Subgroups — C₂×C₂₃×#1348 #19159328

Order 5888.

506 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#340	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1348	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
8	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂²×C₁₆×C₂₃	—	2	1	no	—
10	C₂₃×#467	—	2	1	no	—
11	C₂₃×#467	—	2	1	no	—
12	C₂₃×#467	C₄	1	2	yes	complemented
13	C₂₃×#467	C₂²	1	2	yes	—
14	C₂×C₂₃×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₂₃×#103	—	2	1	no	—
16	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
18	C₂₃×#340	—	2	1	no	—
19	C₂₃×#340	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	—	4	1	no	—
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂×Q₁₆×C₂₃	—	2	1	no	—
24	C₂×Q₁₆×C₂₃	—	4	2	no	—
25	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
26	C₂²×C₈×C₂₃	—	4	2	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	D₄	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	C₂³	1	1	yes	characteristic
33	C₂₃×C₂²⋊Q₈	—	2	2	no	—
34	C₂₃×C₂²⋊Q₈	—	2	2	no	—
35	C₂₃×C₂²⋊Q₈	—	4	1	no	—
36	C₂₃×C₂²⋊Q₈	—	4	1	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	4	1	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×#103	—	2	2	no	—
49	C₂₃×#103	—	2	2	no	—
50	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
51	C₂×Q₈×C₂₃	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	4	1	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	4	4	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	—	8	2	no	—
62	C₂²×C₄×C₂₃	—	8	2	no	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	Q₁₆×C₂₃	—	4	2	no	—
70	Q₁₆×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	4	1	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₄⋊C₄×C₂₃	—	4	4	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	1	no	—
86	C₄⋊C₄×C₂₃	—	4	2	no	—
87	C₄⋊C₄×C₂₃	—	4	2	no	—
88	C₂²⋊C₄×C₂₃	—	4	2	no	—
89	C₂²⋊C₄×C₂₃	—	4	2	no	—
90	C₁₆×C₂₃	—	4	2	no	—
91	C₂×#1348	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	#102	1	1	yes	characteristic
95	C₂³×C₂₃	#102	1	1	yes	characteristic
96	C₂³×C₂₃	SD₃₂	1	2	yes	—
97	C₂³×C₂₃	Q₃₂	1	2	yes	—
98	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₂₃	—	4	2	no	—
100	Q₈×C₂₃	—	4	2	no	—
101	Q₈×C₂₃	—	8	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	4	4	no	—
110	C₂×C₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	4	2	no	—
112	C₂×C₄×C₂₃	—	8	8	no	—
113	C₂×C₄×C₂₃	—	8	2	no	—
114	C₂×C₄×C₂₃	—	8	2	no	—
115	C₂×C₄×C₂₃	—	8	2	no	—
116	C₂×C₄×C₂₃	—	8	2	no	—
117	C₂×C₄×C₂₃	#99	1	2	yes	—
118	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#104	1	2	yes	—
121	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₈×C₂₃	—	2	1	no	—
124	C₈×C₂₃	—	2	1	no	—
125	C₈×C₂₃	—	2	1	no	—
126	C₈×C₂₃	—	2	1	no	—
127	C₂×#467	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
129	C₂×#340	C₂×C₂₃	1	1	yes	characteristic
130	#1348	C₂×C₂₃	1	4	yes	direct product factor
131	C₂²×C₂₃	—	2	4	no	—
132	C₂²×C₂₃	—	2	4	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	#327	1	2	yes	—
138	C₂²×C₂₃	#358	1	2	yes	—
139	C₂²×C₂₃	#358	1	2	yes	—
140	C₂²×C₂₃	#360	1	2	yes	—
141	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
143	C₂²×C₂₃	C₂×Q₃₂	1	1	yes	characteristic
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	1	no	—
146	C₄×C₂₃	—	8	4	no	—
147	C₄×C₂₃	—	8	2	no	—
148	C₄×C₂₃	—	8	2	no	—
149	C₄×C₂₃	#327	1	2	yes	—
150	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
151	C₄×C₂₃	C₂×#104	1	1	yes	characteristic
152	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
153	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₂²⋊Q₈	—	2	1	no	—
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₁₆	—	2	1	no	—
157	#467	—	2	1	no	—
158	#467	—	2	1	no	—
159	#467	C₄×C₂₃	1	2	yes	complemented
160	#467	C₂²×C₂₃	1	2	yes	—
161	C₂×#107	C₄×C₂₃	1	1	yes	characteristic, complemented
162	C₂×#103	—	2	1	no	—
163	C₂²⋊C₁₆	—	2	1	no	—
164	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
165	#340	—	2	1	no	—
166	#340	C₂²×C₂₃	1	2	yes	—
167	C₂×C₂₃	—	2	1	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	#1348	1	4	yes	direct product factor
171	C₂×C₂₃	C₂×#327	1	1	yes	characteristic
172	C₂×C₂₃	C₂×#358	1	1	yes	characteristic
173	C₂×C₂₃	C₂×#360	1	1	yes	characteristic
174	C₂²×Q₈	—	2	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
177	C₂×Q₁₆	—	2	1	no	—
178	C₂×Q₁₆	—	4	2	no	—
179	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
185	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	C₂²⋊Q₈	—	2	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₂²⋊Q₈	—	4	1	no	—
190	C₂²⋊Q₈	—	4	1	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	4	2	no	—
195	C₂×C₄⋊C₄	—	4	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	4	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂×C₄×C₂₃	1	2	yes	complemented
202	#103	—	2	2	no	—
203	#103	—	2	2	no	—
204	C₂₃	C₂×#1348	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	2	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	8	2	no	—
217	C₂²×C₄	—	8	2	no	—
218	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	SD₁₆×C₂₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
224	Q₁₆	—	4	2	no	—
225	Q₁₆	—	4	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₁₆	—	4	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂₃×#102	1	1	yes	characteristic
249	C₂³	C₂₃×#102	1	1	yes	characteristic
250	C₂³	C₂₃×SD₃₂	1	2	yes	—
251	C₂³	C₂₃×Q₃₂	1	2	yes	—
252	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	8	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	8	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	C₂₃×#99	1	2	yes	—
272	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
273	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
274	C₂×C₄	C₂₃×#104	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	C₂₃×#327	1	2	yes	—
288	C₂²	C₂₃×#358	1	2	yes	—
289	C₂²	C₂₃×#358	1	2	yes	—
290	C₂²	C₂₃×#360	1	2	yes	—
291	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
292	C₂²	C₂×C₂₃×SD₃₂	1	1	yes	characteristic
293	C₂²	C₂×C₂₃×Q₃₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	4	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₂₃×#327	1	2	yes	—
300	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
301	C₄	C₂×C₂₃×#104	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	C₂₃×#1348	1	4	yes	direct product factor
306	C₂	C₂×C₂₃×#327	1	1	yes	characteristic
307	C₂	C₂×C₂₃×#358	1	1	yes	characteristic
308	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	C₂⁴×C₂₃	D₈
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
4. C₂₃×#1348	C₂	direct (N × H)
5. C₂²×Q₁₆×C₂₃	C₄	semidirect
12. C₂₃×#467	C₄	semidirect
14. C₂×C₂₃×#107	C₄	semidirect
25. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
47. C₂₃×#107	C₂×C₄	semidirect
91. C₂×#1348	C₂₃	direct (N × H)
130. #1348	C₂×C₂₃	direct (N × H)
152. C₂²×Q₁₆	C₄×C₂₃	semidirect
159. #467	C₄×C₂₃	semidirect
161. C₂×#107	C₄×C₂₃	semidirect
170. C₂×C₂₃	#1348	direct (N × H)
179. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
201. #107	C₂×C₄×C₂₃	semidirect
204. C₂₃	C₂×#1348	direct (N × H)
305. C₂	C₂₃×#1348	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×#467	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂×C₂₃×#340	2
C₂₃×#1348	2
C₂×#1348	23

Subgroups — C2×C23×#1348 #19159328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#1348 #19159328