Subgroups — C₄×C₂₃×H(3,𝔽₄) #19181979

Order 5888.

2664 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2379	C₂	1	9	yes	maximal, complemented
3	C₂₃×#2375	C₂	1	6	yes	maximal, complemented
4	C₄×C₂₃×#124	C₂	1	9	yes	maximal, complemented
5	C₄×C₂₃×#120	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	complemented
7	C₂₃×H(3,𝔽₄)	C₄	1	16	yes	direct product factor
8	C₂×C₂₃×#124	C₂²	1	9	yes	—
9	C₂×C₂₃×#120	C₂²	1	6	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	3	yes	complemented
11	C₂×C₄×D₄×C₂₃	C₂²	1	9	yes	—
12	C₂₃×#391	C₂²	1	9	yes	complemented
13	C₂₃×#390	C₂²	1	9	yes	—
14	C₂₃×#388	C₂²	1	36	yes	complemented
15	C₂₃×#386	C₂²	1	18	yes	—
16	C₂₃×#385	C₂²	1	18	yes	complemented
17	C₂₃×#382	C₂²	1	9	yes	complemented
18	C₂₃×#379	C₂²	1	6	yes	complemented
19	C₄×C₂²⋊C₄×C₂₃	C₂²	1	18	yes	complemented
20	C₄³×C₂₃	C₂²	1	3	yes	complemented
21	C₂⁵×C₂₃	C₂³	1	2	yes	—
22	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
23	C₂²×D₄×C₂₃	C₂³	1	9	yes	—
24	C₂³×C₄×C₂₃	—	4	12	no	—
25	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
26	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
27	C₂₃×#124	C₂×C₄	1	72	yes	complemented
28	C₂₃×#120	C₂×C₄	1	48	yes	complemented
29	C₄×Q₈×C₂₃	—	4	6	no	—
30	C₄×D₄×C₂₃	—	4	18	no	—
31	C₂×C₄⋊C₄×C₂₃	C₂³	1	9	yes	—
32	C₂×C₄⋊C₄×C₂₃	C₂³	1	9	yes	—
33	C₂×C₂²⋊C₄×C₂₃	C₂³	1	18	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂³	1	18	yes	—
35	C₂×C₂²⋊C₄×C₂₃	C₂³	1	18	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	9	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	9	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
39	C₂₃×#95	C₂³	1	36	yes	—
40	C₂⁴×C₂₃	—	4	6	no	—
41	C₂⁴×C₂₃	—	4	6	no	—
42	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	complemented
43	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
44	C₂×Q₈×C₂₃	—	4	3	no	—
45	C₂×Q₈×C₂₃	—	4	3	no	—
46	C₂×Q₈×C₂₃	C₂²×C₄	1	12	yes	complemented
47	C₂×D₄×C₂₃	—	4	9	no	—
48	C₂×D₄×C₂₃	—	4	9	no	—
49	C₂×D₄×C₂₃	C₂²×C₄	1	36	yes	—
50	C₂²×C₄×C₂₃	—	2	36	no	—
51	C₂²×C₄×C₂₃	—	4	18	no	—
52	C₂²×C₄×C₂₃	—	4	32	no	—
53	C₂²×C₄×C₂₃	C₂⁴	1	9	yes	—
54	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
55	C₂²×C₄×C₂₃	C₂⁴	1	9	yes	—
56	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	4	9	no	—
58	C₄⋊C₄×C₂₃	—	4	9	no	—
59	C₂²⋊C₄×C₂₃	—	4	18	no	—
60	C₂²⋊C₄×C₂₃	—	4	18	no	—
61	C₂²⋊C₄×C₂₃	C₂²×C₄	1	72	yes	complemented
62	C₄²×C₂₃	—	2	18	no	—
63	C₄²×C₂₃	—	4	9	no	—
64	C₄²×C₂₃	C₂²×C₄	1	12	yes	complemented
65	C₄×H(3,𝔽₄)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	2	18	no	—
67	C₂³×C₂₃	—	2	18	no	—
68	C₂³×C₂₃	—	4	24	no	—
69	C₂³×C₂₃	—	4	8	no	—
70	C₂³×C₂₃	—	4	24	no	—
71	C₂³×C₂₃	C₂³×C₄	1	12	yes	—
72	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
73	Q₈×C₂₃	—	4	12	no	—
74	D₄×C₂₃	—	4	36	no	—
75	C₂×C₄×C₂₃	—	2	9	no	—
76	C₂×C₄×C₂₃	—	2	9	no	—
77	C₂×C₄×C₂₃	—	2	9	no	—
78	C₂×C₄×C₂₃	—	2	18	no	—
79	C₂×C₄×C₂₃	—	2	18	no	—
80	C₂×C₄×C₂₃	—	4	24	no	—
81	C₂×C₄×C₂₃	—	4	24	no	—
82	C₂×C₄×C₂₃	C₂³×C₄	1	18	yes	—
83	C₂×C₄×C₂₃	H(5,𝔽₂)	1	6	yes	—
84	C₂×H(3,𝔽₄)	C₂×C₂₃	1	1	yes	characteristic
85	#2379	C₂×C₂₃	1	9	yes	complemented
86	#2375	C₂×C₂₃	1	6	yes	complemented
87	C₄×#124	C₂×C₂₃	1	9	yes	complemented
88	C₄×#120	C₂×C₂₃	1	6	yes	complemented
89	C₂²×C₂₃	—	2	36	no	—
90	C₂²×C₂₃	—	4	32	no	—
91	C₂²×C₂₃	—	4	6	no	—
92	C₂²×C₂₃	—	4	18	no	—
93	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	3	yes	—
95	C₂²×C₂₃	#585	1	3	yes	—
96	C₄×C₂₃	—	2	18	no	—
97	C₄×C₂₃	—	4	6	no	—
98	C₄×C₂₃	—	4	9	no	—
99	C₄×C₂₃	H(3,𝔽₄)	1	4	yes	direct product factor
100	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	complemented
101	H(3,𝔽₄)	C₄×C₂₃	1	16	yes	direct product factor
102	C₂×#124	C₂²×C₂₃	1	9	yes	—
103	C₂×#120	C₂²×C₂₃	1	6	yes	—
104	C₂×C₄×Q₈	C₂²×C₂₃	1	3	yes	complemented
105	C₂×C₄×D₄	C₂²×C₂₃	1	9	yes	—
106	#391	C₂²×C₂₃	1	9	yes	complemented
107	#390	C₂²×C₂₃	1	9	yes	—
108	#388	C₂²×C₂₃	1	36	yes	complemented
109	#386	C₂²×C₂₃	1	18	yes	—
110	#385	C₂²×C₂₃	1	18	yes	complemented
111	#382	C₂²×C₂₃	1	9	yes	complemented
112	#379	C₂²×C₂₃	1	6	yes	complemented
113	C₄×C₂²⋊C₄	C₂²×C₂₃	1	18	yes	complemented
114	C₄³	C₂²×C₂₃	1	3	yes	complemented
115	C₂×C₂₃	—	4	12	no	—
116	C₂×C₂₃	C₄×H(5,𝔽₂)	1	3	yes	—
117	C₂×C₂₃	C₂×H(3,𝔽₄)	1	1	yes	characteristic
118	C₂×C₂₃	#3529	1	3	yes	—
119	C₂⁵	C₂³×C₂₃	1	2	yes	—
120	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
121	C₂²×D₄	C₂³×C₂₃	1	9	yes	—
122	C₂³×C₄	—	4	12	no	—
123	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
124	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
125	#124	C₂×C₄×C₂₃	1	72	yes	complemented
126	#120	C₂×C₄×C₂₃	1	48	yes	complemented
127	C₄×Q₈	—	4	6	no	—
128	C₄×D₄	—	4	18	no	—
129	C₂×C₄⋊C₄	C₂³×C₂₃	1	9	yes	—
130	C₂×C₄⋊C₄	C₂³×C₂₃	1	9	yes	—
131	C₂×C₂²⋊C₄	C₂³×C₂₃	1	18	yes	—
132	C₂×C₂²⋊C₄	C₂³×C₂₃	1	18	yes	—
133	C₂×C₂²⋊C₄	C₂³×C₂₃	1	18	yes	—
134	C₂×C₄²	C₂³×C₂₃	1	9	yes	—
135	C₂×C₄²	C₂³×C₂₃	1	9	yes	—
136	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
137	#95	C₂³×C₂₃	1	36	yes	—
138	C₂₃	C₄×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	—	4	6	no	—
140	C₂⁴	—	4	6	no	—
141	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	complemented
142	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
143	C₂×Q₈	—	4	3	no	—
144	C₂×Q₈	—	4	3	no	—
145	C₂×Q₈	C₂²×C₄×C₂₃	1	12	yes	complemented
146	C₂×D₄	—	4	9	no	—
147	C₂×D₄	—	4	9	no	—
148	C₂×D₄	C₂²×C₄×C₂₃	1	36	yes	—
149	C₂²×C₄	—	2	36	no	—
150	C₂²×C₄	—	4	18	no	—
151	C₂²×C₄	—	4	32	no	—
152	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
153	C₂²×C₄	C₂⁴×C₂₃	1	9	yes	—
154	C₂²×C₄	C₂⁴×C₂₃	1	9	yes	—
155	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
156	C₄⋊C₄	—	4	9	no	—
157	C₄⋊C₄	—	4	9	no	—
158	C₂²⋊C₄	—	4	18	no	—
159	C₂²⋊C₄	—	4	18	no	—
160	C₂²⋊C₄	C₂²×C₄×C₂₃	1	72	yes	complemented
161	C₄²	—	2	18	no	—
162	C₄²	—	4	9	no	—
163	C₄²	C₂²×C₄×C₂₃	1	12	yes	complemented
164	C₂³	—	2	18	no	—
165	C₂³	—	2	18	no	—
166	C₂³	—	4	24	no	—
167	C₂³	—	4	8	no	—
168	C₂³	—	4	24	no	—
169	C₂³	C₂³×C₄×C₂₃	1	12	yes	—
170	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
171	Q₈	—	4	12	no	—
172	D₄	—	4	36	no	—
173	C₂×C₄	—	2	9	no	—
174	C₂×C₄	—	2	9	no	—
175	C₂×C₄	—	2	9	no	—
176	C₂×C₄	—	2	18	no	—
177	C₂×C₄	—	2	18	no	—
178	C₂×C₄	—	4	24	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	C₂³×C₄×C₂₃	1	18	yes	—
181	C₂×C₄	C₂₃×H(5,𝔽₂)	1	6	yes	—
182	C₂²	—	2	36	no	—
183	C₂²	—	4	32	no	—
184	C₂²	—	4	6	no	—
185	C₂²	—	4	18	no	—
186	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
187	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	3	yes	—
188	C₂²	C₂₃×#585	1	3	yes	—
189	C₄	—	2	18	no	—
190	C₄	—	4	6	no	—
191	C₄	—	4	9	no	—
192	C₄	C₂₃×H(3,𝔽₄)	1	4	yes	direct product factor
193	C₂	—	4	12	no	—
194	C₂	C₄×C₂₃×H(5,𝔽₂)	1	3	yes	—
195	C₂	C₂×C₂₃×H(3,𝔽₄)	1	1	yes	characteristic
196	C₂	C₂₃×#3529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×H(3,𝔽₄)	C₂
LCS	G ▷ C₂² ▷ 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×#2379	C₂	semidirect
3. C₂₃×#2375	C₂	semidirect
4. C₄×C₂₃×#124	C₂	semidirect
5. C₄×C₂₃×#120	C₂	semidirect
6. C₂⁴×C₄×C₂₃	C₂²	semidirect
7. C₂₃×H(3,𝔽₄)	C₄	direct (N × H)
10. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
12. C₂₃×#391	C₂²	semidirect
14. C₂₃×#388	C₂²	semidirect
16. C₂₃×#385	C₂²	semidirect
17. C₂₃×#382	C₂²	semidirect
18. C₂₃×#379	C₂²	semidirect
19. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
20. C₄³×C₂₃	C₂²	semidirect
27. C₂₃×#124	C₂×C₄	semidirect
28. C₂₃×#120	C₂×C₄	semidirect
42. C₂⁴×C₂₃	C₂²×C₄	semidirect
46. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
61. C₂²⋊C₄×C₂₃	C₂²×C₄	semidirect
64. C₄²×C₂₃	C₂²×C₄	semidirect
65. C₄×H(3,𝔽₄)	C₂₃	direct (N × H)
85. #2379	C₂×C₂₃	semidirect
86. #2375	C₂×C₂₃	semidirect
87. C₄×#124	C₂×C₂₃	semidirect
88. C₄×#120	C₂×C₂₃	semidirect
99. C₄×C₂₃	H(3,𝔽₄)	direct (N × H)
100. C₂⁴×C₄	C₂²×C₂₃	semidirect
101. H(3,𝔽₄)	C₄×C₂₃	direct (N × H)
104. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
106. #391	C₂²×C₂₃	semidirect
108. #388	C₂²×C₂₃	semidirect
110. #385	C₂²×C₂₃	semidirect
111. #382	C₂²×C₂₃	semidirect
112. #379	C₂²×C₂₃	semidirect
113. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
114. C₄³	C₂²×C₂₃	semidirect
125. #124	C₂×C₄×C₂₃	semidirect
126. #120	C₂×C₄×C₂₃	semidirect
138. C₂₃	C₄×H(3,𝔽₄)	direct (N × H)
141. C₂⁴	C₂²×C₄×C₂₃	semidirect
145. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
160. C₂²⋊C₄	C₂²×C₄×C₂₃	semidirect
163. C₄²	C₂²×C₄×C₂₃	semidirect
192. C₄	C₂₃×H(3,𝔽₄)	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×H(3,𝔽₄)	2
C₂₃×#2379	2
C₂₃×#2375	2
C₄×C₂₃×#124	2
C₄×C₂₃×#120	2
C₄×H(3,𝔽₄)	23

Subgroups — C4×C23×H(3,𝔽4) #19181979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₃×H(3,𝔽₄) #19181979