Subgroups — C₂×C₂₃×#3394 #19209016

Order 5888.

2022 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3394	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#495	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#464	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic
10	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#559	—	2	4	no	—
19	C₂₃×#559	C₂²	1	8	yes	complemented
20	C₂₃×#556	—	2	4	no	—
21	C₂₃×#556	C₂²	1	8	yes	complemented
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#126	—	2	2	no	—
25	C₂×C₂₃×#124	—	2	1	no	—
26	C₂×C₂₃×#124	C₂²	1	2	yes	—
27	C₂×C₂₃×#124	C₂²	1	2	yes	—
28	C₂×C₂₃×#123	—	2	2	no	—
29	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₂₃	—	2	1	no	—
34	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
36	C₂×C₂₃×#117	—	2	1	no	—
37	C₂₃×#495	C₂²	1	16	yes	complemented
38	C₂₃×#492	C₂²	1	8	yes	complemented
39	C₂₃×#464	C₂²	1	16	yes	complemented
40	C₂₃×#463	C₂²	1	16	yes	complemented
41	C₂₃×#462	C₂²	1	8	yes	complemented
42	C₂₃×#459	C₂²	1	8	yes	—
43	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
44	C₄×SD₁₆×C₂₃	C₂²	1	16	yes	complemented
45	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
46	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
47	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
49	C₂×C₂₃×#103	C₂²	1	2	yes	—
50	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#102	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
53	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
55	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
57	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	—	2	1	no	—
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×C₂₃	—	2	2	no	—
64	C₂×Q₁₆×C₂₃	—	4	4	no	—
65	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
66	C₂×SD₁₆×C₂₃	—	2	8	no	—
67	C₂×SD₁₆×C₂₃	—	2	4	no	—
68	C₂×SD₁₆×C₂₃	—	4	4	no	—
69	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
70	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
71	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
72	C₂×D₈×C₂₃	—	4	4	no	—
73	C₂×D₈×C₂₃	C₂³	1	8	yes	—
74	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	—	2	2	no	—
78	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₂₃×#127	—	2	2	no	—
80	C₂₃×#127	C₂³	1	4	yes	—
81	C₂₃×#126	—	2	16	no	—
82	C₂₃×#124	—	2	8	no	—
83	C₂₃×#124	—	2	4	no	—
84	C₂₃×#124	—	2	4	no	—
85	C₂₃×#124	C₂³	1	8	yes	—
86	C₂₃×#124	C₂³	1	8	yes	—
87	C₂₃×#123	—	2	16	no	—
88	C₂₃×C₂²⋊Q₈	—	2	8	no	—
89	C₄×Q₈×C₂₃	—	2	4	no	—
90	C₄×Q₈×C₂₃	—	2	2	no	—
91	C₄×Q₈×C₂₃	D₄	1	4	yes	—
92	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
93	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
94	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
95	C₄×D₄×C₂₃	—	2	8	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	—	2	4	no	—
98	C₄×D₄×C₂₃	C₂³	1	4	yes	—
99	C₄×D₄×C₂₃	C₂³	1	8	yes	—
100	C₂₃×#117	—	2	8	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂×C₄²×C₂₃	—	2	2	no	—
116	C₂×C₄²×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
118	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
119	C₂₃×#106	C₂³	1	4	yes	—
120	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
121	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
122	C₂₃×#103	C₂³	1	8	yes	—
123	C₂₃×#103	C₂³	1	4	yes	—
124	C₂₃×#102	C₂³	1	8	yes	—
125	C₂₃×#102	C₂³	1	4	yes	—
126	C₄×C₈×C₂₃	C₂³	1	4	yes	—
127	C₄×C₈×C₂₃	C₂³	1	8	yes	—
128	C₂⁴×C₂₃	—	2	1	no	—
129	C₂⁴×C₂₃	—	2	2	no	—
130	C₂×Q₈×C₂₃	—	2	4	no	—
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	2	2	no	—
133	C₂×Q₈×C₂₃	—	2	4	no	—
134	C₂×Q₈×C₂₃	—	2	2	no	—
135	C₂×Q₈×C₂₃	—	2	4	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
138	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
139	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
140	C₂×D₄×C₂₃	—	2	1	no	—
141	C₂×D₄×C₂₃	—	2	2	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	4	no	—
144	C₂×D₄×C₂₃	—	2	2	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	4	no	—
148	C₂×D₄×C₂₃	—	4	4	no	—
149	C₂×D₄×C₂₃	—	8	2	no	—
150	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	2	no	—
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	8	no	—
158	C₂²×C₄×C₂₃	—	4	4	no	—
159	C₂²×C₄×C₂₃	—	8	4	no	—
160	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
165	Q₁₆×C₂₃	—	4	8	no	—
166	SD₁₆×C₂₃	—	2	16	no	—
167	SD₁₆×C₂₃	—	2	8	no	—
168	SD₁₆×C₂₃	—	4	8	no	—
169	D₈×C₂₃	—	4	8	no	—
170	C₂×C₈×C₂₃	—	2	4	no	—
171	C₂×C₈×C₂₃	—	2	2	no	—
172	C₂×C₈×C₂₃	#41	1	4	yes	—
173	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
174	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
175	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
176	C₄⋊C₄×C₂₃	—	2	4	no	—
177	C₄⋊C₄×C₂₃	—	2	8	no	—
178	C₄⋊C₄×C₂₃	—	2	4	no	—
179	C₄⋊C₄×C₂₃	—	2	4	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
184	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
186	C₂²⋊C₄×C₂₃	—	2	8	no	—
187	C₂²⋊C₄×C₂₃	—	2	8	no	—
188	C₂²⋊C₄×C₂₃	—	2	8	no	—
189	C₂²⋊C₄×C₂₃	—	2	4	no	—
190	C₂²⋊C₄×C₂₃	—	2	8	no	—
191	C₂²⋊C₄×C₂₃	—	2	8	no	—
192	C₄²×C₂₃	—	2	8	no	—
193	C₄²×C₂₃	—	2	4	no	—
194	C₄²×C₂₃	C₂×D₄	1	4	yes	—
195	C₄²×C₂₃	C₂×D₄	1	2	yes	—
196	C₄²×C₂₃	C₂⁴	1	4	yes	—
197	C₄²×C₂₃	C₂⁴	1	2	yes	—
198	C₂×#3394	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	4	no	—
201	C₂³×C₂₃	—	4	2	no	—
202	C₂³×C₂₃	—	4	4	no	—
203	C₂³×C₂₃	—	4	2	no	—
204	C₂³×C₂₃	—	4	4	no	—
205	C₂³×C₂₃	—	4	4	no	—
206	C₂³×C₂₃	—	4	2	no	—
207	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	8	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	2	4	no	—
211	Q₈×C₂₃	—	2	4	no	—
212	Q₈×C₂₃	—	2	2	no	—
213	Q₈×C₂₃	—	4	2	no	—
214	Q₈×C₂₃	—	4	4	no	—
215	D₄×C₂₃	—	2	2	no	—
216	D₄×C₂₃	—	2	4	no	—
217	D₄×C₂₃	—	2	2	no	—
218	D₄×C₂₃	—	2	4	no	—
219	D₄×C₂₃	—	4	2	no	—
220	D₄×C₂₃	—	4	4	no	—
221	D₄×C₂₃	—	4	8	no	—
222	D₄×C₂₃	—	8	4	no	—
223	C₂×C₄×C₂₃	—	2	4	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	2	2	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	4	4	no	—
232	C₂×C₄×C₂₃	—	4	2	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	8	no	—
235	C₂×C₄×C₂₃	—	8	8	no	—
236	C₂×C₄×C₂₃	#135	1	4	yes	—
237	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
243	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
244	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
245	C₈×C₂₃	—	2	4	no	—
246	C₈×C₂₃	—	2	2	no	—
247	C₈×C₂₃	C₂×#41	1	4	yes	—
248	C₂×#559	C₂×C₂₃	1	1	yes	characteristic, complemented
249	C₂×#556	C₂×C₂₃	1	1	yes	characteristic, complemented
250	#3394	C₂×C₂₃	1	16	yes	direct product factor
251	C₂×#495	C₂×C₂₃	1	2	yes	complemented
252	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₂×#464	C₂×C₂₃	1	2	yes	complemented
254	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
255	C₂×#463	C₂×C₂₃	1	2	yes	complemented
256	C₂×#459	C₂×C₂₃	1	1	yes	characteristic
257	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
258	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₂×C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
260	C₂²×C₂₃	—	4	2	no	—
261	C₂²×C₂₃	—	4	4	no	—
262	C₂²×C₂₃	—	4	2	no	—
263	C₂²×C₂₃	—	4	4	no	—
264	C₂²×C₂₃	—	4	4	no	—
265	C₂²×C₂₃	—	4	2	no	—
266	C₂²×C₂₃	—	8	4	no	—
267	C₂²×C₂₃	—	8	8	no	—
268	C₂²×C₂₃	#550	1	2	yes	—
269	C₂²×C₂₃	C₂×#135	1	2	yes	—
270	C₂²×C₂₃	#577	1	2	yes	—
271	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
272	C₄×C₂₃	—	2	4	no	—
273	C₄×C₂₃	—	2	4	no	—
274	C₄×C₂₃	—	2	2	no	—
275	C₄×C₂₃	—	4	2	no	—
276	C₄×C₂₃	—	4	4	no	—
277	C₄×C₂₃	—	4	2	no	—
278	C₄×C₂₃	—	4	2	no	—
279	C₄×C₂₃	#550	1	2	yes	—
280	C₄×C₂₃	C₂×#135	1	4	yes	—
281	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
282	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
284	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
285	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
286	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
287	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
288	#559	—	2	4	no	—
289	#559	C₂²×C₂₃	1	8	yes	complemented
290	#556	—	2	4	no	—
291	#556	C₂²×C₂₃	1	8	yes	complemented
292	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
293	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
294	C₂×#126	—	2	2	no	—
295	C₂×#124	—	2	1	no	—
296	C₂×#124	C₂²×C₂₃	1	2	yes	—
297	C₂×#124	C₂²×C₂₃	1	2	yes	—
298	C₂×#123	—	2	2	no	—
299	C₂×C₂²⋊Q₈	—	2	1	no	—
300	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
301	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×C₄×D₄	—	2	1	no	—
304	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
306	C₂×#117	—	2	1	no	—
307	#495	C₂²×C₂₃	1	16	yes	complemented
308	#492	C₂²×C₂₃	1	8	yes	complemented
309	#464	C₂²×C₂₃	1	16	yes	complemented
310	#463	C₂²×C₂₃	1	16	yes	complemented
311	#462	C₂²×C₂₃	1	8	yes	complemented
312	#459	C₂²×C₂₃	1	8	yes	—
313	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
314	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
315	C₄×SD₁₆	C₂²×C₂₃	1	16	yes	complemented
316	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
319	C₂×#103	C₂²×C₂₃	1	2	yes	—
320	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#102	C₂²×C₂₃	1	2	yes	—
322	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
325	C₂×C₂₃	—	8	2	no	—
326	C₂×C₂₃	—	8	4	no	—
327	C₂×C₂₃	#3394	1	4	yes	direct product factor
328	C₂×C₂₃	C₂×#550	1	1	yes	characteristic
329	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
330	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
331	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
332	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
334	C₂²×D₄	—	2	1	no	—
335	C₂²×D₄	—	2	1	no	—
336	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
338	C₂³×C₄	—	2	1	no	—
339	C₂³×C₄	—	2	2	no	—
340	C₂×Q₁₆	—	4	4	no	—
341	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
342	C₂×SD₁₆	—	2	8	no	—
343	C₂×SD₁₆	—	2	4	no	—
344	C₂×SD₁₆	—	4	4	no	—
345	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
346	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
347	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
348	C₂×D₈	—	4	4	no	—
349	C₂×D₈	C₂³×C₂₃	1	8	yes	—
350	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
351	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
353	C₄⋊Q₈	—	2	2	no	—
354	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
355	#127	—	2	2	no	—
356	#127	C₂³×C₂₃	1	4	yes	—
357	#126	—	2	16	no	—
358	#124	—	2	8	no	—
359	#124	—	2	4	no	—
360	#124	—	2	4	no	—
361	#124	C₂³×C₂₃	1	8	yes	—
362	#124	C₂³×C₂₃	1	8	yes	—
363	#123	—	2	16	no	—
364	C₂²⋊Q₈	—	2	8	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	D₄×C₂₃	1	4	yes	—
368	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
369	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
370	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
371	C₄×D₄	—	2	8	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	4	no	—
374	C₄×D₄	C₂³×C₂₃	1	4	yes	—
375	C₄×D₄	C₂³×C₂₃	1	8	yes	—
376	#117	—	2	8	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
394	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
395	#106	C₂³×C₂₃	1	4	yes	—
396	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
397	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
398	#103	C₂³×C₂₃	1	8	yes	—
399	#103	C₂³×C₂₃	1	4	yes	—
400	#102	C₂³×C₂₃	1	8	yes	—
401	#102	C₂³×C₂₃	1	4	yes	—
402	C₄×C₈	C₂³×C₂₃	1	4	yes	—
403	C₄×C₈	C₂³×C₂₃	1	8	yes	—
404	C₂₃	C₂×#3394	1	1	yes	characteristic, direct product factor, Sylow
405	C₂⁴	—	2	1	no	—
406	C₂⁴	—	2	2	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	2	4	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
415	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
416	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	8	2	no	—
427	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
428	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	8	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	8	4	no	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
438	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
439	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
442	Q₁₆	—	4	8	no	—
443	SD₁₆	—	2	16	no	—
444	SD₁₆	—	2	8	no	—
445	SD₁₆	—	4	8	no	—
446	D₈	—	4	8	no	—
447	C₂×C₈	—	2	4	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	#41×C₂₃	1	4	yes	—
450	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
451	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
452	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
460	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
461	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
462	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₄²	—	2	8	no	—
470	C₄²	—	2	4	no	—
471	C₄²	C₂×D₄×C₂₃	1	4	yes	—
472	C₄²	C₂×D₄×C₂₃	1	2	yes	—
473	C₄²	C₂⁴×C₂₃	1	4	yes	—
474	C₄²	C₂⁴×C₂₃	1	2	yes	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	2	no	—
483	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
484	Q₈	—	2	8	no	—
485	Q₈	—	2	2	no	—
486	Q₈	—	2	4	no	—
487	Q₈	—	2	4	no	—
488	Q₈	—	2	2	no	—
489	Q₈	—	4	2	no	—
490	Q₈	—	4	4	no	—
491	D₄	—	2	2	no	—
492	D₄	—	2	4	no	—
493	D₄	—	2	2	no	—
494	D₄	—	2	4	no	—
495	D₄	—	4	2	no	—
496	D₄	—	4	4	no	—
497	D₄	—	4	8	no	—
498	D₄	—	8	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	4	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	8	no	—
511	C₂×C₄	—	8	8	no	—
512	C₂×C₄	C₂₃×#135	1	4	yes	—
513	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
514	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
515	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
518	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
519	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
520	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
521	C₈	—	2	4	no	—
522	C₈	—	2	2	no	—
523	C₈	C₂×#41×C₂₃	1	4	yes	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	4	no	—
531	C₂²	—	8	8	no	—
532	C₂²	C₂₃×#550	1	2	yes	—
533	C₂²	C₂×C₂₃×#135	1	2	yes	—
534	C₂²	C₂₃×#577	1	2	yes	—
535	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
536	C₄	—	2	4	no	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	4	2	no	—
540	C₄	—	4	4	no	—
541	C₄	—	4	2	no	—
542	C₄	—	4	2	no	—
543	C₄	C₂₃×#550	1	2	yes	—
544	C₄	C₂×C₂₃×#135	1	4	yes	—
545	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
546	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
547	C₂	—	8	2	no	—
548	C₂	—	8	4	no	—
549	C₂	C₂₃×#3394	1	4	yes	direct product factor
550	C₂	C₂×C₂₃×#550	1	1	yes	characteristic
551	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
552	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#459	C₂
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×C₂₃×#559	C₂	semidirect
2. C₂×C₂₃×#556	C₂	semidirect
3. C₂₃×#3394	C₂	direct (N × H)
4. C₂×C₂₃×#495	C₂	semidirect
5. C₂×C₂₃×#492	C₂	semidirect
6. C₂×C₂₃×#464	C₂	semidirect
7. C₂×C₂₃×#462	C₂	semidirect
8. C₂×C₂₃×#463	C₂	semidirect
10. C₂×C₈×Q₈×C₂₃	C₂	semidirect
11. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
12. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
19. C₂₃×#559	C₂²	semidirect
21. C₂₃×#556	C₂²	semidirect
37. C₂₃×#495	C₂²	semidirect
38. C₂₃×#492	C₂²	semidirect
39. C₂₃×#464	C₂²	semidirect
40. C₂₃×#463	C₂²	semidirect
41. C₂₃×#462	C₂²	semidirect
43. C₈×Q₈×C₂₃	C₂²	semidirect
44. C₄×SD₁₆×C₂₃	C₂²	semidirect
45. C₄×SD₁₆×C₂₃	C₂²	semidirect
198. C₂×#3394	C₂₃	direct (N × H)
248. C₂×#559	C₂×C₂₃	semidirect
249. C₂×#556	C₂×C₂₃	semidirect
250. #3394	C₂×C₂₃	direct (N × H)
251. C₂×#495	C₂×C₂₃	semidirect
252. C₂×#492	C₂×C₂₃	semidirect
253. C₂×#464	C₂×C₂₃	semidirect
254. C₂×#462	C₂×C₂₃	semidirect
255. C₂×#463	C₂×C₂₃	semidirect
257. C₂×C₈×Q₈	C₂×C₂₃	semidirect
258. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
259. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
289. #559	C₂²×C₂₃	semidirect
291. #556	C₂²×C₂₃	semidirect
307. #495	C₂²×C₂₃	semidirect
308. #492	C₂²×C₂₃	semidirect
309. #464	C₂²×C₂₃	semidirect
310. #463	C₂²×C₂₃	semidirect
311. #462	C₂²×C₂₃	semidirect
313. C₈×Q₈	C₂²×C₂₃	semidirect
314. C₄×SD₁₆	C₂²×C₂₃	semidirect
315. C₄×SD₁₆	C₂²×C₂₃	semidirect
327. C₂×C₂₃	#3394	direct (N × H)
404. C₂₃	C₂×#3394	direct (N × H)
549. C₂	C₂₃×#3394	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×#559	2
C₂×C₂₃×#556	2
C₂₃×#3394	2
C₂×C₂₃×#495	2
C₂×C₂₃×#492	2
C₂×C₂₃×#464	2
C₂×C₂₃×#462	2
C₂×C₂₃×#463	2
C₂×C₂₃×#459	2
C₂×C₈×Q₈×C₂₃	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×#3394	23

Subgroups — C2×C23×#3394 #19209016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3394 #19209016