Subgroups — C₄²×H(5,𝔽₂) #7625216

Order 512.

14250 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×H(5,𝔽₂)	C₂	1	3	yes	maximal
2	C₄×#520	C₂	1	18	yes	maximal, complemented
3	C₄×#518	C₂	1	27	yes	maximal, complemented
4	C₄²×#41	C₂	1	6	yes	maximal, complemented
5	C₂×C₄²×D₄	C₂	1	9	yes	maximal, complemented
6	C₂²×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
7	C₄×H(5,𝔽₂)	C₄	1	96	yes	direct product factor
8	C₂×#520	C₂²	1	18	yes	—
9	C₂×#518	C₂²	1	27	yes	—
10	C₂×C₄×#41	C₂²	1	18	yes	—
11	C₂²×C₄×D₄	C₂²	1	27	yes	—
12	C₂³×C₄²	C₂²	1	6	yes	complemented
13	#2276	C₂²	1	12	yes	complemented
14	#2275	C₂²	1	108	yes	complemented
15	#2274	C₂²	1	54	yes	complemented
16	#2273	C₂²	1	36	yes	complemented
17	C₄²×Q₈	C₂²	1	2	yes	complemented
18	C₄²×Q₈	C₂²	1	18	yes	complemented
19	C₄²×D₄	C₂²	1	18	yes	complemented
20	C₄²×D₄	C₂²	1	108	yes	complemented
21	C₄²×D₄	C₂²	1	54	yes	complemented
22	C₄×#117	C₂²	1	54	yes	complemented
23	C₂×C₄×C₄⋊C₄	C₂²	1	27	yes	complemented
24	C₂×C₄×C₂²⋊C₄	C₂²	1	54	yes	complemented
25	C₂×C₄³	C₂²	1	9	yes	complemented
26	C₂×H(5,𝔽₂)	C₂×C₄	1	48	yes	—
27	C₂²×#41	C₂³	1	6	yes	—
28	C₂³×D₄	C₂³	1	9	yes	—
29	C₂⁴×C₄	C₂³	1	18	yes	—
30	#520	C₂×C₄	1	288	yes	complemented
31	#518	C₂×C₄	1	432	yes	complemented
32	C₄×#41	C₂×C₄	1	288	yes	complemented
33	C₂×C₄×Q₈	C₂³	1	6	yes	—
34	C₂×C₄×Q₈	C₂³	1	18	yes	—
35	C₂×C₄×D₄	C₂×C₄	1	432	yes	complemented
36	C₂×C₄×D₄	C₂³	1	54	yes	—
37	C₂×C₄×D₄	C₂³	1	108	yes	—
38	C₂×C₄×D₄	C₂³	1	54	yes	—
39	C₂×#117	C₂³	1	54	yes	—
40	C₂²×C₄⋊C₄	C₂³	1	27	yes	—
41	C₂²×C₂²⋊C₄	C₂³	1	54	yes	—
42	C₂²×C₄²	—	4	24	no	—
43	C₂²×C₄²	C₂³	1	9	yes	—
44	C₂²×C₄²	C₂³	1	54	yes	—
45	C₂²×C₄²	C₂³	1	27	yes	—
46	C₂²×C₄²	C₂³	1	27	yes	—
47	C₄×C₄⋊C₄	C₂³	1	54	yes	—
48	C₄×C₄⋊C₄	C₂³	1	108	yes	—
49	C₄×C₄⋊C₄	C₂³	1	18	yes	—
50	C₄×C₄⋊C₄	C₂³	1	108	yes	—
51	C₄×C₂²⋊C₄	C₂³	1	108	yes	—
52	C₄×C₂²⋊C₄	C₂³	1	216	yes	—
53	C₄×C₂²⋊C₄	C₂³	1	108	yes	—
54	#376	C₂³	1	36	yes	—
55	C₂×#95	C₂³	1	54	yes	—
56	C₄³	C₂³	1	54	yes	—
57	C₄³	C₂³	1	6	yes	—
58	C₂⁵	C₂⁴	1	6	yes	—
59	H(5,𝔽₂)	C₄²	1	256	yes	direct product factor
60	C₂×#41	C₂²×C₄	1	144	yes	—
61	C₂²×Q₈	C₂⁴	1	2	yes	—
62	C₂²×D₄	C₂²×C₄	1	216	yes	—
63	C₂²×D₄	C₂⁴	1	18	yes	—
64	C₂³×C₄	—	4	36	no	—
65	C₂³×C₄	—	4	36	no	—
66	C₂³×C₄	C₂²×C₄	1	144	yes	complemented
67	C₂³×C₄	C₂⁴	1	27	yes	—
68	C₂³×C₄	C₂⁴	1	27	yes	—
69	C₂³×C₄	C₂⁴	1	54	yes	—
70	C₂³×C₄	C₂⁴	1	9	yes	—
71	C₄×Q₈	C₂²×C₄	1	48	yes	complemented
72	C₄×Q₈	C₂²×C₄	1	144	yes	complemented
73	C₄×D₄	C₂²×C₄	1	432	yes	complemented
74	C₄×D₄	C₂²×C₄	1	864	yes	complemented
75	C₄×D₄	C₂²×C₄	1	432	yes	complemented
76	#117	C₂²×C₄	1	432	yes	complemented
77	C₂×C₄⋊C₄	C₂²×C₄	1	216	yes	complemented
78	C₂×C₄⋊C₄	C₂⁴	1	54	yes	—
79	C₂×C₄⋊C₄	C₂⁴	1	18	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₄	1	432	yes	complemented
81	C₂×C₂²⋊C₄	C₂⁴	1	108	yes	—
82	C₂×C₄²	—	2	36	no	—
83	C₂×C₄²	—	4	108	no	—
84	C₂×C₄²	—	4	54	no	—
85	C₂×C₄²	C₂²×C₄	1	216	yes	complemented
86	C₂×C₄²	C₂⁴	1	18	yes	—
87	C₂×C₄²	C₂⁴	1	27	yes	—
88	C₂×C₄²	C₂⁴	1	54	yes	—
89	C₂×C₄²	C₂⁴	1	36	yes	—
90	C₂×C₄²	C₂⁴	1	54	yes	—
91	C₂×C₄²	C₂⁴	1	18	yes	—
92	C₂×C₄²	C₂⁴	1	1	yes	characteristic
93	#95	C₂⁴	1	108	yes	—
94	#95	C₂⁴	1	12	yes	—
95	C₂⁴	—	4	6	no	—
96	C₂⁴	—	4	18	no	—
97	C₂⁴	C₂³×C₄	1	72	yes	—
98	C₂⁴	C₂⁵	1	9	yes	—
99	#41	C₂×C₄²	1	384	yes	complemented
100	C₂×Q₈	C₂³×C₄	1	24	yes	—
101	C₂×D₄	C₂×C₄²	1	576	yes	complemented
102	C₂×D₄	C₂³×C₄	1	216	yes	—
103	C₂²×C₄	—	2	54	no	—
104	C₂²×C₄	—	2	54	no	—
105	C₂²×C₄	—	4	288	no	—
106	C₂²×C₄	—	4	54	no	—
107	C₂²×C₄	—	4	54	no	—
108	C₂²×C₄	—	4	54	no	—
109	C₂²×C₄	C₂³×C₄	1	108	yes	—
110	C₂²×C₄	C₂³×C₄	1	108	yes	—
111	C₂²×C₄	C₂³×C₄	1	216	yes	—
112	C₂²×C₄	C₂³×C₄	1	108	yes	—
113	C₂²×C₄	C₂⁵	1	18	yes	—
114	C₂²×C₄	C₂⁵	1	27	yes	—
115	C₂²×C₄	C₂⁵	1	6	yes	—
116	C₂²×C₄	C₂⁵	1	3	yes	—
117	C₄⋊C₄	C₂³×C₄	1	216	yes	—
118	C₄⋊C₄	C₂³×C₄	1	72	yes	—
119	C₂²⋊C₄	C₂³×C₄	1	432	yes	—
120	C₄²	—	2	54	no	—
121	C₄²	—	2	36	no	—
122	C₄²	—	4	36	no	—
123	C₄²	—	4	54	no	—
124	C₄²	C₂³×C₄	1	72	yes	—
125	C₄²	C₂³×C₄	1	216	yes	—
126	C₄²	H(5,𝔽₂)	1	4	yes	direct product factor
127	C₂³	—	2	9	no	—
128	C₂³	—	2	27	no	—
129	C₂³	—	4	72	no	—
130	C₂³	—	4	72	no	—
131	C₂³	C₂²×C₄²	1	96	yes	complemented
132	C₂³	C₂⁴×C₄	1	54	yes	—
133	C₂³	C₂⁶	1	1	yes	characteristic
134	Q₈	C₂²×C₄²	1	32	yes	complemented
135	D₄	C₂²×C₄²	1	288	yes	complemented
136	C₂×C₄	—	2	27	no	—
137	C₂×C₄	—	2	36	no	—
138	C₂×C₄	—	2	27	no	—
139	C₂×C₄	—	2	216	no	—
140	C₂×C₄	—	4	216	no	—
141	C₂×C₄	—	4	108	no	—
142	C₂×C₄	C₂²×C₄²	1	144	yes	complemented
143	C₂×C₄	C₂⁴×C₄	1	36	yes	—
144	C₂×C₄	C₂⁴×C₄	1	54	yes	—
145	C₂×C₄	C₂⁴×C₄	1	36	yes	—
146	C₂×C₄	C₂⁴×C₄	1	6	yes	—
147	C₂×C₄	C₂×H(5,𝔽₂)	1	6	yes	—
148	C₂×C₄	#585	1	6	yes	—
149	C₂²	—	2	54	no	—
150	C₂²	—	2	54	no	—
151	C₂²	—	4	96	no	—
152	C₂²	C₂³×C₄²	1	36	yes	—
153	C₂²	C₂⁵×C₄	1	3	yes	—
154	C₂²	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²	C₂×#585	1	3	yes	—
156	C₄	—	2	54	no	—
157	C₄	—	2	36	no	—
158	C₄	C₂³×C₄²	1	24	yes	—
159	C₄	C₄×H(5,𝔽₂)	1	12	yes	direct product factor
160	C₂	—	2	36	no	—
161	C₂	C₂⁴×C₄²	1	1	yes	characteristic
162	C₂	C₂×C₄×H(5,𝔽₂)	1	3	yes	—
163	C₂	C₄×#585	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂	C₂⁴×C₄²
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×#520	C₂	semidirect
3. C₄×#518	C₂	semidirect
4. C₄²×#41	C₂	semidirect
5. C₂×C₄²×D₄	C₂	semidirect
7. C₄×H(5,𝔽₂)	C₄	direct (N × H)
12. C₂³×C₄²	C₂²	semidirect
13. #2276	C₂²	semidirect
14. #2275	C₂²	semidirect
15. #2274	C₂²	semidirect
16. #2273	C₂²	semidirect
17. C₄²×Q₈	C₂²	semidirect
18. C₄²×Q₈	C₂²	semidirect
19. C₄²×D₄	C₂²	semidirect
20. C₄²×D₄	C₂²	semidirect
21. C₄²×D₄	C₂²	semidirect
22. C₄×#117	C₂²	semidirect
23. C₂×C₄×C₄⋊C₄	C₂²	semidirect
24. C₂×C₄×C₂²⋊C₄	C₂²	semidirect
25. C₂×C₄³	C₂²	semidirect
30. #520	C₂×C₄	semidirect
31. #518	C₂×C₄	semidirect
32. C₄×#41	C₂×C₄	semidirect
35. C₂×C₄×D₄	C₂×C₄	semidirect
59. H(5,𝔽₂)	C₄²	direct (N × H)
66. C₂³×C₄	C₂²×C₄	semidirect
71. C₄×Q₈	C₂²×C₄	semidirect
72. C₄×Q₈	C₂²×C₄	semidirect
73. C₄×D₄	C₂²×C₄	semidirect
74. C₄×D₄	C₂²×C₄	semidirect
75. C₄×D₄	C₂²×C₄	semidirect
76. #117	C₂²×C₄	semidirect
77. C₂×C₄⋊C₄	C₂²×C₄	semidirect
80. C₂×C₂²⋊C₄	C₂²×C₄	semidirect
85. C₂×C₄²	C₂²×C₄	semidirect
99. #41	C₂×C₄²	semidirect
101. C₂×D₄	C₂×C₄²	semidirect
126. C₄²	H(5,𝔽₂)	direct (N × H)
131. C₂³	C₂²×C₄²	semidirect
134. Q₈	C₂²×C₄²	semidirect
135. D₄	C₂²×C₄²	semidirect
142. C₂×C₄	C₂²×C₄²	semidirect
159. C₄	C₄×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
C₂×C₄×H(5,𝔽₂)	2
C₄×#520	2
C₄×#518	2
C₄²×#41	2
C₂×C₄²×D₄	2

Subgroups — C42×H(5,𝔽2) #7625216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×H(5,𝔽₂) #7625216