Subgroups — C₂₃×#12742 #19160368

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#578	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2145	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2141	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
7	C₂₃×#578	—	2	4	no	—
8	C₂₃×#578	—	2	4	no	—
9	C₂×C₂₃×#137	—	2	1	no	—
10	C₂×C₂₃×#137	—	2	2	no	—
11	C₂×C₂₃×#135	C₄	1	2	yes	complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₂₃	C₄	1	2	yes	complemented
14	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂₃×#427	C₂²	1	2	yes	—
18	C₂₃×#410	—	2	1	no	—
19	C₂₃×#360	—	2	4	no	—
20	C₂₃×#358	—	2	4	no	—
21	C₂₃×#349	—	2	2	no	—
22	C₂₃×H(3,ℤ/4)	—	2	1	no	—
23	C₂₃×2₋¹⁺⁴	—	2	8	no	—
24	C₂₃×2₋¹⁺⁴	—	4	4	no	—
25	C₂×#41×C₂₃	—	2	2	no	—
26	C₂×#41×C₂₃	—	2	1	no	—
27	C₂×#41×C₂₃	—	4	2	no	—
28	C₂×#41×C₂₃	—	4	1	no	—
29	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₂₃	—	2	2	no	—
31	C₂²×Q₈×C₂₃	—	2	1	no	—
32	C₂²×Q₈×C₂₃	—	4	1	no	—
33	C₂₃×#137	—	2	8	no	—
34	C₂₃×#137	—	2	16	no	—
35	C₂₃×#135	—	2	4	no	—
36	C₂₃×#135	—	2	4	no	—
37	C₂₃×#135	—	2	2	no	—
38	C₂₃×#135	D₄	1	4	yes	—
39	C₂×Q₁₆×C₂₃	—	2	4	no	—
40	C₂×Q₁₆×C₂₃	—	2	4	no	—
41	C₂×Q₁₆×C₂₃	—	2	2	no	—
42	C₂×Q₁₆×C₂₃	—	2	1	no	—
43	C₂×Q₁₆×C₂₃	—	2	2	no	—
44	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×Q₁₆×C₂₃	D₄	1	4	yes	—
47	C₂×SD₁₆×C₂₃	—	2	2	no	—
48	C₂×SD₁₆×C₂₃	—	2	1	no	—
49	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂₃×#131	—	2	2	no	—
51	C₂₃×#131	—	2	2	no	—
52	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#117	—	2	2	no	—
57	C₂₃×M₃₂	—	2	2	no	—
58	C₂₃×M₃₂	—	2	2	no	—
59	C₂×C₁₆×C₂₃	—	2	2	no	—
60	C₂₃×#107	—	2	2	no	—
61	C₂₃×#106	—	2	2	no	—
62	C₂₃×#99	—	4	2	no	—
63	#41×C₂₃	—	2	2	no	—
64	#41×C₂₃	—	2	2	no	—
65	#41×C₂₃	—	2	8	no	—
66	#41×C₂₃	—	2	4	no	—
67	#41×C₂₃	—	4	4	no	—
68	#41×C₂₃	—	4	16	no	—
69	#41×C₂₃	—	4	8	no	—
70	#41×C₂₃	—	4	2	no	—
71	C₂×Q₈×C₂₃	—	2	8	no	—
72	C₂×Q₈×C₂₃	—	2	2	no	—
73	C₂×Q₈×C₂₃	—	2	1	no	—
74	C₂×Q₈×C₂₃	—	2	4	no	—
75	C₂×Q₈×C₂₃	—	4	4	no	—
76	C₂×Q₈×C₂₃	—	4	8	no	—
77	C₂×Q₈×C₂₃	—	4	2	no	—
78	C₂×Q₈×C₂₃	—	4	1	no	—
79	C₂×Q₈×C₂₃	—	4	2	no	—
80	C₂×Q₈×C₂₃	SD₁₆	1	1	yes	characteristic
81	C₂×D₄×C₂₃	—	2	1	no	—
82	C₂×D₄×C₂₃	—	4	2	no	—
83	C₂×D₄×C₂₃	—	4	1	no	—
84	C₂×D₄×C₂₃	D₈	1	2	yes	—
85	C₂×D₄×C₂₃	SD₁₆	1	1	yes	characteristic
86	C₂²×C₄×C₂₃	—	4	2	no	—
87	C₂²×C₄×C₂₃	—	4	1	no	—
88	C₂²×C₄×C₂₃	—	4	2	no	—
89	C₂²×C₄×C₂₃	—	4	1	no	—
90	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₂₃	—	2	8	no	—
93	Q₁₆×C₂₃	—	2	1	no	—
94	Q₁₆×C₂₃	—	2	2	no	—
95	Q₁₆×C₂₃	—	2	4	no	—
96	Q₁₆×C₂₃	C₂²⋊C₄	1	2	yes	—
97	Q₁₆×C₂₃	C₂²⋊C₄	1	4	yes	—
98	SD₁₆×C₂₃	—	2	8	no	—
99	SD₁₆×C₂₃	—	2	4	no	—
100	D₈×C₂₃	—	2	1	no	—
101	D₈×C₂₃	C₂²⋊C₄	1	2	yes	—
102	M₁₆×C₂₃	—	2	2	no	—
103	M₁₆×C₂₃	—	2	4	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	2	2	no	—
106	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₂₃	—	4	2	no	—
112	C₂²⋊C₄×C₂₃	—	4	2	no	—
113	C₄²×C₂₃	—	4	2	no	—
114	C₁₆×C₂₃	—	4	2	no	—
115	#12742	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₃	—	4	1	no	—
117	C₂³×C₂₃	#102	1	2	yes	—
118	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	2	4	no	—
121	Q₈×C₂₃	—	2	2	no	—
122	Q₈×C₂₃	—	4	8	no	—
123	Q₈×C₂₃	—	4	1	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	Q₈×C₂₃	—	4	4	no	—
126	D₄×C₂₃	—	2	4	no	—
127	D₄×C₂₃	—	2	2	no	—
128	D₄×C₂₃	—	2	2	no	—
129	D₄×C₂₃	—	4	8	no	—
130	D₄×C₂₃	—	4	4	no	—
131	D₄×C₂₃	—	4	1	no	—
132	C₂×C₄×C₂₃	—	2	2	no	—
133	C₂×C₄×C₂₃	—	2	2	no	—
134	C₂×C₄×C₂₃	—	4	8	no	—
135	C₂×C₄×C₂₃	—	4	8	no	—
136	C₂×C₄×C₂₃	—	4	1	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	4	no	—
139	C₂×C₄×C₂₃	—	4	4	no	—
140	C₂×C₄×C₂₃	—	8	2	no	—
141	C₂×C₄×C₂₃	#102	1	2	yes	—
142	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₂₃	#120	1	2	yes	—
144	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
146	C₈×C₂₃	—	2	2	no	—
147	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₈×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₈×C₂₃	#120	1	2	yes	—
150	C₂×#578	C₂×C₂₃	1	1	yes	characteristic
151	#2145	C₂×C₂₃	1	2	yes	complemented
152	#2141	C₂×C₂₃	1	2	yes	complemented
153	#2115	C₂×C₂₃	1	1	yes	characteristic, complemented
154	#1812	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂²×C₂₃	—	2	2	no	—
156	C₂²×C₂₃	—	4	1	no	—
157	C₂²×C₂₃	—	4	2	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	—	8	2	no	—
160	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₂₃	#447	1	2	yes	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	4	4	no	—
164	C₄×C₂₃	—	4	2	no	—
165	C₄×C₂₃	—	16	2	no	—
166	C₄×C₂₃	#379	1	1	yes	characteristic
167	C₄×C₂₃	#418	1	1	yes	characteristic
168	C₄×C₂₃	#450	1	2	yes	—
169	C₂×2₋¹⁺⁴	—	2	1	no	—
170	#578	—	2	4	no	—
171	#578	—	2	4	no	—
172	C₂×#137	—	2	1	no	—
173	C₂×#137	—	2	2	no	—
174	C₂×#135	C₄×C₂₃	1	2	yes	complemented
175	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
176	C₂²×Q₁₆	C₄×C₂₃	1	2	yes	complemented
177	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
180	#427	C₂²×C₂₃	1	2	yes	—
181	#410	—	2	1	no	—
182	#360	—	2	4	no	—
183	#358	—	2	4	no	—
184	#349	—	2	2	no	—
185	H(3,ℤ/4)	—	2	1	no	—
186	C₂×C₂₃	—	2	2	no	—
187	C₂×C₂₃	—	2	1	no	—
188	C₂×C₂₃	—	4	2	no	—
189	C₂×C₂₃	—	8	2	no	—
190	C₂×C₂₃	#1786	1	1	yes	characteristic
191	2₋¹⁺⁴	—	2	8	no	—
192	2₋¹⁺⁴	—	4	4	no	—
193	C₂×#41	—	2	2	no	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	—	4	2	no	—
196	C₂×#41	—	4	1	no	—
197	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
198	C₂²×Q₈	—	2	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	4	1	no	—
201	#137	—	2	8	no	—
202	#137	—	2	16	no	—
203	#135	—	2	4	no	—
204	#135	—	2	4	no	—
205	#135	—	2	2	no	—
206	#135	D₄×C₂₃	1	4	yes	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×Q₁₆	—	2	4	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
213	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
214	C₂×Q₁₆	D₄×C₂₃	1	4	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
222	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
224	#117	—	2	2	no	—
225	M₃₂	—	2	2	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	#107	—	2	2	no	—
229	#106	—	2	2	no	—
230	#99	—	4	2	no	—
231	C₂₃	#12742	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	8	no	—
235	#41	—	2	4	no	—
236	#41	—	4	4	no	—
237	#41	—	4	16	no	—
238	#41	—	4	8	no	—
239	#41	—	4	2	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	8	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	SD₁₆×C₂₃	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	D₈×C₂₃	1	2	yes	—
254	C₂×D₄	SD₁₆×C₂₃	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
261	Q₁₆	—	2	8	no	—
262	Q₁₆	—	2	1	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	C₂²⋊C₄×C₂₃	1	2	yes	—
266	Q₁₆	C₂²⋊C₄×C₂₃	1	4	yes	—
267	SD₁₆	—	2	8	no	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	2	1	no	—
270	D₈	C₂²⋊C₄×C₂₃	1	2	yes	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₁₆	—	4	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	C₂₃×#102	1	2	yes	—
286	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	8	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	4	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	8	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	C₂₃×#102	1	2	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
311	C₂×C₄	C₂₃×#120	1	2	yes	—
312	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
316	C₈	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
317	C₈	C₂₃×#120	1	2	yes	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
324	C₂²	C₂₃×#447	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	16	2	no	—
329	C₄	C₂₃×#379	1	1	yes	characteristic
330	C₄	C₂₃×#418	1	1	yes	characteristic
331	C₄	C₂₃×#450	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₂₃×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1786
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂×C₂₃×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2145	C₂	semidirect
3. C₂₃×#2141	C₂	semidirect
4. C₂₃×#2115	C₂	semidirect
5. C₂₃×#1812	C₂	semidirect
11. C₂×C₂₃×#135	C₄	semidirect
13. C₂²×Q₁₆×C₂₃	C₄	semidirect
115. #12742	C₂₃	direct (N × H)
151. #2145	C₂×C₂₃	semidirect
152. #2141	C₂×C₂₃	semidirect
153. #2115	C₂×C₂₃	semidirect
154. #1812	C₂×C₂₃	semidirect
174. C₂×#135	C₄×C₂₃	semidirect
176. C₂²×Q₁₆	C₄×C₂₃	semidirect
231. C₂₃	#12742	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×#578	2
C₂₃×#2145	2
C₂₃×#2141	2
C₂₃×#2115	2
C₂₃×#1812	2
#12742	23

Subgroups — C23×#12742 #19160368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12742 #19160368