Subgroups — C₂₃×#9774 #19157400

Order 5888.

458 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#486	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂₃×#486	—	2	2	no	—
11	C₂₃×#486	C₄	1	4	yes	—
12	C₂₃×#431	C₂²	1	1	yes	characteristic
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#108	—	2	2	no	—
15	C₂₃×#423	C₂²	1	1	yes	characteristic
16	C₂₃×#423	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#102	—	2	2	no	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#333	—	2	2	no	—
20	C₂₃×#332	—	2	2	no	—
21	C₂₃×#331	—	2	2	no	—
22	C₂₃×#331	—	2	2	no	—
23	C₂₃×#322	—	2	2	no	—
24	C₂²×Q₈×C₂₃	—	2	1	no	—
25	C₂²×D₄×C₂₃	—	2	1	no	—
26	C₂²×D₄×C₂₃	—	2	1	no	—
27	C₂²×D₄×C₂₃	—	2	1	no	—
28	C₂×M₁₆×C₂₃	—	2	1	no	—
29	C₂×M₁₆×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂₃×C₄⋊Q₈	—	2	1	no	—
32	C₂₃×C₄⋊Q₈	—	2	1	no	—
33	C₂₃×C₄⋊Q₈	—	2	1	no	—
34	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂₃×#127	—	2	1	no	—
36	C₂₃×#127	—	2	1	no	—
37	C₂₃×#127	—	2	1	no	—
38	C₂₃×#127	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×#108	—	4	4	no	—
43	C₂₃×C₄⋊C₈	—	2	1	no	—
44	C₂₃×C₄⋊C₈	—	2	1	no	—
45	C₂₃×C₄⋊C₈	—	2	1	no	—
46	C₂₃×C₄⋊C₈	—	2	1	no	—
47	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
49	C₂₃×#102	—	2	8	no	—
50	C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×C₂²⋊C₈	—	2	1	no	—
52	C₄×C₈×C₂₃	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₂₃	D₄	1	2	yes	—
56	C₂⁴×C₂₃	—	4	1	no	—
57	C₂×Q₈×C₂₃	—	2	2	no	—
58	C₂×Q₈×C₂₃	—	4	1	no	—
59	C₂×Q₈×C₂₃	—	8	2	no	—
60	C₂×D₄×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	—	4	1	no	—
64	C₂×D₄×C₂₃	—	4	1	no	—
65	C₂×D₄×C₂₃	—	4	1	no	—
66	C₂×D₄×C₂₃	—	4	2	no	—
67	C₂×D₄×C₂₃	—	4	2	no	—
68	C₂×D₄×C₂₃	—	8	2	no	—
69	C₂×D₄×C₂₃	—	8	2	no	—
70	C₂²×C₄×C₂₃	—	4	1	no	—
71	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
74	M₁₆×C₂₃	—	4	2	no	—
75	M₁₆×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	4	1	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	4	1	no	—
86	C₄⋊C₄×C₂₃	—	4	1	no	—
87	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
91	#9774	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	4	2	no	—
93	C₂³×C₂₃	—	8	2	no	—
94	C₂³×C₂₃	—	8	2	no	—
95	C₂³×C₂₃	—	8	2	no	—
96	C₂³×C₂₃	#124	1	1	yes	characteristic
97	Q₈×C₂₃	—	8	2	no	—
98	Q₈×C₂₃	—	8	2	no	—
99	D₄×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	8	2	no	—
101	D₄×C₂₃	—	8	2	no	—
102	D₄×C₂₃	—	8	2	no	—
103	D₄×C₂₃	—	8	2	no	—
104	D₄×C₂₃	—	8	1	no	—
105	D₄×C₂₃	—	8	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	8	2	no	—
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₂₃	#137	1	2	yes	—
118	C₈×C₂₃	—	2	4	no	—
119	C₈×C₂₃	—	4	2	no	—
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	—	8	1	no	—
122	C₈×C₂₃	—	8	1	no	—
123	C₂×#486	C₂×C₂₃	1	1	yes	characteristic
124	#1848	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#1545	C₂×C₂₃	1	1	yes	characteristic
128	#1545	C₂×C₂₃	1	1	yes	characteristic
129	#1452	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₂²×C₂₃	—	2	1	no	—
131	C₂²×C₂₃	—	2	1	no	—
132	C₂²×C₂₃	—	8	2	no	—
133	C₂²×C₂₃	—	8	2	no	—
134	C₂²×C₂₃	—	8	2	no	—
135	C₂²×C₂₃	—	16	4	no	—
136	C₂²×C₂₃	#390	1	1	yes	characteristic
137	C₂²×C₂₃	#489	1	2	yes	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	8	2	no	—
141	C₄×C₂₃	#417	1	2	yes	—
142	C₄×C₂₃	#471	1	2	yes	—
143	C₄×C₂₃	#472	1	2	yes	—
144	C₄×C₂₃	#496	1	1	yes	characteristic
145	C₄×C₂₃	#497	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
148	#486	—	2	2	no	—
149	#486	C₄×C₂₃	1	4	yes	—
150	#431	C₂²×C₂₃	1	1	yes	characteristic
151	#431	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#108	—	2	2	no	—
153	#423	C₂²×C₂₃	1	1	yes	characteristic
154	#423	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×#102	—	2	2	no	—
156	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	#333	—	2	2	no	—
158	#332	—	2	2	no	—
159	#331	—	2	2	no	—
160	#331	—	2	2	no	—
161	#322	—	2	2	no	—
162	C₂×C₂₃	—	2	2	no	—
163	C₂×C₂₃	—	16	2	no	—
164	C₂×C₂₃	#1959	1	1	yes	characteristic
165	C₂×C₂₃	#1977	1	1	yes	characteristic
166	C₂×C₂₃	#1978	1	1	yes	characteristic
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×D₄	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂²×C₈	D₄×C₂₃	1	2	yes	—
174	C₄⋊Q₈	—	2	1	no	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
178	#127	—	2	1	no	—
179	#127	—	2	1	no	—
180	#127	—	2	1	no	—
181	#127	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	#108	—	4	4	no	—
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	2	1	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
191	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
192	#102	—	2	8	no	—
193	C₂²⋊C₈	—	2	1	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
198	C₄×C₈	D₄×C₂₃	1	2	yes	—
199	C₂₃	#9774	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	—	4	1	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	1	no	—
203	C₂×Q₈	—	8	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	2	no	—
213	C₂×D₄	—	8	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
233	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂³	—	4	2	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂₃×#124	1	1	yes	characteristic
240	Q₈	—	8	2	no	—
241	Q₈	—	8	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	2	no	—
245	D₄	—	8	2	no	—
246	D₄	—	8	2	no	—
247	D₄	—	8	1	no	—
248	D₄	—	8	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
259	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
260	C₂×C₄	C₂₃×#137	1	2	yes	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	8	1	no	—
265	C₈	—	8	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	16	4	no	—
272	C₂²	C₂₃×#390	1	1	yes	characteristic
273	C₂²	C₂₃×#489	1	2	yes	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	8	2	no	—
277	C₄	C₂₃×#417	1	2	yes	—
278	C₄	C₂₃×#471	1	2	yes	—
279	C₄	C₂₃×#472	1	2	yes	—
280	C₄	C₂₃×#496	1	1	yes	characteristic
281	C₄	C₂₃×#497	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	16	2	no	—
284	C₂	C₂₃×#1959	1	1	yes	characteristic
285	C₂	C₂₃×#1977	1	1	yes	characteristic
286	C₂	C₂₃×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	#390
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1848	C₂	semidirect
3. C₂₃×#1549	C₂	semidirect
4. C₂₃×#1548	C₂	semidirect
7. C₂₃×#1452	C₂	semidirect
91. #9774	C₂₃	direct (N × H)
124. #1848	C₂×C₂₃	semidirect
125. #1549	C₂×C₂₃	semidirect
126. #1548	C₂×C₂₃	semidirect
129. #1452	C₂×C₂₃	semidirect
199. C₂₃	#9774	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×#486	2
C₂₃×#1848	2
C₂₃×#1549	2
C₂₃×#1548	2
C₂₃×#1545	2
C₂₃×#1452	2
#9774	23

Subgroups — C23×#9774 #19157400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9774 #19157400