Subgroups — C₂₃×#36608 #19184234

Order 5888.

1626 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2862	C₂	1	3	yes	maximal
2	C₂₃×#2581	C₂	1	6	yes	maximal
3	C₂₃×#2514	C₂	1	12	yes	maximal
4	C₂₃×#2356	C₂	1	6	yes	maximal
5	C₄×C₂₃×C₄⋊Q₈	C₂	1	3	yes	maximal
6	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
8	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	—
9	C₂×C₄×Q₈×C₂₃	C₂²	1	6	yes	—
10	C₂₃×#401	C₂²	1	8	yes	—
11	C₂₃×#400	C₂²	1	12	yes	—
12	C₂₃×#395	C₂²	1	4	yes	—
13	C₂₃×#391	C₂²	1	6	yes	—
14	C₂₃×#391	C₂²	1	12	yes	—
15	C₂₃×#391	C₂²	1	12	yes	—
16	C₂₃×#389	C₂²	1	12	yes	—
17	C₂₃×#387	C₂²	1	24	yes	—
18	C₂₃×#384	C₂²	1	1	yes	characteristic
19	C₂₃×#383	C₂²	1	12	yes	—
20	C₂₃×#382	C₂²	1	6	yes	—
21	C₂₃×#382	C₂²	1	6	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₂²	1	6	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
25	C₂₃×#376	C₂²	1	4	yes	—
26	C₄³×C₂₃	C₂²	1	3	yes	—
27	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
28	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂₃×C₄⋊Q₈	—	2	12	no	—
30	C₄×Q₈×C₂₃	—	2	12	no	—
31	C₄×Q₈×C₂₃	—	2	24	no	—
32	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
36	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
43	C₂₃×#95	C₂³	1	1	yes	characteristic
44	C₂₃×#95	C₂³	1	12	yes	—
45	C₂₃×#95	C₂³	1	3	yes	—
46	C₂₃×#95	C₂³	1	2	yes	—
47	C₂₃×#95	C₂³	1	24	yes	—
48	C₂₃×#95	C₂³	1	24	yes	—
49	C₂₃×#95	C₂³	1	12	yes	—
50	C₂×Q₈×C₂₃	—	2	6	no	—
51	C₂×Q₈×C₂₃	—	2	12	no	—
52	C₂×Q₈×C₂₃	—	2	12	no	—
53	C₂×Q₈×C₂₃	—	2	12	no	—
54	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
57	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
60	C₄⋊C₄×C₂₃	—	2	12	no	—
61	C₄⋊C₄×C₂₃	—	2	6	no	—
62	C₄⋊C₄×C₂₃	—	2	12	no	—
63	C₄⋊C₄×C₂₃	—	2	24	no	—
64	C₄⋊C₄×C₂₃	—	2	24	no	—
65	C₄²×C₂₃	—	2	12	no	—
66	C₄²×C₂₃	—	2	24	no	—
67	C₄²×C₂₃	—	2	12	no	—
68	C₄²×C₂₃	—	2	6	no	—
69	C₄²×C₂₃	#41	1	12	yes	—
70	C₄²×C₂₃	#41	1	4	yes	—
71	#36608	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
73	Q₈×C₂₃	—	2	8	no	—
74	Q₈×C₂₃	—	4	24	no	—
75	C₂×C₄×C₂₃	—	2	6	no	—
76	C₂×C₄×C₂₃	—	2	6	no	—
77	C₂×C₄×C₂₃	—	2	12	no	—
78	C₂×C₄×C₂₃	—	2	6	no	—
79	C₂×C₄×C₂₃	—	2	12	no	—
80	C₂×C₄×C₂₃	—	2	12	no	—
81	C₂×C₄×C₂₃	—	2	6	no	—
82	C₂×C₄×C₂₃	—	2	6	no	—
83	C₂×C₄×C₂₃	—	2	12	no	—
84	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
85	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
86	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
87	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
88	C₂×C₄×C₂₃	2₋¹⁺⁴	1	6	yes	—
89	#2862	C₂×C₂₃	1	3	yes	—
90	#2581	C₂×C₂₃	1	6	yes	—
91	#2514	C₂×C₂₃	1	12	yes	—
92	#2356	C₂×C₂₃	1	6	yes	—
93	C₄×C₄⋊Q₈	C₂×C₂₃	1	3	yes	—
94	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
95	C₂²×C₂₃	C₂²×#41	1	3	yes	—
96	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
97	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	3	yes	—
98	C₄×C₂₃	—	2	6	no	—
99	C₄×C₂₃	—	2	6	no	—
100	C₄×C₂₃	—	4	4	no	—
101	C₄×C₂₃	—	4	12	no	—
102	C₄×C₂₃	—	4	6	no	—
103	C₄×C₂₃	#556	1	12	yes	—
104	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
105	C₂×C₄×Q₈	C₂²×C₂₃	1	6	yes	—
106	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	—
107	#401	C₂²×C₂₃	1	8	yes	—
108	#400	C₂²×C₂₃	1	12	yes	—
109	#395	C₂²×C₂₃	1	4	yes	—
110	#391	C₂²×C₂₃	1	6	yes	—
111	#391	C₂²×C₂₃	1	12	yes	—
112	#391	C₂²×C₂₃	1	12	yes	—
113	#389	C₂²×C₂₃	1	12	yes	—
114	#387	C₂²×C₂₃	1	24	yes	—
115	#384	C₂²×C₂₃	1	1	yes	characteristic
116	#383	C₂²×C₂₃	1	12	yes	—
117	#382	C₂²×C₂₃	1	6	yes	—
118	#382	C₂²×C₂₃	1	6	yes	—
119	C₄×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
120	C₄×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
121	C₄×C₄⋊C₄	C₂²×C₂₃	1	6	yes	—
122	#376	C₂²×C₂₃	1	4	yes	—
123	C₄³	C₂²×C₂₃	1	3	yes	—
124	C₂×C₂₃	#3444	1	1	yes	characteristic
125	C₂×C₂₃	C₂×#556	1	3	yes	—
126	C₂×C₂₃	#3509	1	3	yes	—
127	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
128	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
129	C₄⋊Q₈	—	2	12	no	—
130	C₄×Q₈	—	2	12	no	—
131	C₄×Q₈	—	2	24	no	—
132	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
133	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
134	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
135	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
136	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
137	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
138	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
139	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
140	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
141	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
142	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
143	#95	C₂³×C₂₃	1	1	yes	characteristic
144	#95	C₂³×C₂₃	1	12	yes	—
145	#95	C₂³×C₂₃	1	3	yes	—
146	#95	C₂³×C₂₃	1	2	yes	—
147	#95	C₂³×C₂₃	1	24	yes	—
148	#95	C₂³×C₂₃	1	24	yes	—
149	#95	C₂³×C₂₃	1	12	yes	—
150	C₂₃	#36608	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×Q₈	—	2	6	no	—
152	C₂×Q₈	—	2	12	no	—
153	C₂×Q₈	—	2	12	no	—
154	C₂×Q₈	—	2	12	no	—
155	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
156	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
157	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
158	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
159	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
160	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
161	C₄⋊C₄	—	2	12	no	—
162	C₄⋊C₄	—	2	6	no	—
163	C₄⋊C₄	—	2	12	no	—
164	C₄⋊C₄	—	2	24	no	—
165	C₄⋊C₄	—	2	24	no	—
166	C₄²	—	2	12	no	—
167	C₄²	—	2	24	no	—
168	C₄²	—	2	12	no	—
169	C₄²	—	2	6	no	—
170	C₄²	#41×C₂₃	1	12	yes	—
171	C₄²	#41×C₂₃	1	4	yes	—
172	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	Q₈	—	4	24	no	—
175	C₂×C₄	—	2	6	no	—
176	C₂×C₄	—	2	6	no	—
177	C₂×C₄	—	2	12	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	12	no	—
184	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
185	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
186	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
187	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
188	C₂×C₄	C₂₃×2₋¹⁺⁴	1	6	yes	—
189	C₂²	C₂²×#41×C₂₃	1	3	yes	—
190	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
191	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	3	yes	—
192	C₄	—	2	6	no	—
193	C₄	—	2	6	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	12	no	—
196	C₄	—	4	6	no	—
197	C₄	C₂₃×#556	1	12	yes	—
198	C₂	C₂₃×#3444	1	1	yes	characteristic
199	C₂	C₂×C₂₃×#556	1	3	yes	—
200	C₂	C₂₃×#3509	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
71. #36608	C₂₃	direct (N × H)
150. C₂₃	#36608	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2862	2
C₂₃×#2581	2
C₂₃×#2514	2
C₂₃×#2356	2
C₄×C₂₃×C₄⋊Q₈	2
C₄²×Q₈×C₂₃	2
#36608	23

Subgroups — C23×#36608 #19184234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36608 #19184234