Subgroups — C₂×C₂₃×#3228 #19208469

Order 5888.

2058 conjugacy classes of subgroups.

The subgroup lattice has 488 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×#3228	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#482	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#478	C₂	1	4	yes	maximal, complemented
10	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₂₃	—	2	1	no	—
12	C₂₃×#533	—	2	4	no	—
13	C₂₃×#533	C₂²	1	8	yes	complemented
14	C₂₃×#532	—	2	4	no	—
15	C₂₃×#532	C₂²	1	8	yes	—
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#125	C₂²	1	2	yes	—
21	C₂×C₂₃×#124	—	2	1	no	—
22	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
23	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₂₃×#121	—	2	1	no	—
25	C₂×C₂₃×#121	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
27	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
28	C₂×C₄×D₄×C₂₃	—	2	1	no	—
29	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
30	C₂×C₂₃×#117	C₂²	1	2	yes	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#500	C₂²	1	8	yes	complemented
33	C₂₃×#498	C₂²	1	8	yes	complemented
34	C₂₃×#486	C₂²	1	8	yes	complemented
35	C₂₃×#486	C₂²	1	8	yes	complemented
36	C₂₃×#482	C₂²	1	32	yes	complemented
37	C₂₃×#478	C₂²	1	32	yes	complemented
38	C₂₃×#431	C₂²	1	8	yes	complemented
39	C₂×C₂₃×#107	C₂²	1	4	yes	—
40	C₂×C₂₃×#106	C₂²	1	4	yes	—
41	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
42	C₂×C₂₃×#102	C₂²	1	4	yes	—
43	C₂×C₂₃×#102	C₂²	1	4	yes	—
44	C₂×C₂₃×C₂²⋊C₈	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	—	2	1	no	—
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	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
66	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
67	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
68	C₂₃×#127	—	2	2	no	—
69	C₂₃×#127	C₂³	1	4	yes	—
70	C₂₃×#125	—	2	4	no	—
71	C₂₃×#125	C₂³	1	8	yes	—
72	C₂₃×#124	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	8	no	—
74	C₂₃×C₂²⋊Q₈	—	2	8	no	—
75	C₂₃×#121	—	2	8	no	—
76	C₂₃×#121	—	2	4	no	—
77	C₂₃×#121	C₂³	1	8	yes	—
78	C₄×Q₈×C₂₃	—	2	8	no	—
79	C₄×Q₈×C₂₃	—	2	8	no	—
80	C₄×D₄×C₂₃	—	2	8	no	—
81	C₄×D₄×C₂₃	—	2	4	no	—
82	C₄×D₄×C₂₃	C₂³	1	8	yes	—
83	C₂₃×#117	—	2	4	no	—
84	C₂₃×#117	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₄²×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	D₄	1	4	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
103	C₂₃×#107	C₂³	1	16	yes	—
104	C₂₃×#106	C₂³	1	16	yes	—
105	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
106	C₂₃×#102	C₂³	1	16	yes	—
107	C₂₃×#102	C₂³	1	16	yes	—
108	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
109	C₄×C₈×C₂₃	C₂³	1	4	yes	—
110	C₄×C₈×C₂₃	C₂³	1	4	yes	—
111	C₂⁴×C₂₃	—	2	2	no	—
112	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₂₃	—	4	32	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	4	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	—	4	4	no	—
118	C₂×Q₈×C₂₃	—	4	4	no	—
119	C₂×Q₈×C₂₃	—	4	4	no	—
120	C₂×D₄×C₂₃	—	2	8	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	8	no	—
125	C₂×D₄×C₂₃	—	4	8	no	—
126	C₂×D₄×C₂₃	—	4	4	no	—
127	C₂×D₄×C₂₃	—	4	4	no	—
128	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	4	no	—
131	C₂²×C₄×C₂₃	—	2	4	no	—
132	C₂²×C₄×C₂₃	—	2	2	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	8	no	—
136	C₂²×C₄×C₂₃	—	4	8	no	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₂×C₈×C₂₃	—	2	4	no	—
148	C₂×C₈×C₂₃	—	2	4	no	—
149	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
150	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
151	C₄⋊C₄×C₂₃	—	2	4	no	—
152	C₄⋊C₄×C₂₃	—	2	8	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	2	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
160	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
161	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
162	C₂²⋊C₄×C₂₃	—	2	8	no	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	—	2	8	no	—
166	C₄²×C₂₃	C₂×D₄	1	4	yes	—
167	C₄²×C₂₃	C₂×D₄	1	2	yes	—
168	C₄²×C₂₃	C₂×D₄	1	4	yes	—
169	C₄²×C₂₃	C₂×D₄	1	2	yes	—
170	C₄²×C₂₃	C₂⁴	1	2	yes	—
171	C₄²×C₂₃	C₂⁴	1	2	yes	—
172	C₂×#3228	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	2	no	—
175	C₂³×C₂₃	—	2	2	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	4	4	no	—
178	C₂³×C₂₃	—	4	4	no	—
179	C₂³×C₂₃	—	4	4	no	—
180	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
181	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₂₃	—	4	8	no	—
183	Q₈×C₂₃	—	4	8	no	—
184	Q₈×C₂₃	—	4	8	no	—
185	D₄×C₂₃	—	2	8	no	—
186	D₄×C₂₃	—	4	16	no	—
187	D₄×C₂₃	—	4	4	no	—
188	D₄×C₂₃	—	4	8	no	—
189	D₄×C₂₃	—	4	8	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	8	no	—
195	C₂×C₄×C₂₃	—	2	8	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	4	16	no	—
201	C₂×C₄×C₂₃	—	4	4	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	4	no	—
204	C₂×C₄×C₂₃	#135	1	4	yes	—
205	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
206	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
213	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
214	C₈×C₂₃	—	2	4	no	—
215	C₈×C₂₃	—	2	4	no	—
216	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
218	#3228	C₂×C₂₃	1	16	yes	direct product factor
219	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
220	C₂×#498	C₂×C₂₃	1	1	yes	characteristic, complemented
221	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂×#486	C₂×C₂₃	1	1	yes	characteristic, complemented
223	C₂×#482	C₂×C₂₃	1	4	yes	complemented
224	C₂×#478	C₂×C₂₃	1	4	yes	complemented
225	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂²×C₂₃	—	2	2	no	—
227	C₂²×C₂₃	—	2	2	no	—
228	C₂²×C₂₃	—	2	2	no	—
229	C₂²×C₂₃	—	4	4	no	—
230	C₂²×C₂₃	—	4	4	no	—
231	C₂²×C₂₃	—	4	4	no	—
232	C₂²×C₂₃	—	4	4	no	—
233	C₂²×C₂₃	—	8	8	no	—
234	C₂²×C₂₃	#536	1	2	yes	—
235	C₂²×C₂₃	C₂×#135	1	2	yes	—
236	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
237	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₂₃	—	2	2	no	—
239	C₄×C₂₃	—	2	2	no	—
240	C₄×C₂₃	—	2	2	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	—	4	4	no	—
243	C₄×C₂₃	—	4	4	no	—
244	C₄×C₂₃	#536	1	2	yes	—
245	C₄×C₂₃	C₂×#135	1	4	yes	—
246	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
247	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
248	C₂²×#41	—	2	1	no	—
249	#533	—	2	4	no	—
250	#533	C₂²×C₂₃	1	8	yes	complemented
251	#532	—	2	4	no	—
252	#532	C₂²×C₂₃	1	8	yes	—
253	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
254	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×#125	C₂²×C₂₃	1	2	yes	—
258	C₂×#124	—	2	1	no	—
259	C₂×C₂²⋊Q₈	—	2	1	no	—
260	C₂×C₂²⋊Q₈	—	2	1	no	—
261	C₂×#121	—	2	1	no	—
262	C₂×#121	C₂²×C₂₃	1	2	yes	—
263	C₂×C₄×Q₈	—	2	1	no	—
264	C₂×C₄×Q₈	—	2	1	no	—
265	C₂×C₄×D₄	—	2	1	no	—
266	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
267	C₂×#117	C₂²×C₂₃	1	2	yes	—
268	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
269	#500	C₂²×C₂₃	1	8	yes	complemented
270	#498	C₂²×C₂₃	1	8	yes	complemented
271	#486	C₂²×C₂₃	1	8	yes	complemented
272	#486	C₂²×C₂₃	1	8	yes	complemented
273	#482	C₂²×C₂₃	1	32	yes	complemented
274	#478	C₂²×C₂₃	1	32	yes	complemented
275	#431	C₂²×C₂₃	1	8	yes	complemented
276	C₂×#107	C₂²×C₂₃	1	4	yes	—
277	C₂×#106	C₂²×C₂₃	1	4	yes	—
278	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
279	C₂×#102	C₂²×C₂₃	1	4	yes	—
280	C₂×#102	C₂²×C₂₃	1	4	yes	—
281	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
282	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
284	C₂×C₂₃	—	4	2	no	—
285	C₂×C₂₃	—	8	4	no	—
286	C₂×C₂₃	#3228	1	4	yes	direct product factor
287	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
288	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
289	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×#41	—	2	8	no	—
291	C₂×#41	—	4	8	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×D₄	—	2	2	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
299	C₂³×C₄	—	2	2	no	—
300	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
301	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
302	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
304	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
309	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
310	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
311	#127	—	2	2	no	—
312	#127	C₂³×C₂₃	1	4	yes	—
313	#125	—	2	4	no	—
314	#125	C₂³×C₂₃	1	8	yes	—
315	#124	—	2	8	no	—
316	C₂²⋊Q₈	—	2	8	no	—
317	C₂²⋊Q₈	—	2	8	no	—
318	#121	—	2	8	no	—
319	#121	—	2	4	no	—
320	#121	C₂³×C₂₃	1	8	yes	—
321	C₄×Q₈	—	2	8	no	—
322	C₄×Q₈	—	2	8	no	—
323	C₄×D₄	—	2	8	no	—
324	C₄×D₄	—	2	4	no	—
325	C₄×D₄	C₂³×C₂₃	1	8	yes	—
326	#117	—	2	4	no	—
327	#117	C₂³×C₂₃	1	8	yes	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	2	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
341	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
342	C₂×C₄²	D₄×C₂₃	1	4	yes	—
343	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
346	#107	C₂³×C₂₃	1	16	yes	—
347	#106	C₂³×C₂₃	1	16	yes	—
348	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
349	#102	C₂³×C₂₃	1	16	yes	—
350	#102	C₂³×C₂₃	1	16	yes	—
351	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
352	C₄×C₈	C₂³×C₂₃	1	4	yes	—
353	C₄×C₈	C₂³×C₂₃	1	4	yes	—
354	C₂₃	C₂×#3228	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁴	—	2	2	no	—
356	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
357	#41	—	4	32	no	—
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	4	4	no	—
362	C₂×Q₈	—	4	4	no	—
363	C₂×Q₈	—	4	4	no	—
364	C₂×D₄	—	2	8	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	4	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	8	no	—
380	C₂²×C₄	—	4	8	no	—
381	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
386	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
387	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	C₂×C₈	—	2	4	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
394	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	8	no	—
397	C₄⋊C₄	—	2	8	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
404	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
405	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
406	C₂²⋊C₄	—	2	8	no	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	2	8	no	—
409	C₄²	—	2	8	no	—
410	C₄²	C₂×D₄×C₂₃	1	4	yes	—
411	C₄²	C₂×D₄×C₂₃	1	2	yes	—
412	C₄²	C₂×D₄×C₂₃	1	4	yes	—
413	C₄²	C₂×D₄×C₂₃	1	2	yes	—
414	C₄²	C₂⁴×C₂₃	1	2	yes	—
415	C₄²	C₂⁴×C₂₃	1	2	yes	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	4	no	—
421	C₂³	—	4	4	no	—
422	C₂³	—	4	4	no	—
423	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
424	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	Q₈	—	4	8	no	—
426	Q₈	—	4	8	no	—
427	Q₈	—	4	8	no	—
428	D₄	—	2	8	no	—
429	D₄	—	4	16	no	—
430	D₄	—	4	8	no	—
431	D₄	—	4	8	no	—
432	D₄	—	4	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	4	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	4	16	no	—
444	C₂×C₄	—	4	4	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	C₂₃×#135	1	4	yes	—
448	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
449	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
453	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
454	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
455	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
456	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
457	C₈	—	2	4	no	—
458	C₈	—	2	4	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	4	4	no	—
463	C₂²	—	4	4	no	—
464	C₂²	—	4	4	no	—
465	C₂²	—	4	4	no	—
466	C₂²	—	8	8	no	—
467	C₂²	C₂₃×#536	1	2	yes	—
468	C₂²	C₂×C₂₃×#135	1	2	yes	—
469	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
470	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
471	C₄	—	2	2	no	—
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	4	4	no	—
475	C₄	—	4	4	no	—
476	C₄	—	4	4	no	—
477	C₄	C₂₃×#536	1	2	yes	—
478	C₄	C₂×C₂₃×#135	1	4	yes	—
479	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
480	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
481	C₂	—	4	2	no	—
482	C₂	—	8	4	no	—
483	C₂	C₂₃×#3228	1	4	yes	direct product factor
484	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
485	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
486	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	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×#3228	C₂	direct (N × H)
4. C₂×C₂₃×#500	C₂	semidirect
5. C₂×C₂₃×#498	C₂	semidirect
6. C₂×C₂₃×#486	C₂	semidirect
7. C₂×C₂₃×#486	C₂	semidirect
8. C₂×C₂₃×#482	C₂	semidirect
9. C₂×C₂₃×#478	C₂	semidirect
10. C₂×C₂₃×#431	C₂	semidirect
13. C₂₃×#533	C₂²	semidirect
32. C₂₃×#500	C₂²	semidirect
33. C₂₃×#498	C₂²	semidirect
34. C₂₃×#486	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
36. C₂₃×#482	C₂²	semidirect
37. C₂₃×#478	C₂²	semidirect
38. C₂₃×#431	C₂²	semidirect
172. C₂×#3228	C₂₃	direct (N × H)
216. C₂×#533	C₂×C₂₃	semidirect
218. #3228	C₂×C₂₃	direct (N × H)
219. C₂×#500	C₂×C₂₃	semidirect
220. C₂×#498	C₂×C₂₃	semidirect
221. C₂×#486	C₂×C₂₃	semidirect
222. C₂×#486	C₂×C₂₃	semidirect
223. C₂×#482	C₂×C₂₃	semidirect
224. C₂×#478	C₂×C₂₃	semidirect
225. C₂×#431	C₂×C₂₃	semidirect
250. #533	C₂²×C₂₃	semidirect
269. #500	C₂²×C₂₃	semidirect
270. #498	C₂²×C₂₃	semidirect
271. #486	C₂²×C₂₃	semidirect
272. #486	C₂²×C₂₃	semidirect
273. #482	C₂²×C₂₃	semidirect
274. #478	C₂²×C₂₃	semidirect
275. #431	C₂²×C₂₃	semidirect
286. C₂×C₂₃	#3228	direct (N × H)
354. C₂₃	C₂×#3228	direct (N × H)
483. C₂	C₂₃×#3228	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×#3228	2
C₂×C₂₃×#500	2
C₂×C₂₃×#498	2
C₂×C₂₃×#486	2
C₂×C₂₃×#482	2
C₂×C₂₃×#478	2
C₂×C₂₃×#431	2
C₂×#3228	23

Subgroups — C2×C23×#3228 #19208469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3228 #19208469