Subgroups — C₂₃×#7876 #19155502

Order 5888.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1277	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂²×M₁₆×C₂₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#432	—	2	2	no	—
9	C₂₃×#432	C₄	1	4	yes	—
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#423	C₄	1	4	yes	—
12	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
16	C₄×M₁₆×C₂₃	—	2	2	no	—
17	C₄×M₁₆×C₂₃	—	2	2	no	—
18	C₂×C₂₃×#97	—	2	1	no	—
19	C₂×C₂₃×#97	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₂₃	—	2	4	no	—
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	—	2	8	no	—
27	C₂×M₁₆×C₂₃	—	4	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	2	yes	—
37	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×C₄⋊C₈	—	2	4	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₂₃×C₂²⋊C₈	—	2	4	no	—
45	C₂₃×#97	—	2	4	no	—
46	C₂₃×#97	—	2	4	no	—
47	C₄×C₈×C₂₃	—	2	2	no	—
48	C₄×C₈×C₂₃	—	2	2	no	—
49	C₂⁴×C₂₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₂₃	—	2	16	no	—
67	M₁₆×C₂₃	—	4	8	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	4	no	—
74	C₂×C₈×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	M₁₆	1	8	yes	complemented
76	C₄²×C₂₃	C₄²	1	2	yes	—
77	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	#7876	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	#95	1	2	yes	—
91	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	#95	1	2	yes	—
104	C₂×C₄×C₂₃	#95	1	2	yes	—
105	C₂×C₄×C₂₃	#95	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₂₃	#137	1	2	yes	—
115	C₈×C₂₃	—	2	8	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₈×C₂₃	—	4	4	no	—
118	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
119	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
120	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
121	#1277	C₂×C₂₃	1	4	yes	complemented
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
127	C₂²×C₂₃	#423	1	2	yes	—
128	C₂²×C₂₃	#428	1	2	yes	—
129	C₂²×C₂₃	#430	1	2	yes	—
130	C₄×C₂₃	—	2	2	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
136	C₄×C₂₃	#407	1	2	yes	—
137	C₄×C₂₃	#417	1	2	yes	—
138	C₄×C₂₃	#418	1	1	yes	characteristic
139	C₄×C₂₃	#419	1	1	yes	characteristic
140	C₂²×M₁₆	—	2	1	no	—
141	C₂²×M₁₆	C₄×C₂₃	1	2	yes	—
142	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
143	#432	—	2	2	no	—
144	#432	C₄×C₂₃	1	4	yes	—
145	#423	—	2	2	no	—
146	#423	C₄×C₂₃	1	4	yes	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₈	—	2	1	no	—
151	C₄×M₁₆	—	2	2	no	—
152	C₄×M₁₆	—	2	2	no	—
153	C₂×#97	—	2	1	no	—
154	C₂×#97	C₄×C₂₃	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
156	C₂×C₂₃	—	4	2	no	—
157	C₂×C₂₃	#1569	1	4	yes	—
158	C₂×C₂₃	#1728	1	1	yes	characteristic
159	C₂×C₂₃	#1736	1	1	yes	characteristic
160	C₂×C₂₃	#1738	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
162	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	Q₈×C₂₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	8	no	—
167	C₂×M₁₆	—	4	4	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
171	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
173	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
174	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₂₃	1	2	yes	—
176	C₂×C₄²	D₄×C₂₃	1	2	yes	—
177	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
183	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	4	no	—
186	#97	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₂₃	#7876	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₄⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
207	M₁₆	—	2	16	no	—
208	M₁₆	—	4	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	M₁₆×C₂₃	1	8	yes	complemented
217	C₄²	C₄²×C₂₃	1	2	yes	—
218	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
219	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
226	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂₃×#95	1	2	yes	—
231	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₂₃×#95	1	2	yes	—
244	C₂×C₄	C₂₃×#95	1	2	yes	—
245	C₂×C₄	C₂₃×#95	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
253	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
254	C₂×C₄	C₂₃×#137	1	2	yes	—
255	C₈	—	2	8	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
263	C₂²	C₂₃×#423	1	2	yes	—
264	C₂²	C₂₃×#428	1	2	yes	—
265	C₂²	C₂₃×#430	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
272	C₄	C₂₃×#407	1	2	yes	—
273	C₄	C₂₃×#417	1	2	yes	—
274	C₄	C₂₃×#418	1	1	yes	characteristic
275	C₄	C₂₃×#419	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	C₂₃×#1569	1	4	yes	—
278	C₂	C₂₃×#1728	1	1	yes	characteristic
279	C₂	C₂₃×#1736	1	1	yes	characteristic
280	C₂	C₂₃×#1738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₃	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1277	C₂	semidirect
75. C₂×C₈×C₂₃	M₁₆	semidirect
86. #7876	C₂₃	direct (N × H)
121. #1277	C₂×C₂₃	semidirect
189. C₂₃	#7876	direct (N × H)
216. C₂×C₈	M₁₆×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×#432	2
C₂×C₂₃×#423	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1277	2
#7876	23

Subgroups — C23×#7876 #19155502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7876 #19155502