Subgroups — C₂₃×#11355 #19158981

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#343	C₂	1	4	yes	maximal
3	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#382	—	2	2	no	—
10	C₂₃×#382	—	2	2	no	—
11	C₂×C₂₃×#95	C₄	1	4	yes	—
12	C₂₃×#343	—	2	8	no	—
13	C₂₃×#336	—	2	2	no	—
14	C₂₃×#336	—	2	2	no	—
15	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	2	no	—
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	4	no	—
21	C₂×M₁₆×C₂₃	—	2	4	no	—
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	—	2	4	no	—
24	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
25	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄²×C₂₃	—	2	4	no	—
33	C₂×C₄²×C₂₃	D₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂₃×#95	—	2	8	no	—
36	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	—	2	4	no	—
38	C₂²×C₄×C₂₃	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	8	no	—
40	C₂²×C₄×C₂₃	—	4	4	no	—
41	C₂²×C₄×C₂₃	—	4	8	no	—
42	C₂²×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₂₃	#41	1	2	yes	—
48	C₂²×C₄×C₂₃	#41	1	4	yes	—
49	C₂²×C₄×C₂₃	#41	1	2	yes	—
50	M₁₆×C₂₃	—	2	8	no	—
51	M₁₆×C₂₃	—	2	8	no	—
52	M₁₆×C₂₃	—	4	4	no	—
53	M₁₆×C₂₃	—	4	4	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	—	2	4	no	—
56	C₂×C₈×C₂₃	—	4	2	no	—
57	C₂×C₈×C₂₃	—	4	2	no	—
58	C₂×C₈×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₄⋊C₄×C₂₃	—	4	4	no	—
61	C₄⋊C₄×C₂₃	—	4	2	no	—
62	C₄⋊C₄×C₂₃	—	4	2	no	—
63	C₄²×C₂₃	—	2	4	no	—
64	C₄²×C₂₃	—	2	4	no	—
65	#11355	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	2	4	no	—
67	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₂₃	#117	1	2	yes	—
69	C₂³×C₂₃	C₄×D₄	1	2	yes	—
70	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	8	no	—
73	C₂×C₄×C₂₃	—	4	8	no	—
74	C₂×C₄×C₂₃	—	4	4	no	—
75	C₂×C₄×C₂₃	—	4	8	no	—
76	C₂×C₄×C₂₃	—	8	8	no	—
77	C₂×C₄×C₂₃	#121	1	2	yes	—
78	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
79	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₄×C₂₃	#123	1	2	yes	—
81	C₂×C₄×C₂₃	#123	1	4	yes	—
82	C₂×C₄×C₂₃	#123	1	2	yes	—
83	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#125	1	4	yes	—
86	C₂×C₄×C₂₃	#126	1	4	yes	—
87	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
88	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
89	C₈×C₂₃	—	4	4	no	—
90	C₈×C₂₃	—	4	4	no	—
91	C₂×#382	C₂×C₂₃	1	1	yes	characteristic
92	C₂×#343	C₂×C₂₃	1	4	yes	—
93	#1728	C₂×C₂₃	1	1	yes	characteristic
94	#1728	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	—	2	4	no	—
96	C₂²×C₂₃	—	2	4	no	—
97	C₂²×C₂₃	—	2	4	no	—
98	C₂²×C₂₃	#381	1	1	yes	characteristic
99	C₂²×C₂₃	#387	1	4	yes	—
100	C₂²×C₂₃	#390	1	1	yes	characteristic
101	C₂²×C₂₃	#391	1	1	yes	characteristic
102	C₂²×C₂₃	#413	1	4	yes	—
103	C₄×C₂₃	—	4	4	no	—
104	C₄×C₂₃	—	8	4	no	—
105	C₄×C₂₃	#397	1	2	yes	—
106	C₄×C₂₃	#398	1	2	yes	—
107	C₄×C₂₃	#399	1	2	yes	—
108	C₄×C₂₃	#400	1	2	yes	—
109	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
110	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
111	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
112	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
113	#382	—	2	2	no	—
114	#382	—	2	2	no	—
115	C₂×#95	C₄×C₂₃	1	4	yes	—
116	#343	—	2	8	no	—
117	#336	—	2	2	no	—
118	#336	—	2	2	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	#1445	1	1	yes	characteristic
121	C₂×C₂₃	#1822	1	2	yes	—
122	C₂×C₂₃	#1967	1	2	yes	—
123	C₂×C₂₃	#1998	1	2	yes	—
124	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
125	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
126	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	—	2	4	no	—
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
134	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	4	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	D₄×C₂₃	1	2	yes	—
143	C₂×C₄²	D₄×C₂₃	1	2	yes	—
144	#95	—	2	8	no	—
145	C₂₃	#11355	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	—	4	8	no	—
152	C₂²×C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
153	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
157	C₂²×C₄	#41×C₂₃	1	2	yes	—
158	C₂²×C₄	#41×C₂₃	1	4	yes	—
159	C₂²×C₄	#41×C₂₃	1	2	yes	—
160	M₁₆	—	2	8	no	—
161	M₁₆	—	2	8	no	—
162	M₁₆	—	4	4	no	—
163	M₁₆	—	4	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	4	no	—
176	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
177	C₂³	C₂₃×#117	1	2	yes	—
178	C₂³	C₄×D₄×C₂₃	1	2	yes	—
179	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	8	8	no	—
186	C₂×C₄	C₂₃×#121	1	2	yes	—
187	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
188	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
189	C₂×C₄	C₂₃×#123	1	2	yes	—
190	C₂×C₄	C₂₃×#123	1	4	yes	—
191	C₂×C₄	C₂₃×#123	1	2	yes	—
192	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
193	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
194	C₂×C₄	C₂₃×#125	1	4	yes	—
195	C₂×C₄	C₂₃×#126	1	4	yes	—
196	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
197	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
198	C₈	—	4	4	no	—
199	C₈	—	4	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	C₂₃×#381	1	1	yes	characteristic
204	C₂²	C₂₃×#387	1	4	yes	—
205	C₂²	C₂₃×#390	1	1	yes	characteristic
206	C₂²	C₂₃×#391	1	1	yes	characteristic
207	C₂²	C₂₃×#413	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₂₃×#397	1	2	yes	—
211	C₄	C₂₃×#398	1	2	yes	—
212	C₄	C₂₃×#399	1	2	yes	—
213	C₄	C₂₃×#400	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	C₂₃×#1445	1	1	yes	characteristic
216	C₂	C₂₃×#1822	1	2	yes	—
217	C₂	C₂₃×#1967	1	2	yes	—
218	C₂	C₂₃×#1998	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
65. #11355	C₂₃	direct (N × H)
145. C₂₃	#11355	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×#382	2
C₂×C₂₃×#343	2
C₂₃×#1728	2
#11355	23

Subgroups — C23×#11355 #19158981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11355 #19158981