Subgroups — C₂×C₂₃×#3229 #19208465

Order 5888.

2006 conjugacy classes of subgroups.

The subgroup lattice has 408 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×#3229	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#498	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#487	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#483	C₂	1	4	yes	maximal
7	C₂×C₂₃×#475	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₂₃	—	2	1	no	—
10	C₂₃×#533	—	2	4	no	—
11	C₂₃×#533	C₂²	1	8	yes	—
12	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
14	C₂×C₂₃×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
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×#125	—	2	1	no	—
21	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂₃×#117	—	2	1	no	—
26	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#498	C₂²	1	16	yes	complemented
30	C₂₃×#487	C₂²	1	16	yes	complemented
31	C₂₃×#483	C₂²	1	32	yes	—
32	C₂₃×#475	C₂²	1	32	yes	complemented
33	C₂₃×#431	C₂²	1	8	yes	—
34	C₂×C₂₃×#106	C₂²	1	8	yes	—
35	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
36	C₂×C₂₃×#103	C₂²	1	8	yes	—
37	C₂×C₂₃×C₂²⋊C₈	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	—	4	8	no	—
44	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
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	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
50	C₂₃×C₄⋊Q₈	—	2	8	no	—
51	C₂₃×C₄⋊Q₈	—	2	4	no	—
52	C₂₃×C₄⋊Q₈	—	2	8	no	—
53	C₂₃×C₄⋊Q₈	—	2	4	no	—
54	C₂₃×C₄⋊Q₈	—	2	2	no	—
55	C₂₃×C₄⋊Q₈	—	2	2	no	—
56	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
57	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
58	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
59	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
60	C₂₃×#125	—	2	8	no	—
61	C₂₃×C₂²⋊Q₈	—	2	8	no	—
62	C₂₃×C₂²⋊Q₈	—	2	4	no	—
63	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
64	C₄×Q₈×C₂₃	—	2	8	no	—
65	C₄×Q₈×C₂₃	—	2	4	no	—
66	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
67	C₂₃×#117	—	2	8	no	—
68	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₄²×C₂₃	—	2	2	no	—
83	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
85	C₂×C₄²×C₂₃	D₄	1	4	yes	—
86	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
89	C₂₃×#106	C₂³	1	32	yes	—
90	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
91	C₂₃×#103	C₂³	1	32	yes	—
92	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
93	C₄×C₈×C₂₃	C₂³	1	8	yes	—
94	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₂₃	—	2	2	no	—
96	C₂×Q₈×C₂₃	—	2	4	no	—
97	C₂×Q₈×C₂₃	—	2	8	no	—
98	C₂×Q₈×C₂₃	—	2	8	no	—
99	C₂×Q₈×C₂₃	—	2	8	no	—
100	C₂×Q₈×C₂₃	—	4	32	no	—
101	C₂×Q₈×C₂₃	—	4	8	no	—
102	C₂×Q₈×C₂₃	—	4	8	no	—
103	C₂×Q₈×C₂₃	—	4	4	no	—
104	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
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	SD₁₆	1	8	yes	—
113	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
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×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₂₃	—	2	8	no	—
124	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
125	C₄⋊C₄×C₂₃	—	2	8	no	—
126	C₄⋊C₄×C₂₃	—	2	8	no	—
127	C₄⋊C₄×C₂₃	—	2	8	no	—
128	C₄⋊C₄×C₂₃	—	2	4	no	—
129	C₄⋊C₄×C₂₃	—	2	4	no	—
130	C₄⋊C₄×C₂₃	—	2	4	no	—
131	C₄⋊C₄×C₂₃	—	2	4	no	—
132	C₄⋊C₄×C₂₃	—	2	8	no	—
133	C₄⋊C₄×C₂₃	—	2	8	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
136	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
137	C₂²⋊C₄×C₂₃	—	2	8	no	—
138	C₄²×C₂₃	—	2	8	no	—
139	C₄²×C₂₃	C₂×D₄	1	4	yes	—
140	C₄²×C₂₃	C₂×D₄	1	2	yes	—
141	C₄²×C₂₃	C₂×D₄	1	4	yes	—
142	C₄²×C₂₃	C₂×D₄	1	2	yes	—
143	C₄²×C₂₃	C₂⁴	1	2	yes	—
144	C₄²×C₂₃	C₂⁴	1	2	yes	—
145	C₂×#3229	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₂₃	—	2	2	no	—
147	C₂³×C₂₃	—	2	2	no	—
148	C₂³×C₂₃	C₂×SD₁₆	1	4	yes	—
149	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
150	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₂₃	—	2	8	no	—
152	Q₈×C₂₃	—	4	4	no	—
153	Q₈×C₂₃	—	4	16	no	—
154	Q₈×C₂₃	—	4	16	no	—
155	Q₈×C₂₃	—	4	8	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	4	no	—
159	C₂×C₄×C₂₃	—	2	8	no	—
160	C₂×C₄×C₂₃	—	2	4	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	4	no	—
165	C₂×C₄×C₂₃	—	4	16	no	—
166	C₂×C₄×C₂₃	—	4	8	no	—
167	C₂×C₄×C₂₃	—	4	4	no	—
168	C₂×C₄×C₂₃	C₂×SD₁₆	1	16	yes	—
169	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
170	C₂×C₄×C₂₃	C₂×SD₁₆	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	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
179	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
180	C₈×C₂₃	—	2	8	no	—
181	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
182	C₂²×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
183	#3229	C₂×C₂₃	1	16	yes	direct product factor
184	C₂×#498	C₂×C₂₃	1	2	yes	complemented
185	C₂×#487	C₂×C₂₃	1	2	yes	complemented
186	C₂×#483	C₂×C₂₃	1	4	yes	—
187	C₂×#475	C₂×C₂₃	1	4	yes	complemented
188	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
189	C₂²×C₂₃	—	2	2	no	—
190	C₂²×C₂₃	—	2	8	no	—
191	C₂²×C₂₃	—	2	2	no	—
192	C₂²×C₂₃	#536	1	2	yes	—
193	C₂²×C₂₃	C₂²×SD₁₆	1	4	yes	—
194	C₂²×C₂₃	C₂²×SD₁₆	1	2	yes	—
195	C₂²×C₂₃	C₂×#137	1	2	yes	—
196	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
197	C₄×C₂₃	—	2	2	no	—
198	C₄×C₂₃	—	2	2	no	—
199	C₄×C₂₃	—	4	4	no	—
200	C₄×C₂₃	—	4	8	no	—
201	C₄×C₂₃	—	4	4	no	—
202	C₄×C₂₃	#536	1	2	yes	—
203	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
204	C₄×C₂₃	C₂²×SD₁₆	1	4	yes	—
205	C₄×C₂₃	C₂×#137	1	4	yes	—
206	C₄×C₂₃	C₂×2₋¹⁺⁴	1	2	yes	—
207	C₂³×Q₈	—	2	1	no	—
208	#533	—	2	4	no	—
209	#533	C₂²×C₂₃	1	8	yes	—
210	C₂×C₄⋊Q₈	—	2	1	no	—
211	C₂×C₄⋊Q₈	—	2	1	no	—
212	C₂×C₄⋊Q₈	—	2	4	no	—
213	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
214	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
215	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
216	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
217	C₂×C₄⋊Q₈	C₂²×C₂₃	1	8	yes	—
218	C₂×#125	—	2	1	no	—
219	C₂×C₂²⋊Q₈	—	2	1	no	—
220	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
221	C₂×C₄×Q₈	—	2	1	no	—
222	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
223	C₂×#117	—	2	1	no	—
224	C₂²×C₄⋊C₄	—	2	1	no	—
225	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
226	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
227	#498	C₂²×C₂₃	1	16	yes	complemented
228	#487	C₂²×C₂₃	1	16	yes	complemented
229	#483	C₂²×C₂₃	1	32	yes	—
230	#475	C₂²×C₂₃	1	32	yes	complemented
231	#431	C₂²×C₂₃	1	8	yes	—
232	C₂×#106	C₂²×C₂₃	1	8	yes	—
233	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
234	C₂×#103	C₂²×C₂₃	1	8	yes	—
235	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
237	C₂×C₂₃	—	2	4	no	—
238	C₂×C₂₃	#3229	1	4	yes	direct product factor
239	C₂×C₂₃	C₂×#536	1	1	yes	characteristic
240	C₂×C₂₃	C₂³×SD₁₆	1	1	yes	characteristic
241	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×Q₈	—	2	2	no	—
244	C₂²×Q₈	—	2	2	no	—
245	C₂²×Q₈	—	2	8	no	—
246	C₂²×Q₈	—	4	8	no	—
247	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
248	C₂³×C₄	—	2	2	no	—
249	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
250	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
251	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
253	C₄⋊Q₈	—	2	4	no	—
254	C₄⋊Q₈	—	2	8	no	—
255	C₄⋊Q₈	—	2	8	no	—
256	C₄⋊Q₈	—	2	4	no	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	—	2	2	no	—
259	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
260	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
261	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
262	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
263	#125	—	2	8	no	—
264	C₂²⋊Q₈	—	2	8	no	—
265	C₂²⋊Q₈	—	2	4	no	—
266	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
267	C₄×Q₈	—	2	8	no	—
268	C₄×Q₈	—	2	4	no	—
269	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
270	#117	—	2	8	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	8	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	4	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
283	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
287	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄²	D₄×C₂₃	1	4	yes	—
289	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
291	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
292	#106	C₂³×C₂₃	1	32	yes	—
293	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
294	#103	C₂³×C₂₃	1	32	yes	—
295	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
296	C₄×C₈	C₂³×C₂₃	1	8	yes	—
297	C₂₃	C₂×#3229	1	1	yes	characteristic, direct product factor, Sylow
298	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	4	no	—
301	C₂×Q₈	—	2	8	no	—
302	C₂×Q₈	—	2	8	no	—
303	C₂×Q₈	—	2	8	no	—
304	C₂×Q₈	—	4	32	no	—
305	C₂×Q₈	—	4	8	no	—
306	C₂×Q₈	—	4	8	no	—
307	C₂×Q₈	—	4	4	no	—
308	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	4	8	no	—
316	C₂²×C₄	SD₁₆×C₂₃	1	8	yes	—
317	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₂₃	1	2	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	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
324	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
326	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
327	C₂×C₈	—	2	8	no	—
328	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
340	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₄²	—	2	8	no	—
343	C₄²	C₂×D₄×C₂₃	1	4	yes	—
344	C₄²	C₂×D₄×C₂₃	1	2	yes	—
345	C₄²	C₂×D₄×C₂₃	1	4	yes	—
346	C₄²	C₂×D₄×C₂₃	1	2	yes	—
347	C₄²	C₂⁴×C₂₃	1	2	yes	—
348	C₄²	C₂⁴×C₂₃	1	2	yes	—
349	C₂³	—	2	2	no	—
350	C₂³	—	2	2	no	—
351	C₂³	C₂×SD₁₆×C₂₃	1	4	yes	—
352	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
353	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
354	Q₈	—	2	8	no	—
355	Q₈	—	4	16	no	—
356	Q₈	—	4	16	no	—
357	Q₈	—	4	8	no	—
358	Q₈	—	4	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	8	no	—
367	C₂×C₄	—	4	16	no	—
368	C₂×C₄	—	4	8	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	C₂×SD₁₆×C₂₃	1	16	yes	—
372	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
373	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
374	C₂×C₄	C₂₃×#137	1	4	yes	—
375	C₂×C₄	C₂²×D₄×C₂₃	1	2	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	1	yes	characteristic
381	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
382	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
383	C₈	—	2	8	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	2	2	no	—
387	C₂²	C₂₃×#536	1	2	yes	—
388	C₂²	C₂²×SD₁₆×C₂₃	1	4	yes	—
389	C₂²	C₂²×SD₁₆×C₂₃	1	2	yes	—
390	C₂²	C₂×C₂₃×#137	1	2	yes	—
391	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	4	4	no	—
395	C₄	—	4	8	no	—
396	C₄	—	4	4	no	—
397	C₄	C₂₃×#536	1	2	yes	—
398	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
399	C₄	C₂²×SD₁₆×C₂₃	1	4	yes	—
400	C₄	C₂×C₂₃×#137	1	4	yes	—
401	C₄	C₂×C₂₃×2₋¹⁺⁴	1	2	yes	—
402	C₂	—	2	4	no	—
403	C₂	C₂₃×#3229	1	4	yes	direct product factor
404	C₂	C₂×C₂₃×#536	1	1	yes	characteristic
405	C₂	C₂³×SD₁₆×C₂₃	1	1	yes	characteristic
406	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×#3229	C₂	direct (N × H)
4. C₂×C₂₃×#498	C₂	semidirect
5. C₂×C₂₃×#487	C₂	semidirect
7. C₂×C₂₃×#475	C₂	semidirect
29. C₂₃×#498	C₂²	semidirect
30. C₂₃×#487	C₂²	semidirect
32. C₂₃×#475	C₂²	semidirect
145. C₂×#3229	C₂₃	direct (N × H)
183. #3229	C₂×C₂₃	direct (N × H)
184. C₂×#498	C₂×C₂₃	semidirect
185. C₂×#487	C₂×C₂₃	semidirect
187. C₂×#475	C₂×C₂₃	semidirect
227. #498	C₂²×C₂₃	semidirect
228. #487	C₂²×C₂₃	semidirect
230. #475	C₂²×C₂₃	semidirect
238. C₂×C₂₃	#3229	direct (N × H)
297. C₂₃	C₂×#3229	direct (N × H)
403. C₂	C₂₃×#3229	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×#3229	2
C₂×C₂₃×#498	2
C₂×C₂₃×#487	2
C₂×C₂₃×#483	2
C₂×C₂₃×#475	2
C₂×C₂₃×#431	2
C₂×#3229	23

Subgroups — C2×C23×#3229 #19208465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3229 #19208465