Subgroups — C₂×C₂₃×#2896 #19207709

Order 5888.

5198 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2896	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#417	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#419	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#418	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#416	C₂	1	4	yes	maximal, complemented
9	C₂²×#41×C₂₃	—	2	8	no	—
10	C₂²×#41×C₂₃	—	2	1	no	—
11	C₂²×#41×C₂₃	—	2	4	no	—
12	C₂²×#41×C₂₃	—	2	1	no	—
13	C₂²×#41×C₂₃	C₄	1	16	yes	complemented
14	C₂²×#41×C₂₃	C₂²	1	2	yes	—
15	C₂²×#41×C₂₃	C₂²	1	2	yes	—
16	C₂²×#41×C₂₃	C₂²	1	8	yes	—
17	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂³×D₄×C₂₃	—	2	1	no	—
19	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂⁴×C₄×C₂₃	—	2	1	no	—
21	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₂₃	—	2	8	no	—
23	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
24	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	complemented
25	C₂²×M₁₆×C₂₃	C₂²	1	8	yes	complemented
26	C₂³×C₈×C₂₃	C₂²	1	2	yes	complemented
27	C₂×C₂₃×#117	—	2	1	no	—
28	C₂×C₂₃×#117	—	2	4	no	—
29	C₂×C₂₃×#117	—	2	1	no	—
30	C₂×C₂₃×#117	C₂²	1	2	yes	—
31	C₂×C₂₃×#117	C₂²	1	2	yes	complemented
32	C₂×C₂₃×#117	C₂²	1	8	yes	complemented
33	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
34	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
35	C₂²×C₄²×C₂₃	—	2	1	no	—
36	C₂₃×#419	C₂²	1	16	yes	complemented
37	C₂₃×#418	C₂²	1	16	yes	complemented
38	C₂₃×#417	C₂²	1	32	yes	complemented
39	C₂₃×#416	C₂²	1	32	yes	complemented
40	C₂×C₂₃×#103	C₂²	1	8	yes	complemented
41	C₂×C₂₃×#102	C₂²	1	8	yes	complemented
42	C₂⁵×C₂₃	—	2	1	no	—
43	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
44	C₂×#41×C₂₃	—	2	16	no	—
45	C₂×#41×C₂₃	—	2	16	no	—
46	C₂×#41×C₂₃	—	2	64	no	—
47	C₂×#41×C₂₃	—	2	8	no	—
48	C₂×#41×C₂₃	—	2	8	no	—
49	C₂×#41×C₂₃	—	2	32	no	—
50	C₂×#41×C₂₃	—	2	8	no	—
51	C₂×#41×C₂₃	—	2	4	no	—
52	C₂×#41×C₂₃	—	2	4	no	—
53	C₂×#41×C₂₃	—	2	8	no	—
54	C₂×#41×C₂₃	C₂×C₄	1	16	yes	—
55	C₂×#41×C₂₃	C₂×C₄	1	16	yes	complemented
56	C₂×#41×C₂₃	C₂×C₄	1	64	yes	complemented
57	C₂×#41×C₂₃	C₂³	1	8	yes	—
58	C₂×#41×C₂₃	C₂³	1	8	yes	—
59	C₂²×Q₈×C₂₃	—	2	4	no	—
60	C₂²×Q₈×C₂₃	—	2	1	no	—
61	C₂²×Q₈×C₂₃	—	2	2	no	—
62	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
63	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₂₃	—	2	16	no	—
66	C₂²×D₄×C₂₃	—	2	2	no	—
67	C₂²×D₄×C₂₃	—	2	2	no	—
68	C₂²×D₄×C₂₃	—	2	8	no	—
69	C₂²×D₄×C₂₃	—	2	1	no	—
70	C₂²×D₄×C₂₃	—	2	2	no	—
71	C₂²×D₄×C₂₃	—	2	4	no	—
72	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
73	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
74	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
75	C₂³×C₄×C₂₃	—	2	1	no	—
76	C₂³×C₄×C₂₃	—	2	16	no	—
77	C₂³×C₄×C₂₃	—	2	2	no	—
78	C₂³×C₄×C₂₃	—	2	8	no	—
79	C₂³×C₄×C₂₃	—	2	2	no	—
80	C₂³×C₄×C₂₃	—	2	2	no	—
81	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
82	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
83	C₂³×C₄×C₂₃	D₄	1	4	yes	—
84	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
85	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂³×C₄×C₂₃	D₄	1	8	yes	—
87	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
90	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×M₁₆×C₂₃	—	2	8	no	—
92	C₂×M₁₆×C₂₃	—	2	8	no	—
93	C₂×M₁₆×C₂₃	—	2	32	no	—
94	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
95	C₂×M₁₆×C₂₃	C₂³	1	8	yes	complemented
96	C₂²×C₈×C₂₃	—	2	8	no	—
97	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
98	C₂²×C₈×C₂₃	C₂³	1	8	yes	complemented
99	C₂₃×#117	—	2	4	no	—
100	C₂₃×#117	—	2	8	no	—
101	C₂₃×#117	—	2	8	no	—
102	C₂₃×#117	—	2	4	no	—
103	C₂₃×#117	C₂³	1	8	yes	—
104	C₂₃×#117	C₂³	1	8	yes	complemented
105	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
106	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	complemented
109	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
111	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
112	C₂×C₄²×C₂₃	—	2	2	no	—
113	C₂×C₄²×C₂₃	—	2	8	no	—
114	C₂×C₄²×C₂₃	—	2	2	no	—
115	C₂₃×#103	C₂³	1	32	yes	complemented
116	C₂₃×#102	C₂³	1	32	yes	complemented
117	C₂⁴×C₂₃	—	2	8	no	—
118	C₂⁴×C₂₃	—	2	2	no	—
119	C₂⁴×C₂₃	—	2	4	no	—
120	C₂⁴×C₂₃	—	4	4	no	—
121	C₂⁴×C₂₃	—	4	8	no	—
122	C₂⁴×C₂₃	C₂²⋊C₄	1	8	yes	—
123	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
127	#41×C₂₃	—	2	32	no	—
128	#41×C₂₃	—	2	64	no	—
129	#41×C₂₃	—	2	64	no	—
130	#41×C₂₃	—	2	32	no	—
131	#41×C₂₃	—	2	16	no	—
132	#41×C₂₃	—	2	16	no	—
133	#41×C₂₃	C₂²×C₄	1	32	yes	—
134	#41×C₂₃	C₂²×C₄	1	32	yes	complemented
135	C₂×Q₈×C₂₃	—	2	8	no	—
136	C₂×Q₈×C₂₃	—	2	16	no	—
137	C₂×Q₈×C₂₃	—	2	4	no	—
138	C₂×Q₈×C₂₃	—	2	8	no	—
139	C₂×Q₈×C₂₃	—	2	4	no	—
140	C₂×Q₈×C₂₃	—	2	2	no	—
141	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
142	C₂×Q₈×C₂₃	C₂²×C₄	1	16	yes	complemented
143	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
144	C₂×D₄×C₂₃	—	2	16	no	—
145	C₂×D₄×C₂₃	—	2	16	no	—
146	C₂×D₄×C₂₃	—	2	64	no	—
147	C₂×D₄×C₂₃	—	2	8	no	—
148	C₂×D₄×C₂₃	—	2	8	no	—
149	C₂×D₄×C₂₃	—	2	16	no	—
150	C₂×D₄×C₂₃	—	2	8	no	—
151	C₂×D₄×C₂₃	—	2	4	no	—
152	C₂×D₄×C₂₃	—	2	2	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	2	8	no	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
156	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	complemented
157	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₂₃	—	2	8	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	16	no	—
162	C₂²×C₄×C₂₃	—	2	16	no	—
163	C₂²×C₄×C₂₃	—	2	64	no	—
164	C₂²×C₄×C₂₃	—	2	8	no	—
165	C₂²×C₄×C₂₃	—	2	8	no	—
166	C₂²×C₄×C₂₃	—	2	4	no	—
167	C₂²×C₄×C₂₃	—	2	8	no	—
168	C₂²×C₄×C₂₃	—	4	8	no	—
169	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
170	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
171	C₂²×C₄×C₂₃	C₂²⋊C₄	1	32	yes	—
172	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
173	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
174	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
176	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
186	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
188	C₂²×C₄×C₂₃	C₂×D₄	1	16	yes	—
189	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
194	M₁₆×C₂₃	—	2	16	no	—
195	M₁₆×C₂₃	—	2	16	no	—
196	C₂×C₈×C₂₃	—	2	8	no	—
197	C₂×C₈×C₂₃	—	2	16	no	—
198	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	—	2	8	no	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
202	C₂²⋊C₄×C₂₃	—	2	8	no	—
203	C₂²⋊C₄×C₂₃	—	2	8	no	—
204	C₄²×C₂₃	—	2	8	no	—
205	C₄²×C₂₃	—	2	8	no	—
206	C₂×#2896	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂³×C₂₃	—	2	4	no	—
208	C₂³×C₂₃	—	2	4	no	—
209	C₂³×C₂₃	—	2	4	no	—
210	C₂³×C₂₃	—	2	8	no	—
211	C₂³×C₂₃	—	2	16	no	—
212	C₂³×C₂₃	—	2	4	no	—
213	C₂³×C₂₃	—	4	16	no	—
214	C₂³×C₂₃	—	4	8	no	—
215	C₂³×C₂₃	—	4	32	no	—
216	C₂³×C₂₃	—	4	16	no	—
217	C₂³×C₂₃	—	4	8	no	—
218	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
219	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
220	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
221	C₂³×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
222	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
223	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
224	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₂₃	—	2	8	no	—
227	Q₈×C₂₃	—	2	16	no	—
228	Q₈×C₂₃	—	2	4	no	—
229	Q₈×C₂₃	C₂³×C₄	1	8	yes	—
230	D₄×C₂₃	—	2	32	no	—
231	D₄×C₂₃	—	2	32	no	—
232	D₄×C₂₃	—	2	8	no	—
233	D₄×C₂₃	—	2	16	no	—
234	D₄×C₂₃	—	2	4	no	—
235	D₄×C₂₃	C₂³×C₄	1	8	yes	—
236	C₂×C₄×C₂₃	—	2	8	no	—
237	C₂×C₄×C₂₃	—	2	16	no	—
238	C₂×C₄×C₂₃	—	2	4	no	—
239	C₂×C₄×C₂₃	—	2	32	no	—
240	C₂×C₄×C₂₃	—	2	32	no	—
241	C₂×C₄×C₂₃	—	2	8	no	—
242	C₂×C₄×C₂₃	—	4	16	no	—
243	C₂×C₄×C₂₃	—	4	8	no	—
244	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
245	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
246	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
247	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
248	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
249	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
250	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
251	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
252	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	16	yes	—
253	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
254	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
255	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
256	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
265	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
266	C₈×C₂₃	—	2	8	no	—
267	C₂³×#41	C₂×C₂₃	1	1	yes	characteristic
268	C₂³×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
269	C₂²×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#2896	C₂×C₂₃	1	16	yes	direct product factor
271	C₂×#417	C₂×C₂₃	1	4	yes	complemented
272	C₂×#419	C₂×C₂₃	1	2	yes	complemented
273	C₂×#418	C₂×C₂₃	1	2	yes	complemented
274	C₂×#416	C₂×C₂₃	1	4	yes	complemented
275	C₂²×C₂₃	—	2	8	no	—
276	C₂²×C₂₃	—	2	8	no	—
277	C₂²×C₂₃	—	2	8	no	—
278	C₂²×C₂₃	—	2	4	no	—
279	C₂²×C₂₃	—	2	4	no	—
280	C₂²×C₂₃	—	2	4	no	—
281	C₂²×C₂₃	—	2	8	no	—
282	C₂²×C₂₃	—	4	16	no	—
283	C₂²×C₂₃	—	4	32	no	—
284	C₂²×C₂₃	—	4	16	no	—
285	C₂²×C₂₃	—	4	8	no	—
286	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
287	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
288	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
289	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
290	C₂²×C₂₃	#575	1	4	yes	—
291	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
292	C₄×C₂₃	—	2	8	no	—
293	C₄×C₂₃	—	4	8	no	—
294	C₄×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
295	C₄×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
296	C₄×C₂₃	C₂²×C₂²⋊C₄	1	2	yes	—
297	C₄×C₂₃	C₂²×C₂²⋊C₄	1	4	yes	—
298	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
299	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
300	C₂²×#41	—	2	8	no	—
301	C₂²×#41	—	2	1	no	—
302	C₂²×#41	—	2	4	no	—
303	C₂²×#41	—	2	1	no	—
304	C₂²×#41	C₄×C₂₃	1	16	yes	complemented
305	C₂²×#41	C₂²×C₂₃	1	2	yes	—
306	C₂²×#41	C₂²×C₂₃	1	2	yes	—
307	C₂²×#41	C₂²×C₂₃	1	8	yes	—
308	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
309	C₂³×D₄	—	2	1	no	—
310	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
311	C₂⁴×C₄	—	2	1	no	—
312	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
313	C₂²×M₁₆	—	2	8	no	—
314	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
315	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	complemented
316	C₂²×M₁₆	C₂²×C₂₃	1	8	yes	complemented
317	C₂³×C₈	C₂²×C₂₃	1	2	yes	complemented
318	C₂×#117	—	2	1	no	—
319	C₂×#117	—	2	1	no	—
320	C₂×#117	—	2	4	no	—
321	C₂×#117	C₂²×C₂₃	1	2	yes	—
322	C₂×#117	C₂²×C₂₃	1	2	yes	complemented
323	C₂×#117	C₂²×C₂₃	1	8	yes	complemented
324	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
325	C₂²×C₂²⋊C₄	—	2	1	no	—
326	C₂²×C₄²	—	2	1	no	—
327	#419	C₂²×C₂₃	1	16	yes	complemented
328	#418	C₂²×C₂₃	1	16	yes	complemented
329	#417	C₂²×C₂₃	1	32	yes	complemented
330	#416	C₂²×C₂₃	1	32	yes	complemented
331	C₂×#103	C₂²×C₂₃	1	8	yes	complemented
332	C₂×#102	C₂²×C₂₃	1	8	yes	complemented
333	C₂×C₂₃	—	2	4	no	—
334	C₂×C₂₃	—	2	4	no	—
335	C₂×C₂₃	—	2	4	no	—
336	C₂×C₂₃	—	4	8	no	—
337	C₂×C₂₃	#2896	1	4	yes	direct product factor
338	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
339	C₂×C₂₃	C₂×#575	1	2	yes	—
340	C₂⁵	—	2	1	no	—
341	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
342	C₂×#41	—	2	16	no	—
343	C₂×#41	—	2	16	no	—
344	C₂×#41	—	2	64	no	—
345	C₂×#41	—	2	8	no	—
346	C₂×#41	—	2	8	no	—
347	C₂×#41	—	2	32	no	—
348	C₂×#41	—	2	8	no	—
349	C₂×#41	—	2	4	no	—
350	C₂×#41	—	2	4	no	—
351	C₂×#41	—	2	8	no	—
352	C₂×#41	C₂×C₄×C₂₃	1	16	yes	—
353	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
354	C₂×#41	C₂×C₄×C₂₃	1	64	yes	complemented
355	C₂×#41	C₂³×C₂₃	1	8	yes	—
356	C₂×#41	C₂³×C₂₃	1	8	yes	—
357	C₂²×Q₈	—	2	4	no	—
358	C₂²×Q₈	—	2	1	no	—
359	C₂²×Q₈	—	2	2	no	—
360	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
361	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
362	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
363	C₂²×D₄	—	2	16	no	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	8	no	—
367	C₂²×D₄	—	2	1	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	—	2	4	no	—
370	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
371	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
372	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
373	C₂³×C₄	—	2	1	no	—
374	C₂³×C₄	—	2	16	no	—
375	C₂³×C₄	—	2	2	no	—
376	C₂³×C₄	—	2	2	no	—
377	C₂³×C₄	—	2	8	no	—
378	C₂³×C₄	—	2	2	no	—
379	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
380	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
381	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
382	C₂³×C₄	D₄×C₂₃	1	4	yes	—
383	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
384	C₂³×C₄	D₄×C₂₃	1	8	yes	—
385	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
389	C₂×M₁₆	—	2	8	no	—
390	C₂×M₁₆	—	2	8	no	—
391	C₂×M₁₆	—	2	32	no	—
392	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
393	C₂×M₁₆	C₂³×C₂₃	1	8	yes	complemented
394	C₂²×C₈	—	2	8	no	—
395	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
396	C₂²×C₈	C₂³×C₂₃	1	8	yes	complemented
397	#117	—	2	4	no	—
398	#117	—	2	8	no	—
399	#117	—	2	8	no	—
400	#117	—	2	4	no	—
401	#117	C₂³×C₂₃	1	8	yes	—
402	#117	C₂³×C₂₃	1	8	yes	complemented
403	C₂×C₄⋊C₄	—	2	4	no	—
404	C₂×C₄⋊C₄	—	2	2	no	—
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	complemented
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	8	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₄²	—	2	2	no	—
411	C₂×C₄²	—	2	8	no	—
412	C₂×C₄²	—	2	2	no	—
413	#103	C₂³×C₂₃	1	32	yes	complemented
414	#102	C₂³×C₂₃	1	32	yes	complemented
415	C₂₃	C₂×#2896	1	1	yes	characteristic, direct product factor, Sylow
416	C₂⁴	—	2	8	no	—
417	C₂⁴	—	2	2	no	—
418	C₂⁴	—	2	4	no	—
419	C₂⁴	—	4	4	no	—
420	C₂⁴	—	4	8	no	—
421	C₂⁴	C₂²⋊C₄×C₂₃	1	8	yes	—
422	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
423	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
424	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
425	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
426	#41	—	2	32	no	—
427	#41	—	2	64	no	—
428	#41	—	2	64	no	—
429	#41	—	2	32	no	—
430	#41	—	2	16	no	—
431	#41	—	2	16	no	—
432	#41	C₂²×C₄×C₂₃	1	32	yes	—
433	#41	C₂²×C₄×C₂₃	1	32	yes	complemented
434	C₂×Q₈	—	2	8	no	—
435	C₂×Q₈	—	2	16	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	8	no	—
440	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
441	C₂×Q₈	C₂²×C₄×C₂₃	1	16	yes	complemented
442	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
443	C₂×D₄	—	2	16	no	—
444	C₂×D₄	—	2	16	no	—
445	C₂×D₄	—	2	64	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	8	no	—
448	C₂×D₄	—	2	16	no	—
449	C₂×D₄	—	2	8	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
455	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	complemented
456	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
457	C₂²×C₄	—	2	8	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	16	no	—
461	C₂²×C₄	—	2	16	no	—
462	C₂²×C₄	—	2	64	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	4	8	no	—
468	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
469	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
470	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
471	C₂²×C₄	C₂²⋊C₄×C₂₃	1	32	yes	—
472	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
474	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
475	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
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	4	yes	—
482	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
484	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
485	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
486	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
488	C₂²×C₄	C₂×D₄×C₂₃	1	16	yes	—
489	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
493	M₁₆	—	2	16	no	—
494	M₁₆	—	2	16	no	—
495	C₂×C₈	—	2	8	no	—
496	C₂×C₈	—	2	16	no	—
497	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	8	no	—
500	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
501	C₂²⋊C₄	—	2	8	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	8	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	8	no	—
510	C₂³	—	2	16	no	—
511	C₂³	—	4	16	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	16	no	—
514	C₂³	—	4	32	no	—
515	C₂³	—	4	8	no	—
516	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
517	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
518	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
519	C₂³	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
520	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
521	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
522	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
523	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
524	Q₈	—	2	4	no	—
525	Q₈	—	2	8	no	—
526	Q₈	—	2	16	no	—
527	Q₈	C₂³×C₄×C₂₃	1	8	yes	—
528	D₄	—	2	32	no	—
529	D₄	—	2	32	no	—
530	D₄	—	2	4	no	—
531	D₄	—	2	8	no	—
532	D₄	—	2	16	no	—
533	D₄	C₂³×C₄×C₂₃	1	8	yes	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	8	no	—
536	C₂×C₄	—	2	16	no	—
537	C₂×C₄	—	2	32	no	—
538	C₂×C₄	—	2	32	no	—
539	C₂×C₄	—	2	8	no	—
540	C₂×C₄	—	4	16	no	—
541	C₂×C₄	—	4	8	no	—
542	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
543	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
544	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
545	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
546	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
547	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
548	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
549	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
550	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	16	yes	—
551	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
552	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
553	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
554	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
555	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
557	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
558	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
559	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
561	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
564	C₈	—	2	8	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	4	no	—
568	C₂²	—	2	8	no	—
569	C₂²	—	2	8	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	8	no	—
572	C₂²	—	4	16	no	—
573	C₂²	—	4	32	no	—
574	C₂²	—	4	8	no	—
575	C₂²	—	4	16	no	—
576	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
577	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
578	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
579	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
580	C₂²	C₂₃×#575	1	4	yes	—
581	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
582	C₄	—	2	8	no	—
583	C₄	—	4	8	no	—
584	C₄	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
585	C₄	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
586	C₄	C₂²×C₂²⋊C₄×C₂₃	1	2	yes	—
587	C₄	C₂²×C₂²⋊C₄×C₂₃	1	4	yes	—
588	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
589	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
590	C₂	—	2	4	no	—
591	C₂	—	2	4	no	—
592	C₂	—	2	4	no	—
593	C₂	—	4	8	no	—
594	C₂	C₂₃×#2896	1	4	yes	direct product factor
595	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
596	C₂	C₂×C₂₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂³×#41×C₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₂₃	C₂	semidirect
3. C₂²×C₂₃×#117	C₂	semidirect
4. C₂₃×#2896	C₂	direct (N × H)
5. C₂×C₂₃×#417	C₂	semidirect
6. C₂×C₂₃×#419	C₂	semidirect
7. C₂×C₂₃×#418	C₂	semidirect
8. C₂×C₂₃×#416	C₂	semidirect
13. C₂²×#41×C₂₃	C₄	semidirect
24. C₂²×M₁₆×C₂₃	C₂²	semidirect
25. C₂²×M₁₆×C₂₃	C₂²	semidirect
26. C₂³×C₈×C₂₃	C₂²	semidirect
31. C₂×C₂₃×#117	C₂²	semidirect
32. C₂×C₂₃×#117	C₂²	semidirect
33. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
36. C₂₃×#419	C₂²	semidirect
37. C₂₃×#418	C₂²	semidirect
38. C₂₃×#417	C₂²	semidirect
39. C₂₃×#416	C₂²	semidirect
40. C₂×C₂₃×#103	C₂²	semidirect
41. C₂×C₂₃×#102	C₂²	semidirect
55. C₂×#41×C₂₃	C₂×C₄	semidirect
56. C₂×#41×C₂₃	C₂×C₄	semidirect
62. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
72. C₂²×D₄×C₂₃	C₂×C₄	semidirect
95. C₂×M₁₆×C₂₃	C₂³	semidirect
98. C₂²×C₈×C₂₃	C₂³	semidirect
104. C₂₃×#117	C₂³	semidirect
108. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
115. C₂₃×#103	C₂³	semidirect
116. C₂₃×#102	C₂³	semidirect
134. #41×C₂₃	C₂²×C₄	semidirect
142. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
156. C₂×D₄×C₂₃	C₂²×C₄	semidirect
206. C₂×#2896	C₂₃	direct (N × H)
268. C₂³×M₁₆	C₂×C₂₃	semidirect
269. C₂²×#117	C₂×C₂₃	semidirect
270. #2896	C₂×C₂₃	direct (N × H)
271. C₂×#417	C₂×C₂₃	semidirect
272. C₂×#419	C₂×C₂₃	semidirect
273. C₂×#418	C₂×C₂₃	semidirect
274. C₂×#416	C₂×C₂₃	semidirect
304. C₂²×#41	C₄×C₂₃	semidirect
315. C₂²×M₁₆	C₂²×C₂₃	semidirect
316. C₂²×M₁₆	C₂²×C₂₃	semidirect
317. C₂³×C₈	C₂²×C₂₃	semidirect
322. C₂×#117	C₂²×C₂₃	semidirect
323. C₂×#117	C₂²×C₂₃	semidirect
324. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
327. #419	C₂²×C₂₃	semidirect
328. #418	C₂²×C₂₃	semidirect
329. #417	C₂²×C₂₃	semidirect
330. #416	C₂²×C₂₃	semidirect
331. C₂×#103	C₂²×C₂₃	semidirect
332. C₂×#102	C₂²×C₂₃	semidirect
337. C₂×C₂₃	#2896	direct (N × H)
353. C₂×#41	C₂×C₄×C₂₃	semidirect
354. C₂×#41	C₂×C₄×C₂₃	semidirect
360. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
370. C₂²×D₄	C₂×C₄×C₂₃	semidirect
393. C₂×M₁₆	C₂³×C₂₃	semidirect
396. C₂²×C₈	C₂³×C₂₃	semidirect
402. #117	C₂³×C₂₃	semidirect
406. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
413. #103	C₂³×C₂₃	semidirect
414. #102	C₂³×C₂₃	semidirect
415. C₂₃	C₂×#2896	direct (N × H)
433. #41	C₂²×C₄×C₂₃	semidirect
441. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
455. C₂×D₄	C₂²×C₄×C₂₃	semidirect
594. C₂	C₂₃×#2896	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#41×C₂₃	2
C₂³×M₁₆×C₂₃	2
C₂²×C₂₃×#117	2
C₂₃×#2896	2
C₂×C₂₃×#417	2
C₂×C₂₃×#419	2
C₂×C₂₃×#418	2
C₂×C₂₃×#416	2
C₂×#2896	23

Subgroups — C2×C23×#2896 #19207709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2896 #19207709