Subgroups — C₂×C₂₃×#3364 #19208927

Order 5888.

1830 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3364	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#488	C₂	1	2	yes	maximal
5	C₂×C₂₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#479	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#464	C₂	1	2	yes	maximal
9	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
13	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#557	—	2	4	no	—
16	C₂₃×#557	C₂²	1	8	yes	complemented
17	C₂₃×#556	—	2	4	no	—
18	C₂₃×#556	C₂²	1	8	yes	—
19	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#126	—	2	2	no	—
23	C₂×C₂₃×#125	—	2	2	no	—
24	C₂×C₂₃×#125	C₂²	1	2	yes	—
25	C₂×C₂₃×#124	C₂²	1	2	yes	—
26	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
28	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	—	2	1	no	—
34	C₂₃×#488	C₂²	1	16	yes	—
35	C₂₃×#487	C₂²	1	8	yes	complemented
36	C₂₃×#479	C₂²	1	16	yes	complemented
37	C₂₃×#474	C₂²	1	8	yes	—
38	C₂₃×#464	C₂²	1	16	yes	—
39	C₂₃×#462	C₂²	1	8	yes	complemented
40	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
41	C₄×Q₁₆×C₂₃	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	—
43	C₂×C₂₃×#107	C₂²	1	2	yes	—
44	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
45	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
47	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
48	C₂×C₂₃×#103	C₂²	1	2	yes	—
49	C₂×C₂₃×#103	C₂²	1	2	yes	—
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×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
54	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
58	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂×Q₁₆×C₂₃	—	2	8	no	—
62	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
63	C₂×SD₁₆×C₂₃	—	2	4	no	—
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	2	no	—
70	C₂₃×C₄⋊Q₈	—	2	2	no	—
71	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
73	C₂₃×#126	—	2	16	no	—
74	C₂₃×#125	—	2	16	no	—
75	C₂₃×#125	—	2	4	no	—
76	C₂₃×#125	C₂³	1	8	yes	—
77	C₂₃×#124	—	2	4	no	—
78	C₂₃×#124	C₂³	1	8	yes	—
79	C₂₃×C₂²⋊Q₈	—	2	8	no	—
80	C₄×Q₈×C₂₃	—	2	8	no	—
81	C₄×Q₈×C₂₃	—	2	4	no	—
82	C₄×Q₈×C₂₃	—	2	4	no	—
83	C₄×Q₈×C₂₃	—	2	2	no	—
84	C₄×Q₈×C₂₃	D₄	1	4	yes	—
85	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
86	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
87	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
88	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
89	C₄×D₄×C₂₃	—	2	2	no	—
90	C₄×D₄×C₂₃	C₂³	1	4	yes	—
91	C₂₃×#117	—	2	8	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
108	C₂×C₄²×C₂₃	—	2	2	no	—
109	C₂×C₄²×C₂₃	—	2	2	no	—
110	C₂×C₄²×C₂₃	—	2	1	no	—
111	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
113	C₂₃×#107	C₂³	1	8	yes	—
114	C₂₃×#106	C₂³	1	4	yes	—
115	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
116	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
117	C₂₃×#103	C₂³	1	4	yes	—
118	C₂₃×#103	C₂³	1	8	yes	—
119	C₂₃×#103	C₂³	1	8	yes	—
120	C₂₃×#103	C₂³	1	8	yes	—
121	C₂₃×#102	C₂³	1	4	yes	—
122	C₂₃×#102	C₂³	1	8	yes	—
123	C₄×C₈×C₂₃	C₂³	1	4	yes	—
124	C₄×C₈×C₂₃	C₂³	1	8	yes	—
125	C₂⁴×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	—	2	4	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	2	no	—
129	C₂×Q₈×C₂₃	—	2	4	no	—
130	C₂×Q₈×C₂₃	—	2	2	no	—
131	C₂×Q₈×C₂₃	—	2	4	no	—
132	C₂×Q₈×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	—	4	4	no	—
134	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₂₃	#41	1	4	yes	—
136	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
137	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
138	C₂×D₄×C₂₃	—	2	1	no	—
139	C₂×D₄×C₂₃	—	2	2	no	—
140	C₂×D₄×C₂₃	—	2	2	no	—
141	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	2	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	4	no	—
148	C₂²×C₄×C₂₃	—	4	4	no	—
149	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	Q₁₆×C₂₃	—	2	16	no	—
155	SD₁₆×C₂₃	—	2	8	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	4	no	—
158	C₂×C₈×C₂₃	—	2	2	no	—
159	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
160	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
161	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	8	no	—
164	C₄⋊C₄×C₂₃	—	2	8	no	—
165	C₄⋊C₄×C₂₃	—	2	4	no	—
166	C₄⋊C₄×C₂₃	—	2	8	no	—
167	C₄⋊C₄×C₂₃	—	2	4	no	—
168	C₄⋊C₄×C₂₃	—	2	8	no	—
169	C₄⋊C₄×C₂₃	—	2	4	no	—
170	C₄⋊C₄×C₂₃	—	2	1	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	C₂²⋊C₄×C₂₃	—	2	8	no	—
179	C₂²⋊C₄×C₂₃	—	2	4	no	—
180	C₂²⋊C₄×C₂₃	—	2	8	no	—
181	C₄²×C₂₃	—	2	8	no	—
182	C₄²×C₂₃	—	2	8	no	—
183	C₄²×C₂₃	—	2	4	no	—
184	C₄²×C₂₃	C₂×D₄	1	2	yes	—
185	C₄²×C₂₃	C₂×D₄	1	4	yes	—
186	C₄²×C₂₃	C₂⁴	1	2	yes	—
187	C₄²×C₂₃	C₂⁴	1	4	yes	—
188	C₂×#3364	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₂₃	—	2	2	no	—
190	C₂³×C₂₃	—	4	2	no	—
191	C₂³×C₂₃	—	4	2	no	—
192	C₂³×C₂₃	—	4	2	no	—
193	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₂₃	—	2	4	no	—
195	Q₈×C₂₃	—	2	8	no	—
196	Q₈×C₂₃	—	2	2	no	—
197	Q₈×C₂₃	—	2	4	no	—
198	Q₈×C₂₃	—	2	4	no	—
199	Q₈×C₂₃	—	4	4	no	—
200	Q₈×C₂₃	—	4	8	no	—
201	Q₈×C₂₃	C₂×#41	1	4	yes	—
202	D₄×C₂₃	—	2	2	no	—
203	D₄×C₂₃	—	2	2	no	—
204	D₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	2	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	—	2	2	no	—
213	C₂×C₄×C₂₃	—	2	4	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	4	2	no	—
216	C₂×C₄×C₂₃	—	4	4	no	—
217	C₂×C₄×C₂₃	—	4	2	no	—
218	C₂×C₄×C₂₃	—	4	8	no	—
219	C₂×C₄×C₂₃	#135	1	4	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
221	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
222	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
226	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
227	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
228	C₈×C₂₃	—	2	2	no	—
229	C₈×C₂₃	—	2	4	no	—
230	C₈×C₂₃	—	2	2	no	—
231	C₂×#557	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂×#556	C₂×C₂₃	1	1	yes	characteristic
233	#3364	C₂×C₂₃	1	16	yes	direct product factor
234	C₂×#488	C₂×C₂₃	1	2	yes	—
235	C₂×#487	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#479	C₂×C₂₃	1	2	yes	complemented
237	C₂×#474	C₂×C₂₃	1	1	yes	characteristic
238	C₂×#464	C₂×C₂₃	1	2	yes	—
239	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
241	C₂×C₄×Q₁₆	C₂×C₂₃	1	2	yes	complemented
242	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
243	C₂²×C₂₃	—	4	2	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	—	4	2	no	—
246	C₂²×C₂₃	—	8	4	no	—
247	C₂²×C₂₃	#548	1	2	yes	—
248	C₂²×C₂₃	C₂×#135	1	2	yes	—
249	C₂²×C₂₃	#578	1	2	yes	—
250	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	2	4	no	—
253	C₄×C₂₃	—	2	4	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	4	4	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	—	4	2	no	—
259	C₄×C₂₃	#548	1	2	yes	—
260	C₄×C₂₃	C₂×#135	1	4	yes	—
261	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
262	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
264	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
265	#557	—	2	4	no	—
266	#557	C₂²×C₂₃	1	8	yes	complemented
267	#556	—	2	4	no	—
268	#556	C₂²×C₂₃	1	8	yes	—
269	C₂×C₄⋊Q₈	—	2	1	no	—
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×#126	—	2	2	no	—
273	C₂×#125	—	2	2	no	—
274	C₂×#125	C₂²×C₂₃	1	2	yes	—
275	C₂×#124	C₂²×C₂₃	1	2	yes	—
276	C₂×C₂²⋊Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	—	2	1	no	—
278	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
279	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
283	C₂×#117	—	2	1	no	—
284	#488	C₂²×C₂₃	1	16	yes	—
285	#487	C₂²×C₂₃	1	8	yes	complemented
286	#479	C₂²×C₂₃	1	16	yes	complemented
287	#474	C₂²×C₂₃	1	8	yes	—
288	#464	C₂²×C₂₃	1	16	yes	—
289	#462	C₂²×C₂₃	1	8	yes	complemented
290	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
291	C₄×Q₁₆	C₂²×C₂₃	1	16	yes	complemented
292	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	—
293	C₂×#107	C₂²×C₂₃	1	2	yes	—
294	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
296	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
297	C₂×#103	C₂²×C₂₃	1	2	yes	—
298	C₂×#103	C₂²×C₂₃	1	2	yes	—
299	C₂×#103	C₂²×C₂₃	1	2	yes	—
300	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×#102	C₂²×C₂₃	1	2	yes	—
303	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
305	C₂×C₂₃	—	8	2	no	—
306	C₂×C₂₃	#3364	1	4	yes	direct product factor
307	C₂×C₂₃	C₂×#548	1	1	yes	characteristic
308	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
309	C₂×C₂₃	C₂×#578	1	1	yes	characteristic
310	C₂²×Q₈	—	2	1	no	—
311	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
312	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
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	C₂×Q₁₆	—	2	8	no	—
317	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
318	C₂×SD₁₆	—	2	4	no	—
319	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
320	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
322	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
323	C₄⋊Q₈	—	2	8	no	—
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
327	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
328	#126	—	2	16	no	—
329	#125	—	2	16	no	—
330	#125	—	2	4	no	—
331	#125	C₂³×C₂₃	1	8	yes	—
332	#124	—	2	4	no	—
333	#124	C₂³×C₂₃	1	8	yes	—
334	C₂²⋊Q₈	—	2	8	no	—
335	C₄×Q₈	—	2	8	no	—
336	C₄×Q₈	—	2	4	no	—
337	C₄×Q₈	—	2	4	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	D₄×C₂₃	1	4	yes	—
340	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
341	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
342	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
343	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	C₂³×C₂₃	1	4	yes	—
346	#117	—	2	8	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	2	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
357	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	—	2	2	no	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
368	#107	C₂³×C₂₃	1	8	yes	—
369	#106	C₂³×C₂₃	1	4	yes	—
370	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
371	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
372	#103	C₂³×C₂₃	1	4	yes	—
373	#103	C₂³×C₂₃	1	8	yes	—
374	#103	C₂³×C₂₃	1	8	yes	—
375	#103	C₂³×C₂₃	1	8	yes	—
376	#102	C₂³×C₂₃	1	4	yes	—
377	#102	C₂³×C₂₃	1	8	yes	—
378	C₄×C₈	C₂³×C₂₃	1	4	yes	—
379	C₄×C₈	C₂³×C₂₃	1	8	yes	—
380	C₂₃	C₂×#3364	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	—	2	1	no	—
382	C₂×Q₈	—	2	4	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	4	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
391	C₂×Q₈	#41×C₂₃	1	4	yes	—
392	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
393	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	4	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	Q₁₆	—	2	16	no	—
411	SD₁₆	—	2	8	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	4	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
416	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
417	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	8	no	—
420	C₄⋊C₄	—	2	8	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	8	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	8	no	—
425	C₄⋊C₄	—	2	4	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
430	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
431	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
432	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
433	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
434	C₂²⋊C₄	—	2	8	no	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₄²	—	2	8	no	—
438	C₄²	—	2	8	no	—
439	C₄²	—	2	4	no	—
440	C₄²	C₂×D₄×C₂₃	1	2	yes	—
441	C₄²	C₂×D₄×C₂₃	1	4	yes	—
442	C₄²	C₂⁴×C₂₃	1	2	yes	—
443	C₄²	C₂⁴×C₂₃	1	4	yes	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
449	Q₈	—	2	4	no	—
450	Q₈	—	2	8	no	—
451	Q₈	—	2	2	no	—
452	Q₈	—	2	4	no	—
453	Q₈	—	2	4	no	—
454	Q₈	—	4	4	no	—
455	Q₈	—	4	8	no	—
456	Q₈	C₂×#41×C₂₃	1	4	yes	—
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	8	no	—
474	C₂×C₄	C₂₃×#135	1	4	yes	—
475	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
476	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
477	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
478	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
480	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
481	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
482	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	2	4	no	—
485	C₈	—	2	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	8	4	no	—
490	C₂²	C₂₃×#548	1	2	yes	—
491	C₂²	C₂×C₂₃×#135	1	2	yes	—
492	C₂²	C₂₃×#578	1	2	yes	—
493	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
494	C₄	—	2	4	no	—
495	C₄	—	2	4	no	—
496	C₄	—	2	4	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	2	no	—
502	C₄	C₂₃×#548	1	2	yes	—
503	C₄	C₂×C₂₃×#135	1	4	yes	—
504	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
505	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
506	C₂	—	8	2	no	—
507	C₂	C₂₃×#3364	1	4	yes	direct product factor
508	C₂	C₂×C₂₃×#548	1	1	yes	characteristic
509	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
510	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×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×C₂₃×#557	C₂	semidirect
3. C₂₃×#3364	C₂	direct (N × H)
5. C₂×C₂₃×#487	C₂	semidirect
6. C₂×C₂₃×#479	C₂	semidirect
9. C₂×C₂₃×#462	C₂	semidirect
10. C₂×C₈×Q₈×C₂₃	C₂	semidirect
11. C₂×C₄×Q₁₆×C₂₃	C₂	semidirect
16. C₂₃×#557	C₂²	semidirect
35. C₂₃×#487	C₂²	semidirect
36. C₂₃×#479	C₂²	semidirect
39. C₂₃×#462	C₂²	semidirect
40. C₈×Q₈×C₂₃	C₂²	semidirect
41. C₄×Q₁₆×C₂₃	C₂²	semidirect
188. C₂×#3364	C₂₃	direct (N × H)
231. C₂×#557	C₂×C₂₃	semidirect
233. #3364	C₂×C₂₃	direct (N × H)
235. C₂×#487	C₂×C₂₃	semidirect
236. C₂×#479	C₂×C₂₃	semidirect
239. C₂×#462	C₂×C₂₃	semidirect
240. C₂×C₈×Q₈	C₂×C₂₃	semidirect
241. C₂×C₄×Q₁₆	C₂×C₂₃	semidirect
266. #557	C₂²×C₂₃	semidirect
285. #487	C₂²×C₂₃	semidirect
286. #479	C₂²×C₂₃	semidirect
289. #462	C₂²×C₂₃	semidirect
290. C₈×Q₈	C₂²×C₂₃	semidirect
291. C₄×Q₁₆	C₂²×C₂₃	semidirect
306. C₂×C₂₃	#3364	direct (N × H)
380. C₂₃	C₂×#3364	direct (N × H)
507. C₂	C₂₃×#3364	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×#557	2
C₂×C₂₃×#556	2
C₂₃×#3364	2
C₂×C₂₃×#488	2
C₂×C₂₃×#487	2
C₂×C₂₃×#479	2
C₂×C₂₃×#474	2
C₂×C₂₃×#464	2
C₂×C₂₃×#462	2
C₂×C₈×Q₈×C₂₃	2
C₂×C₄×Q₁₆×C₂₃	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×#3364	23

Subgroups — C2×C23×#3364 #19208927

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3364 #19208927