Subgroups — C₂×C₂₃×#3357 #19208876

Order 5888.

1930 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3357	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#480	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#468	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#428	C₂	1	2	yes	maximal, complemented
13	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#555	—	2	4	no	—
16	C₂₃×#555	C₂²	1	8	yes	—
17	C₂₃×#554	—	2	4	no	—
18	C₂₃×#554	C₂²	1	8	yes	complemented
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#125	—	2	1	no	—
22	C₂×C₂₃×#124	—	2	2	no	—
23	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
24	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂×C₂₃×#121	—	2	1	no	—
26	C₂×C₂₃×#121	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₂₃	—	2	1	no	—
29	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#117	—	2	1	no	—
32	C₂×C₂₃×#117	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₂₃×#500	C₂²	1	8	yes	complemented
35	C₂₃×#485	C₂²	1	16	yes	complemented
36	C₂₃×#480	C₂²	1	16	yes	complemented
37	C₂₃×#477	C₂²	1	8	yes	complemented
38	C₂₃×#476	C₂²	1	8	yes	complemented
39	C₂₃×#468	C₂²	1	16	yes	complemented
40	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
41	C₂₃×#436	C₂²	1	8	yes	complemented
42	C₂₃×#428	C₂²	1	16	yes	complemented
43	C₂×C₂₃×#107	C₂²	1	2	yes	—
44	C₂×C₂₃×#107	C₂²	1	2	yes	—
45	C₂×C₂₃×#107	C₂²	1	2	yes	—
46	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×#106	C₂²	1	2	yes	—
48	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#103	C₂²	1	2	yes	—
51	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂²×Q₈×C₂₃	—	2	1	no	—
57	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	—	2	2	no	—
59	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	2	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
64	C₂×SD₁₆×C₂₃	—	2	4	no	—
65	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
66	C₂×M₁₆×C₂₃	—	2	8	no	—
67	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
68	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×C₄⋊Q₈	—	2	2	no	—
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
75	C₂₃×#125	—	2	8	no	—
76	C₂₃×#124	—	2	16	no	—
77	C₂₃×C₂²⋊Q₈	—	2	4	no	—
78	C₂₃×C₂²⋊Q₈	—	2	16	no	—
79	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
80	C₂₃×#121	—	2	8	no	—
81	C₂₃×#121	—	2	4	no	—
82	C₂₃×#121	C₂³	1	8	yes	—
83	C₄×Q₈×C₂₃	—	2	2	no	—
84	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
85	C₄×D₄×C₂₃	—	2	8	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	D₄	1	4	yes	—
88	C₄×D₄×C₂₃	C₂³	1	4	yes	—
89	C₄×D₄×C₂₃	C₂³	1	4	yes	—
90	C₂₃×#117	—	2	4	no	—
91	C₂₃×#117	—	2	8	no	—
92	C₂₃×#117	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
110	C₂×C₄²×C₂₃	—	2	2	no	—
111	C₂×C₄²×C₂₃	—	2	1	no	—
112	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
113	C₂₃×#107	C₂³	1	8	yes	—
114	C₂₃×#107	C₂³	1	8	yes	—
115	C₂₃×#107	C₂³	1	8	yes	—
116	C₂₃×#106	C₂³	1	4	yes	—
117	C₂₃×#106	C₂³	1	8	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
119	C₂₃×#103	C₂³	1	4	yes	—
120	C₂₃×#103	C₂³	1	8	yes	—
121	C₂₃×#102	C₂³	1	4	yes	—
122	C₂₃×#102	C₂³	1	8	yes	—
123	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
124	C₄×C₈×C₂₃	C₂³	1	4	yes	—
125	C₂⁴×C₂₃	—	2	1	no	—
126	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	—	2	4	no	—
130	C₂×Q₈×C₂₃	—	2	2	no	—
131	C₂×Q₈×C₂₃	—	2	2	no	—
132	C₂×Q₈×C₂₃	—	8	2	no	—
133	C₂×Q₈×C₂₃	—	8	2	no	—
134	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
135	C₂×D₄×C₂₃	—	2	8	no	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	2	no	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	—	8	4	no	—
141	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
142	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₂₃	—	2	4	no	—
144	C₂²×C₄×C₂₃	—	2	4	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	2	2	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	2	no	—
150	C₂²×C₄×C₂₃	—	2	8	no	—
151	C₂²×C₄×C₂₃	—	2	4	no	—
152	C₂²×C₄×C₂₃	—	4	4	no	—
153	C₂²×C₄×C₂₃	—	8	4	no	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
160	SD₁₆×C₂₃	—	2	8	no	—
161	M₁₆×C₂₃	—	2	16	no	—
162	C₂×C₈×C₂₃	—	2	4	no	—
163	C₂×C₈×C₂₃	—	2	2	no	—
164	C₂×C₈×C₂₃	#41	1	4	yes	—
165	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
166	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
167	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	8	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	4	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	4	no	—
174	C₄⋊C₄×C₂₃	—	2	8	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
182	C₂²⋊C₄×C₂₃	—	2	8	no	—
183	C₂²⋊C₄×C₂₃	—	2	8	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	—	2	4	no	—
186	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
187	C₄²×C₂₃	—	2	8	no	—
188	C₄²×C₂₃	—	2	4	no	—
189	C₄²×C₂₃	C₂×D₄	1	2	yes	—
190	C₄²×C₂₃	C₂⁴	1	2	yes	—
191	C₂×#3357	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₂₃	—	2	4	no	—
193	C₂³×C₂₃	—	2	4	no	—
194	C₂³×C₂₃	—	2	4	no	—
195	C₂³×C₂₃	—	2	2	no	—
196	C₂³×C₂₃	—	4	2	no	—
197	C₂³×C₂₃	—	4	2	no	—
198	C₂³×C₂₃	—	4	2	no	—
199	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
200	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₂₃	—	2	2	no	—
202	Q₈×C₂₃	—	2	2	no	—
203	Q₈×C₂₃	—	4	2	no	—
204	Q₈×C₂₃	—	8	4	no	—
205	Q₈×C₂₃	—	8	4	no	—
206	D₄×C₂₃	—	2	8	no	—
207	D₄×C₂₃	—	2	2	no	—
208	D₄×C₂₃	—	2	2	no	—
209	D₄×C₂₃	—	4	2	no	—
210	D₄×C₂₃	—	8	8	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	8	no	—
214	C₂×C₄×C₂₃	—	2	4	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	2	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	4	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	4	no	—
221	C₂×C₄×C₂₃	—	4	2	no	—
222	C₂×C₄×C₂₃	—	4	2	no	—
223	C₂×C₄×C₂₃	—	4	2	no	—
224	C₂×C₄×C₂₃	—	4	4	no	—
225	C₂×C₄×C₂₃	—	4	4	no	—
226	C₂×C₄×C₂₃	—	4	8	no	—
227	C₂×C₄×C₂₃	—	8	8	no	—
228	C₂×C₄×C₂₃	#137	1	4	yes	—
229	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
234	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
235	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
236	C₈×C₂₃	—	2	4	no	—
237	C₈×C₂₃	—	2	2	no	—
238	C₈×C₂₃	C₂×#41	1	4	yes	—
239	C₂×#555	C₂×C₂₃	1	1	yes	characteristic
240	C₂×#554	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#3357	C₂×C₂₃	1	16	yes	direct product factor
242	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
243	C₂×#485	C₂×C₂₃	1	2	yes	complemented
244	C₂×#480	C₂×C₂₃	1	2	yes	complemented
245	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
246	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
247	C₂×#468	C₂×C₂₃	1	2	yes	complemented
248	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×#436	C₂×C₂₃	1	1	yes	characteristic, complemented
250	C₂×#428	C₂×C₂₃	1	2	yes	complemented
251	C₂²×C₂₃	—	2	4	no	—
252	C₂²×C₂₃	—	2	4	no	—
253	C₂²×C₂₃	—	2	4	no	—
254	C₂²×C₂₃	—	4	8	no	—
255	C₂²×C₂₃	—	4	2	no	—
256	C₂²×C₂₃	—	4	2	no	—
257	C₂²×C₂₃	—	4	2	no	—
258	C₂²×C₂₃	—	8	4	no	—
259	C₂²×C₂₃	#547	1	2	yes	—
260	C₂²×C₂₃	C₂×#137	1	2	yes	—
261	C₂²×C₂₃	#576	1	2	yes	—
262	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₂₃	—	2	4	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	4	2	no	—
266	C₄×C₂₃	—	4	2	no	—
267	C₄×C₂₃	—	4	2	no	—
268	C₄×C₂₃	—	4	4	no	—
269	C₄×C₂₃	—	4	4	no	—
270	C₄×C₂₃	#547	1	2	yes	—
271	C₄×C₂₃	C₂×#137	1	4	yes	—
272	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
273	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
274	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
275	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
276	#555	—	2	4	no	—
277	#555	C₂²×C₂₃	1	8	yes	—
278	#554	—	2	4	no	—
279	#554	C₂²×C₂₃	1	8	yes	complemented
280	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×#125	—	2	1	no	—
283	C₂×#124	—	2	2	no	—
284	C₂×C₂²⋊Q₈	—	2	2	no	—
285	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
286	C₂×#121	—	2	1	no	—
287	C₂×#121	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×C₄×D₄	—	2	1	no	—
290	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
292	C₂×#117	—	2	1	no	—
293	C₂×#117	C₂²×C₂₃	1	2	yes	—
294	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
295	#500	C₂²×C₂₃	1	8	yes	complemented
296	#485	C₂²×C₂₃	1	16	yes	complemented
297	#480	C₂²×C₂₃	1	16	yes	complemented
298	#477	C₂²×C₂₃	1	8	yes	complemented
299	#476	C₂²×C₂₃	1	8	yes	complemented
300	#468	C₂²×C₂₃	1	16	yes	complemented
301	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
302	#436	C₂²×C₂₃	1	8	yes	complemented
303	#428	C₂²×C₂₃	1	16	yes	complemented
304	C₂×#107	C₂²×C₂₃	1	2	yes	—
305	C₂×#107	C₂²×C₂₃	1	2	yes	—
306	C₂×#107	C₂²×C₂₃	1	2	yes	—
307	C₂×#106	C₂²×C₂₃	1	2	yes	—
308	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#103	C₂²×C₂₃	1	2	yes	—
311	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#102	C₂²×C₂₃	1	2	yes	—
314	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₂₃	—	4	4	no	—
317	C₂×C₂₃	—	8	2	no	—
318	C₂×C₂₃	#3357	1	4	yes	direct product factor
319	C₂×C₂₃	C₂×#547	1	1	yes	characteristic
320	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
321	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
322	C₂²×Q₈	—	2	1	no	—
323	C₂²×Q₈	—	2	1	no	—
324	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
327	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	—	2	1	no	—
330	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
331	C₂×SD₁₆	—	2	4	no	—
332	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
333	C₂×M₁₆	—	2	8	no	—
334	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
335	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
336	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
338	C₄⋊Q₈	—	2	2	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
341	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
342	#125	—	2	8	no	—
343	#124	—	2	16	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	16	no	—
346	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
347	#121	—	2	8	no	—
348	#121	—	2	4	no	—
349	#121	C₂³×C₂₃	1	8	yes	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
352	C₄×D₄	—	2	8	no	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	D₄×C₂₃	1	4	yes	—
355	C₄×D₄	C₂³×C₂₃	1	4	yes	—
356	C₄×D₄	C₂³×C₂₃	1	4	yes	—
357	#117	—	2	4	no	—
358	#117	—	2	8	no	—
359	#117	C₂³×C₂₃	1	8	yes	—
360	C₂×C₄⋊C₄	—	2	2	no	—
361	C₂×C₄⋊C₄	—	2	4	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
370	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
377	C₂×C₄²	—	2	2	no	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
380	#107	C₂³×C₂₃	1	8	yes	—
381	#107	C₂³×C₂₃	1	8	yes	—
382	#107	C₂³×C₂₃	1	8	yes	—
383	#106	C₂³×C₂₃	1	4	yes	—
384	#106	C₂³×C₂₃	1	8	yes	—
385	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
386	#103	C₂³×C₂₃	1	4	yes	—
387	#103	C₂³×C₂₃	1	8	yes	—
388	#102	C₂³×C₂₃	1	4	yes	—
389	#102	C₂³×C₂₃	1	8	yes	—
390	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
391	C₄×C₈	C₂³×C₂₃	1	4	yes	—
392	C₂₃	C₂×#3357	1	1	yes	characteristic, direct product factor, Sylow
393	C₂⁴	—	2	1	no	—
394	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	8	2	no	—
401	C₂×Q₈	—	8	2	no	—
402	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	8	4	no	—
409	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
410	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	8	4	no	—
422	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
428	SD₁₆	—	2	8	no	—
429	M₁₆	—	2	16	no	—
430	C₂×C₈	—	2	4	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	#41×C₂₃	1	4	yes	—
433	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
434	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
435	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	8	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
449	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
455	C₄²	—	2	8	no	—
456	C₄²	—	2	4	no	—
457	C₄²	C₂×D₄×C₂₃	1	2	yes	—
458	C₄²	C₂⁴×C₂₃	1	2	yes	—
459	C₂³	—	2	4	no	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	4	2	no	—
466	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
467	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	Q₈	—	2	2	no	—
469	Q₈	—	2	2	no	—
470	Q₈	—	4	2	no	—
471	Q₈	—	8	4	no	—
472	Q₈	—	8	4	no	—
473	D₄	—	2	8	no	—
474	D₄	—	2	2	no	—
475	D₄	—	2	2	no	—
476	D₄	—	4	2	no	—
477	D₄	—	8	8	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	8	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	4	no	—
492	C₂×C₄	—	4	4	no	—
493	C₂×C₄	—	4	8	no	—
494	C₂×C₄	—	8	8	no	—
495	C₂×C₄	C₂₃×#137	1	4	yes	—
496	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
497	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
498	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
500	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
501	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
502	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
503	C₈	—	2	4	no	—
504	C₈	—	2	2	no	—
505	C₈	C₂×#41×C₂₃	1	4	yes	—
506	C₂²	—	2	4	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	2	4	no	—
509	C₂²	—	4	8	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	8	4	no	—
514	C₂²	C₂₃×#547	1	2	yes	—
515	C₂²	C₂×C₂₃×#137	1	2	yes	—
516	C₂²	C₂₃×#576	1	2	yes	—
517	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
518	C₄	—	2	4	no	—
519	C₄	—	2	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	4	no	—
524	C₄	—	4	4	no	—
525	C₄	C₂₃×#547	1	2	yes	—
526	C₄	C₂×C₂₃×#137	1	4	yes	—
527	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
528	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
529	C₂	—	4	4	no	—
530	C₂	—	8	2	no	—
531	C₂	C₂₃×#3357	1	4	yes	direct product factor
532	C₂	C₂×C₂₃×#547	1	1	yes	characteristic
533	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
534	C₂	C₂×C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#555	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#554	C₂	semidirect
3. C₂₃×#3357	C₂	direct (N × H)
4. C₂×C₂₃×#500	C₂	semidirect
5. C₂×C₂₃×#485	C₂	semidirect
6. C₂×C₂₃×#480	C₂	semidirect
7. C₂×C₂₃×#477	C₂	semidirect
8. C₂×C₂₃×#476	C₂	semidirect
9. C₂×C₂₃×#468	C₂	semidirect
10. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂×C₂₃×#436	C₂	semidirect
12. C₂×C₂₃×#428	C₂	semidirect
18. C₂₃×#554	C₂²	semidirect
34. C₂₃×#500	C₂²	semidirect
35. C₂₃×#485	C₂²	semidirect
36. C₂₃×#480	C₂²	semidirect
37. C₂₃×#477	C₂²	semidirect
38. C₂₃×#476	C₂²	semidirect
39. C₂₃×#468	C₂²	semidirect
40. C₄×SD₁₆×C₂₃	C₂²	semidirect
41. C₂₃×#436	C₂²	semidirect
42. C₂₃×#428	C₂²	semidirect
191. C₂×#3357	C₂₃	direct (N × H)
240. C₂×#554	C₂×C₂₃	semidirect
241. #3357	C₂×C₂₃	direct (N × H)
242. C₂×#500	C₂×C₂₃	semidirect
243. C₂×#485	C₂×C₂₃	semidirect
244. C₂×#480	C₂×C₂₃	semidirect
245. C₂×#477	C₂×C₂₃	semidirect
246. C₂×#476	C₂×C₂₃	semidirect
247. C₂×#468	C₂×C₂₃	semidirect
248. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
249. C₂×#436	C₂×C₂₃	semidirect
250. C₂×#428	C₂×C₂₃	semidirect
279. #554	C₂²×C₂₃	semidirect
295. #500	C₂²×C₂₃	semidirect
296. #485	C₂²×C₂₃	semidirect
297. #480	C₂²×C₂₃	semidirect
298. #477	C₂²×C₂₃	semidirect
299. #476	C₂²×C₂₃	semidirect
300. #468	C₂²×C₂₃	semidirect
301. C₄×SD₁₆	C₂²×C₂₃	semidirect
302. #436	C₂²×C₂₃	semidirect
303. #428	C₂²×C₂₃	semidirect
318. C₂×C₂₃	#3357	direct (N × H)
392. C₂₃	C₂×#3357	direct (N × H)
531. C₂	C₂₃×#3357	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×#555	2
C₂×C₂₃×#554	2
C₂₃×#3357	2
C₂×C₂₃×#500	2
C₂×C₂₃×#485	2
C₂×C₂₃×#480	2
C₂×C₂₃×#477	2
C₂×C₂₃×#476	2
C₂×C₂₃×#468	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₂₃×#436	2
C₂×C₂₃×#428	2
C₂×#3357	23

Subgroups — C2×C23×#3357 #19208876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3357 #19208876