Subgroups — C₂×C₂₃×#3137 #19208237

Order 5888.

2070 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3137	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#491	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#488	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#487	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#417	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
11	C₂²×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#544	—	2	4	no	—
18	C₂₃×#544	C₂²	1	8	yes	complemented
19	C₂₃×#536	—	2	4	no	—
20	C₂₃×#536	C₂²	1	8	yes	—
21	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
24	C₂×C₂₃×#125	—	2	1	no	—
25	C₂×C₂₃×#125	C₂²	1	2	yes	—
26	C₂×C₂₃×#124	C₂²	1	2	yes	—
27	C₂×C₂₃×#123	—	2	2	no	—
28	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
29	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
30	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
31	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#491	C₂²	1	16	yes	complemented
35	C₂₃×#489	C₂²	1	16	yes	complemented
36	C₂₃×#488	C₂²	1	16	yes	complemented
37	C₂₃×#487	C₂²	1	16	yes	complemented
38	C₂₃×#419	C₂²	1	8	yes	complemented
39	C₂₃×#417	C₂²	1	8	yes	—
40	C₂₃×#416	C₂²	1	8	yes	—
41	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
42	C₂×C₂₃×#103	C₂²	1	2	yes	—
43	C₂×C₂₃×#103	C₂²	1	2	yes	—
44	C₂×C₂₃×#103	C₂²	1	2	yes	—
45	C₂×C₂₃×#103	C₂²	1	2	yes	—
46	C₂×C₂₃×#103	C₂²	1	2	yes	—
47	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#103	C₂²	1	8	yes	complemented
49	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#102	C₂²	1	2	yes	—
51	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
52	C₂₃×#405	C₂²	1	8	yes	complemented
53	C₂×C₂₃×#97	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
55	C₂×#41×C₂₃	—	2	2	no	—
56	C₂×#41×C₂₃	—	2	4	no	—
57	C₂×#41×C₂₃	—	2	4	no	—
58	C₂×#41×C₂₃	C₂³	1	4	yes	—
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	—	2	2	no	—
61	C₂²×Q₈×C₂₃	—	2	2	no	—
62	C₂²×Q₈×C₂₃	—	2	4	no	—
63	C₂²×Q₈×C₂₃	—	2	4	no	—
64	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
66	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₂₃	—	2	1	no	—
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₂₃	—	2	1	no	—
70	C₂³×C₄×C₂₃	—	2	1	no	—
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₂₃	—	2	4	no	—
74	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
75	C₂²×C₈×C₂₃	—	2	4	no	—
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
79	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
80	C₂₃×C₄⋊Q₈	—	2	8	no	—
81	C₂₃×C₄⋊Q₈	—	2	4	no	—
82	C₂₃×C₄⋊Q₈	—	2	4	no	—
83	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
84	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
85	C₂₃×#125	—	2	8	no	—
86	C₂₃×#125	—	2	4	no	—
87	C₂₃×#125	C₂³	1	8	yes	—
88	C₂₃×#124	—	2	4	no	—
89	C₂₃×#124	C₂³	1	8	yes	—
90	C₂₃×#123	—	2	16	no	—
91	C₂₃×C₂²⋊Q₈	—	2	8	no	—
92	C₂₃×C₂²⋊Q₈	—	2	8	no	—
93	C₂₃×C₂²⋊Q₈	—	2	16	no	—
94	C₂₃×#117	—	2	2	no	—
95	C₂₃×#117	D₄	1	4	yes	—
96	C₂₃×#117	C₂³	1	4	yes	—
97	C₂₃×#117	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
115	C₂×C₄²×C₂₃	—	2	1	no	—
116	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
117	C₂₃×#103	C₂³	1	8	yes	—
118	C₂₃×#103	C₂³	1	8	yes	—
119	C₂₃×#103	C₂³	1	4	yes	—
120	C₂₃×#103	C₂³	1	4	yes	—
121	C₂₃×#103	C₂³	1	8	yes	—
122	C₂₃×#103	C₂³	1	8	yes	—
123	C₂₃×#103	C₂³	1	8	yes	—
124	C₂₃×#102	C₂³	1	4	yes	—
125	C₂₃×#102	C₂³	1	4	yes	—
126	C₂₃×#102	C₂³	1	8	yes	—
127	C₂₃×#97	C₂³	1	8	yes	—
128	C₄×C₈×C₂₃	C₂³	1	8	yes	—
129	C₂⁴×C₂₃	—	2	1	no	—
130	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
131	#41×C₂₃	—	2	8	no	—
132	#41×C₂₃	—	2	8	no	—
133	#41×C₂₃	—	4	8	no	—
134	C₂×Q₈×C₂₃	—	2	4	no	—
135	C₂×Q₈×C₂₃	—	2	2	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	8	no	—
138	C₂×Q₈×C₂₃	—	2	4	no	—
139	C₂×Q₈×C₂₃	—	2	4	no	—
140	C₂×Q₈×C₂₃	—	2	8	no	—
141	C₂×Q₈×C₂₃	—	2	8	no	—
142	C₂×Q₈×C₂₃	—	2	2	no	—
143	C₂×Q₈×C₂₃	—	2	2	no	—
144	C₂×Q₈×C₂₃	—	4	4	no	—
145	C₂×Q₈×C₂₃	—	4	8	no	—
146	C₂×Q₈×C₂₃	—	8	4	no	—
147	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
148	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
149	C₂×D₄×C₂₃	—	2	4	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	2	2	no	—
152	C₂×D₄×C₂₃	—	2	2	no	—
153	C₂×D₄×C₂₃	—	4	4	no	—
154	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₂₃	—	2	4	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	2	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	4	no	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	4	4	no	—
164	C₂²×C₄×C₂₃	—	4	4	no	—
165	C₂²×C₄×C₂₃	—	8	2	no	—
166	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	#41	1	4	yes	—
171	C₂²×C₄×C₂₃	#41	1	4	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	M₁₆×C₂₃	—	2	8	no	—
175	C₂×C₈×C₂₃	—	2	2	no	—
176	C₂×C₈×C₂₃	—	2	2	no	—
177	C₂×C₈×C₂₃	—	2	8	no	—
178	C₂×C₈×C₂₃	—	2	4	no	—
179	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
180	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
181	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
182	C₄⋊C₄×C₂₃	—	2	4	no	—
183	C₄⋊C₄×C₂₃	—	2	2	no	—
184	C₄⋊C₄×C₂₃	—	2	8	no	—
185	C₄⋊C₄×C₂₃	—	2	8	no	—
186	C₄⋊C₄×C₂₃	—	2	8	no	—
187	C₄⋊C₄×C₂₃	—	2	8	no	—
188	C₄⋊C₄×C₂₃	—	2	8	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
196	C₂²⋊C₄×C₂₃	—	2	4	no	—
197	C₂²⋊C₄×C₂₃	—	2	8	no	—
198	C₂²⋊C₄×C₂₃	—	2	8	no	—
199	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
200	C₄²×C₂₃	—	2	4	no	—
201	C₄²×C₂₃	C₂×D₄	1	4	yes	—
202	C₄²×C₂₃	C₂⁴	1	4	yes	—
203	C₂×#3137	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₂₃	—	2	2	no	—
205	C₂³×C₂₃	—	2	2	no	—
206	C₂³×C₂₃	—	2	2	no	—
207	C₂³×C₂₃	—	4	2	no	—
208	C₂³×C₂₃	—	4	2	no	—
209	C₂³×C₂₃	—	4	2	no	—
210	C₂³×C₂₃	#124	1	4	yes	—
211	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
212	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
213	Q₈×C₂₃	—	2	4	no	—
214	Q₈×C₂₃	—	2	4	no	—
215	Q₈×C₂₃	—	2	2	no	—
216	Q₈×C₂₃	—	2	2	no	—
217	Q₈×C₂₃	—	4	8	no	—
218	Q₈×C₂₃	—	4	4	no	—
219	Q₈×C₂₃	—	4	2	no	—
220	Q₈×C₂₃	—	8	8	no	—
221	D₄×C₂₃	—	2	2	no	—
222	D₄×C₂₃	—	2	2	no	—
223	D₄×C₂₃	—	4	8	no	—
224	D₄×C₂₃	—	4	2	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	4	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	2	no	—
229	C₂×C₄×C₂₃	—	2	4	no	—
230	C₂×C₄×C₂₃	—	2	4	no	—
231	C₂×C₄×C₂₃	—	2	2	no	—
232	C₂×C₄×C₂₃	—	4	2	no	—
233	C₂×C₄×C₂₃	—	4	8	no	—
234	C₂×C₄×C₂₃	—	4	4	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	2	no	—
237	C₂×C₄×C₂₃	—	4	2	no	—
238	C₂×C₄×C₂₃	—	4	8	no	—
239	C₂×C₄×C₂₃	—	8	4	no	—
240	C₂×C₄×C₂₃	#124	1	4	yes	—
241	C₂×C₄×C₂₃	#124	1	8	yes	—
242	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
243	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
247	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
248	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
249	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
250	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
251	C₈×C₂₃	—	2	2	no	—
252	C₈×C₂₃	—	2	2	no	—
253	C₈×C₂₃	—	2	4	no	—
254	C₂×#544	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#536	C₂×C₂₃	1	1	yes	characteristic
256	#3137	C₂×C₂₃	1	16	yes	direct product factor
257	C₂×#491	C₂×C₂₃	1	2	yes	complemented
258	C₂×#489	C₂×C₂₃	1	2	yes	complemented
259	C₂×#488	C₂×C₂₃	1	2	yes	complemented
260	C₂×#487	C₂×C₂₃	1	2	yes	complemented
261	C₂×#417	C₂×C₂₃	1	1	yes	characteristic
262	C₂×#419	C₂×C₂₃	1	1	yes	characteristic, complemented
263	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
264	C₂²×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
266	C₂²×C₂₃	—	2	8	no	—
267	C₂²×C₂₃	—	2	2	no	—
268	C₂²×C₂₃	—	2	2	no	—
269	C₂²×C₂₃	—	4	2	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	4	2	no	—
272	C₂²×C₂₃	—	8	4	no	—
273	C₂²×C₂₃	C₂×#124	1	4	yes	—
274	C₂²×C₂₃	C₂×#124	1	2	yes	—
275	C₂²×C₂₃	#577	1	2	yes	—
276	C₂²×C₂₃	#578	1	2	yes	—
277	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
278	C₄×C₂₃	—	2	4	no	—
279	C₄×C₂₃	—	2	4	no	—
280	C₄×C₂₃	—	4	2	no	—
281	C₄×C₂₃	—	4	4	no	—
282	C₄×C₂₃	—	4	4	no	—
283	C₄×C₂₃	—	4	2	no	—
284	C₄×C₂₃	—	4	2	no	—
285	C₄×C₂₃	C₂×#124	1	4	yes	—
286	C₄×C₂₃	C₂×#124	1	2	yes	—
287	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
288	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
289	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
290	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
291	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
292	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
293	#544	—	2	4	no	—
294	#544	C₂²×C₂₃	1	8	yes	complemented
295	#536	—	2	4	no	—
296	#536	C₂²×C₂₃	1	8	yes	—
297	C₂×C₄⋊Q₈	—	2	1	no	—
298	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
299	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
300	C₂×#125	—	2	1	no	—
301	C₂×#125	C₂²×C₂₃	1	2	yes	—
302	C₂×#124	C₂²×C₂₃	1	2	yes	—
303	C₂×#123	—	2	2	no	—
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	1	no	—
306	C₂×C₂²⋊Q₈	—	2	2	no	—
307	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
309	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
310	#491	C₂²×C₂₃	1	16	yes	complemented
311	#489	C₂²×C₂₃	1	16	yes	complemented
312	#488	C₂²×C₂₃	1	16	yes	complemented
313	#487	C₂²×C₂₃	1	16	yes	complemented
314	#419	C₂²×C₂₃	1	8	yes	complemented
315	#417	C₂²×C₂₃	1	8	yes	—
316	#416	C₂²×C₂₃	1	8	yes	—
317	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#103	C₂²×C₂₃	1	2	yes	—
319	C₂×#103	C₂²×C₂₃	1	2	yes	—
320	C₂×#103	C₂²×C₂₃	1	2	yes	—
321	C₂×#103	C₂²×C₂₃	1	2	yes	—
322	C₂×#103	C₂²×C₂₃	1	2	yes	—
323	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#103	C₂²×C₂₃	1	8	yes	complemented
325	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#102	C₂²×C₂₃	1	2	yes	—
327	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
328	#405	C₂²×C₂₃	1	8	yes	complemented
329	C₂×#97	C₂²×C₂₃	1	2	yes	—
330	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
331	C₂×C₂₃	—	2	4	no	—
332	C₂×C₂₃	—	8	2	no	—
333	C₂×C₂₃	#3137	1	4	yes	direct product factor
334	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
335	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
336	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
337	C₂×#41	—	2	2	no	—
338	C₂×#41	—	2	4	no	—
339	C₂×#41	—	2	4	no	—
340	C₂×#41	C₂³×C₂₃	1	4	yes	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	—	2	2	no	—
344	C₂²×Q₈	—	2	4	no	—
345	C₂²×Q₈	—	2	4	no	—
346	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
347	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
348	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
349	C₂²×D₄	—	2	1	no	—
350	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	—	2	1	no	—
353	C₂³×C₄	—	2	1	no	—
354	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
355	C₂×M₁₆	—	2	4	no	—
356	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
357	C₂²×C₈	—	2	4	no	—
358	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
359	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
360	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
361	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
362	C₄⋊Q₈	—	2	8	no	—
363	C₄⋊Q₈	—	2	4	no	—
364	C₄⋊Q₈	—	2	4	no	—
365	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
366	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
367	#125	—	2	8	no	—
368	#125	—	2	4	no	—
369	#125	C₂³×C₂₃	1	8	yes	—
370	#124	—	2	4	no	—
371	#124	C₂³×C₂₃	1	8	yes	—
372	#123	—	2	16	no	—
373	C₂²⋊Q₈	—	2	8	no	—
374	C₂²⋊Q₈	—	2	8	no	—
375	C₂²⋊Q₈	—	2	16	no	—
376	#117	—	2	2	no	—
377	#117	D₄×C₂₃	1	4	yes	—
378	#117	C₂³×C₂₃	1	4	yes	—
379	#117	C₂³×C₂₃	1	4	yes	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	2	no	—
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
399	#103	C₂³×C₂₃	1	8	yes	—
400	#103	C₂³×C₂₃	1	8	yes	—
401	#103	C₂³×C₂₃	1	4	yes	—
402	#103	C₂³×C₂₃	1	4	yes	—
403	#103	C₂³×C₂₃	1	8	yes	—
404	#103	C₂³×C₂₃	1	8	yes	—
405	#103	C₂³×C₂₃	1	8	yes	—
406	#102	C₂³×C₂₃	1	4	yes	—
407	#102	C₂³×C₂₃	1	4	yes	—
408	#102	C₂³×C₂₃	1	8	yes	—
409	#97	C₂³×C₂₃	1	8	yes	—
410	C₄×C₈	C₂³×C₂₃	1	8	yes	—
411	C₂₃	C₂×#3137	1	1	yes	characteristic, direct product factor, Sylow
412	C₂⁴	—	2	1	no	—
413	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	#41	—	4	8	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	8	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	2	4	no	—
423	C₂×Q₈	—	2	8	no	—
424	C₂×Q₈	—	2	8	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	4	4	no	—
428	C₂×Q₈	—	4	8	no	—
429	C₂×Q₈	—	8	4	no	—
430	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
431	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	4	4	no	—
437	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	8	2	no	—
449	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
453	C₂²×C₄	#41×C₂₃	1	4	yes	—
454	C₂²×C₄	#41×C₂₃	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
457	M₁₆	—	2	8	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	8	no	—
461	C₂×C₈	—	2	4	no	—
462	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
463	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
464	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	8	no	—
468	C₄⋊C₄	—	2	8	no	—
469	C₄⋊C₄	—	2	8	no	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
475	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
476	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
478	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
479	C₂²⋊C₄	—	2	4	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
483	C₄²	—	2	4	no	—
484	C₄²	C₂×D₄×C₂₃	1	4	yes	—
485	C₄²	C₂⁴×C₂₃	1	4	yes	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	4	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	2	no	—
492	C₂³	C₂₃×#124	1	4	yes	—
493	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
495	Q₈	—	2	4	no	—
496	Q₈	—	2	4	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	2	2	no	—
499	Q₈	—	4	8	no	—
500	Q₈	—	4	4	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	8	8	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	2	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	8	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	C₂₃×#124	1	4	yes	—
523	C₂×C₄	C₂₃×#124	1	8	yes	—
524	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
526	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
527	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
528	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
529	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
530	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
531	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
532	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	—	2	4	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	8	4	no	—
543	C₂²	C₂×C₂₃×#124	1	4	yes	—
544	C₂²	C₂×C₂₃×#124	1	2	yes	—
545	C₂²	C₂₃×#577	1	2	yes	—
546	C₂²	C₂₃×#578	1	2	yes	—
547	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
548	C₄	—	2	4	no	—
549	C₄	—	2	4	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	C₂×C₂₃×#124	1	4	yes	—
556	C₄	C₂×C₂₃×#124	1	2	yes	—
557	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
558	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
559	C₂	—	2	4	no	—
560	C₂	—	8	2	no	—
561	C₂	C₂₃×#3137	1	4	yes	direct product factor
562	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
563	C₂	C₂×C₂₃×#577	1	1	yes	characteristic
564	C₂	C₂×C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#544	C₂	semidirect
3. C₂₃×#3137	C₂	direct (N × H)
4. C₂×C₂₃×#491	C₂	semidirect
5. C₂×C₂₃×#489	C₂	semidirect
6. C₂×C₂₃×#488	C₂	semidirect
7. C₂×C₂₃×#487	C₂	semidirect
9. C₂×C₂₃×#419	C₂	semidirect
11. C₂²×C₂₃×#103	C₂	semidirect
12. C₂×C₂₃×#405	C₂	semidirect
18. C₂₃×#544	C₂²	semidirect
34. C₂₃×#491	C₂²	semidirect
35. C₂₃×#489	C₂²	semidirect
36. C₂₃×#488	C₂²	semidirect
37. C₂₃×#487	C₂²	semidirect
38. C₂₃×#419	C₂²	semidirect
48. C₂×C₂₃×#103	C₂²	semidirect
52. C₂₃×#405	C₂²	semidirect
203. C₂×#3137	C₂₃	direct (N × H)
254. C₂×#544	C₂×C₂₃	semidirect
256. #3137	C₂×C₂₃	direct (N × H)
257. C₂×#491	C₂×C₂₃	semidirect
258. C₂×#489	C₂×C₂₃	semidirect
259. C₂×#488	C₂×C₂₃	semidirect
260. C₂×#487	C₂×C₂₃	semidirect
262. C₂×#419	C₂×C₂₃	semidirect
264. C₂²×#103	C₂×C₂₃	semidirect
265. C₂×#405	C₂×C₂₃	semidirect
294. #544	C₂²×C₂₃	semidirect
310. #491	C₂²×C₂₃	semidirect
311. #489	C₂²×C₂₃	semidirect
312. #488	C₂²×C₂₃	semidirect
313. #487	C₂²×C₂₃	semidirect
314. #419	C₂²×C₂₃	semidirect
324. C₂×#103	C₂²×C₂₃	semidirect
328. #405	C₂²×C₂₃	semidirect
333. C₂×C₂₃	#3137	direct (N × H)
411. C₂₃	C₂×#3137	direct (N × H)
561. C₂	C₂₃×#3137	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×#544	2
C₂×C₂₃×#536	2
C₂₃×#3137	2
C₂×C₂₃×#491	2
C₂×C₂₃×#489	2
C₂×C₂₃×#488	2
C₂×C₂₃×#487	2
C₂×C₂₃×#417	2
C₂×C₂₃×#419	2
C₂×C₂₃×#416	2
C₂²×C₂₃×#103	2
C₂×C₂₃×#405	2
C₂×#3137	23

Subgroups — C2×C23×#3137 #19208237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3137 #19208237