Subgroups — C₂₃×#7856 #19155482

Order 5888.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2920	C₂	1	2	yes	maximal
2	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1274	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#432	—	2	2	no	—
7	C₂₃×#432	—	2	2	no	—
8	C₂₃×#431	C₄	1	4	yes	complemented
9	C₂₃×#431	C₂²	1	4	yes	—
10	C₂₃×#423	—	2	2	no	—
11	C₂₃×#407	—	2	2	no	—
12	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
13	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
15	C₂₃×#376	C₂²	1	2	yes	—
16	C₄³×C₂₃	C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	D₄	1	2	yes	—
19	C₂×M₁₆×C₂₃	—	2	4	no	—
20	C₂₃×#117	—	2	8	no	—
21	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄²×C₂₃	—	2	4	no	—
24	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₃	D₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	—	2	4	no	—
35	C₂₃×C₄⋊C₈	—	2	4	no	—
36	C₂₃×C₂²⋊C₈	—	2	4	no	—
37	C₂₃×C₂²⋊C₈	—	2	4	no	—
38	C₂₃×#97	—	2	4	no	—
39	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
40	C₂₃×#95	—	2	4	no	—
41	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	4	4	no	—
45	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	M₁₆×C₂₃	—	4	8	no	—
58	C₂×C₈×C₂₃	—	2	8	no	—
59	C₄⋊C₄×C₂₃	—	2	8	no	—
60	C₂²⋊C₄×C₂₃	—	2	8	no	—
61	C₄²×C₂₃	—	2	4	no	—
62	C₄²×C₂₃	—	4	8	no	—
63	C₄²×C₂₃	C₄²	1	2	yes	—
64	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₂₃	M₁₆	1	8	yes	complemented
71	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
72	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
74	#7856	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	4	2	no	—
77	C₂³×C₂₃	#95	1	1	yes	characteristic
78	C₂³×C₂₃	#95	1	1	yes	characteristic
79	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	8	no	—
89	C₂×C₄×C₂₃	—	4	2	no	—
90	C₂×C₄×C₂₃	—	4	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	8	no	—
93	C₂×C₄×C₂₃	#95	1	2	yes	—
94	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#95	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
102	C₈×C₂₃	—	4	8	no	—
103	#2920	C₂×C₂₃	1	2	yes	—
104	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
105	#1274	C₂×C₂₃	1	4	yes	complemented
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	2	1	no	—
108	C₂²×C₂₃	—	2	1	no	—
109	C₂²×C₂₃	—	2	2	no	—
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	4	4	no	—
112	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
113	C₂²×C₂₃	#407	1	2	yes	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	2	no	—
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	4	8	no	—
120	C₄×C₂₃	H(3,ℤ/4)	1	4	yes	—
121	C₄×C₂₃	C₄×M₁₆	1	2	yes	—
122	C₄×C₂₃	#423	1	2	yes	—
123	C₂×#117	C₄×C₂₃	1	2	yes	—
124	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
125	#432	—	2	2	no	—
126	#432	—	2	2	no	—
127	#431	C₄×C₂₃	1	4	yes	complemented
128	#431	C₂²×C₂₃	1	4	yes	—
129	#423	—	2	2	no	—
130	#407	—	2	2	no	—
131	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
132	C₄×C₄⋊C₄	—	2	2	no	—
133	C₄×C₂²⋊C₄	—	2	2	no	—
134	#376	C₂²×C₂₃	1	2	yes	—
135	C₄³	C₄×C₂₃	1	2	yes	—
136	C₂×C₂₃	—	2	2	no	—
137	C₂×C₂₃	—	4	1	no	—
138	C₂×C₂₃	—	4	1	no	—
139	C₂×C₂₃	#1728	1	1	yes	characteristic
140	C₂×C₂₃	C₂×H(3,ℤ/4)	1	2	yes	—
141	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	D₄×C₂₃	1	2	yes	—
143	C₂×M₁₆	—	2	4	no	—
144	#117	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
149	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
151	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
152	C₂×C₄²	D₄×C₂₃	1	2	yes	—
153	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
155	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
156	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
164	#95	—	2	4	no	—
165	C₂₃	#7856	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
177	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
182	M₁₆	—	4	8	no	—
183	C₂×C₈	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄²×C₂₃	1	2	yes	—
189	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
191	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
192	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
193	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
194	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
195	C₄²	M₁₆×C₂₃	1	8	yes	complemented
196	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
197	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
198	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	C₂₃×#95	1	1	yes	characteristic
202	C₂³	C₂₃×#95	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₂₃×#95	1	2	yes	—
218	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#95	1	2	yes	—
221	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
226	C₈	—	4	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
234	C₂²	C₂₃×#407	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	C₂₃×H(3,ℤ/4)	1	4	yes	—
242	C₄	C₄×M₁₆×C₂₃	1	2	yes	—
243	C₄	C₂₃×#423	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	C₂₃×#1728	1	1	yes	characteristic
248	C₂	C₂×C₂₃×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#95
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1274	C₂	semidirect
8. C₂₃×#431	C₄	semidirect
12. C₄×M₁₆×C₂₃	C₄	semidirect
70. C₄²×C₂₃	M₁₆	semidirect
74. #7856	C₂₃	direct (N × H)
105. #1274	C₂×C₂₃	semidirect
127. #431	C₄×C₂₃	semidirect
131. C₄×M₁₆	C₄×C₂₃	semidirect
165. C₂₃	#7856	direct (N × H)
195. C₄²	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2920	2
C₄×C₂₃×#117	2
C₂₃×#1274	2
#7856	23

Subgroups — C23×#7856 #19155482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7856 #19155482