439	D₄	—	4	16	no	—
440	D₄	—	4	8	no	—
441	D₄	—	4	8	no	—
442	D₄	—	4	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	4	16	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
458	C₂×C₄	C₂₃×#137	1	4	yes	—
459	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
466	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
467	C₈	—	2	4	no	—
468	C₈	C₂×#41×C₂₃	1	8	yes	complemented
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	8	8	no	—
477	C₂²	C₂₃×#532	1	2	yes	—
478	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
479	C₂²	C₂×C₂₃×#137	1	2	yes	—
480	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	4	4	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	4	no	—
487	C₄	C₂₃×#532	1	2	yes	—
488	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
489	C₄	C₂×C₂₃×#137	1	4	yes	—
490	C₄	C₂²×#41×C₂₃	1	2	yes	—
491	C₂	—	4	2	no	—
492	C₂	—	8	4	no	—
493	C₂	C₂₃×#3182	1	4	yes	direct product factor
494	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
495	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
496	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3182	C₂	direct (N × H)
4. C₂×C₂₃×#500	C₂	semidirect
5. C₂×C₂₃×#492	C₂	semidirect
6. C₂×C₂₃×#487	C₂	semidirect
7. C₂×C₂₃×#486	C₂	semidirect
8. C₂×C₂₃×#468	C₂	semidirect
9. C₂×C₂₃×#440	C₂	semidirect
10. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
15. C₂₃×#533	C₂²	semidirect
34. C₂₃×#500	C₂²	semidirect
35. C₂₃×#492	C₂²	semidirect
36. C₂₃×#487	C₂²	semidirect
37. C₂₃×#486	C₂²	semidirect
38. C₂₃×#468	C₂²	semidirect
39. C₂₃×#440	C₂²	semidirect
43. C₄×M₁₆×C₂₃	C₂²	semidirect
153. C₂×C₈×C₂₃	#41	semidirect
176. C₂×#3182	C₂₃	direct (N × H)
220. C₈×C₂₃	C₂×#41	semidirect
221. C₂×#533	C₂×C₂₃	semidirect
223. #3182	C₂×C₂₃	direct (N × H)
224. C₂×#500	C₂×C₂₃	semidirect
225. C₂×#492	C₂×C₂₃	semidirect
226. C₂×#487	C₂×C₂₃	semidirect
227. C₂×#486	C₂×C₂₃	semidirect
228. C₂×#468	C₂×C₂₃	semidirect
229. C₂×#440	C₂×C₂₃	semidirect
230. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
257. #533	C₂²×C₂₃	semidirect
276. #500	C₂²×C₂₃	semidirect
277. #492	C₂²×C₂₃	semidirect
278. #487	C₂²×C₂₃	semidirect
279. #486	C₂²×C₂₃	semidirect
280. #468	C₂²×C₂₃	semidirect
281. #440	C₂²×C₂₃	semidirect
285. C₄×M₁₆	C₂²×C₂₃	semidirect
291. C₂×C₂₃	#3182	direct (N × H)
361. C₂₃	C₂×#3182	direct (N × H)
402. C₂×C₈	#41×C₂₃	semidirect
468. C₈	C₂×#41×C₂₃	semidirect
493. C₂	C₂₃×#3182	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×#533	2
C₂×C₂₃×#532	2
C₂₃×#3182	2
C₂×C₂₃×#500	2
C₂×C₂₃×#492	2
C₂×C₂₃×#487	2
C₂×C₂₃×#486	2
C₂×C₂₃×#468	2
C₂×C₂₃×#440	2
C₂×C₄×M₁₆×C₂₃	2
C₂×#3182	23

Subgroups — C2×C23×#3182 #19208315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3182 #19208315