Subgroups — C₂×C₂₃×#3234 #19208473

Order 5888.

2050 conjugacy classes of subgroups.

The subgroup lattice has 472 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	C₂²×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3234	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#501	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#490	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#481	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#480	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#478	C₂	1	4	yes	maximal, complemented
9	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	—	2	1	no	—
11	C₂₃×#532	—	2	4	no	—
12	C₂₃×#532	C₂²	1	8	yes	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#125	—	2	1	no	—
16	C₂×C₂₃×#125	—	2	1	no	—
17	C₂×C₂₃×#125	—	2	4	no	—
18	C₂×C₂₃×#125	C₂²	1	2	yes	—
19	C₂×C₂₃×#125	C₂²	1	2	yes	—
20	C₂×C₂₃×#125	C₂²	1	8	yes	complemented
21	C₂×C₂₃×#124	—	2	1	no	—
22	C₂×C₂₃×#121	—	2	1	no	—
23	C₂×C₂₃×#121	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₂₃	—	2	1	no	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#501	C₂²	1	16	yes	complemented
32	C₂₃×#490	C₂²	1	16	yes	complemented
33	C₂₃×#481	C₂²	1	16	yes	complemented
34	C₂₃×#480	C₂²	1	16	yes	complemented
35	C₂₃×#478	C₂²	1	32	yes	complemented
36	C₂₃×#432	C₂²	1	8	yes	complemented
37	C₂×C₂₃×#107	C₂²	1	4	yes	—
38	C₂×C₂₃×#106	C₂²	1	4	yes	—
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂×C₂₃×#102	C₂²	1	4	yes	—
41	C₂×C₂₃×#102	C₂²	1	4	yes	—
42	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₂₃×#97	C₂²	1	2	yes	—
44	C₂×#41×C₂₃	—	2	8	no	—
45	C₂×#41×C₂₃	—	4	8	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	—	2	1	no	—
50	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₂₃	—	2	1	no	—
52	C₂³×C₄×C₂₃	—	2	1	no	—
53	C₂³×C₄×C₂₃	—	2	2	no	—
54	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
60	C₂₃×#127	—	2	2	no	—
61	C₂₃×#127	C₂³	1	4	yes	—
62	C₂₃×#125	—	2	4	no	—
63	C₂₃×#125	—	2	4	no	—
64	C₂₃×#125	—	2	8	no	—
65	C₂₃×#125	—	2	8	no	—
66	C₂₃×#125	C₂³	1	8	yes	—
67	C₂₃×#125	C₂³	1	8	yes	—
68	C₂₃×#124	—	2	8	no	—
69	C₂₃×#121	—	2	8	no	—
70	C₂₃×#121	—	2	4	no	—
71	C₂₃×#121	C₂³	1	8	yes	—
72	C₄×D₄×C₂₃	—	2	8	no	—
73	C₄×D₄×C₂₃	—	2	4	no	—
74	C₄×D₄×C₂₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	D₄	1	4	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	4	yes	—
99	C₂₃×#107	C₂³	1	16	yes	—
100	C₂₃×#106	C₂³	1	16	yes	—
101	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
102	C₂₃×#102	C₂³	1	16	yes	—
103	C₂₃×#102	C₂³	1	16	yes	—
104	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
105	C₂₃×#97	C₂³	1	8	yes	—
106	C₂⁴×C₂₃	—	2	2	no	—
107	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
108	#41×C₂₃	—	4	32	no	—
109	C₂×Q₈×C₂₃	—	2	4	no	—
110	C₂×Q₈×C₂₃	—	4	4	no	—
111	C₂×D₄×C₂₃	—	2	8	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	2	4	no	—
114	C₂×D₄×C₂₃	—	2	4	no	—
115	C₂×D₄×C₂₃	—	2	8	no	—
116	C₂×D₄×C₂₃	—	4	4	no	—
117	C₂×D₄×C₂₃	—	4	4	no	—
118	C₂×D₄×C₂₃	—	4	8	no	—
119	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₂₃	—	2	4	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	2	4	no	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	2	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	8	no	—
128	C₂²×C₄×C₂₃	—	4	2	no	—
129	C₂²×C₄×C₂₃	—	4	8	no	—
130	C₂²×C₄×C₂₃	—	4	4	no	—
131	C₂²×C₄×C₂₃	—	4	4	no	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
142	C₂×C₈×C₂₃	—	2	8	no	—
143	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	2	8	no	—
146	C₄⋊C₄×C₂₃	—	2	8	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	4	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
155	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	8	no	—
159	C₄²×C₂₃	C₂×D₄	1	4	yes	—
160	C₄²×C₂₃	C₂×D₄	1	2	yes	—
161	C₄²×C₂₃	C₂×D₄	1	4	yes	—
162	C₄²×C₂₃	C₂×D₄	1	2	yes	—
163	C₄²×C₂₃	C₂⁴	1	2	yes	—
164	C₄²×C₂₃	C₂⁴	1	2	yes	—
165	C₂×#3234	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₂₃	—	2	2	no	—
167	C₂³×C₂₃	—	2	2	no	—
168	C₂³×C₂₃	—	2	4	no	—
169	C₂³×C₂₃	—	4	4	no	—
170	C₂³×C₂₃	—	4	4	no	—
171	C₂³×C₂₃	—	4	4	no	—
172	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
173	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
174	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₂₃	—	4	8	no	—
176	D₄×C₂₃	—	2	8	no	—
177	D₄×C₂₃	—	4	8	no	—
178	D₄×C₂₃	—	4	8	no	—
179	D₄×C₂₃	—	4	16	no	—
180	D₄×C₂₃	—	4	4	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	8	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	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	—	4	4	no	—
193	C₂×C₄×C₂₃	—	4	16	no	—
194	C₂×C₄×C₂₃	—	4	8	no	—
195	C₂×C₄×C₂₃	—	4	8	no	—
196	C₂×C₄×C₂₃	—	4	4	no	—
197	C₂×C₄×C₂₃	—	4	4	no	—
198	C₂×C₄×C₂₃	—	4	4	no	—
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
202	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
206	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
207	C₈×C₂₃	—	2	8	no	—
208	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
209	C₂²×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#3234	C₂×C₂₃	1	16	yes	direct product factor
211	C₂×#501	C₂×C₂₃	1	2	yes	complemented
212	C₂×#490	C₂×C₂₃	1	2	yes	complemented
213	C₂×#481	C₂×C₂₃	1	2	yes	complemented
214	C₂×#480	C₂×C₂₃	1	2	yes	complemented
215	C₂×#478	C₂×C₂₃	1	4	yes	complemented
216	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	8	no	—
219	C₂²×C₂₃	—	2	2	no	—
220	C₂²×C₂₃	—	4	4	no	—
221	C₂²×C₂₃	—	4	4	no	—
222	C₂²×C₂₃	—	4	4	no	—
223	C₂²×C₂₃	—	8	8	no	—
224	C₂²×C₂₃	#536	1	2	yes	—
225	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
226	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
227	C₂²×C₂₃	#575	1	2	yes	—
228	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
229	C₄×C₂₃	—	2	2	no	—
230	C₄×C₂₃	—	2	2	no	—
231	C₄×C₂₃	—	2	2	no	—
232	C₄×C₂₃	—	2	2	no	—
233	C₄×C₂₃	—	2	2	no	—
234	C₄×C₂₃	—	4	4	no	—
235	C₄×C₂₃	—	4	4	no	—
236	C₄×C₂₃	—	4	4	no	—
237	C₄×C₂₃	#536	1	2	yes	—
238	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
239	C₂²×#41	—	2	1	no	—
240	#532	—	2	4	no	—
241	#532	C₂²×C₂₃	1	8	yes	—
242	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×#125	—	2	1	no	—
245	C₂×#125	—	2	1	no	—
246	C₂×#125	—	2	4	no	—
247	C₂×#125	C₂²×C₂₃	1	2	yes	—
248	C₂×#125	C₂²×C₂₃	1	2	yes	—
249	C₂×#125	C₂²×C₂₃	1	8	yes	complemented
250	C₂×#124	—	2	1	no	—
251	C₂×#121	—	2	1	no	—
252	C₂×#121	C₂²×C₂₃	1	2	yes	—
253	C₂×C₄×D₄	—	2	1	no	—
254	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
255	C₂²×C₄⋊C₄	—	2	1	no	—
256	C₂²×C₄⋊C₄	—	2	1	no	—
257	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
258	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
259	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
260	#501	C₂²×C₂₃	1	16	yes	complemented
261	#490	C₂²×C₂₃	1	16	yes	complemented
262	#481	C₂²×C₂₃	1	16	yes	complemented
263	#480	C₂²×C₂₃	1	16	yes	complemented
264	#478	C₂²×C₂₃	1	32	yes	complemented
265	#432	C₂²×C₂₃	1	8	yes	complemented
266	C₂×#107	C₂²×C₂₃	1	4	yes	—
267	C₂×#106	C₂²×C₂₃	1	4	yes	—
268	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
269	C₂×#102	C₂²×C₂₃	1	4	yes	—
270	C₂×#102	C₂²×C₂₃	1	4	yes	—
271	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
272	C₂×#97	C₂²×C₂₃	1	2	yes	—
273	C₂×C₂₃	—	2	4	no	—
274	C₂×C₂₃	—	8	4	no	—
275	C₂×C₂₃	#3234	1	4	yes	direct product factor
276	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
277	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
279	C₂×#41	—	2	8	no	—
280	C₂×#41	—	4	8	no	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
286	C₂³×C₄	—	2	1	no	—
287	C₂³×C₄	—	2	1	no	—
288	C₂³×C₄	—	2	2	no	—
289	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
290	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
291	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
295	#127	—	2	2	no	—
296	#127	C₂³×C₂₃	1	4	yes	—
297	#125	—	2	4	no	—
298	#125	—	2	4	no	—
299	#125	—	2	8	no	—
300	#125	—	2	8	no	—
301	#125	C₂³×C₂₃	1	8	yes	—
302	#125	C₂³×C₂₃	1	8	yes	—
303	#124	—	2	8	no	—
304	#121	—	2	8	no	—
305	#121	—	2	4	no	—
306	#121	C₂³×C₂₃	1	8	yes	—
307	C₄×D₄	—	2	8	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	C₂³×C₂₃	1	8	yes	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	8	no	—
315	C₂×C₄⋊C₄	—	2	8	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
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	2	yes	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
329	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
330	C₂×C₄²	D₄×C₂₃	1	4	yes	—
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
334	#107	C₂³×C₂₃	1	16	yes	—
335	#106	C₂³×C₂₃	1	16	yes	—
336	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
337	#102	C₂³×C₂₃	1	16	yes	—
338	#102	C₂³×C₂₃	1	16	yes	—
339	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
340	#97	C₂³×C₂₃	1	8	yes	—
341	C₂₃	C₂×#3234	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	—	2	2	no	—
343	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
344	#41	—	4	32	no	—
345	C₂×Q₈	—	2	4	no	—
346	C₂×Q₈	—	4	4	no	—
347	C₂×D₄	—	2	8	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	4	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	4	4	no	—
353	C₂×D₄	—	4	4	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	8	no	—
366	C₂²×C₄	—	4	4	no	—
367	C₂²×C₄	—	4	4	no	—
368	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
377	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
378	C₂×C₈	—	2	8	no	—
379	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
380	C₄⋊C₄	—	2	8	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
391	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
392	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	8	no	—
395	C₄²	C₂×D₄×C₂₃	1	4	yes	—
396	C₄²	C₂×D₄×C₂₃	1	2	yes	—
397	C₄²	C₂×D₄×C₂₃	1	4	yes	—
398	C₄²	C₂×D₄×C₂₃	1	2	yes	—
399	C₄²	C₂⁴×C₂₃	1	2	yes	—
400	C₄²	C₂⁴×C₂₃	1	2	yes	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	4	no	—
406	C₂³	—	4	4	no	—
407	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
408	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
409	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
410	Q₈	—	4	8	no	—
411	D₄	—	2	8	no	—
412	D₄	—	4	8	no	—
413	D₄	—	4	8	no	—
414	D₄	—	4	16	no	—
415	D₄	—	4	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	16	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	8	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	—	4	4	no	—
434	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
435	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
437	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
441	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
442	C₈	—	2	8	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	8	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	8	8	no	—
450	C₂²	C₂₃×#536	1	2	yes	—
451	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
452	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
453	C₂²	C₂₃×#575	1	2	yes	—
454	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
455	C₄	—	2	2	no	—
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	2	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	4	no	—
462	C₄	—	4	4	no	—
463	C₄	C₂₃×#536	1	2	yes	—
464	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
465	C₂	—	2	4	no	—
466	C₂	—	8	4	no	—
467	C₂	C₂₃×#3234	1	4	yes	direct product factor
468	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
469	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
470	C₂	C₂×C₂₃×#575	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
2. C₂²×C₂₃×#125	C₂	semidirect
3. C₂₃×#3234	C₂	direct (N × H)
4. C₂×C₂₃×#501	C₂	semidirect
5. C₂×C₂₃×#490	C₂	semidirect
6. C₂×C₂₃×#481	C₂	semidirect
7. C₂×C₂₃×#480	C₂	semidirect
8. C₂×C₂₃×#478	C₂	semidirect
9. C₂×C₂₃×#432	C₂	semidirect
20. C₂×C₂₃×#125	C₂²	semidirect
31. C₂₃×#501	C₂²	semidirect
32. C₂₃×#490	C₂²	semidirect
33. C₂₃×#481	C₂²	semidirect
34. C₂₃×#480	C₂²	semidirect
35. C₂₃×#478	C₂²	semidirect
36. C₂₃×#432	C₂²	semidirect
165. C₂×#3234	C₂₃	direct (N × H)
209. C₂²×#125	C₂×C₂₃	semidirect
210. #3234	C₂×C₂₃	direct (N × H)
211. C₂×#501	C₂×C₂₃	semidirect
212. C₂×#490	C₂×C₂₃	semidirect
213. C₂×#481	C₂×C₂₃	semidirect
214. C₂×#480	C₂×C₂₃	semidirect
215. C₂×#478	C₂×C₂₃	semidirect
216. C₂×#432	C₂×C₂₃	semidirect
249. C₂×#125	C₂²×C₂₃	semidirect
260. #501	C₂²×C₂₃	semidirect
261. #490	C₂²×C₂₃	semidirect
262. #481	C₂²×C₂₃	semidirect
263. #480	C₂²×C₂₃	semidirect
264. #478	C₂²×C₂₃	semidirect
265. #432	C₂²×C₂₃	semidirect
275. C₂×C₂₃	#3234	direct (N × H)
341. C₂₃	C₂×#3234	direct (N × H)
467. C₂	C₂₃×#3234	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×#532	2
C₂²×C₂₃×#125	2
C₂₃×#3234	2
C₂×C₂₃×#501	2
C₂×C₂₃×#490	2
C₂×C₂₃×#481	2
C₂×C₂₃×#480	2
C₂×C₂₃×#478	2
C₂×C₂₃×#432	2
C₂×#3234	23

Subgroups — C2×C23×#3234 #19208473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3234 #19208473