Subgroups — C₂₃×#3013 #15670632

Order 2944.

712 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#537	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#452	C₂	1	2	yes	maximal, complemented
6	C₂₃×#451	C₂	1	2	yes	maximal, complemented
7	C₂₃×#449	C₂	1	2	yes	maximal, complemented
8	C₂₃×#448	C₂	1	2	yes	maximal, complemented
9	C₂₃×#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×2₋¹⁺⁴	—	2	4	no	—
13	C₂₃×2₋¹⁺⁴	—	2	4	no	—
14	C₂×#41×C₂₃	—	2	1	no	—
15	C₂×#41×C₂₃	—	2	2	no	—
16	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×#41×C₂₃	C₂²	1	2	yes	—
18	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
22	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#137	—	2	4	no	—
24	C₂₃×#135	—	2	4	no	—
25	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
27	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
30	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×C₂²⋊Q₈	—	2	1	no	—
34	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂₃×#121	—	2	1	no	—
36	C₂₃×#121	—	2	2	no	—
37	C₂₃×#121	C₂²	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#103	C₂²	1	2	yes	—
40	C₂₃×#103	C₂²	1	2	yes	—
41	C₂₃×#103	C₂²	1	2	yes	—
42	C₂₃×#102	C₂²	1	2	yes	—
43	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
44	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
45	#41×C₂₃	—	2	16	no	—
46	#41×C₂₃	—	2	8	no	—
47	#41×C₂₃	—	2	4	no	—
48	#41×C₂₃	—	2	4	no	—
49	#41×C₂₃	—	2	2	no	—
50	#41×C₂₃	—	2	2	no	—
51	#41×C₂₃	—	2	2	no	—
52	#41×C₂₃	—	2	2	no	—
53	#41×C₂₃	—	2	2	no	—
54	#41×C₂₃	D₄	1	4	yes	complemented
55	C₂×Q₈×C₂₃	—	2	8	no	—
56	C₂×Q₈×C₂₃	—	2	4	no	—
57	C₂×Q₈×C₂₃	—	2	4	no	—
58	C₂×Q₈×C₂₃	—	2	2	no	—
59	C₂×Q₈×C₂₃	—	2	1	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
62	C₂×Q₈×C₂₃	D₄	1	4	yes	complemented
63	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	2	1	no	—
68	C₂×D₄×C₂₃	—	2	1	no	—
69	C₂×D₄×C₂₃	—	2	2	no	—
70	C₂×D₄×C₂₃	D₄	1	2	yes	—
71	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
72	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	2	1	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	1	no	—
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	—	2	1	no	—
80	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
82	Q₁₆×C₂₃	—	2	2	no	—
83	Q₁₆×C₂₃	—	2	4	no	—
84	SD₁₆×C₂₃	—	2	2	no	—
85	SD₁₆×C₂₃	—	2	4	no	—
86	SD₁₆×C₂₃	—	2	2	no	—
87	D₈×C₂₃	—	2	2	no	—
88	M₁₆×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	C₂³	1	2	yes	—
91	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂²⋊C₄×C₂₃	—	2	2	no	—
97	C₂²⋊C₄×C₂₃	—	2	2	no	—
98	C₂³×C₂₃	—	2	1	no	—
99	C₂³×C₂₃	—	2	1	no	—
100	C₂³×C₂₃	C₂×D₄	1	2	yes	—
101	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₂₃	—	2	8	no	—
103	Q₈×C₂₃	—	2	2	no	—
104	Q₈×C₂₃	—	2	4	no	—
105	Q₈×C₂₃	—	2	1	no	—
106	Q₈×C₂₃	—	2	2	no	—
107	Q₈×C₂₃	—	2	1	no	—
108	Q₈×C₂₃	—	2	1	no	—
109	Q₈×C₂₃	C₂×D₄	1	2	yes	—
110	Q₈×C₂₃	C₂×D₄	1	4	yes	—
111	D₄×C₂₃	—	2	8	no	—
112	D₄×C₂₃	—	2	4	no	—
113	D₄×C₂₃	—	2	4	no	—
114	D₄×C₂₃	—	2	2	no	—
115	D₄×C₂₃	—	2	1	no	—
116	D₄×C₂₃	—	2	1	no	—
117	D₄×C₂₃	—	2	1	no	—
118	D₄×C₂₃	—	4	2	no	—
119	D₄×C₂₃	—	8	2	no	—
120	D₄×C₂₃	C₂×D₄	1	2	yes	—
121	C₂×C₄×C₂₃	—	2	8	no	—
122	C₂×C₄×C₂₃	—	2	8	no	—
123	C₂×C₄×C₂₃	—	2	4	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	1	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	8	1	no	—
133	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
134	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
138	C₈×C₂₃	—	2	1	no	—
139	C₈×C₂₃	—	2	1	no	—
140	C₈×C₂₃	—	2	2	no	—
141	#3013	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	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	—	4	1	no	—
149	C₂²×C₂₃	—	4	1	no	—
150	C₂²×C₂₃	—	4	1	no	—
151	C₂²×C₂₃	—	4	1	no	—
152	C₂²×C₂₃	—	4	1	no	—
153	C₂²×C₂₃	#120	1	2	yes	—
154	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₂₃	—	2	4	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	2	2	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	#120	1	2	yes	—
161	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
162	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic, complemented
164	C₂×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
165	C₂×#135	C₂×C₂₃	1	1	yes	characteristic, complemented
166	#537	C₂×C₂₃	1	1	yes	characteristic, complemented
167	#452	C₂×C₂₃	1	2	yes	complemented
168	#451	C₂×C₂₃	1	2	yes	complemented
169	#449	C₂×C₂₃	1	2	yes	complemented
170	#448	C₂×C₂₃	1	2	yes	complemented
171	#417	C₂×C₂₃	1	1	yes	characteristic, complemented
172	C₂×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
173	#408	C₂×C₂₃	1	1	yes	characteristic, complemented
174	C₂×C₂₃	—	2	2	no	—
175	C₂×C₂₃	—	4	2	no	—
176	C₂×C₂₃	—	4	2	no	—
177	C₂×C₂₃	—	8	1	no	—
178	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
179	C₂×C₂₃	#577	1	1	yes	characteristic
180	C₂×C₂₃	#578	1	1	yes	characteristic
181	2₋¹⁺⁴	—	2	4	no	—
182	2₋¹⁺⁴	—	2	4	no	—
183	C₂×#41	—	2	1	no	—
184	C₂×#41	—	2	2	no	—
185	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
186	C₂×#41	C₂²×C₂₃	1	2	yes	—
187	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
188	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
189	C₂²×Q₈	—	2	1	no	—
190	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
191	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
192	#137	—	2	4	no	—
193	#135	—	2	4	no	—
194	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
195	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
196	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
197	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
198	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
199	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
201	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
202	C₂²⋊Q₈	—	2	1	no	—
203	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
204	#121	—	2	1	no	—
205	#121	—	2	2	no	—
206	#121	C₂²×C₂₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
208	#103	C₂²×C₂₃	1	2	yes	—
209	#103	C₂²×C₂₃	1	2	yes	—
210	#103	C₂²×C₂₃	1	2	yes	—
211	#102	C₂²×C₂₃	1	2	yes	—
212	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
213	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
214	C₂₃	#3013	1	1	yes	characteristic, direct product factor, Sylow
215	#41	—	2	16	no	—
216	#41	—	2	8	no	—
217	#41	—	2	4	no	—
218	#41	—	2	4	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	—	2	2	no	—
224	#41	D₄×C₂₃	1	4	yes	complemented
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
232	C₂×Q₈	D₄×C₂₃	1	4	yes	complemented
233	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
234	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	D₄×C₂₃	1	2	yes	—
241	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
242	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
243	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
251	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	2	4	no	—
254	SD₁₆	—	2	2	no	—
255	SD₁₆	—	2	4	no	—
256	SD₁₆	—	2	2	no	—
257	D₈	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂³×C₂₃	1	2	yes	—
261	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
262	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	C₂×D₄×C₂₃	1	2	yes	—
271	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
272	Q₈	—	2	8	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	2	4	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	C₂×D₄×C₂₃	1	2	yes	—
280	Q₈	C₂×D₄×C₂₃	1	4	yes	—
281	D₄	—	2	8	no	—
282	D₄	—	2	4	no	—
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	2	no	—
290	D₄	C₂×D₄×C₂₃	1	2	yes	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
304	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
306	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
307	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	C₂₃×#120	1	2	yes	—
323	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	1	no	—
329	C₄	C₂₃×#120	1	2	yes	—
330	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
331	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	2	no	—
335	C₂	—	8	1	no	—
336	C₂	C₂×C₂₃×#120	1	1	yes	characteristic
337	C₂	C₂₃×#577	1	1	yes	characteristic
338	C₂	C₂₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₂₃×#137	C₂	semidirect
3. C₂×C₂₃×#135	C₂	semidirect
4. C₂₃×#537	C₂	semidirect
5. C₂₃×#452	C₂	semidirect
6. C₂₃×#451	C₂	semidirect
7. C₂₃×#449	C₂	semidirect
8. C₂₃×#448	C₂	semidirect
9. C₂₃×#417	C₂	semidirect
10. C₂×C₂₃×#103	C₂	semidirect
11. C₂₃×#408	C₂	semidirect
54. #41×C₂₃	D₄	semidirect
62. C₂×Q₈×C₂₃	D₄	semidirect
141. #3013	C₂₃	direct (N × H)
163. C₂×2₋¹⁺⁴	C₂×C₂₃	semidirect
164. C₂×#137	C₂×C₂₃	semidirect
165. C₂×#135	C₂×C₂₃	semidirect
166. #537	C₂×C₂₃	semidirect
167. #452	C₂×C₂₃	semidirect
168. #451	C₂×C₂₃	semidirect
169. #449	C₂×C₂₃	semidirect
170. #448	C₂×C₂₃	semidirect
171. #417	C₂×C₂₃	semidirect
172. C₂×#103	C₂×C₂₃	semidirect
173. #408	C₂×C₂₃	semidirect
214. C₂₃	#3013	direct (N × H)
224. #41	D₄×C₂₃	semidirect
232. C₂×Q₈	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×2₋¹⁺⁴	2
C₂×C₂₃×#137	2
C₂×C₂₃×#135	2
C₂₃×#537	2
C₂₃×#452	2
C₂₃×#451	2
C₂₃×#449	2
C₂₃×#448	2
C₂₃×#417	2
C₂×C₂₃×#103	2
C₂₃×#408	2
#3013	23

Subgroups — C23×#3013 #15670632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3013 #15670632