Subgroups — C₂×C₂₃×#3116 #19208165

Order 5888.

2150 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3116	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#485	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#452	C₂	1	2	yes	maximal
10	C₂×C₂₃×#441	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#533	—	2	4	no	—
17	C₂₃×#533	C₂²	1	8	yes	complemented
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#125	—	2	1	no	—
21	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
22	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₄×#41×C₂₃	—	2	4	no	—
24	C₄×#41×C₂₃	C₂²	1	8	yes	—
25	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
26	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₂₃	—	2	1	no	—
30	C₂×C₄×D₄×C₂₃	—	2	1	no	—
31	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#117	—	2	1	no	—
34	C₂×C₂₃×#117	—	2	1	no	—
35	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₂₃	—	2	1	no	—
37	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#485	C₂²	1	16	yes	complemented
39	C₂₃×#477	C₂²	1	8	yes	complemented
40	C₂₃×#476	C₂²	1	8	yes	complemented
41	C₂₃×#462	C₂²	1	8	yes	complemented
42	C₂₃×#461	C₂²	1	8	yes	complemented
43	C₂₃×#452	C₂²	1	16	yes	—
44	C₂₃×#441	C₂²	1	16	yes	complemented
45	C₂₃×#440	C₂²	1	16	yes	complemented
46	C₂₃×#432	C₂²	1	8	yes	complemented
47	C₂×C₂₃×#107	C₂²	1	2	yes	—
48	C₂×C₂₃×#106	C₂²	1	2	yes	—
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#103	C₂²	1	2	yes	—
52	C₂×C₂₃×#103	C₂²	1	2	yes	—
53	C₂×C₂₃×#103	C₂²	1	2	yes	—
54	C₂×C₂₃×#102	C₂²	1	2	yes	—
55	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
56	C₂×C₂₃×#97	C₂²	1	2	yes	—
57	C₂×#41×C₂₃	—	2	4	no	—
58	C₂×#41×C₂₃	—	2	4	no	—
59	C₂×#41×C₂₃	—	2	2	no	—
60	C₂×#41×C₂₃	C₂³	1	4	yes	—
61	C₂²×Q₈×C₂₃	—	2	1	no	—
62	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×Q₁₆×C₂₃	—	2	8	no	—
73	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₂₃	—	2	8	no	—
75	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
76	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
78	C₂₃×C₄⋊Q₈	—	2	2	no	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
81	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
82	C₂₃×#125	—	2	8	no	—
83	C₂₃×C₂²⋊Q₈	—	2	8	no	—
84	C₂₃×C₂²⋊Q₈	—	2	4	no	—
85	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
86	C₄×Q₈×C₂₃	—	2	8	no	—
87	C₄×Q₈×C₂₃	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	4	no	—
90	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
91	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
92	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
93	C₄×D₄×C₂₃	—	2	8	no	—
94	C₄×D₄×C₂₃	—	2	8	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	C₂³	1	4	yes	—
98	C₄×D₄×C₂₃	C₂³	1	4	yes	—
99	C₂₃×#117	—	2	8	no	—
100	C₂₃×#117	—	2	2	no	—
101	C₂₃×#117	—	2	8	no	—
102	C₂₃×#117	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₄²×C₂₃	—	2	1	no	—
118	C₂×C₄²×C₂₃	—	2	1	no	—
119	C₂×C₄²×C₂₃	—	2	1	no	—
120	C₂×C₄²×C₂₃	—	2	2	no	—
121	C₂×C₄²×C₂₃	—	2	8	no	—
122	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₂₃	D₄	1	4	yes	—
125	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
128	C₂₃×#107	C₂³	1	8	yes	—
129	C₂₃×#106	C₂³	1	8	yes	—
130	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
131	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
132	C₂₃×#103	C₂³	1	8	yes	—
133	C₂₃×#103	C₂³	1	8	yes	—
134	C₂₃×#103	C₂³	1	8	yes	—
135	C₂₃×#102	C₂³	1	8	yes	—
136	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
137	C₂₃×#97	C₂³	1	8	yes	—
138	C₂⁴×C₂₃	—	2	1	no	—
139	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
140	#41×C₂₃	—	2	16	no	—
141	#41×C₂₃	—	2	8	no	—
142	#41×C₂₃	—	2	8	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	2	4	no	—
146	C₂×Q₈×C₂₃	—	2	2	no	—
147	C₂×Q₈×C₂₃	—	2	4	no	—
148	C₂×Q₈×C₂₃	—	2	4	no	—
149	C₂×Q₈×C₂₃	—	4	4	no	—
150	C₂×Q₈×C₂₃	#41	1	4	yes	—
151	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
152	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
153	C₂×D₄×C₂₃	—	2	8	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	2	no	—
157	C₂×D₄×C₂₃	#41	1	4	yes	—
158	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	—	2	8	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	—	2	2	no	—
164	C₂²×C₄×C₂₃	—	2	2	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	2	no	—
169	C₂²×C₄×C₂₃	—	2	4	no	—
170	C₂²×C₄×C₂₃	—	2	4	no	—
171	C₂²×C₄×C₂₃	—	4	2	no	—
172	C₂²×C₄×C₂₃	—	4	4	no	—
173	C₂²×C₄×C₂₃	—	4	4	no	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
176	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₂₃	—	2	16	no	—
185	SD₁₆×C₂₃	—	2	16	no	—
186	C₂×C₈×C₂₃	—	2	4	no	—
187	C₂×C₈×C₂₃	—	2	4	no	—
188	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
189	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	2	4	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	8	no	—
196	C₄⋊C₄×C₂₃	—	2	8	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	2	no	—
200	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
203	C₂²⋊C₄×C₂₃	—	2	8	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₄²×C₂₃	—	2	4	no	—
208	C₄²×C₂₃	—	2	4	no	—
209	C₄²×C₂₃	—	2	4	no	—
210	C₄²×C₂₃	—	2	8	no	—
211	C₄²×C₂₃	C₂×D₄	1	4	yes	—
212	C₄²×C₂₃	C₂×D₄	1	4	yes	—
213	C₄²×C₂₃	C₂×D₄	1	2	yes	—
214	C₄²×C₂₃	C₂×D₄	1	2	yes	—
215	C₄²×C₂₃	C₂⁴	1	2	yes	—
216	C₄²×C₂₃	C₂⁴	1	2	yes	—
217	C₂×#3116	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	2	4	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	—	4	2	no	—
225	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
226	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₂₃	—	2	4	no	—
228	Q₈×C₂₃	—	2	2	no	—
229	Q₈×C₂₃	—	2	4	no	—
230	Q₈×C₂₃	—	2	4	no	—
231	Q₈×C₂₃	—	4	8	no	—
232	Q₈×C₂₃	—	4	4	no	—
233	Q₈×C₂₃	C₂×#41	1	4	yes	—
234	D₄×C₂₃	—	2	16	no	—
235	D₄×C₂₃	—	2	4	no	—
236	D₄×C₂₃	—	2	2	no	—
237	D₄×C₂₃	C₂×#41	1	4	yes	—
238	C₂×C₄×C₂₃	—	2	16	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	8	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	4	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	4	4	no	—
254	C₂×C₄×C₂₃	—	4	8	no	—
255	C₂×C₄×C₂₃	—	4	4	no	—
256	C₂×C₄×C₂₃	—	4	2	no	—
257	C₂×C₄×C₂₃	—	4	2	no	—
258	C₂×C₄×C₂₃	—	4	4	no	—
259	C₂×C₄×C₂₃	—	4	8	no	—
260	C₂×C₄×C₂₃	#137	1	4	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
268	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
269	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
270	C₈×C₂₃	—	2	4	no	—
271	C₈×C₂₃	—	2	4	no	—
272	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
273	C₂×C₄×#41	C₂×C₂₃	1	1	yes	characteristic
274	#3116	C₂×C₂₃	1	16	yes	direct product factor
275	C₂×#485	C₂×C₂₃	1	2	yes	complemented
276	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂×#476	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
279	C₂×#461	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₂×#452	C₂×C₂₃	1	2	yes	—
281	C₂×#441	C₂×C₂₃	1	2	yes	complemented
282	C₂×#440	C₂×C₂₃	1	2	yes	complemented
283	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂²×C₂₃	—	2	2	no	—
285	C₂²×C₂₃	—	2	4	no	—
286	C₂²×C₂₃	—	2	2	no	—
287	C₂²×C₂₃	—	2	2	no	—
288	C₂²×C₂₃	—	4	4	no	—
289	C₂²×C₂₃	—	4	8	no	—
290	C₂²×C₂₃	—	4	2	no	—
291	C₂²×C₂₃	—	4	2	no	—
292	C₂²×C₂₃	#525	1	2	yes	—
293	C₂²×C₂₃	C₂×#137	1	2	yes	—
294	C₂²×C₂₃	#575	1	2	yes	—
295	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	4	no	—
298	C₄×C₂₃	—	2	2	no	—
299	C₄×C₂₃	—	2	2	no	—
300	C₄×C₂₃	—	2	2	no	—
301	C₄×C₂₃	—	4	4	no	—
302	C₄×C₂₃	—	4	2	no	—
303	C₄×C₂₃	—	4	2	no	—
304	C₄×C₂₃	—	4	4	no	—
305	C₄×C₂₃	#525	1	2	yes	—
306	C₄×C₂₃	C₂×#137	1	4	yes	—
307	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
308	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
309	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
310	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
311	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
312	#533	—	2	4	no	—
313	#533	C₂²×C₂₃	1	8	yes	complemented
314	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#125	—	2	1	no	—
317	C₂×C₂²⋊Q₈	—	2	1	no	—
318	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
319	C₄×#41	—	2	4	no	—
320	C₄×#41	C₂²×C₂₃	1	8	yes	—
321	C₂×C₄×Q₈	—	2	1	no	—
322	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
325	C₂×C₄×D₄	—	2	1	no	—
326	C₂×C₄×D₄	—	2	1	no	—
327	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#117	—	2	1	no	—
330	C₂×#117	—	2	1	no	—
331	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²×C₄²	—	2	1	no	—
333	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
334	#485	C₂²×C₂₃	1	16	yes	complemented
335	#477	C₂²×C₂₃	1	8	yes	complemented
336	#476	C₂²×C₂₃	1	8	yes	complemented
337	#462	C₂²×C₂₃	1	8	yes	complemented
338	#461	C₂²×C₂₃	1	8	yes	complemented
339	#452	C₂²×C₂₃	1	16	yes	—
340	#441	C₂²×C₂₃	1	16	yes	complemented
341	#440	C₂²×C₂₃	1	16	yes	complemented
342	#432	C₂²×C₂₃	1	8	yes	complemented
343	C₂×#107	C₂²×C₂₃	1	2	yes	—
344	C₂×#106	C₂²×C₂₃	1	2	yes	—
345	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#103	C₂²×C₂₃	1	2	yes	—
348	C₂×#103	C₂²×C₂₃	1	2	yes	—
349	C₂×#103	C₂²×C₂₃	1	2	yes	—
350	C₂×#102	C₂²×C₂₃	1	2	yes	—
351	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
352	C₂×#97	C₂²×C₂₃	1	2	yes	—
353	C₂×C₂₃	—	4	2	no	—
354	C₂×C₂₃	—	4	4	no	—
355	C₂×C₂₃	#3116	1	4	yes	direct product factor
356	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
357	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
358	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
359	C₂×#41	—	2	4	no	—
360	C₂×#41	—	2	4	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	C₂³×C₂₃	1	4	yes	—
363	C₂²×Q₈	—	2	1	no	—
364	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
365	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₂²×D₄	—	2	1	no	—
367	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
372	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
374	C₂×Q₁₆	—	2	8	no	—
375	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
376	C₂×SD₁₆	—	2	8	no	—
377	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
378	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
379	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
380	C₄⋊Q₈	—	2	2	no	—
381	C₄⋊Q₈	—	2	2	no	—
382	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
383	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
384	#125	—	2	8	no	—
385	C₂²⋊Q₈	—	2	8	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
388	C₄×Q₈	—	2	8	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	4	no	—
392	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
393	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
394	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
395	C₄×D₄	—	2	8	no	—
396	C₄×D₄	—	2	8	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	C₂³×C₂₃	1	4	yes	—
400	C₄×D₄	C₂³×C₂₃	1	4	yes	—
401	#117	—	2	8	no	—
402	#117	—	2	2	no	—
403	#117	—	2	8	no	—
404	#117	C₂³×C₂₃	1	4	yes	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
415	C₂×C₂²⋊C₄	—	2	2	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	—	2	2	no	—
421	C₂×C₄²	—	2	1	no	—
422	C₂×C₄²	—	2	1	no	—
423	C₂×C₄²	—	2	8	no	—
424	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄²	D₄×C₂₃	1	4	yes	—
426	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
430	#107	C₂³×C₂₃	1	8	yes	—
431	#106	C₂³×C₂₃	1	8	yes	—
432	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
433	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
434	#103	C₂³×C₂₃	1	8	yes	—
435	#103	C₂³×C₂₃	1	8	yes	—
436	#103	C₂³×C₂₃	1	8	yes	—
437	#102	C₂³×C₂₃	1	8	yes	—
438	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
439	#97	C₂³×C₂₃	1	8	yes	—
440	C₂₃	C₂×#3116	1	1	yes	characteristic, direct product factor, Sylow
441	C₂⁴	—	2	1	no	—
442	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
443	#41	—	2	16	no	—
444	#41	—	2	8	no	—
445	#41	—	2	8	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	4	no	—
451	C₂×Q₈	—	2	4	no	—
452	C₂×Q₈	—	4	4	no	—
453	C₂×Q₈	#41×C₂₃	1	4	yes	—
454	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
455	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
456	C₂×D₄	—	2	8	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	#41×C₂₃	1	4	yes	—
461	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
462	C₂²×C₄	—	2	8	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
481	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
483	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
484	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
487	Q₁₆	—	2	16	no	—
488	SD₁₆	—	2	16	no	—
489	C₂×C₈	—	2	4	no	—
490	C₂×C₈	—	2	4	no	—
491	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
492	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	2	8	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
504	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
505	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
506	C₂²⋊C₄	—	2	8	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	8	no	—
514	C₄²	C₂×D₄×C₂₃	1	4	yes	—
515	C₄²	C₂×D₄×C₂₃	1	4	yes	—
516	C₄²	C₂×D₄×C₂₃	1	2	yes	—
517	C₄²	C₂×D₄×C₂₃	1	2	yes	—
518	C₄²	C₂⁴×C₂₃	1	2	yes	—
519	C₄²	C₂⁴×C₂₃	1	2	yes	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	4	2	no	—
527	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
528	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
529	Q₈	—	2	2	no	—
530	Q₈	—	2	4	no	—
531	Q₈	—	2	4	no	—
532	Q₈	—	2	4	no	—
533	Q₈	—	4	8	no	—
534	Q₈	—	4	4	no	—
535	Q₈	C₂×#41×C₂₃	1	4	yes	—
536	D₄	—	2	16	no	—
537	D₄	—	2	2	no	—
538	D₄	—	2	4	no	—
539	D₄	C₂×#41×C₂₃	1	4	yes	—
540	C₂×C₄	—	2	16	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	4	4	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	C₂₃×#137	1	4	yes	—
563	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
569	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
570	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
571	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
572	C₈	—	2	4	no	—
573	C₈	—	2	4	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	8	no	—
582	C₂²	C₂₃×#525	1	2	yes	—
583	C₂²	C₂×C₂₃×#137	1	2	yes	—
584	C₂²	C₂₃×#575	1	2	yes	—
585	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
586	C₄	—	2	2	no	—
587	C₄	—	2	4	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	2	no	—
594	C₄	—	4	4	no	—
595	C₄	C₂₃×#525	1	2	yes	—
596	C₄	C₂×C₂₃×#137	1	4	yes	—
597	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
598	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
599	C₂	—	4	2	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂₃×#3116	1	4	yes	direct product factor
602	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
603	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
604	C₂	C₂×C₂₃×#575	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 ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#533	C₂	semidirect
3. C₂₃×#3116	C₂	direct (N × H)
4. C₂×C₂₃×#485	C₂	semidirect
5. C₂×C₂₃×#477	C₂	semidirect
6. C₂×C₂₃×#476	C₂	semidirect
7. C₂×C₂₃×#462	C₂	semidirect
8. C₂×C₂₃×#461	C₂	semidirect
10. C₂×C₂₃×#441	C₂	semidirect
11. C₂×C₂₃×#440	C₂	semidirect
12. C₂×C₂₃×#432	C₂	semidirect
17. C₂₃×#533	C₂²	semidirect
38. C₂₃×#485	C₂²	semidirect
39. C₂₃×#477	C₂²	semidirect
40. C₂₃×#476	C₂²	semidirect
41. C₂₃×#462	C₂²	semidirect
42. C₂₃×#461	C₂²	semidirect
44. C₂₃×#441	C₂²	semidirect
45. C₂₃×#440	C₂²	semidirect
46. C₂₃×#432	C₂²	semidirect
217. C₂×#3116	C₂₃	direct (N × H)
272. C₂×#533	C₂×C₂₃	semidirect
274. #3116	C₂×C₂₃	direct (N × H)
275. C₂×#485	C₂×C₂₃	semidirect
276. C₂×#477	C₂×C₂₃	semidirect
277. C₂×#476	C₂×C₂₃	semidirect
278. C₂×#462	C₂×C₂₃	semidirect
279. C₂×#461	C₂×C₂₃	semidirect
281. C₂×#441	C₂×C₂₃	semidirect
282. C₂×#440	C₂×C₂₃	semidirect
283. C₂×#432	C₂×C₂₃	semidirect
313. #533	C₂²×C₂₃	semidirect
334. #485	C₂²×C₂₃	semidirect
335. #477	C₂²×C₂₃	semidirect
336. #476	C₂²×C₂₃	semidirect
337. #462	C₂²×C₂₃	semidirect
338. #461	C₂²×C₂₃	semidirect
340. #441	C₂²×C₂₃	semidirect
341. #440	C₂²×C₂₃	semidirect
342. #432	C₂²×C₂₃	semidirect
355. C₂×C₂₃	#3116	direct (N × H)
440. C₂₃	C₂×#3116	direct (N × H)
601. C₂	C₂₃×#3116	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×#533	2
C₂×C₄×#41×C₂₃	2
C₂₃×#3116	2
C₂×C₂₃×#485	2
C₂×C₂₃×#477	2
C₂×C₂₃×#476	2
C₂×C₂₃×#462	2
C₂×C₂₃×#461	2
C₂×C₂₃×#452	2
C₂×C₂₃×#441	2
C₂×C₂₃×#440	2
C₂×C₂₃×#432	2
C₂×#3116	23

Subgroups — C2×C23×#3116 #19208165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3116 #19208165