Subgroups — C₂×C₂₃×#3385 #19208985

Order 5888.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#554	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3385	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#501	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#479	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
9	C₂×C₂₃×#474	C₂	1	2	yes	maximal
10	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#440	C₂	1	2	yes	maximal
12	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
13	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	Q₈²×C₂₃	—	2	4	no	—
15	Q₈²×C₂₃	C₂²	1	8	yes	complemented
16	C₂₃×#554	—	2	4	no	—
17	C₂₃×#554	C₂²	1	8	yes	—
18	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#125	C₂²	1	2	yes	—
24	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
27	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₂₃×#501	C₂²	1	16	yes	complemented
34	C₂₃×#500	C₂²	1	8	yes	complemented
35	C₂₃×#479	C₂²	1	16	yes	complemented
36	C₂₃×#477	C₂²	1	8	yes	complemented
37	C₂₃×#476	C₂²	1	8	yes	—
38	C₂₃×#474	C₂²	1	16	yes	—
39	C₂₃×#446	C₂²	1	8	yes	complemented
40	C₂₃×#440	C₂²	1	16	yes	—
41	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	—
42	C₂×C₂₃×#107	C₂²	1	2	yes	—
43	C₂×C₂₃×#107	C₂²	1	2	yes	—
44	C₂×C₂₃×#107	C₂²	1	2	yes	—
45	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#106	C₂²	1	2	yes	—
47	C₂×C₂₃×C₄⋊C₈	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×#97	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	—	2	2	no	—
58	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	2	no	—
63	C₂×SD₁₆×C₂₃	Q₈	1	8	yes	complemented
64	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	—	2	8	no	—
69	C₂₃×C₄⋊Q₈	—	2	16	no	—
70	C₂₃×C₄⋊Q₈	—	2	2	no	—
71	C₂₃×C₄⋊Q₈	—	2	4	no	—
72	C₂₃×C₄⋊Q₈	—	2	2	no	—
73	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
76	C₂₃×#125	—	2	4	no	—
77	C₂₃×#125	C₂³	1	8	yes	—
78	C₂₃×C₂²⋊Q₈	—	2	8	no	—
79	C₂₃×C₂²⋊Q₈	—	2	16	no	—
80	C₄×Q₈×C₂₃	—	2	8	no	—
81	C₄×Q₈×C₂₃	—	2	4	no	—
82	C₄×Q₈×C₂₃	—	2	2	no	—
83	C₄×Q₈×C₂₃	D₄	1	4	yes	—
84	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
85	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
86	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
87	C₄×D₄×C₂₃	—	2	4	no	—
88	C₄×D₄×C₂₃	—	2	2	no	—
89	C₄×D₄×C₂₃	C₂³	1	4	yes	—
90	C₄×D₄×C₂₃	C₂³	1	8	yes	—
91	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
99	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	D₄	1	2	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	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄²×C₂₃	—	2	2	no	—
108	C₂×C₄²×C₂₃	—	2	1	no	—
109	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
111	C₂₃×#107	C₂³	1	8	yes	—
112	C₂₃×#107	C₂³	1	8	yes	—
113	C₂₃×#107	C₂³	1	8	yes	—
114	C₂₃×#106	C₂³	1	4	yes	—
115	C₂₃×#106	C₂³	1	8	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
117	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
118	C₂₃×#103	C₂³	1	4	yes	—
119	C₂₃×#103	C₂³	1	8	yes	—
120	C₂₃×#102	C₂³	1	4	yes	—
121	C₂₃×#102	C₂³	1	8	yes	—
122	C₂₃×#97	C₂³	1	8	yes	—
123	C₄×C₈×C₂₃	C₂³	1	4	yes	—
124	C₂⁴×C₂₃	—	2	1	no	—
125	C₂×Q₈×C₂₃	—	2	4	no	—
126	C₂×Q₈×C₂₃	—	2	1	no	—
127	C₂×Q₈×C₂₃	—	2	4	no	—
128	C₂×Q₈×C₂₃	—	2	4	no	—
129	C₂×Q₈×C₂₃	—	2	8	no	—
130	C₂×Q₈×C₂₃	—	2	2	no	—
131	C₂×Q₈×C₂₃	—	8	4	no	—
132	C₂×Q₈×C₂₃	—	8	2	no	—
133	C₂×Q₈×C₂₃	—	8	2	no	—
134	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
136	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	—	2	2	no	—
139	C₂×D₄×C₂₃	C₂×Q₈	1	4	yes	—
140	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	2	no	—
146	C₂²×C₄×C₂₃	—	2	4	no	—
147	C₂²×C₄×C₂₃	—	2	8	no	—
148	C₂²×C₄×C₂₃	—	4	4	no	—
149	C₂²×C₄×C₂₃	—	8	4	no	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
155	SD₁₆×C₂₃	C₂×Q₈	1	16	yes	complemented
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	4	no	—
158	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
159	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
160	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₂₃	—	2	8	no	—
163	C₄⋊C₄×C₂₃	—	2	4	no	—
164	C₄⋊C₄×C₂₃	—	2	8	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	4	no	—
167	C₄⋊C₄×C₂₃	—	2	8	no	—
168	C₄⋊C₄×C₂₃	—	2	8	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
174	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
175	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
179	C₂²⋊C₄×C₂₃	—	2	4	no	—
180	C₂²⋊C₄×C₂₃	—	2	8	no	—
181	C₄²×C₂₃	—	2	8	no	—
182	C₄²×C₂₃	—	2	4	no	—
183	C₄²×C₂₃	C₂×D₄	1	4	yes	—
184	C₄²×C₂₃	C₂×D₄	1	2	yes	—
185	C₄²×C₂₃	C₂⁴	1	4	yes	—
186	C₄²×C₂₃	C₂⁴	1	2	yes	—
187	C₂×#3385	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	4	no	—
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	—	4	2	no	—
191	C₂³×C₂₃	—	4	2	no	—
192	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
193	Q₈×C₂₃	—	2	4	no	—
194	Q₈×C₂₃	—	2	8	no	—
195	Q₈×C₂₃	—	2	2	no	—
196	Q₈×C₂₃	—	8	8	no	—
197	Q₈×C₂₃	—	8	4	no	—
198	Q₈×C₂₃	—	8	4	no	—
199	Q₈×C₂₃	C₂²×Q₈	1	4	yes	—
200	D₄×C₂₃	—	2	4	no	—
201	D₄×C₂₃	—	2	2	no	—
202	D₄×C₂₃	C₂²×Q₈	1	4	yes	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	4	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	—	4	2	no	—
215	C₂×C₄×C₂₃	—	4	4	no	—
216	C₂×C₄×C₂₃	—	4	8	no	—
217	C₂×C₄×C₂₃	—	8	8	no	—
218	C₂×C₄×C₂₃	#137	1	4	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
225	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
226	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
227	C₈×C₂₃	—	2	4	no	—
228	C₈×C₂₃	—	2	2	no	—
229	C₈×C₂₃	C₂²×Q₈	1	4	yes	—
230	C₂×Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
231	C₂×#554	C₂×C₂₃	1	1	yes	characteristic
232	#3385	C₂×C₂₃	1	16	yes	direct product factor
233	C₂×#501	C₂×C₂₃	1	2	yes	complemented
234	C₂×#500	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂×#479	C₂×C₂₃	1	2	yes	complemented
236	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
237	C₂×#476	C₂×C₂₃	1	1	yes	characteristic
238	C₂×#474	C₂×C₂₃	1	2	yes	—
239	C₂×#446	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂×#440	C₂×C₂₃	1	2	yes	—
241	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	—	4	8	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	—	4	2	no	—
246	C₂²×C₂₃	D₄×Q₈	1	2	yes	—
247	C₂²×C₂₃	C₂×#137	1	2	yes	—
248	C₂²×C₂₃	#576	1	2	yes	—
249	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
250	C₄×C₂₃	—	2	4	no	—
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	—	2	2	no	—
254	C₄×C₂₃	—	4	2	no	—
255	C₄×C₂₃	—	4	4	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	D₄×Q₈	1	2	yes	—
259	C₄×C₂₃	C₂×#137	1	4	yes	—
260	C₄×C₂₃	C₂³×Q₈	1	1	yes	characteristic
261	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
262	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
263	Q₈²	—	2	4	no	—
264	Q₈²	C₂²×C₂₃	1	8	yes	complemented
265	#554	—	2	4	no	—
266	#554	C₂²×C₂₃	1	8	yes	—
267	C₂×C₄⋊Q₈	—	2	1	no	—
268	C₂×C₄⋊Q₈	—	2	2	no	—
269	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
272	C₂×#125	C₂²×C₂₃	1	2	yes	—
273	C₂×C₂²⋊Q₈	—	2	1	no	—
274	C₂×C₂²⋊Q₈	—	2	2	no	—
275	C₂×C₄×Q₈	—	2	1	no	—
276	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
278	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
281	C₂²×C₄⋊C₄	—	2	1	no	—
282	#501	C₂²×C₂₃	1	16	yes	complemented
283	#500	C₂²×C₂₃	1	8	yes	complemented
284	#479	C₂²×C₂₃	1	16	yes	complemented
285	#477	C₂²×C₂₃	1	8	yes	complemented
286	#476	C₂²×C₂₃	1	8	yes	—
287	#474	C₂²×C₂₃	1	16	yes	—
288	#446	C₂²×C₂₃	1	8	yes	complemented
289	#440	C₂²×C₂₃	1	16	yes	—
290	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	—
291	C₂×#107	C₂²×C₂₃	1	2	yes	—
292	C₂×#107	C₂²×C₂₃	1	2	yes	—
293	C₂×#107	C₂²×C₂₃	1	2	yes	—
294	C₂×#106	C₂²×C₂₃	1	2	yes	—
295	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
296	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
298	C₂×#103	C₂²×C₂₃	1	2	yes	—
299	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×#102	C₂²×C₂₃	1	2	yes	—
302	C₂×#97	C₂²×C₂₃	1	2	yes	—
303	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₂₃	—	4	4	no	—
305	C₂×C₂₃	#3385	1	4	yes	direct product factor
306	C₂×C₂₃	C₂×D₄×Q₈	1	1	yes	characteristic
307	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
308	C₂×C₂₃	C₂×#576	1	1	yes	characteristic
309	C₂²×Q₈	—	2	1	no	—
310	C₂²×Q₈	—	2	1	no	—
311	C₂²×Q₈	—	2	2	no	—
312	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	—	2	2	no	—
317	C₂×SD₁₆	Q₈×C₂₃	1	8	yes	complemented
318	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
319	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
321	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₄⋊Q₈	—	2	8	no	—
323	C₄⋊Q₈	—	2	16	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	4	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
328	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
329	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
330	#125	—	2	4	no	—
331	#125	C₂³×C₂₃	1	8	yes	—
332	C₂²⋊Q₈	—	2	8	no	—
333	C₂²⋊Q₈	—	2	16	no	—
334	C₄×Q₈	—	2	8	no	—
335	C₄×Q₈	—	2	4	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	D₄×C₂₃	1	4	yes	—
338	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
339	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
340	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
341	C₄×D₄	—	2	4	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	C₂³×C₂₃	1	4	yes	—
344	C₄×D₄	C₂³×C₂₃	1	8	yes	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	8	no	—
353	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
354	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₄²	—	2	2	no	—
362	C₂×C₄²	—	2	1	no	—
363	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
364	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
365	#107	C₂³×C₂₃	1	8	yes	—
366	#107	C₂³×C₂₃	1	8	yes	—
367	#107	C₂³×C₂₃	1	8	yes	—
368	#106	C₂³×C₂₃	1	4	yes	—
369	#106	C₂³×C₂₃	1	8	yes	—
370	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
371	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
372	#103	C₂³×C₂₃	1	4	yes	—
373	#103	C₂³×C₂₃	1	8	yes	—
374	#102	C₂³×C₂₃	1	4	yes	—
375	#102	C₂³×C₂₃	1	8	yes	—
376	#97	C₂³×C₂₃	1	8	yes	—
377	C₄×C₈	C₂³×C₂₃	1	4	yes	—
378	C₂₃	C₂×#3385	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	2	1	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	4	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	8	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	8	4	no	—
387	C₂×Q₈	—	8	2	no	—
388	C₂×Q₈	—	8	2	no	—
389	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
390	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
391	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	C₂×Q₈×C₂₃	1	4	yes	—
395	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	8	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	8	4	no	—
405	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
408	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
410	SD₁₆	C₂×Q₈×C₂₃	1	16	yes	complemented
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	4	no	—
413	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
414	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
415	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
416	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
429	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
430	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
431	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
432	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
433	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
434	C₂²⋊C₄	—	2	4	no	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₄²	—	2	8	no	—
437	C₄²	—	2	4	no	—
438	C₄²	C₂×D₄×C₂₃	1	4	yes	—
439	C₄²	C₂×D₄×C₂₃	1	2	yes	—
440	C₄²	C₂⁴×C₂₃	1	2	yes	—
441	C₄²	C₂⁴×C₂₃	1	4	yes	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
447	Q₈	—	2	4	no	—
448	Q₈	—	2	8	no	—
449	Q₈	—	2	2	no	—
450	Q₈	—	8	8	no	—
451	Q₈	—	8	4	no	—
452	Q₈	—	8	4	no	—
453	Q₈	C₂²×Q₈×C₂₃	1	4	yes	—
454	D₄	—	2	4	no	—
455	D₄	—	2	2	no	—
456	D₄	C₂²×Q₈×C₂₃	1	4	yes	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	C₂₃×#137	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
474	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
476	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
478	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
479	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
480	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
481	C₈	—	2	2	no	—
482	C₈	—	2	4	no	—
483	C₈	C₂²×Q₈×C₂₃	1	4	yes	—
484	C₂²	—	2	4	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	8	no	—
488	C₂²	D₄×Q₈×C₂₃	1	2	yes	—
489	C₂²	C₂×C₂₃×#137	1	2	yes	—
490	C₂²	C₂₃×#576	1	2	yes	—
491	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
492	C₄	—	2	4	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	4	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	4	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	4	no	—
500	C₄	D₄×Q₈×C₂₃	1	2	yes	—
501	C₄	C₂×C₂₃×#137	1	4	yes	—
502	C₄	C₂³×Q₈×C₂₃	1	1	yes	characteristic
503	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
504	C₂	—	4	4	no	—
505	C₂	C₂₃×#3385	1	4	yes	direct product factor
506	C₂	C₂×D₄×Q₈×C₂₃	1	1	yes	characteristic
507	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
508	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	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₂₃	C₂	semidirect
3. C₂₃×#3385	C₂	direct (N × H)
4. C₂×C₂₃×#501	C₂	semidirect
5. C₂×C₂₃×#500	C₂	semidirect
6. C₂×C₂₃×#479	C₂	semidirect
7. C₂×C₂₃×#477	C₂	semidirect
10. C₂×C₂₃×#446	C₂	semidirect
15. Q₈²×C₂₃	C₂²	semidirect
33. C₂₃×#501	C₂²	semidirect
34. C₂₃×#500	C₂²	semidirect
35. C₂₃×#479	C₂²	semidirect
36. C₂₃×#477	C₂²	semidirect
39. C₂₃×#446	C₂²	semidirect
63. C₂×SD₁₆×C₂₃	Q₈	semidirect
155. SD₁₆×C₂₃	C₂×Q₈	semidirect
187. C₂×#3385	C₂₃	direct (N × H)
230. C₂×Q₈²	C₂×C₂₃	semidirect
232. #3385	C₂×C₂₃	direct (N × H)
233. C₂×#501	C₂×C₂₃	semidirect
234. C₂×#500	C₂×C₂₃	semidirect
235. C₂×#479	C₂×C₂₃	semidirect
236. C₂×#477	C₂×C₂₃	semidirect
239. C₂×#446	C₂×C₂₃	semidirect
264. Q₈²	C₂²×C₂₃	semidirect
282. #501	C₂²×C₂₃	semidirect
283. #500	C₂²×C₂₃	semidirect
284. #479	C₂²×C₂₃	semidirect
285. #477	C₂²×C₂₃	semidirect
288. #446	C₂²×C₂₃	semidirect
305. C₂×C₂₃	#3385	direct (N × H)
317. C₂×SD₁₆	Q₈×C₂₃	semidirect
378. C₂₃	C₂×#3385	direct (N × H)
410. SD₁₆	C₂×Q₈×C₂₃	semidirect
505. C₂	C₂₃×#3385	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×Q₈²×C₂₃	2
C₂×C₂₃×#554	2
C₂₃×#3385	2
C₂×C₂₃×#501	2
C₂×C₂₃×#500	2
C₂×C₂₃×#479	2
C₂×C₂₃×#477	2
C₂×C₂₃×#476	2
C₂×C₂₃×#474	2
C₂×C₂₃×#446	2
C₂×C₂₃×#440	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×#3385	23

Subgroups — C2×C23×#3385 #19208985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3385 #19208985