Subgroups — C₂×C₂₃×#3165 #19208287

Order 5888.

2118 conjugacy classes of subgroups.

The subgroup lattice has 410 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
2	C₂²×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3165	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#501	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×#500	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#498	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#428	C₂	1	4	yes	maximal
8	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₂₃	—	2	1	no	—
10	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#533	—	2	4	no	—
12	C₂₃×#533	C₂²	1	8	yes	—
13	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂×C₂₃×C₄⋊Q₈	—	2	4	no	—
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	8	yes	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#125	C₂²	1	2	yes	—
22	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
23	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
25	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
26	C₂×C₂₃×#117	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#501	C₂²	1	32	yes	complemented
31	C₂₃×#500	C₂²	1	16	yes	complemented
32	C₂₃×#498	C₂²	1	16	yes	complemented
33	C₂₃×#428	C₂²	1	32	yes	—
34	C₂×C₂₃×#107	C₂²	1	8	yes	—
35	C₂×C₂₃×#106	C₂²	1	8	yes	—
36	C₄×M₁₆×C₂₃	C₂²	1	8	yes	—
37	C₂×C₂₃×#97	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
39	C₂²×Q₈×C₂₃	—	2	8	no	—
40	C₂²×Q₈×C₂₃	—	2	2	no	—
41	C₂²×Q₈×C₂₃	—	2	2	no	—
42	C₂²×Q₈×C₂₃	—	2	1	no	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	—	4	8	no	—
45	C₂³×C₄×C₂₃	—	2	2	no	—
46	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₂₃	Q₈	1	16	yes	complemented
50	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
51	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
52	C₂₃×C₄⋊Q₈	—	2	8	no	—
53	C₂₃×C₄⋊Q₈	—	2	8	no	—
54	C₂₃×C₄⋊Q₈	—	2	4	no	—
55	C₂₃×C₄⋊Q₈	—	2	4	no	—
56	C₂₃×C₄⋊Q₈	—	2	2	no	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
59	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
61	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
62	C₂₃×#125	—	2	4	no	—
63	C₂₃×#125	C₂³	1	8	yes	—
64	C₂₃×C₂²⋊Q₈	—	2	8	no	—
65	C₂₃×C₂²⋊Q₈	—	2	8	no	—
66	C₄×Q₈×C₂₃	—	2	8	no	—
67	C₄×Q₈×C₂₃	—	2	8	no	—
68	C₂₃×#117	—	2	4	no	—
69	C₂₃×#117	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄²×C₂₃	—	2	2	no	—
85	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₂₃	D₄	1	4	yes	—
88	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
91	C₂₃×#107	C₂³	1	32	yes	—
92	C₂₃×#106	C₂³	1	32	yes	—
93	C₂₃×#97	C₂³	1	8	yes	—
94	C₄×C₈×C₂₃	C₂³	1	8	yes	—
95	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₂₃	—	2	4	no	—
97	C₂×Q₈×C₂₃	—	2	4	no	—
98	C₂×Q₈×C₂₃	—	2	8	no	—
99	C₂×Q₈×C₂₃	—	2	8	no	—
100	C₂×Q₈×C₂₃	—	4	8	no	—
101	C₂×Q₈×C₂₃	—	4	32	no	—
102	C₂×Q₈×C₂₃	—	4	4	no	—
103	C₂×Q₈×C₂₃	—	4	8	no	—
104	C₂×Q₈×C₂₃	—	4	4	no	—
105	C₂²×C₄×C₂₃	—	2	4	no	—
106	C₂²×C₄×C₂₃	—	2	4	no	—
107	C₂²×C₄×C₂₃	—	2	8	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	4	no	—
111	C₂²×C₄×C₂₃	—	4	8	no	—
112	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂×Q₈	1	8	yes	—
121	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₂₃	C₂×Q₈	1	32	yes	complemented
124	C₂×C₈×C₂₃	C₂×Q₈	1	16	yes	complemented
125	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
126	C₄⋊C₄×C₂₃	—	2	8	no	—
127	C₄⋊C₄×C₂₃	—	2	8	no	—
128	C₄⋊C₄×C₂₃	—	2	8	no	—
129	C₄⋊C₄×C₂₃	—	2	4	no	—
130	C₄⋊C₄×C₂₃	—	2	8	no	—
131	C₄⋊C₄×C₂₃	—	2	8	no	—
132	C₄⋊C₄×C₂₃	—	2	4	no	—
133	C₄⋊C₄×C₂₃	—	2	4	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
137	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
138	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
139	C₂²⋊C₄×C₂₃	—	2	8	no	—
140	C₄²×C₂₃	—	2	8	no	—
141	C₄²×C₂₃	C₂×D₄	1	4	yes	—
142	C₄²×C₂₃	C₂×D₄	1	4	yes	—
143	C₄²×C₂₃	C₂×D₄	1	2	yes	—
144	C₄²×C₂₃	C₂×D₄	1	2	yes	—
145	C₄²×C₂₃	C₂⁴	1	2	yes	—
146	C₄²×C₂₃	C₂⁴	1	2	yes	—
147	C₂×#3165	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₂₃	—	2	2	no	—
149	C₂³×C₂₃	—	2	2	no	—
150	C₂³×C₂₃	C₄⋊Q₈	1	4	yes	—
151	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
152	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₂₃	—	4	16	no	—
154	Q₈×C₂₃	—	4	8	no	—
155	Q₈×C₂₃	—	4	16	no	—
156	Q₈×C₂₃	—	4	8	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	2	8	no	—
160	C₂×C₄×C₂₃	—	2	8	no	—
161	C₂×C₄×C₂₃	—	2	4	no	—
162	C₂×C₄×C₂₃	—	2	4	no	—
163	C₂×C₄×C₂₃	—	2	8	no	—
164	C₂×C₄×C₂₃	—	4	16	no	—
165	C₂×C₄×C₂₃	—	4	8	no	—
166	C₂×C₄×C₂₃	—	4	4	no	—
167	C₂×C₄×C₂₃	—	4	4	no	—
168	C₂×C₄×C₂₃	C₄⋊Q₈	1	8	yes	—
169	C₂×C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
170	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
171	C₂×C₄×C₂₃	#137	1	4	yes	—
172	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₂₃	C₂²×D₄	1	2	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	2	yes	—
177	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₂₃	C₂²×Q₈	1	4	yes	—
179	C₂×C₄×C₂₃	C₂²×Q₈	1	8	yes	—
180	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
181	C₈×C₂₃	C₂²×Q₈	1	16	yes	complemented
182	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
183	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
184	#3165	C₂×C₂₃	1	16	yes	direct product factor
185	C₂×#501	C₂×C₂₃	1	4	yes	complemented
186	C₂×#500	C₂×C₂₃	1	2	yes	complemented
187	C₂×#498	C₂×C₂₃	1	2	yes	complemented
188	C₂×#428	C₂×C₂₃	1	4	yes	—
189	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
190	C₂²×C₂₃	—	2	8	no	—
191	C₂²×C₂₃	—	2	2	no	—
192	C₂²×C₂₃	—	2	2	no	—
193	C₂²×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
194	C₂²×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
195	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
196	C₂²×C₂₃	C₂×#137	1	2	yes	—
197	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
198	C₄×C₂₃	—	2	2	no	—
199	C₄×C₂₃	—	2	2	no	—
200	C₄×C₂₃	—	4	8	no	—
201	C₄×C₂₃	—	4	4	no	—
202	C₄×C₂₃	—	4	4	no	—
203	C₄×C₂₃	C₂×C₄⋊Q₈	1	4	yes	—
204	C₄×C₂₃	C₂×C₄⋊Q₈	1	2	yes	—
205	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
206	C₄×C₂₃	C₂×#137	1	4	yes	—
207	C₄×C₂₃	C₂³×Q₈	1	2	yes	—
208	C₂³×Q₈	—	2	1	no	—
209	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
210	#533	—	2	4	no	—
211	#533	C₂²×C₂₃	1	8	yes	—
212	C₂×C₄⋊Q₈	—	2	1	no	—
213	C₂×C₄⋊Q₈	—	2	1	no	—
214	C₂×C₄⋊Q₈	—	2	4	no	—
215	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
216	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
217	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
219	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	—
220	C₂×#125	C₂²×C₂₃	1	2	yes	—
221	C₂×C₂²⋊Q₈	—	2	1	no	—
222	C₂×C₂²⋊Q₈	—	2	1	no	—
223	C₂×C₄×Q₈	—	2	1	no	—
224	C₂×C₄×Q₈	—	2	1	no	—
225	C₂×#117	C₂²×C₂₃	1	2	yes	—
226	C₂²×C₄⋊C₄	—	2	1	no	—
227	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
228	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
229	#501	C₂²×C₂₃	1	32	yes	complemented
230	#500	C₂²×C₂₃	1	16	yes	complemented
231	#498	C₂²×C₂₃	1	16	yes	complemented
232	#428	C₂²×C₂₃	1	32	yes	—
233	C₂×#107	C₂²×C₂₃	1	8	yes	—
234	C₂×#106	C₂²×C₂₃	1	8	yes	—
235	C₄×M₁₆	C₂²×C₂₃	1	8	yes	—
236	C₂×#97	C₂²×C₂₃	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
238	C₂×C₂₃	—	2	4	no	—
239	C₂×C₂₃	#3165	1	4	yes	direct product factor
240	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
241	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
243	C₂²×Q₈	—	2	1	no	—
244	C₂²×Q₈	—	2	1	no	—
245	C₂²×Q₈	—	2	2	no	—
246	C₂²×Q₈	—	2	8	no	—
247	C₂²×Q₈	—	2	2	no	—
248	C₂²×Q₈	—	4	8	no	—
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
251	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
252	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
253	C₂×M₁₆	Q₈×C₂₃	1	16	yes	complemented
254	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
255	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
256	C₄⋊Q₈	—	2	8	no	—
257	C₄⋊Q₈	—	2	8	no	—
258	C₄⋊Q₈	—	2	4	no	—
259	C₄⋊Q₈	—	2	4	no	—
260	C₄⋊Q₈	—	2	2	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
263	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
264	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
265	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
266	#125	—	2	4	no	—
267	#125	C₂³×C₂₃	1	8	yes	—
268	C₂²⋊Q₈	—	2	8	no	—
269	C₂²⋊Q₈	—	2	8	no	—
270	C₄×Q₈	—	2	8	no	—
271	C₄×Q₈	—	2	8	no	—
272	#117	—	2	4	no	—
273	#117	C₂³×C₂₃	1	8	yes	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	8	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
285	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
287	C₂×C₂²⋊C₄	—	2	2	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₄²	D₄×C₂₃	1	4	yes	—
292	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
295	#107	C₂³×C₂₃	1	32	yes	—
296	#106	C₂³×C₂₃	1	32	yes	—
297	#97	C₂³×C₂₃	1	8	yes	—
298	C₄×C₈	C₂³×C₂₃	1	8	yes	—
299	C₂₃	C₂×#3165	1	1	yes	characteristic, direct product factor, Sylow
300	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	2	8	no	—
304	C₂×Q₈	—	2	8	no	—
305	C₂×Q₈	—	4	8	no	—
306	C₂×Q₈	—	4	32	no	—
307	C₂×Q₈	—	4	8	no	—
308	C₂×Q₈	—	4	4	no	—
309	C₂×Q₈	—	4	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	8	no	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	4	8	no	—
317	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
319	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
321	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
325	C₂²×C₄	C₂×Q₈×C₂₃	1	8	yes	—
326	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
327	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
328	M₁₆	C₂×Q₈×C₂₃	1	32	yes	complemented
329	C₂×C₈	C₂×Q₈×C₂₃	1	16	yes	complemented
330	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
342	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
343	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
344	C₂²⋊C₄	—	2	8	no	—
345	C₄²	—	2	8	no	—
346	C₄²	C₂×D₄×C₂₃	1	4	yes	—
347	C₄²	C₂×D₄×C₂₃	1	4	yes	—
348	C₄²	C₂×D₄×C₂₃	1	2	yes	—
349	C₄²	C₂×D₄×C₂₃	1	2	yes	—
350	C₄²	C₂⁴×C₂₃	1	2	yes	—
351	C₄²	C₂⁴×C₂₃	1	2	yes	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	C₂₃×C₄⋊Q₈	1	4	yes	—
355	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
356	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
357	Q₈	—	4	16	no	—
358	Q₈	—	4	16	no	—
359	Q₈	—	4	8	no	—
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	8	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	C₂₃×C₄⋊Q₈	1	8	yes	—
373	C₂×C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
374	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
375	C₂×C₄	C₂₃×#137	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
377	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
378	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
379	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
380	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂²×Q₈×C₂₃	1	4	yes	—
383	C₂×C₄	C₂²×Q₈×C₂₃	1	8	yes	—
384	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
385	C₈	C₂²×Q₈×C₂₃	1	16	yes	complemented
386	C₂²	—	2	8	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
390	C₂²	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
391	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
392	C₂²	C₂×C₂₃×#137	1	2	yes	—
393	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	8	no	—
397	C₄	—	4	4	no	—
398	C₄	—	4	4	no	—
399	C₄	C₂×C₂₃×C₄⋊Q₈	1	4	yes	—
400	C₄	C₂×C₂₃×C₄⋊Q₈	1	2	yes	—
401	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
402	C₄	C₂×C₂₃×#137	1	4	yes	—
403	C₄	C₂³×Q₈×C₂₃	1	2	yes	—
404	C₂	—	2	4	no	—
405	C₂	C₂₃×#3165	1	4	yes	direct product factor
406	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
407	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
408	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	C₂⁴×C₂₃	C₂×D₄
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
3. C₂₃×#3165	C₂	direct (N × H)
4. C₂×C₂₃×#501	C₂	semidirect
5. C₂×C₂₃×#500	C₂	semidirect
6. C₂×C₂₃×#498	C₂	semidirect
30. C₂₃×#501	C₂²	semidirect
31. C₂₃×#500	C₂²	semidirect
32. C₂₃×#498	C₂²	semidirect
49. C₂×M₁₆×C₂₃	Q₈	semidirect
123. M₁₆×C₂₃	C₂×Q₈	semidirect
124. C₂×C₈×C₂₃	C₂×Q₈	semidirect
147. C₂×#3165	C₂₃	direct (N × H)
181. C₈×C₂₃	C₂²×Q₈	semidirect
184. #3165	C₂×C₂₃	direct (N × H)
185. C₂×#501	C₂×C₂₃	semidirect
186. C₂×#500	C₂×C₂₃	semidirect
187. C₂×#498	C₂×C₂₃	semidirect
229. #501	C₂²×C₂₃	semidirect
230. #500	C₂²×C₂₃	semidirect
231. #498	C₂²×C₂₃	semidirect
239. C₂×C₂₃	#3165	direct (N × H)
253. C₂×M₁₆	Q₈×C₂₃	semidirect
299. C₂₃	C₂×#3165	direct (N × H)
328. M₁₆	C₂×Q₈×C₂₃	semidirect
329. C₂×C₈	C₂×Q₈×C₂₃	semidirect
385. C₈	C₂²×Q₈×C₂₃	semidirect
405. C₂	C₂₃×#3165	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
C₂²×C₂₃×C₄⋊Q₈	2
C₂₃×#3165	2
C₂×C₂₃×#501	2
C₂×C₂₃×#500	2
C₂×C₂₃×#498	2
C₂×C₂₃×#428	2
C₂×C₄×M₁₆×C₂₃	2
C₂×#3165	23

Subgroups — C2×C23×#3165 #19208287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3165 #19208287