Subgroups — C₂₃×#33999 #19181625

Order 5888.

3598 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₂₃	C₂	1	3	yes	maximal
2	C₂₃×#2276	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2275	C₂	1	12	yes	maximal, complemented
4	C₄×C₂₃×#117	C₂	1	9	yes	maximal, complemented
5	C₂×C₂₃×#376	C₂	1	3	yes	maximal, complemented
6	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₄×#41×C₂₃	C₄	1	48	yes	complemented
8	C₂×C₄×Q₈×C₂₃	C₂²	1	6	yes	—
9	C₂×C₄×D₄×C₂₃	C₂²	1	18	yes	—
10	C₂×C₂₃×#117	C₂²	1	9	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	9	yes	—
12	C₄×C₄⋊C₄×C₂₃	C₂²	1	36	yes	—
13	C₄×C₂²⋊C₄×C₂₃	C₂²	1	36	yes	—
14	C₂₃×#376	C₂²	1	1	yes	characteristic
15	C₂₃×#376	C₂²	1	9	yes	—
16	C₂₃×#376	C₂²	1	12	yes	—
17	C₂×C₂₃×#95	C₂²	1	12	yes	—
18	C₄³×C₂₃	C₂²	1	6	yes	—
19	C₂×#41×C₂₃	C₂×C₄	1	24	yes	—
20	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
22	C₂³×C₄×C₂₃	C₂³	1	18	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
24	C₄×Q₈×C₂₃	C₂×C₄	1	48	yes	complemented
25	C₄×D₄×C₂₃	C₂×C₄	1	144	yes	complemented
26	C₂₃×#117	C₂×C₄	1	72	yes	complemented
27	C₂×C₄⋊C₄×C₂₃	C₂³	1	18	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₂³	1	18	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	72	yes	complemented
30	C₂×C₄²×C₂₃	C₂³	1	24	yes	—
31	C₂×C₄²×C₂₃	C₂³	1	18	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	9	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
34	C₂₃×#95	C₂³	1	4	yes	—
35	C₂₃×#95	C₂³	1	36	yes	—
36	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
37	#41×C₂₃	C₄²	1	64	yes	complemented
38	C₂×Q₈×C₂₃	C₂²×C₄	1	12	yes	—
39	C₂×D₄×C₂₃	C₂²×C₄	1	36	yes	—
40	C₂²×C₄×C₂₃	—	4	36	no	—
41	C₂²×C₄×C₂₃	C₂²×C₄	1	72	yes	—
42	C₂²×C₄×C₂₃	C₂²×C₄	1	36	yes	—
43	C₂²×C₄×C₂₃	C₂⁴	1	18	yes	—
44	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
45	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₂₃	C₂²×C₄	1	72	yes	—
48	C₂²⋊C₄×C₂₃	C₂²×C₄	1	72	yes	—
49	C₄²×C₂₃	—	4	24	no	—
50	C₄²×C₂₃	C₂²×C₄	1	12	yes	—
51	C₄²×C₂₃	C₂²×C₄	1	36	yes	—
52	C₄²×C₂₃	C₂²×C₄	1	72	yes	—
53	#33999	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₂₃	—	2	12	no	—
55	C₂³×C₂₃	C₂³×C₄	1	18	yes	—
56	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
57	Q₈×C₂₃	C₂×C₄²	1	16	yes	—
58	D₄×C₂₃	C₂×C₄²	1	48	yes	—
59	C₂×C₄×C₂₃	—	2	36	no	—
60	C₂×C₄×C₂₃	—	2	12	no	—
61	C₂×C₄×C₂₃	—	4	72	no	—
62	C₂×C₄×C₂₃	C₂×C₄²	1	48	yes	—
63	C₂×C₄×C₂₃	C₂³×C₄	1	12	yes	—
64	C₂×C₄×C₂₃	C₂³×C₄	1	36	yes	—
65	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
66	C₂×C₄×C₂₃	C₂³×C₄	1	18	yes	—
67	C₂×C₄×#41	C₂×C₂₃	1	3	yes	—
68	#2276	C₂×C₂₃	1	4	yes	complemented
69	#2275	C₂×C₂₃	1	12	yes	complemented
70	C₄×#117	C₂×C₂₃	1	9	yes	complemented
71	C₂×#376	C₂×C₂₃	1	3	yes	complemented
72	C₂²×C₂₃	—	2	36	no	—
73	C₂²×C₂₃	C₂²×C₄²	1	12	yes	—
74	C₂²×C₂₃	C₂⁴×C₄	1	3	yes	—
75	C₂²×C₂₃	#585	1	4	yes	—
76	C₄×C₂₃	—	2	12	no	—
77	C₄×C₂₃	—	2	36	no	—
78	C₄×C₂₃	C₂²×C₄²	1	4	yes	—
79	C₄×C₂₃	C₂²×C₄²	1	12	yes	—
80	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
81	C₄×#41	C₄×C₂₃	1	48	yes	complemented
82	C₂×C₄×Q₈	C₂²×C₂₃	1	6	yes	—
83	C₂×C₄×D₄	C₂²×C₂₃	1	18	yes	—
84	C₂×#117	C₂²×C₂₃	1	9	yes	—
85	C₂²×C₄²	C₂²×C₂₃	1	9	yes	—
86	C₄×C₄⋊C₄	C₂²×C₂₃	1	36	yes	—
87	C₄×C₂²⋊C₄	C₂²×C₂₃	1	36	yes	—
88	#376	C₂²×C₂₃	1	1	yes	characteristic
89	#376	C₂²×C₂₃	1	9	yes	—
90	#376	C₂²×C₂₃	1	12	yes	—
91	C₂×#95	C₂²×C₂₃	1	12	yes	—
92	C₄³	C₂²×C₂₃	1	6	yes	—
93	C₂×C₂₃	—	2	12	no	—
94	C₂×C₂₃	C₂³×C₄²	1	1	yes	characteristic
95	C₂×C₂₃	#3428	1	6	yes	—
96	C₂×#41	C₂×C₄×C₂₃	1	24	yes	—
97	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
98	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
99	C₂³×C₄	C₂³×C₂₃	1	18	yes	—
100	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
101	C₄×Q₈	C₂×C₄×C₂₃	1	48	yes	complemented
102	C₄×D₄	C₂×C₄×C₂₃	1	144	yes	complemented
103	#117	C₂×C₄×C₂₃	1	72	yes	complemented
104	C₂×C₄⋊C₄	C₂³×C₂₃	1	18	yes	—
105	C₂×C₂²⋊C₄	C₂³×C₂₃	1	18	yes	—
106	C₂×C₄²	C₂×C₄×C₂₃	1	72	yes	complemented
107	C₂×C₄²	C₂³×C₂₃	1	24	yes	—
108	C₂×C₄²	C₂³×C₂₃	1	9	yes	—
109	C₂×C₄²	C₂³×C₂₃	1	18	yes	—
110	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
111	#95	C₂³×C₂₃	1	4	yes	—
112	#95	C₂³×C₂₃	1	36	yes	—
113	C₂₃	#33999	1	1	yes	characteristic, direct product factor, Sylow
114	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
115	#41	C₄²×C₂₃	1	64	yes	complemented
116	C₂×Q₈	C₂²×C₄×C₂₃	1	12	yes	—
117	C₂×D₄	C₂²×C₄×C₂₃	1	36	yes	—
118	C₂²×C₄	—	4	36	no	—
119	C₂²×C₄	C₂²×C₄×C₂₃	1	72	yes	—
120	C₂²×C₄	C₂²×C₄×C₂₃	1	36	yes	—
121	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
122	C₂²×C₄	C₂⁴×C₂₃	1	18	yes	—
123	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
125	C₄⋊C₄	C₂²×C₄×C₂₃	1	72	yes	—
126	C₂²⋊C₄	C₂²×C₄×C₂₃	1	72	yes	—
127	C₄²	—	4	24	no	—
128	C₄²	C₂²×C₄×C₂₃	1	12	yes	—
129	C₄²	C₂²×C₄×C₂₃	1	36	yes	—
130	C₄²	C₂²×C₄×C₂₃	1	72	yes	—
131	C₂³	—	2	12	no	—
132	C₂³	C₂³×C₄×C₂₃	1	18	yes	—
133	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
134	Q₈	C₂×C₄²×C₂₃	1	16	yes	—
135	D₄	C₂×C₄²×C₂₃	1	48	yes	—
136	C₂×C₄	—	2	36	no	—
137	C₂×C₄	—	2	12	no	—
138	C₂×C₄	—	4	72	no	—
139	C₂×C₄	C₂×C₄²×C₂₃	1	48	yes	—
140	C₂×C₄	C₂³×C₄×C₂₃	1	12	yes	—
141	C₂×C₄	C₂³×C₄×C₂₃	1	36	yes	—
142	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
143	C₂×C₄	C₂³×C₄×C₂₃	1	18	yes	—
144	C₂²	—	2	36	no	—
145	C₂²	C₂²×C₄²×C₂₃	1	12	yes	—
146	C₂²	C₂⁴×C₄×C₂₃	1	3	yes	—
147	C₂²	C₂₃×#585	1	4	yes	—
148	C₄	—	2	12	no	—
149	C₄	—	2	36	no	—
150	C₄	C₂²×C₄²×C₂₃	1	4	yes	—
151	C₄	C₂²×C₄²×C₂₃	1	12	yes	—
152	C₂	—	2	12	no	—
153	C₂	C₂³×C₄²×C₂₃	1	1	yes	characteristic
154	C₂	C₂₃×#3428	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂	C₂³×C₄²×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2276	C₂	semidirect
3. C₂₃×#2275	C₂	semidirect
4. C₄×C₂₃×#117	C₂	semidirect
5. C₂×C₂₃×#376	C₂	semidirect
7. C₄×#41×C₂₃	C₄	semidirect
24. C₄×Q₈×C₂₃	C₂×C₄	semidirect
25. C₄×D₄×C₂₃	C₂×C₄	semidirect
26. C₂₃×#117	C₂×C₄	semidirect
29. C₂×C₄²×C₂₃	C₂×C₄	semidirect
37. #41×C₂₃	C₄²	semidirect
53. #33999	C₂₃	direct (N × H)
68. #2276	C₂×C₂₃	semidirect
69. #2275	C₂×C₂₃	semidirect
70. C₄×#117	C₂×C₂₃	semidirect
71. C₂×#376	C₂×C₂₃	semidirect
81. C₄×#41	C₄×C₂₃	semidirect
101. C₄×Q₈	C₂×C₄×C₂₃	semidirect
102. C₄×D₄	C₂×C₄×C₂₃	semidirect
103. #117	C₂×C₄×C₂₃	semidirect
106. C₂×C₄²	C₂×C₄×C₂₃	semidirect
113. C₂₃	#33999	direct (N × H)
115. #41	C₄²×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₂₃	2
C₂₃×#2276	2
C₂₃×#2275	2
C₄×C₂₃×#117	2
C₂×C₂₃×#376	2
#33999	23

Subgroups — C23×#33999 #19181625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33999 #19181625