Subgroups — C₂×C₂₃×#3151 #19208262

Order 5888.

3950 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3151	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#496	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#493	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#492	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₂₃	C₂²	1	2	yes	—
10	C₂³×D₄×C₂₃	—	2	2	no	—
11	C₂³×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
13	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	32	yes	complemented
14	C₂²×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
15	C₂²×D₈×C₂₃	C₂²	1	8	yes	complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#532	—	2	4	no	—
18	C₂₃×#532	C₂²	1	8	yes	complemented
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂₃×#127	—	2	2	no	—
21	C₂×C₂₃×#127	—	2	4	no	—
22	C₂×C₂₃×#127	C₂²	1	2	yes	—
23	C₂×C₂₃×#127	C₂²	1	2	yes	—
24	C₂×C₂₃×#127	C₂²	1	8	yes	complemented
25	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#121	—	2	2	no	—
28	C₂×C₄×D₄×C₂₃	—	2	2	no	—
29	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#496	C₂²	1	32	yes	complemented
31	C₂₃×#493	C₂²	1	16	yes	complemented
32	C₂₃×#492	C₂²	1	16	yes	complemented
33	C₄×M₁₆×C₂₃	C₂²	1	8	yes	complemented
34	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
35	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
36	C₂⁵×C₂₃	—	2	2	no	—
37	C₂×#41×C₂₃	—	2	4	no	—
38	C₂×#41×C₂₃	—	2	16	no	—
39	C₂×#41×C₂₃	C₂³	1	8	yes	—
40	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×D₄×C₂₃	—	2	2	no	—
42	C₂²×D₄×C₂₃	—	2	16	no	—
43	C₂²×D₄×C₂₃	—	2	4	no	—
44	C₂²×D₄×C₂₃	—	2	4	no	—
45	C₂²×D₄×C₂₃	—	2	16	no	—
46	C₂²×D₄×C₂₃	—	2	4	no	—
47	C₂²×D₄×C₂₃	—	2	2	no	—
48	C₂²×D₄×C₂₃	—	2	2	no	—
49	C₂²×D₄×C₂₃	—	4	16	no	—
50	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
52	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
53	C₂³×C₄×C₂₃	—	2	2	no	—
54	C₂³×C₄×C₂₃	D₄	1	2	yes	—
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×AGL(1,ℤ/8)	—	2	128	no	—
57	C₂×SD₁₆×C₂₃	—	2	32	no	—
58	C₂×SD₁₆×C₂₃	C₂³	1	32	yes	complemented
59	C₂×D₈×C₂₃	—	2	32	no	—
60	C₂×D₈×C₂₃	C₂³	1	32	yes	complemented
61	C₂×M₁₆×C₂₃	D₄	1	16	yes	complemented
62	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
66	C₂₃×#127	—	2	4	no	—
67	C₂₃×#127	—	2	4	no	—
68	C₂₃×#127	—	2	16	no	—
69	C₂₃×#127	—	2	2	no	—
70	C₂₃×#127	C₂³	1	8	yes	—
71	C₂₃×#127	C₂³	1	8	yes	—
72	C₂₃×#127	C₂³	1	4	yes	complemented
73	C₂₃×#124	—	2	4	no	—
74	C₂₃×#124	C₂³	1	8	yes	complemented
75	C₂₃×#121	—	2	16	no	—
76	C₄×D₄×C₂₃	—	2	16	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
79	C₂×C₄²×C₂₃	D₄	1	2	yes	—
80	C₂×C₄²×C₂₃	D₄	1	4	yes	—
81	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
84	C₂₃×#97	C₂³	1	8	yes	complemented
85	C₄×C₈×C₂₃	C₂³	1	8	yes	complemented
86	C₂⁴×C₂₃	—	2	8	no	—
87	C₂⁴×C₂₃	—	2	2	no	—
88	C₂⁴×C₂₃	—	2	4	no	—
89	C₂⁴×C₂₃	—	4	16	no	—
90	C₂⁴×C₂₃	—	4	8	no	—
91	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
92	#41×C₂₃	—	2	32	no	—
93	#41×C₂₃	—	4	16	no	—
94	C₂×Q₈×C₂₃	—	2	2	no	—
95	C₂×Q₈×C₂₃	—	2	8	no	—
96	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
97	C₂×D₄×C₂₃	—	2	8	no	—
98	C₂×D₄×C₂₃	—	2	8	no	—
99	C₂×D₄×C₂₃	—	2	16	no	—
100	C₂×D₄×C₂₃	—	2	32	no	—
101	C₂×D₄×C₂₃	—	2	8	no	—
102	C₂×D₄×C₂₃	—	2	2	no	—
103	C₂×D₄×C₂₃	—	2	8	no	—
104	C₂×D₄×C₂₃	—	2	4	no	—
105	C₂×D₄×C₂₃	—	2	16	no	—
106	C₂×D₄×C₂₃	—	2	16	no	—
107	C₂×D₄×C₂₃	—	4	16	no	—
108	C₂×D₄×C₂₃	—	4	64	no	—
109	C₂×D₄×C₂₃	—	4	8	no	—
110	C₂×D₄×C₂₃	—	4	8	no	—
111	C₂×D₄×C₂₃	—	4	16	no	—
112	C₂×D₄×C₂₃	—	4	8	no	—
113	C₂×D₄×C₂₃	—	4	16	no	—
114	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
115	C₂×D₄×C₂₃	C₂⁴	1	8	yes	—
116	C₂²×C₄×C₂₃	—	2	8	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	2	8	no	—
119	C₂²×C₄×C₂₃	—	4	8	no	—
120	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
121	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
128	SD₁₆×C₂₃	—	2	64	no	—
129	D₈×C₂₃	—	2	64	no	—
130	M₁₆×C₂₃	C₂×D₄	1	32	yes	complemented
131	C₂×C₈×C₂₃	C₂×D₄	1	16	yes	complemented
132	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
133	C₄⋊C₄×C₂₃	—	2	8	no	—
134	C₂²⋊C₄×C₂₃	—	2	16	no	—
135	C₄²×C₂₃	C₂×D₄	1	8	yes	—
136	C₄²×C₂₃	C₂×D₄	1	2	yes	—
137	C₄²×C₂₃	C₂×D₄	1	2	yes	—
138	C₄²×C₂₃	C₂⁴	1	2	yes	—
139	C₄²×C₂₃	C₂⁴	1	2	yes	—
140	C₂×#3151	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₂₃	—	2	4	no	—
142	C₂³×C₂₃	—	2	4	no	—
143	C₂³×C₂₃	—	2	8	no	—
144	C₂³×C₂₃	—	4	32	no	—
145	C₂³×C₂₃	—	4	4	no	—
146	C₂³×C₂₃	—	4	16	no	—
147	C₂³×C₂₃	—	4	8	no	—
148	C₂³×C₂₃	—	4	16	no	—
149	C₂³×C₂₃	—	4	8	no	—
150	C₂³×C₂₃	—	8	32	no	—
151	C₂³×C₂₃	#127	1	4	yes	—
152	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
153	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₂₃	—	2	8	no	—
155	Q₈×C₂₃	—	4	4	no	—
156	D₄×C₂₃	—	2	8	no	—
157	D₄×C₂₃	—	2	16	no	—
158	D₄×C₂₃	—	4	32	no	—
159	D₄×C₂₃	—	4	16	no	—
160	D₄×C₂₃	—	4	16	no	—
161	D₄×C₂₃	—	4	8	no	—
162	D₄×C₂₃	—	4	4	no	—
163	D₄×C₂₃	—	4	16	no	—
164	D₄×C₂₃	—	4	32	no	—
165	C₂×C₄×C₂₃	—	2	4	no	—
166	C₂×C₄×C₂₃	—	2	16	no	—
167	C₂×C₄×C₂₃	—	2	4	no	—
168	C₂×C₄×C₂₃	—	4	4	no	—
169	C₂×C₄×C₂₃	—	4	16	no	—
170	C₂×C₄×C₂₃	#127	1	4	yes	—
171	C₂×C₄×C₂₃	#127	1	8	yes	—
172	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	8	yes	complemented
173	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
174	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
177	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
178	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
180	C₈×C₂₃	C₂²×D₄	1	16	yes	complemented
181	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	4	yes	complemented
182	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
183	C₂²×#127	C₂×C₂₃	1	1	yes	characteristic, complemented
184	#3151	C₂×C₂₃	1	16	yes	direct product factor
185	C₂×#496	C₂×C₂₃	1	4	yes	complemented
186	C₂×#493	C₂×C₂₃	1	2	yes	complemented
187	C₂×#492	C₂×C₂₃	1	2	yes	complemented
188	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
189	C₂²×C₂₃	—	2	8	no	—
190	C₂²×C₂₃	—	2	4	no	—
191	C₂²×C₂₃	—	4	4	no	—
192	C₂²×C₂₃	—	4	8	no	—
193	C₂²×C₂₃	—	4	16	no	—
194	C₂²×C₂₃	—	4	8	no	—
195	C₂²×C₂₃	—	8	32	no	—
196	C₂²×C₂₃	—	8	8	no	—
197	C₂²×C₂₃	—	8	16	no	—
198	C₂²×C₂₃	C₂×#127	1	4	yes	—
199	C₂²×C₂₃	C₂×#127	1	2	yes	—
200	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
201	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
202	C₄×C₂₃	—	2	4	no	—
203	C₄×C₂₃	—	4	4	no	—
204	C₄×C₂₃	C₂×#127	1	4	yes	—
205	C₄×C₂₃	C₂×#127	1	2	yes	—
206	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	8	yes	complemented
207	C₄×C₂₃	C₂³×D₄	1	2	yes	—
208	C₂²×#41	C₂²×C₂₃	1	2	yes	—
209	C₂³×D₄	—	2	2	no	—
210	C₂³×D₄	C₂²×C₂₃	1	2	yes	—
211	C₂×AGL(1,ℤ/8)	—	2	32	no	—
212	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	32	yes	complemented
213	C₂²×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
214	C₂²×D₈	C₂²×C₂₃	1	8	yes	complemented
215	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
216	#532	—	2	4	no	—
217	#532	C₂²×C₂₃	1	8	yes	complemented
218	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
219	C₂×#127	—	2	2	no	—
220	C₂×#127	—	2	4	no	—
221	C₂×#127	C₂²×C₂₃	1	2	yes	—
222	C₂×#127	C₂²×C₂₃	1	2	yes	—
223	C₂×#127	C₂²×C₂₃	1	8	yes	complemented
224	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
225	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
226	C₂×#121	—	2	2	no	—
227	C₂×C₄×D₄	—	2	2	no	—
228	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
229	#496	C₂²×C₂₃	1	32	yes	complemented
230	#493	C₂²×C₂₃	1	16	yes	complemented
231	#492	C₂²×C₂₃	1	16	yes	complemented
232	C₄×M₁₆	C₂²×C₂₃	1	8	yes	complemented
233	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
234	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
235	C₂×C₂₃	—	2	4	no	—
236	C₂×C₂₃	—	8	4	no	—
237	C₂×C₂₃	—	8	8	no	—
238	C₂×C₂₃	#3151	1	4	yes	direct product factor
239	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
240	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
241	C₂⁵	—	2	2	no	—
242	C₂×#41	—	2	4	no	—
243	C₂×#41	—	2	16	no	—
244	C₂×#41	C₂³×C₂₃	1	8	yes	—
245	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	—	2	16	no	—
248	C₂²×D₄	—	2	2	no	—
249	C₂²×D₄	—	2	2	no	—
250	C₂²×D₄	—	2	4	no	—
251	C₂²×D₄	—	2	4	no	—
252	C₂²×D₄	—	2	4	no	—
253	C₂²×D₄	—	2	16	no	—
254	C₂²×D₄	—	4	16	no	—
255	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
256	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
257	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
258	C₂³×C₄	—	2	2	no	—
259	C₂³×C₄	D₄×C₂₃	1	2	yes	—
260	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
261	AGL(1,ℤ/8)	—	2	128	no	—
262	C₂×SD₁₆	—	2	32	no	—
263	C₂×SD₁₆	C₂³×C₂₃	1	32	yes	complemented
264	C₂×D₈	—	2	32	no	—
265	C₂×D₈	C₂³×C₂₃	1	32	yes	complemented
266	C₂×M₁₆	D₄×C₂₃	1	16	yes	complemented
267	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
268	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
271	#127	—	2	4	no	—
272	#127	—	2	4	no	—
273	#127	—	2	16	no	—
274	#127	—	2	2	no	—
275	#127	C₂³×C₂₃	1	8	yes	—
276	#127	C₂³×C₂₃	1	8	yes	—
277	#127	C₂³×C₂₃	1	4	yes	complemented
278	#124	—	2	4	no	—
279	#124	C₂³×C₂₃	1	8	yes	complemented
280	#121	—	2	16	no	—
281	C₄×D₄	—	2	16	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	4	no	—
284	C₂×C₄²	D₄×C₂₃	1	2	yes	—
285	C₂×C₄²	D₄×C₂₃	1	4	yes	—
286	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
288	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
289	#97	C₂³×C₂₃	1	8	yes	complemented
290	C₄×C₈	C₂³×C₂₃	1	8	yes	complemented
291	C₂₃	C₂×#3151	1	1	yes	characteristic, direct product factor, Sylow
292	C₂⁴	—	2	2	no	—
293	C₂⁴	—	2	8	no	—
294	C₂⁴	—	2	4	no	—
295	C₂⁴	—	4	16	no	—
296	C₂⁴	—	4	8	no	—
297	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
298	#41	—	2	32	no	—
299	#41	—	4	16	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	8	no	—
302	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
303	C₂×D₄	—	2	8	no	—
304	C₂×D₄	—	2	8	no	—
305	C₂×D₄	—	2	16	no	—
306	C₂×D₄	—	2	32	no	—
307	C₂×D₄	—	2	8	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	8	no	—
310	C₂×D₄	—	2	4	no	—
311	C₂×D₄	—	2	16	no	—
312	C₂×D₄	—	2	16	no	—
313	C₂×D₄	—	4	16	no	—
314	C₂×D₄	—	4	64	no	—
315	C₂×D₄	—	4	8	no	—
316	C₂×D₄	—	4	8	no	—
317	C₂×D₄	—	4	16	no	—
318	C₂×D₄	—	4	16	no	—
319	C₂×D₄	—	4	8	no	—
320	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
321	C₂×D₄	C₂⁴×C₂₃	1	8	yes	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	4	8	no	—
326	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
327	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
328	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
331	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
333	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
334	SD₁₆	—	2	64	no	—
335	D₈	—	2	64	no	—
336	M₁₆	C₂×D₄×C₂₃	1	32	yes	complemented
337	C₂×C₈	C₂×D₄×C₂₃	1	16	yes	complemented
338	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
339	C₄⋊C₄	—	2	8	no	—
340	C₂²⋊C₄	—	2	16	no	—
341	C₄²	C₂×D₄×C₂₃	1	8	yes	—
342	C₄²	C₂×D₄×C₂₃	1	2	yes	—
343	C₄²	C₂×D₄×C₂₃	1	2	yes	—
344	C₄²	C₂⁴×C₂₃	1	2	yes	—
345	C₄²	C₂⁴×C₂₃	1	2	yes	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	4	no	—
348	C₂³	—	2	8	no	—
349	C₂³	—	4	32	no	—
350	C₂³	—	4	16	no	—
351	C₂³	—	4	8	no	—
352	C₂³	—	4	4	no	—
353	C₂³	—	4	16	no	—
354	C₂³	—	4	8	no	—
355	C₂³	—	8	32	no	—
356	C₂³	C₂₃×#127	1	4	yes	—
357	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
358	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
359	Q₈	—	2	8	no	—
360	Q₈	—	4	4	no	—
361	D₄	—	2	8	no	—
362	D₄	—	2	16	no	—
363	D₄	—	4	32	no	—
364	D₄	—	4	16	no	—
365	D₄	—	4	32	no	—
366	D₄	—	4	16	no	—
367	D₄	—	4	16	no	—
368	D₄	—	4	8	no	—
369	D₄	—	4	4	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	16	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	4	16	no	—
375	C₂×C₄	C₂₃×#127	1	4	yes	—
376	C₂×C₄	C₂₃×#127	1	8	yes	—
377	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	8	yes	complemented
378	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
379	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
380	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
381	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
382	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
383	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
385	C₈	C₂²×D₄×C₂₃	1	16	yes	complemented
386	C₂²	—	2	4	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	4	4	no	—
389	C₂²	—	4	8	no	—
390	C₂²	—	4	16	no	—
391	C₂²	—	4	8	no	—
392	C₂²	—	8	32	no	—
393	C₂²	—	8	16	no	—
394	C₂²	—	8	8	no	—
395	C₂²	C₂×C₂₃×#127	1	4	yes	—
396	C₂²	C₂×C₂₃×#127	1	2	yes	—
397	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
398	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
399	C₄	—	2	4	no	—
400	C₄	—	4	4	no	—
401	C₄	C₂×C₂₃×#127	1	4	yes	—
402	C₄	C₂×C₂₃×#127	1	2	yes	—
403	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	8	yes	complemented
404	C₄	C₂³×D₄×C₂₃	1	2	yes	—
405	C₂	—	2	4	no	—
406	C₂	—	8	8	no	—
407	C₂	—	8	4	no	—
408	C₂	C₂₃×#3151	1	4	yes	direct product factor
409	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
410	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
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. C₂²×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₂₃×#532	C₂	semidirect
3. C₂²×C₂₃×#127	C₂	semidirect
4. C₂₃×#3151	C₂	direct (N × H)
5. C₂×C₂₃×#496	C₂	semidirect
6. C₂×C₂₃×#493	C₂	semidirect
7. C₂×C₂₃×#492	C₂	semidirect
8. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
13. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂²×SD₁₆×C₂₃	C₂²	semidirect
15. C₂²×D₈×C₂₃	C₂²	semidirect
18. C₂₃×#532	C₂²	semidirect
19. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂×C₂₃×#127	C₂²	semidirect
25. C₂×C₂₃×#127	C₂²	semidirect
26. C₂×C₂₃×#124	C₂²	semidirect
30. C₂₃×#496	C₂²	semidirect
31. C₂₃×#493	C₂²	semidirect
32. C₂₃×#492	C₂²	semidirect
33. C₄×M₁₆×C₂₃	C₂²	semidirect
34. C₂×C₂₃×#97	C₂²	semidirect
35. C₂×C₄×C₈×C₂₃	C₂²	semidirect
58. C₂×SD₁₆×C₂₃	C₂³	semidirect
60. C₂×D₈×C₂₃	C₂³	semidirect
61. C₂×M₁₆×C₂₃	D₄	semidirect
65. C₂₃×C₄⋊Q₈	C₂³	semidirect
72. C₂₃×#127	C₂³	semidirect
74. C₂₃×#124	C₂³	semidirect
84. C₂₃×#97	C₂³	semidirect
85. C₄×C₈×C₂₃	C₂³	semidirect
130. M₁₆×C₂₃	C₂×D₄	semidirect
131. C₂×C₈×C₂₃	C₂×D₄	semidirect
140. C₂×#3151	C₂₃	direct (N × H)
172. C₂×C₄×C₂₃	AGL(1,ℤ/8)	semidirect
180. C₈×C₂₃	C₂²×D₄	semidirect
181. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
182. C₂×#532	C₂×C₂₃	semidirect
183. C₂²×#127	C₂×C₂₃	semidirect
184. #3151	C₂×C₂₃	direct (N × H)
185. C₂×#496	C₂×C₂₃	semidirect
186. C₂×#493	C₂×C₂₃	semidirect
187. C₂×#492	C₂×C₂₃	semidirect
188. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
206. C₄×C₂₃	C₂×AGL(1,ℤ/8)	semidirect
212. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
213. C₂²×SD₁₆	C₂²×C₂₃	semidirect
214. C₂²×D₈	C₂²×C₂₃	semidirect
217. #532	C₂²×C₂₃	semidirect
218. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
223. C₂×#127	C₂²×C₂₃	semidirect
224. C₂×#127	C₂²×C₂₃	semidirect
225. C₂×#124	C₂²×C₂₃	semidirect
229. #496	C₂²×C₂₃	semidirect
230. #493	C₂²×C₂₃	semidirect
231. #492	C₂²×C₂₃	semidirect
232. C₄×M₁₆	C₂²×C₂₃	semidirect
233. C₂×#97	C₂²×C₂₃	semidirect
234. C₂×C₄×C₈	C₂²×C₂₃	semidirect
238. C₂×C₂₃	#3151	direct (N × H)
263. C₂×SD₁₆	C₂³×C₂₃	semidirect
265. C₂×D₈	C₂³×C₂₃	semidirect
266. C₂×M₁₆	D₄×C₂₃	semidirect
270. C₄⋊Q₈	C₂³×C₂₃	semidirect
277. #127	C₂³×C₂₃	semidirect
279. #124	C₂³×C₂₃	semidirect
289. #97	C₂³×C₂₃	semidirect
290. C₄×C₈	C₂³×C₂₃	semidirect
291. C₂₃	C₂×#3151	direct (N × H)
336. M₁₆	C₂×D₄×C₂₃	semidirect
337. C₂×C₈	C₂×D₄×C₂₃	semidirect
377. C₂×C₄	C₂₃×AGL(1,ℤ/8)	semidirect
385. C₈	C₂²×D₄×C₂₃	semidirect
403. C₄	C₂×C₂₃×AGL(1,ℤ/8)	semidirect
408. C₂	C₂₃×#3151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×AGL(1,ℤ/8)	2
C₂×C₂₃×#532	2
C₂²×C₂₃×#127	2
C₂₃×#3151	2
C₂×C₂₃×#496	2
C₂×C₂₃×#493	2
C₂×C₂₃×#492	2
C₂×C₄×M₁₆×C₂₃	2
C₂×#3151	23

Subgroups — C2×C23×#3151 #19208262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3151 #19208262