Subgroups — C₈×C₂₃×#100 #19155788

Order 5888.

542 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#100	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1460	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1283	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₃	C₄	1	2	yes	—
7	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₂₃	—	2	2	no	—
9	C₈×D₄×C₂₃	—	2	2	no	—
10	C₂₃×#431	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#100	C₄	1	2	yes	—
13	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
14	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#343	C₄	1	2	yes	—
17	C₂₃×#323	C₄	1	4	yes	—
18	C₂₃×#322	—	2	2	no	—
19	C₈²×C₂₃	—	2	2	no	—
20	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	4	4	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×D₄×C₂₃	—	2	2	no	—
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×#100	C₈	1	8	yes	direct product factor
37	C₂₃×C₂²⋊C₈	—	2	4	no	—
38	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
39	C₂₃×#97	—	2	2	no	—
40	C₄×C₈×C₂₃	—	2	2	no	—
41	C₄×C₈×C₂₃	—	2	2	no	—
42	C₄×C₈×C₂₃	D₄	1	2	yes	—
43	C₄×C₈×C₂₃	D₄	1	2	yes	—
44	C₂⁴×C₂₃	C₄²	1	2	yes	—
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	4	2	no	—
51	C₂²×C₄×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₂₃	—	2	4	no	—
57	M₁₆×C₂₃	C₂×C₈	1	8	yes	complemented
58	C₂×C₈×C₂₃	—	2	4	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	4	1	no	—
63	C₂×C₈×C₂₃	—	4	4	no	—
64	C₂×C₈×C₂₃	—	4	4	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	4	no	—
71	C₄²×C₂₃	C₂×D₄	1	2	yes	—
72	C₄²×C₂₃	#41	1	2	yes	—
73	C₈×#100	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	4	2	no	—
77	C₂³×C₂₃	—	4	2	no	—
78	C₂³×C₂₃	C₄×C₈	1	4	yes	—
79	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
80	D₄×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	4	1	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	4	2	no	—
89	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₂₃	#131	1	2	yes	—
95	C₈×C₂₃	—	2	1	no	—
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	2	8	no	—
98	C₈×C₂₃	—	4	4	no	—
99	C₈×C₂₃	—	4	2	no	—
100	C₈×C₂₃	#100	1	2	yes	direct product factor
101	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
102	C₄×#100	C₂×C₂₃	1	1	yes	characteristic
103	#1460	C₂×C₂₃	1	1	yes	characteristic
104	C₈×M₁₆	C₂×C₂₃	1	2	yes	complemented
105	#1283	C₂×C₂₃	1	2	yes	complemented
106	C₂²×C₂₃	—	2	1	no	—
107	C₂²×C₂₃	—	2	1	no	—
108	C₂²×C₂₃	—	2	4	no	—
109	C₂²×C₂₃	—	4	4	no	—
110	C₂²×C₂₃	—	4	4	no	—
111	C₂²×C₂₃	—	4	2	no	—
112	C₂²×C₂₃	—	4	2	no	—
113	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₂₃	#405	1	1	yes	characteristic
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	C₂×#100	1	1	yes	characteristic
120	C₄×C₂₃	#413	1	1	yes	characteristic
121	C₄×C₂₃	C₈×D₄	1	4	yes	—
122	C₂³×C₈	C₄×C₂₃	1	2	yes	—
123	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
124	C₈×D₄	—	2	2	no	—
125	C₈×D₄	—	2	2	no	—
126	#431	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×#100	C₄×C₂₃	1	2	yes	—
129	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
130	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
132	#343	C₄×C₂₃	1	2	yes	—
133	#323	C₄×C₂₃	1	4	yes	—
134	#322	—	2	2	no	—
135	C₈²	—	2	2	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	4	4	no	—
139	C₂×C₂₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₂₃	C₄×#100	1	1	yes	characteristic
141	C₂×C₂₃	#1757	1	1	yes	characteristic
142	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
143	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
144	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	4	4	no	—
148	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
151	C₄×D₄	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	#100	C₈×C₂₃	1	8	yes	direct product factor
159	C₂²⋊C₈	—	2	4	no	—
160	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
161	#97	—	2	2	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	D₄×C₂₃	1	2	yes	—
165	C₄×C₈	D₄×C₂₃	1	2	yes	—
166	C₂₃	C₈×#100	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₄²×C₂₃	1	2	yes	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	C₂×C₈×C₂₃	1	8	yes	complemented
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	4	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	C₂×D₄×C₂₃	1	2	yes	—
195	C₄²	#41×C₂₃	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	—	4	2	no	—
200	C₂³	C₄×C₈×C₂₃	1	4	yes	—
201	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
202	D₄	—	2	8	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
214	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
215	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
216	C₂×C₄	C₂₃×#131	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	2	8	no	—
220	C₈	—	4	4	no	—
221	C₈	—	4	2	no	—
222	C₈	C₂₃×#100	1	2	yes	direct product factor
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	4	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
232	C₂²	C₂₃×#405	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂×C₂₃×#100	1	1	yes	characteristic
237	C₄	C₂₃×#413	1	1	yes	characteristic
238	C₄	C₈×D₄×C₂₃	1	4	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	1	no	—
241	C₂	—	4	4	no	—
242	C₂	C₈×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
243	C₂	C₄×C₂₃×#100	1	1	yes	characteristic
244	C₂	C₂₃×#1757	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈×M₁₆×C₂₃	C₂	semidirect
5. C₂₃×#1283	C₂	semidirect
36. C₂₃×#100	C₈	direct (N × H)
57. M₁₆×C₂₃	C₂×C₈	semidirect
73. C₈×#100	C₂₃	direct (N × H)
100. C₈×C₂₃	#100	direct (N × H)
104. C₈×M₁₆	C₂×C₂₃	semidirect
105. #1283	C₂×C₂₃	semidirect
158. #100	C₈×C₂₃	direct (N × H)
166. C₂₃	C₈×#100	direct (N × H)
180. M₁₆	C₂×C₈×C₂₃	semidirect
222. C₈	C₂₃×#100	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×C₈×D₄×C₂₃	2
C₄×C₂₃×#100	2
C₂₃×#1460	2
C₈×M₁₆×C₂₃	2
C₂₃×#1283	2
C₈×#100	23

Subgroups — C8×C23×#100 #19155788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂₃×#100 #19155788