Subgroups — C₈×C₂₃×H(5,𝔽₂) #19207929

Order 5888.

2422 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2988	C₂	1	6	yes	maximal, complemented
3	C₂₃×#2972	C₂	1	9	yes	maximal, complemented
4	C₈×#41×C₂₃	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×D₄×C₂₃	C₂	1	9	yes	maximal, complemented
6	C₂×C₂₃×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	6	yes	complemented
8	C₂₃×#520	C₄	1	6	yes	—
9	C₂₃×#518	C₄	1	9	yes	—
10	C₄×#41×C₂₃	C₂²	1	6	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	9	yes	—
12	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
13	C₈×Q₈×C₂₃	C₂²	1	6	yes	complemented
14	C₈×D₄×C₂₃	C₂²	1	18	yes	complemented
15	C₈×D₄×C₂₃	C₂²	1	36	yes	complemented
16	C₈×D₄×C₂₃	C₂²	1	18	yes	complemented
17	C₂₃×#431	C₂²	1	18	yes	complemented
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	9	yes	complemented
19	C₂×C₂₃×C₂²⋊C₈	C₂²	1	18	yes	complemented
20	C₂×C₄×C₈×C₂₃	C₂²	1	9	yes	complemented
21	C₂₃×H(5,𝔽₂)	C₈	1	16	yes	direct product factor
22	C₂×#41×C₂₃	C₂×C₄	1	6	yes	—
23	C₂²×D₄×C₂₃	C₂×C₄	1	9	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
25	C₂²×C₈×C₂₃	—	4	12	no	—
26	C₂²×C₈×C₂₃	C₂³	1	9	yes	—
27	C₂²×C₈×C₂₃	C₂³	1	18	yes	—
28	C₂²×C₈×C₂₃	C₂³	1	9	yes	—
29	C₄×Q₈×C₂₃	C₂×C₄	1	6	yes	—
30	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
31	C₄×D₄×C₂₃	C₂×C₄	1	36	yes	—
32	C₄×D₄×C₂₃	C₂×C₄	1	18	yes	—
33	C₄×D₄×C₂₃	C₂³	1	18	yes	—
34	C₂₃×#117	C₂×C₄	1	18	yes	—
35	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	9	yes	—
36	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	18	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	9	yes	—
38	C₂₃×C₄⋊C₈	C₂³	1	18	yes	—
39	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
40	C₂₃×C₂²⋊C₈	C₂³	1	36	yes	—
41	C₄×C₈×C₂₃	C₂³	1	18	yes	—
42	C₄×C₈×C₂₃	C₂³	1	6	yes	—
43	C₂⁴×C₂₃	C₂²×C₄	1	6	yes	—
44	#41×C₂₃	C₂×C₈	1	48	yes	complemented
45	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
46	C₂×D₄×C₂₃	C₂×C₈	1	72	yes	complemented
47	C₂×D₄×C₂₃	C₂²×C₄	1	18	yes	—
48	C₂²×C₄×C₂₃	—	4	6	no	—
49	C₂²×C₄×C₂₃	—	4	6	no	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	9	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	18	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	9	yes	—
53	C₂²×C₄×C₂₃	C₂⁴	1	9	yes	—
54	C₂×C₈×C₂₃	—	2	18	no	—
55	C₂×C₈×C₂₃	—	4	18	no	—
56	C₂×C₈×C₂₃	—	4	9	no	—
57	C₂×C₈×C₂₃	C₂⁴	1	9	yes	—
58	C₂×C₈×C₂₃	C₂⁴	1	6	yes	—
59	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂²×C₄	1	18	yes	—
61	C₄⋊C₄×C₂₃	C₂²×C₄	1	6	yes	—
62	C₂²⋊C₄×C₂₃	C₂²×C₄	1	36	yes	—
63	C₄²×C₂₃	C₂²×C₄	1	18	yes	—
64	C₄²×C₂₃	C₂⁴	1	6	yes	—
65	C₈×H(5,𝔽₂)	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₂₃	—	4	6	no	—
67	C₂³×C₂₃	—	4	6	no	—
68	C₂³×C₂₃	C₂²×C₈	1	24	yes	complemented
69	C₂³×C₂₃	C₂³×C₄	1	9	yes	—
70	Q₈×C₂₃	C₂²×C₈	1	8	yes	complemented
71	D₄×C₂₃	C₂²×C₈	1	72	yes	complemented
72	C₂×C₄×C₂₃	—	2	9	no	—
73	C₂×C₄×C₂₃	—	2	9	no	—
74	C₂×C₄×C₂₃	—	4	18	no	—
75	C₂×C₄×C₂₃	—	4	9	no	—
76	C₂×C₄×C₂₃	C₂²×C₈	1	36	yes	complemented
77	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
78	C₂×C₄×C₂₃	C₂³×C₄	1	9	yes	—
79	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
80	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
81	C₈×C₂₃	—	2	9	no	—
82	C₈×C₂₃	—	2	6	no	—
83	C₈×C₂₃	H(5,𝔽₂)	1	2	yes	direct product factor
84	C₄×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic
85	#2988	C₂×C₂₃	1	6	yes	complemented
86	#2972	C₂×C₂₃	1	9	yes	complemented
87	C₈×#41	C₂×C₂₃	1	6	yes	complemented
88	C₂×C₈×D₄	C₂×C₂₃	1	9	yes	complemented
89	C₂²×C₂₃	—	2	9	no	—
90	C₂²×C₂₃	—	2	9	no	—
91	C₂²×C₂₃	—	4	24	no	—
92	C₂²×C₂₃	C₂³×C₈	1	18	yes	—
93	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
94	C₄×C₂₃	—	2	9	no	—
95	C₄×C₂₃	—	2	6	no	—
96	C₄×C₂₃	C₂³×C₈	1	12	yes	—
97	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₂₃	#585	1	1	yes	characteristic
99	C₂×H(5,𝔽₂)	C₄×C₂₃	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₂₃	1	6	yes	complemented
101	#520	C₄×C₂₃	1	6	yes	—
102	#518	C₄×C₂₃	1	9	yes	—
103	C₄×#41	C₂²×C₂₃	1	6	yes	—
104	C₂×C₄×D₄	C₂²×C₂₃	1	9	yes	—
105	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
106	C₈×Q₈	C₂²×C₂₃	1	6	yes	complemented
107	C₈×D₄	C₂²×C₂₃	1	18	yes	complemented
108	C₈×D₄	C₂²×C₂₃	1	36	yes	complemented
109	C₈×D₄	C₂²×C₂₃	1	18	yes	complemented
110	#431	C₂²×C₂₃	1	18	yes	complemented
111	C₂×C₄⋊C₈	C₂²×C₂₃	1	9	yes	complemented
112	C₂×C₂²⋊C₈	C₂²×C₂₃	1	18	yes	complemented
113	C₂×C₄×C₈	C₂²×C₂₃	1	9	yes	complemented
114	C₂×C₂₃	—	2	18	no	—
115	C₂×C₂₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×C₂₃	C₂⁴×C₈	1	1	yes	characteristic
117	C₂×C₂₃	#3573	1	1	yes	characteristic
118	H(5,𝔽₂)	C₈×C₂₃	1	16	yes	direct product factor
119	C₂×#41	C₂×C₄×C₂₃	1	6	yes	—
120	C₂²×D₄	C₂×C₄×C₂₃	1	9	yes	—
121	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
122	C₂²×C₈	—	4	12	no	—
123	C₂²×C₈	C₂³×C₂₃	1	9	yes	—
124	C₂²×C₈	C₂³×C₂₃	1	18	yes	—
125	C₂²×C₈	C₂³×C₂₃	1	9	yes	—
126	C₄×Q₈	C₂×C₄×C₂₃	1	6	yes	—
127	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
128	C₄×D₄	C₂×C₄×C₂₃	1	36	yes	—
129	C₄×D₄	C₂×C₄×C₂₃	1	18	yes	—
130	C₄×D₄	C₂³×C₂₃	1	18	yes	—
131	#117	C₂×C₄×C₂₃	1	18	yes	—
132	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	9	yes	—
133	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	18	yes	—
134	C₂×C₄²	C₂³×C₂₃	1	9	yes	—
135	C₄⋊C₈	C₂³×C₂₃	1	18	yes	—
136	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
137	C₂²⋊C₈	C₂³×C₂₃	1	36	yes	—
138	C₄×C₈	C₂³×C₂₃	1	6	yes	—
139	C₄×C₈	C₂³×C₂₃	1	18	yes	—
140	C₂₃	C₈×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	C₂²×C₄×C₂₃	1	6	yes	—
142	#41	C₂×C₈×C₂₃	1	48	yes	complemented
143	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
144	C₂×D₄	C₂×C₈×C₂₃	1	72	yes	complemented
145	C₂×D₄	C₂²×C₄×C₂₃	1	18	yes	—
146	C₂²×C₄	—	4	6	no	—
147	C₂²×C₄	—	4	6	no	—
148	C₂²×C₄	C₂²×C₄×C₂₃	1	9	yes	—
149	C₂²×C₄	C₂²×C₄×C₂₃	1	18	yes	—
150	C₂²×C₄	C₂²×C₄×C₂₃	1	9	yes	—
151	C₂²×C₄	C₂⁴×C₂₃	1	9	yes	—
152	C₂×C₈	—	2	18	no	—
153	C₂×C₈	—	4	18	no	—
154	C₂×C₈	—	4	9	no	—
155	C₂×C₈	C₂⁴×C₂₃	1	6	yes	—
156	C₂×C₈	C₂⁴×C₂₃	1	9	yes	—
157	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	C₂²×C₄×C₂₃	1	18	yes	—
159	C₄⋊C₄	C₂²×C₄×C₂₃	1	6	yes	—
160	C₂²⋊C₄	C₂²×C₄×C₂₃	1	36	yes	—
161	C₄²	C₂²×C₄×C₂₃	1	18	yes	—
162	C₄²	C₂⁴×C₂₃	1	6	yes	—
163	C₂³	—	4	6	no	—
164	C₂³	—	4	6	no	—
165	C₂³	C₂²×C₈×C₂₃	1	24	yes	complemented
166	C₂³	C₂³×C₄×C₂₃	1	9	yes	—
167	Q₈	C₂²×C₈×C₂₃	1	8	yes	complemented
168	D₄	C₂²×C₈×C₂₃	1	72	yes	complemented
169	C₂×C₄	—	2	9	no	—
170	C₂×C₄	—	2	9	no	—
171	C₂×C₄	—	4	18	no	—
172	C₂×C₄	—	4	9	no	—
173	C₂×C₄	C₂²×C₈×C₂₃	1	36	yes	complemented
174	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
175	C₂×C₄	C₂³×C₄×C₂₃	1	9	yes	—
176	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
177	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
178	C₈	—	2	9	no	—
179	C₈	—	2	6	no	—
180	C₈	C₂₃×H(5,𝔽₂)	1	2	yes	direct product factor
181	C₂²	—	2	9	no	—
182	C₂²	—	2	9	no	—
183	C₂²	—	4	24	no	—
184	C₂²	C₂³×C₈×C₂₃	1	18	yes	—
185	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
186	C₄	—	2	9	no	—
187	C₄	—	2	6	no	—
188	C₄	C₂³×C₈×C₂₃	1	12	yes	—
189	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
190	C₄	C₂₃×#585	1	1	yes	characteristic
191	C₂	—	2	18	no	—
192	C₂	C₄×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
193	C₂	C₂⁴×C₈×C₂₃	1	1	yes	characteristic
194	C₂	C₂₃×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂	C₂⁴×C₈×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂⁴×C₄
prime radical	C₂×C₂₃×H(5,𝔽₂)	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2988	C₂	semidirect
3. C₂₃×#2972	C₂	semidirect
4. C₈×#41×C₂₃	C₂	semidirect
5. C₂×C₈×D₄×C₂₃	C₂	semidirect
7. C₂³×C₈×C₂₃	C₂²	semidirect
12. C₈×Q₈×C₂₃	C₂²	semidirect
13. C₈×Q₈×C₂₃	C₂²	semidirect
14. C₈×D₄×C₂₃	C₂²	semidirect
15. C₈×D₄×C₂₃	C₂²	semidirect
16. C₈×D₄×C₂₃	C₂²	semidirect
17. C₂₃×#431	C₂²	semidirect
18. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
19. C₂×C₂₃×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₄×C₈×C₂₃	C₂²	semidirect
21. C₂₃×H(5,𝔽₂)	C₈	direct (N × H)
44. #41×C₂₃	C₂×C₈	semidirect
46. C₂×D₄×C₂₃	C₂×C₈	semidirect
65. C₈×H(5,𝔽₂)	C₂₃	direct (N × H)
68. C₂³×C₂₃	C₂²×C₈	semidirect
70. Q₈×C₂₃	C₂²×C₈	semidirect
71. D₄×C₂₃	C₂²×C₈	semidirect
76. C₂×C₄×C₂₃	C₂²×C₈	semidirect
83. C₈×C₂₃	H(5,𝔽₂)	direct (N × H)
85. #2988	C₂×C₂₃	semidirect
86. #2972	C₂×C₂₃	semidirect
87. C₈×#41	C₂×C₂₃	semidirect
88. C₂×C₈×D₄	C₂×C₂₃	semidirect
100. C₂³×C₈	C₂²×C₂₃	semidirect
105. C₈×Q₈	C₂²×C₂₃	semidirect
106. C₈×Q₈	C₂²×C₂₃	semidirect
107. C₈×D₄	C₂²×C₂₃	semidirect
108. C₈×D₄	C₂²×C₂₃	semidirect
109. C₈×D₄	C₂²×C₂₃	semidirect
110. #431	C₂²×C₂₃	semidirect
111. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
112. C₂×C₂²⋊C₈	C₂²×C₂₃	semidirect
113. C₂×C₄×C₈	C₂²×C₂₃	semidirect
118. H(5,𝔽₂)	C₈×C₂₃	direct (N × H)
140. C₂₃	C₈×H(5,𝔽₂)	direct (N × H)
142. #41	C₂×C₈×C₂₃	semidirect
144. C₂×D₄	C₂×C₈×C₂₃	semidirect
165. C₂³	C₂²×C₈×C₂₃	semidirect
167. Q₈	C₂²×C₈×C₂₃	semidirect
168. D₄	C₂²×C₈×C₂₃	semidirect
173. C₂×C₄	C₂²×C₈×C₂₃	semidirect
180. C₈	C₂₃×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×H(5,𝔽₂)	2
C₂₃×#2988	2
C₂₃×#2972	2
C₈×#41×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₈×H(5,𝔽₂)	23

Subgroups — C8×C23×H(5,𝔽2) #19207929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂₃×H(5,𝔽₂) #19207929