Subgroups — C₂×C₂₃×#1297 #19155673

Order 5888.

1226 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#407	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
3	C₂₃×#1297	C₂	1	4	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₂₃	C₄	1	2	yes	complemented
6	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
7	C₂²×C₄²×C₂₃	—	2	2	no	—
8	C₂₃×#407	—	2	2	no	—
9	C₂₃×#407	C₄	1	2	yes	complemented
10	C₂₃×#407	C₂²	1	2	yes	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	2	no	—
12	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
13	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	complemented
15	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
16	C₂×C₂₃×#95	C₄	1	4	yes	complemented
17	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	2	4	no	—
19	C₂³×C₄×C₂₃	—	4	4	no	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₂₃	D₄	1	2	yes	—
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₃	D₄	1	2	yes	—
27	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	—	4	4	no	—
32	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
33	C₂²×C₈×C₂₃	—	2	2	no	—
34	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄²×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	—	2	4	no	—
39	C₂×C₄²×C₂₃	—	4	8	no	—
40	C₂₃×C₂²⋊C₈	—	2	4	no	—
41	C₂₃×C₂²⋊C₈	—	2	4	no	—
42	C₂₃×#95	—	2	4	no	—
43	C₂₃×#95	C₂×C₄	1	8	yes	complemented
44	C₂⁴×C₂₃	—	2	8	no	—
45	C₂⁴×C₂₃	—	4	2	no	—
46	C₂⁴×C₂₃	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	—	2	8	no	—
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	2	16	no	—
54	C₂²×C₄×C₂₃	—	2	8	no	—
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	—	4	4	no	—
59	C₂²×C₄×C₂₃	—	4	2	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	—	4	4	no	—
62	C₂²×C₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	4	8	no	—
64	C₂²×C₄×C₂₃	—	4	8	no	—
65	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
72	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
83	M₁₆×C₂₃	—	4	4	no	—
84	M₁₆×C₂₃	—	4	4	no	—
85	C₂×C₈×C₂₃	—	2	4	no	—
86	C₂×C₈×C₂₃	—	4	4	no	—
87	C₂²⋊C₄×C₂₃	—	2	16	no	—
88	C₂²⋊C₄×C₂₃	—	2	16	no	—
89	C₄²×C₂₃	—	4	8	no	—
90	C₄²×C₂₃	—	4	8	no	—
91	C₂×#1297	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	4	no	—
93	C₂³×C₂₃	—	2	4	no	—
94	C₂³×C₂₃	—	2	4	no	—
95	C₂³×C₂₃	—	2	4	no	—
96	C₂³×C₂₃	—	2	4	no	—
97	C₂³×C₂₃	—	2	2	no	—
98	C₂³×C₂₃	—	2	4	no	—
99	C₂³×C₂₃	—	2	4	no	—
100	C₂³×C₂₃	—	4	2	no	—
101	C₂³×C₂₃	—	4	1	no	—
102	C₂³×C₂₃	—	4	1	no	—
103	C₂³×C₂₃	—	4	16	no	—
104	C₂³×C₂₃	#95	1	1	yes	characteristic
105	C₂³×C₂₃	#95	1	2	yes	—
106	C₂³×C₂₃	#95	1	1	yes	characteristic
107	C₂³×C₂₃	#104	1	8	yes	—
108	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
109	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	16	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	1	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	8	no	—
129	C₂×C₄×C₂₃	—	4	32	no	—
130	C₂×C₄×C₂₃	—	4	8	no	—
131	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
132	C₂×C₄×C₂₃	#95	1	2	yes	—
133	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
134	C₂×C₄×C₂₃	#99	1	4	yes	—
135	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₂₃	—	4	4	no	—
139	C₂×#407	C₂×C₂₃	1	1	yes	characteristic
140	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
141	#1297	C₂×C₂₃	1	4	yes	direct product factor
142	C₂²×C₂₃	—	2	16	no	—
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	2	2	no	—
146	C₂²×C₂₃	—	2	2	no	—
147	C₂²×C₂₃	—	2	2	no	—
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	2	1	no	—
150	C₂²×C₂₃	—	2	1	no	—
151	C₂²×C₂₃	—	4	4	no	—
152	C₂²×C₂₃	—	4	8	no	—
153	C₂²×C₂₃	—	4	4	no	—
154	C₂²×C₂₃	—	4	4	no	—
155	C₂²×C₂₃	#339	1	4	yes	—
156	C₂²×C₂₃	#339	1	4	yes	—
157	C₂²×C₂₃	#342	1	2	yes	—
158	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
159	C₂²×C₂₃	C₂×#104	1	4	yes	—
160	C₄×C₂₃	—	2	1	no	—
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	2	2	no	—
164	C₄×C₂₃	—	4	16	no	—
165	C₄×C₂₃	#342	1	2	yes	—
166	C₄×C₂₃	C₂×#99	1	2	yes	—
167	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
168	C₂²×M₁₆	C₄×C₂₃	1	2	yes	complemented
169	C₂²×C₂²⋊C₄	—	2	2	no	—
170	C₂²×C₄²	—	2	2	no	—
171	#407	—	2	2	no	—
172	#407	C₄×C₂₃	1	2	yes	complemented
173	#407	C₂²×C₂₃	1	2	yes	—
174	C₂×C₂²⋊C₈	—	2	2	no	—
175	C₄×C₂²⋊C₄	—	2	2	no	—
176	C₄×C₂²⋊C₄	—	2	2	no	—
177	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	complemented
178	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
179	C₂×#95	C₄×C₂₃	1	4	yes	complemented
180	C₂×C₂₃	—	2	4	no	—
181	C₂×C₂₃	—	2	4	no	—
182	C₂×C₂₃	—	4	1	no	—
183	C₂×C₂₃	—	4	1	no	—
184	C₂×C₂₃	#1297	1	4	yes	direct product factor
185	C₂×C₂₃	C₂×#339	1	2	yes	—
186	C₂×C₂₃	C₂×#342	1	1	yes	characteristic
187	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
188	C₂³×C₄	—	2	4	no	—
189	C₂³×C₄	—	4	4	no	—
190	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
193	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
194	C₂³×C₄	D₄×C₂₃	1	2	yes	—
195	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
196	C₂³×C₄	D₄×C₂₃	1	2	yes	—
197	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
198	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	4	4	no	—
202	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
203	C₂²×C₈	—	2	2	no	—
204	C₂×C₂²⋊C₄	—	2	16	no	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	—	2	4	no	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	4	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₂²⋊C₈	—	2	4	no	—
212	#95	—	2	4	no	—
213	#95	C₂×C₄×C₂₃	1	8	yes	complemented
214	C₂₃	C₂×#1297	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	—	2	8	no	—
216	C₂⁴	—	4	2	no	—
217	C₂⁴	C₂²⋊C₄×C₂₃	1	2	yes	—
218	C₂⁴	C₂²⋊C₄×C₂₃	1	2	yes	—
219	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	8	no	—
235	C₂²×C₄	—	4	8	no	—
236	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
240	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
242	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
243	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
244	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
245	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
246	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
248	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
249	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
253	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
254	M₁₆	—	4	4	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	4	4	no	—
258	C₂²⋊C₄	—	2	16	no	—
259	C₂²⋊C₄	—	2	16	no	—
260	C₄²	—	4	8	no	—
261	C₄²	—	4	8	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	16	no	—
274	C₂³	C₂₃×#95	1	1	yes	characteristic
275	C₂³	C₂₃×#95	1	2	yes	—
276	C₂³	C₂₃×#95	1	1	yes	characteristic
277	C₂³	C₂₃×#104	1	8	yes	—
278	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
279	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	16	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	32	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
302	C₂×C₄	C₂₃×#95	1	2	yes	—
303	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
304	C₂×C₄	C₂₃×#99	1	4	yes	—
305	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
308	C₈	—	4	4	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	16	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	4	4	no	—
322	C₂²	C₂₃×#339	1	4	yes	—
323	C₂²	C₂₃×#339	1	4	yes	—
324	C₂²	C₂₃×#342	1	2	yes	—
325	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
326	C₂²	C₂×C₂₃×#104	1	4	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	16	no	—
332	C₄	C₂₃×#342	1	2	yes	—
333	C₄	C₂×C₂₃×#99	1	2	yes	—
334	C₂	—	2	4	no	—
335	C₂	—	2	4	no	—
336	C₂	—	4	1	no	—
337	C₂	—	4	1	no	—
338	C₂	C₂₃×#1297	1	4	yes	direct product factor
339	C₂	C₂×C₂₃×#339	1	2	yes	—
340	C₂	C₂×C₂₃×#342	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	#95
prime radical	C₂⁵×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
3. C₂₃×#1297	C₂	direct (N × H)
5. C₂²×M₁₆×C₂₃	C₄	semidirect
9. C₂₃×#407	C₄	semidirect
14. C₄×C₂²⋊C₄×C₂₃	C₄	semidirect
16. C₂×C₂₃×#95	C₄	semidirect
32. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
43. C₂₃×#95	C₂×C₄	semidirect
91. C₂×#1297	C₂₃	direct (N × H)
141. #1297	C₂×C₂₃	direct (N × H)
168. C₂²×M₁₆	C₄×C₂₃	semidirect
172. #407	C₄×C₂₃	semidirect
177. C₄×C₂²⋊C₄	C₄×C₂₃	semidirect
179. C₂×#95	C₄×C₂₃	semidirect
184. C₂×C₂₃	#1297	direct (N × H)
202. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
213. #95	C₂×C₄×C₂₃	semidirect
214. C₂₃	C₂×#1297	direct (N × H)
338. C₂	C₂₃×#1297	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×#407	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₂₃×#1297	2
C₂×#1297	23

Subgroups — C2×C23×#1297 #19155673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1297 #19155673