Subgroups — C₂₃×#7199 #19154825

Order 5888.

314 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×C₁₆×C₂₃	—	2	1	no	—
6	C₂²×C₁₆×C₂₃	C₄	1	2	yes	complemented
7	C₂×C₂₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	—	2	1	no	—
9	C₄×C₂²⋊C₄×C₂₃	C₄	1	1	yes	characteristic
10	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
11	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
12	C₂₃×C₂²⋊C₁₆	—	2	1	no	—
13	C₂₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
14	C₂₃×#336	—	2	1	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₂₃	—	2	1	no	—
17	C₂²×C₈×C₂₃	—	4	2	no	—
18	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	Q₈	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
25	C₂×C₄²×C₂₃	—	2	1	no	—
26	C₂×C₁₆×C₂₃	—	2	2	no	—
27	C₂×C₁₆×C₂₃	—	2	2	no	—
28	C₂×C₁₆×C₂₃	—	4	2	no	—
29	C₂×C₁₆×C₂₃	—	4	2	no	—
30	C₂₃×C₂²⋊C₈	—	2	1	no	—
31	C₂₃×C₂²⋊C₈	C₈	1	2	yes	—
32	C₄×C₈×C₂₃	—	4	2	no	—
33	C₂₃×#95	—	2	1	no	—
34	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	4	1	no	—
37	C₂²×C₄×C₂₃	—	4	1	no	—
38	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	M₁₆	1	1	yes	characteristic
44	C₂×C₈×C₂₃	—	2	2	no	—
45	C₂×C₈×C₂₃	—	2	2	no	—
46	C₂×C₈×C₂₃	—	2	2	no	—
47	C₂×C₈×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	4	1	no	—
49	C₂×C₈×C₂₃	—	4	1	no	—
50	C₂×C₈×C₂₃	—	4	1	no	—
51	C₂×C₈×C₂₃	—	4	1	no	—
52	C₂×C₈×C₂₃	—	4	1	no	—
53	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	C₁₆	1	4	yes	complemented
59	C₄²×C₂₃	—	4	1	no	—
60	C₄²×C₂₃	—	4	1	no	—
61	C₁₆×C₂₃	—	4	2	no	—
62	C₁₆×C₂₃	—	4	2	no	—
63	#7199	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	—	4	1	no	—
65	C₂³×C₂₃	—	4	1	no	—
66	C₂³×C₂₃	C₄×C₈	1	1	yes	characteristic
67	C₂³×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂³×C₂₃	C₄⋊C₈	1	1	yes	characteristic
69	C₂³×C₂₃	C₂×C₁₆	1	2	yes	—
70	C₂³×C₂₃	M₃₂	1	2	yes	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	4	1	no	—
79	C₂×C₄×C₂₃	—	4	1	no	—
80	C₂×C₄×C₂₃	—	4	1	no	—
81	C₂×C₄×C₂₃	—	4	2	no	—
82	C₂×C₄×C₂₃	—	4	2	no	—
83	C₂×C₄×C₂₃	#95	1	1	yes	characteristic
84	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
87	C₈×C₂₃	—	2	1	no	—
88	C₈×C₂₃	—	2	1	no	—
89	C₈×C₂₃	—	2	1	no	—
90	C₈×C₂₃	—	4	2	no	—
91	C₈×C₂₃	#99	1	2	yes	—
92	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
93	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic
94	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	—	2	1	no	—
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	2	1	no	—
99	C₂²×C₂₃	—	2	1	no	—
100	C₂²×C₂₃	—	2	1	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	#336	1	1	yes	characteristic
104	C₂²×C₂₃	C₄×C₁₆	1	1	yes	characteristic
105	C₂²×C₂₃	#346	1	1	yes	characteristic
106	C₂²×C₂₃	C₂²⋊C₁₆	1	2	yes	—
107	C₂²×C₂₃	C₄⋊C₁₆	1	2	yes	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	4	4	no	—
112	C₄×C₂₃	#342	1	1	yes	characteristic
113	C₄×C₂₃	#344	1	1	yes	characteristic
114	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
115	C₂²×C₁₆	—	2	1	no	—
116	C₂²×C₁₆	C₄×C₂₃	1	2	yes	complemented
117	C₂×C₂²⋊C₈	C₄×C₂₃	1	1	yes	characteristic
118	C₂×C₄×C₈	—	2	1	no	—
119	C₄×C₂²⋊C₄	C₄×C₂₃	1	1	yes	characteristic
120	C₂²⋊C₁₆	—	2	1	no	—
121	C₂²⋊C₁₆	—	2	1	no	—
122	C₂²⋊C₁₆	—	2	1	no	—
123	C₂²⋊C₁₆	C₄×C₂₃	1	2	yes	complemented
124	#336	—	2	1	no	—
125	C₂×C₂₃	—	2	2	no	—
126	C₂×C₂₃	—	2	1	no	—
127	C₂×C₂₃	—	2	1	no	—
128	C₂×C₂₃	—	2	2	no	—
129	C₂×C₂₃	#1282	1	1	yes	characteristic
130	C₂×C₂₃	#1374	1	1	yes	characteristic
131	C₂×C₂₃	#1378	1	1	yes	characteristic
132	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
133	C₂²×C₈	—	2	1	no	—
134	C₂²×C₈	—	4	2	no	—
135	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
137	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
140	C₂²×C₈	Q₈×C₂₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₈×C₂₃	1	2	yes	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	4	2	no	—
146	C₂×C₁₆	—	4	2	no	—
147	C₂²⋊C₈	—	2	1	no	—
148	C₂²⋊C₈	C₈×C₂₃	1	2	yes	—
149	C₄×C₈	—	4	2	no	—
150	#95	—	2	1	no	—
151	C₂₃	#7199	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	4	1	no	—
156	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
160	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	M₁₆×C₂₃	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
174	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	C₁₆×C₂₃	1	4	yes	complemented
177	C₄²	—	4	1	no	—
178	C₄²	—	4	1	no	—
179	C₁₆	—	4	2	no	—
180	C₁₆	—	4	2	no	—
181	C₂³	—	4	1	no	—
182	C₂³	—	4	1	no	—
183	C₂³	C₄×C₈×C₂₃	1	1	yes	characteristic
184	C₂³	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
185	C₂³	C₂₃×C₄⋊C₈	1	1	yes	characteristic
186	C₂³	C₂×C₁₆×C₂₃	1	2	yes	—
187	C₂³	C₂₃×M₃₂	1	2	yes	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₂₃×#95	1	1	yes	characteristic
201	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
202	C₂×C₄	C₂₃×C₂²⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	4	2	no	—
208	C₈	C₂₃×#99	1	2	yes	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂₃×#336	1	1	yes	characteristic
218	C₂²	C₄×C₁₆×C₂₃	1	1	yes	characteristic
219	C₂²	C₂₃×#346	1	1	yes	characteristic
220	C₂²	C₂₃×C₂²⋊C₁₆	1	2	yes	—
221	C₂²	C₂₃×C₄⋊C₁₆	1	2	yes	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	4	no	—
226	C₄	C₂₃×#342	1	1	yes	characteristic
227	C₄	C₂₃×#344	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	2	no	—
232	C₂	C₂₃×#1282	1	1	yes	characteristic
233	C₂	C₂₃×#1374	1	1	yes	characteristic
234	C₂	C₂₃×#1378	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂²⋊C₄
commutator	C₂²	C₄×C₁₆×C₂₃
Frattini, cosocle	C₂³×C₈	C₂²×C₂₃
socle	C₂²×C₂₃	#336
prime radical	C₂⁴×C₂₃	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂²×C₁₆×C₂₃	C₄	semidirect
13. C₂₃×C₂²⋊C₁₆	C₄	semidirect
58. C₂²⋊C₄×C₂₃	C₁₆	semidirect
63. #7199	C₂₃	direct (N × H)
116. C₂²×C₁₆	C₄×C₂₃	semidirect
123. C₂²⋊C₁₆	C₄×C₂₃	semidirect
151. C₂₃	#7199	direct (N × H)
176. C₂²⋊C₄	C₁₆×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×C₂²⋊C₁₆	2
C₈×C₂²⋊C₄×C₂₃	2
#7199	23

Subgroups — C23×#7199 #19154825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7199 #19154825