Subgroups — C₂×C₂₃×#1347 #19159327

Order 5888.

558 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1347	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#121	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂²×C₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#466	—	2	1	no	—
11	C₂₃×#466	—	2	1	no	—
12	C₂₃×#466	C₄	1	2	yes	complemented
13	C₂₃×#466	C₂²	1	2	yes	—
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₂₃×#102	—	2	1	no	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
18	C₂₃×#340	—	2	1	no	—
19	C₂₃×#340	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₂₃	—	2	1	no	—
21	C₂²×D₄×C₂₃	—	4	1	no	—
22	C₂³×C₄×C₂₃	—	4	1	no	—
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂×D₈×C₂₃	—	2	1	no	—
25	C₂×D₈×C₂₃	—	4	2	no	—
26	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	complemented
27	C₂²×C₈×C₂₃	—	4	2	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#121	—	2	2	no	—
35	C₂₃×#121	—	2	2	no	—
36	C₂₃×#121	—	4	1	no	—
37	C₂₃×#121	—	4	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
44	C₂×C₁₆×C₂₃	—	2	2	no	—
45	C₂×C₁₆×C₂₃	—	4	2	no	—
46	C₂₃×#107	—	2	1	no	—
47	C₂₃×#107	C₂×C₄	1	2	yes	complemented
48	C₂₃×#102	—	2	2	no	—
49	C₂₃×#102	—	2	2	no	—
50	C₂⁴×C₂₃	—	4	1	no	—
51	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
52	C₂×D₄×C₂₃	—	2	2	no	—
53	C₂×D₄×C₂₃	—	4	2	no	—
54	C₂×D₄×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	4	2	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂×D₄×C₂₃	—	8	2	no	—
59	C₂×D₄×C₂₃	—	8	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	2	2	no	—
62	C₂²×C₄×C₂₃	—	4	4	no	—
63	C₂²×C₄×C₂₃	—	4	1	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	—	8	2	no	—
66	C₂²×C₄×C₂₃	—	8	2	no	—
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	D₈×C₂₃	—	4	2	no	—
74	D₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	1	no	—
80	C₂×C₈×C₂₃	—	4	1	no	—
81	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	4	4	no	—
88	C₄⋊C₄×C₂₃	—	4	2	no	—
89	C₄⋊C₄×C₂₃	—	4	1	no	—
90	C₂²⋊C₄×C₂₃	—	4	2	no	—
91	C₂²⋊C₄×C₂₃	—	4	2	no	—
92	C₁₆×C₂₃	—	4	2	no	—
93	C₂×#1347	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	2	2	no	—
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	—	8	2	no	—
98	C₂³×C₂₃	—	8	2	no	—
99	C₂³×C₂₃	—	8	2	no	—
100	C₂³×C₂₃	#102	1	1	yes	characteristic
101	C₂³×C₂₃	#102	1	1	yes	characteristic
102	C₂³×C₂₃	D₁₆	1	2	yes	—
103	C₂³×C₂₃	SD₃₂	1	2	yes	—
104	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
105	D₄×C₂₃	—	4	2	no	—
106	D₄×C₂₃	—	4	2	no	—
107	D₄×C₂₃	—	8	4	no	—
108	D₄×C₂₃	—	8	2	no	—
109	D₄×C₂₃	—	8	2	no	—
110	D₄×C₂₃	—	8	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	1	no	—
118	C₂×C₄×C₂₃	—	4	4	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	8	8	no	—
121	C₂×C₄×C₂₃	—	8	2	no	—
122	C₂×C₄×C₂₃	—	8	2	no	—
123	C₂×C₄×C₂₃	—	8	2	no	—
124	C₂×C₄×C₂₃	#99	1	2	yes	—
125	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
126	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
127	C₂×C₄×C₂₃	#104	1	2	yes	—
128	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₈×C₂₃	—	2	1	no	—
131	C₈×C₂₃	—	2	1	no	—
132	C₈×C₂₃	—	2	1	no	—
133	C₈×C₂₃	—	2	1	no	—
134	C₂×#466	C₂×C₂₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂×#340	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#1347	C₂×C₂₃	1	4	yes	direct product factor
138	C₂²×C₂₃	—	2	4	no	—
139	C₂²×C₂₃	—	2	4	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	8	2	no	—
145	C₂²×C₂₃	—	8	2	no	—
146	C₂²×C₂₃	—	8	2	no	—
147	C₂²×C₂₃	—	16	4	no	—
148	C₂²×C₂₃	#327	1	2	yes	—
149	C₂²×C₂₃	#357	1	2	yes	—
150	C₂²×C₂₃	#357	1	2	yes	—
151	C₂²×C₂₃	#360	1	2	yes	—
152	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
153	C₂²×C₂₃	C₂×D₁₆	1	1	yes	characteristic
154	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	8	4	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	#327	1	2	yes	—
160	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
161	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
162	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
163	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×#121	—	2	1	no	—
165	C₂²×C₄⋊C₄	—	2	1	no	—
166	C₂²×C₁₆	—	2	1	no	—
167	#466	—	2	1	no	—
168	#466	—	2	1	no	—
169	#466	C₄×C₂₃	1	2	yes	complemented
170	#466	C₂²×C₂₃	1	2	yes	—
171	C₂×#107	C₄×C₂₃	1	1	yes	characteristic, complemented
172	C₂×#102	—	2	1	no	—
173	C₂²⋊C₁₆	—	2	1	no	—
174	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
175	#340	—	2	1	no	—
176	#340	C₂²×C₂₃	1	2	yes	complemented
177	C₂×C₂₃	—	2	1	no	—
178	C₂×C₂₃	—	2	2	no	—
179	C₂×C₂₃	—	2	1	no	—
180	C₂×C₂₃	—	16	2	no	—
181	C₂×C₂₃	#1347	1	4	yes	direct product factor
182	C₂×C₂₃	C₂×#327	1	1	yes	characteristic
183	C₂×C₂₃	C₂×#357	1	1	yes	characteristic
184	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
185	C₂²×D₄	—	2	1	no	—
186	C₂²×D₄	—	4	1	no	—
187	C₂³×C₄	—	4	1	no	—
188	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	4	2	no	—
191	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	complemented
192	C₂²×C₈	—	4	2	no	—
193	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	#121	—	4	1	no	—
202	#121	—	4	1	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	4	1	no	—
208	C₂×C₂²⋊C₄	—	4	1	no	—
209	C₂×C₁₆	—	2	2	no	—
210	C₂×C₁₆	—	4	2	no	—
211	#107	—	2	1	no	—
212	#107	C₂×C₄×C₂₃	1	2	yes	complemented
213	#102	—	2	2	no	—
214	#102	—	2	2	no	—
215	C₂₃	C₂×#1347	1	1	yes	characteristic, direct product factor, Sylow
216	C₂⁴	—	4	1	no	—
217	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂×D₄	—	8	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	—	8	2	no	—
233	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	D₈	—	4	2	no	—
240	D₈	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₁₆	—	4	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	8	2	no	—
263	C₂³	—	8	2	no	—
264	C₂³	—	8	2	no	—
265	C₂³	C₂₃×#102	1	1	yes	characteristic
266	C₂³	C₂₃×#102	1	1	yes	characteristic
267	C₂³	C₂₃×D₁₆	1	2	yes	—
268	C₂³	C₂₃×SD₃₂	1	2	yes	—
269	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	2	no	—
274	D₄	—	8	2	no	—
275	D₄	—	8	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	8	8	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₂₃×#99	1	2	yes	—
290	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
291	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
292	C₂×C₄	C₂₃×#104	1	2	yes	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
294	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	4	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	16	4	no	—
309	C₂²	C₂₃×#327	1	2	yes	—
310	C₂²	C₂₃×#357	1	2	yes	—
311	C₂²	C₂₃×#357	1	2	yes	—
312	C₂²	C₂₃×#360	1	2	yes	—
313	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
314	C₂²	C₂×C₂₃×D₁₆	1	1	yes	characteristic
315	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	4	no	—
319	C₄	—	8	2	no	—
320	C₄	C₂₃×#327	1	2	yes	—
321	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
322	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	1	no	—
326	C₂	—	16	2	no	—
327	C₂	C₂₃×#1347	1	4	yes	direct product factor
328	C₂	C₂×C₂₃×#327	1	1	yes	characteristic
329	C₂	C₂×C₂₃×#357	1	1	yes	characteristic
330	C₂	C₂×C₂₃×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#102
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
3. C₂×C₂₃×#340	C₂	semidirect
4. C₂₃×#1347	C₂	direct (N × H)
5. C₂²×D₈×C₂₃	C₄	semidirect
12. C₂₃×#466	C₄	semidirect
14. C₂×C₂₃×#107	C₄	semidirect
17. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
19. C₂₃×#340	C₂²	semidirect
26. C₂×D₈×C₂₃	C₂×C₄	semidirect
47. C₂₃×#107	C₂×C₄	semidirect
93. C₂×#1347	C₂₃	direct (N × H)
135. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
136. C₂×#340	C₂×C₂₃	semidirect
137. #1347	C₂×C₂₃	direct (N × H)
162. C₂²×D₈	C₄×C₂₃	semidirect
169. #466	C₄×C₂₃	semidirect
171. C₂×#107	C₄×C₂₃	semidirect
174. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
176. #340	C₂²×C₂₃	semidirect
181. C₂×C₂₃	#1347	direct (N × H)
191. C₂×D₈	C₂×C₄×C₂₃	semidirect
212. #107	C₂×C₄×C₂₃	semidirect
215. C₂₃	C₂×#1347	direct (N × H)
327. C₂	C₂₃×#1347	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×#466	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂×C₂₃×#340	2
C₂₃×#1347	2
C₂×#1347	23

Subgroups — C2×C23×#1347 #19159327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1347 #19159327