Subgroups — C₅²×#548 #13107406

Order 1600.

1160 conjugacy classes of subgroups.

The subgroup lattice has 159 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#124	C₂	1	3	yes	maximal, complemented
4	C₅²×C₂²⋊Q₈	C₂	1	3	yes	maximal, complemented
5	C₅²×#121	C₂	1	3	yes	maximal, complemented
6	C₄×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅²×D₄	C₂	1	3	yes	maximal, complemented
8	C₅²×#41	—	2	4	no	—
9	C₂×C₅²×Q₈	—	2	4	no	—
10	C₂×C₅²×Q₈	C₂²	1	3	yes	—
11	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅²×D₄	C₂²	1	3	yes	—
13	C₂×C₅²×D₄	C₂²	1	3	yes	—
14	C₂²×C₄×C₅²	C₂²	1	3	yes	—
15	C₂²×C₄×C₅²	C₂²	1	3	yes	—
16	C₅²×C₄⋊C₄	C₂²	1	3	yes	—
17	C₅²×C₄⋊C₄	C₂²	1	3	yes	—
18	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₅²×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₅²×C₂²⋊C₄	C₂²	1	6	yes	—
21	C₄²×C₅²	C₂²	1	3	yes	—
22	C₅×#548	C₅	1	6	yes	maximal, direct product factor
23	C₂³×C₅²	C₂³	1	3	yes	—
24	C₂³×C₅²	C₂³	1	1	yes	characteristic
25	C₅²×Q₈	—	2	6	no	—
26	C₅²×Q₈	D₄	1	4	yes	complemented
27	C₅²×D₄	—	2	6	no	—
28	C₅²×D₄	—	2	6	no	—
29	C₂×C₄×C₅²	—	2	6	no	—
30	C₂×C₄×C₅²	—	2	6	no	—
31	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅²	C₂³	1	3	yes	—
33	C₂×C₄×C₅²	C₂³	1	3	yes	—
34	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅²	C₂³	1	3	yes	—
36	C₂×C₅×#41	C₂×C₅	1	6	yes	complemented
37	C₂²×C₅×Q₈	C₂×C₅	1	6	yes	complemented
38	C₅×#124	C₂×C₅	1	18	yes	complemented
39	C₅×C₂²⋊Q₈	C₂×C₅	1	18	yes	complemented
40	C₅×#121	C₂×C₅	1	18	yes	complemented
41	C₄×C₅×Q₈	C₂×C₅	1	6	yes	complemented
42	C₄×C₅×D₄	C₂×C₅	1	18	yes	complemented
43	C₂²×C₅²	—	2	1	no	—
44	C₂²×C₅²	—	2	1	no	—
45	C₂²×C₅²	—	2	3	no	—
46	C₂²×C₅²	—	2	3	no	—
47	C₂²×C₅²	—	2	3	no	—
48	C₂²×C₅²	#41	1	2	yes	complemented
49	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
50	C₄×C₅²	—	2	1	no	—
51	C₄×C₅²	—	2	3	no	—
52	C₄×C₅²	—	2	3	no	—
53	C₄×C₅²	—	2	1	no	—
54	C₄×C₅²	C₂×D₄	1	6	yes	—
55	C₅×#41	—	2	24	no	—
56	C₂×C₅×Q₈	—	2	24	no	—
57	C₂×C₅×Q₈	C₂²×C₅	1	18	yes	—
58	C₂×C₅×Q₈	C₂²×C₅	1	6	yes	—
59	C₂×C₅×D₄	C₂²×C₅	1	18	yes	—
60	C₂×C₅×D₄	C₂²×C₅	1	18	yes	—
61	C₂²×C₄×C₅	C₂²×C₅	1	18	yes	—
62	C₂²×C₄×C₅	C₂²×C₅	1	18	yes	—
63	C₅×C₄⋊C₄	C₂²×C₅	1	18	yes	—
64	C₅×C₄⋊C₄	C₂²×C₅	1	18	yes	—
65	C₅×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
66	C₅×C₂²⋊C₄	C₂²×C₅	1	18	yes	—
67	C₅×C₂²⋊C₄	C₂²×C₅	1	36	yes	—
68	C₄²×C₅	C₂²×C₅	1	18	yes	—
69	#548	C₅²	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₅²	—	2	2	no	—
71	C₂×C₅²	—	4	3	no	—
72	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
73	C₂×C₅²	C₂×#41	1	1	yes	characteristic
74	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
75	C₂³×C₅	C₂³×C₅	1	18	yes	—
76	C₂³×C₅	C₂³×C₅	1	6	yes	—
77	C₅×Q₈	—	2	36	no	—
78	C₅×Q₈	C₅×D₄	1	24	yes	complemented
79	C₅×D₄	—	2	36	no	—
80	C₅×D₄	—	2	36	no	—
81	C₂×C₄×C₅	—	2	36	no	—
82	C₂×C₄×C₅	—	2	36	no	—
83	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
84	C₂×C₄×C₅	C₂³×C₅	1	18	yes	—
85	C₂×C₄×C₅	C₂³×C₅	1	18	yes	—
86	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
87	C₂×C₄×C₅	C₂³×C₅	1	18	yes	—
88	C₂×#41	C₂×C₅²	1	1	yes	characteristic, complemented
89	C₂²×Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
90	#124	C₂×C₅²	1	3	yes	complemented
91	C₂²⋊Q₈	C₂×C₅²	1	3	yes	complemented
92	#121	C₂×C₅²	1	3	yes	complemented
93	C₄×Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
94	C₄×D₄	C₂×C₅²	1	3	yes	complemented
95	C₅²	#548	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×C₅	—	2	6	no	—
97	C₂²×C₅	—	2	6	no	—
98	C₂²×C₅	—	2	18	no	—
99	C₂²×C₅	—	2	18	no	—
100	C₂²×C₅	—	2	18	no	—
101	C₂²×C₅	C₅×#41	1	12	yes	complemented
102	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
103	C₄×C₅	—	2	6	no	—
104	C₄×C₅	—	2	18	no	—
105	C₄×C₅	—	2	18	no	—
106	C₄×C₅	—	2	6	no	—
107	C₄×C₅	C₂×C₅×D₄	1	36	yes	—
108	#41	—	2	4	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	C₂²×C₅²	1	3	yes	—
111	C₂×Q₈	C₂²×C₅²	1	1	yes	characteristic
112	C₂×D₄	C₂²×C₅²	1	3	yes	—
113	C₂×D₄	C₂²×C₅²	1	3	yes	—
114	C₂²×C₄	C₂²×C₅²	1	3	yes	—
115	C₂²×C₄	C₂²×C₅²	1	3	yes	—
116	C₄⋊C₄	C₂²×C₅²	1	3	yes	—
117	C₄⋊C₄	C₂²×C₅²	1	3	yes	—
118	C₂²⋊C₄	C₂²×C₅²	1	1	yes	characteristic
119	C₂²⋊C₄	C₂²×C₅²	1	3	yes	—
120	C₂²⋊C₄	C₂²×C₅²	1	6	yes	—
121	C₄²	C₂²×C₅²	1	3	yes	—
122	C₂×C₅	—	2	12	no	—
123	C₂×C₅	—	4	18	no	—
124	C₂×C₅	C₂²×C₅×D₄	1	6	yes	—
125	C₂×C₅	C₂×C₅×#41	1	6	yes	—
126	C₂×C₅	C₅×2₋¹⁺⁴	1	6	yes	—
127	C₂³	C₂³×C₅²	1	3	yes	—
128	C₂³	C₂³×C₅²	1	1	yes	characteristic
129	Q₈	—	2	6	no	—
130	Q₈	C₅²×D₄	1	4	yes	complemented
131	D₄	—	2	6	no	—
132	D₄	—	2	6	no	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
136	C₂×C₄	C₂³×C₅²	1	3	yes	—
137	C₂×C₄	C₂³×C₅²	1	3	yes	—
138	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
139	C₂×C₄	C₂³×C₅²	1	3	yes	—
140	C₅	C₅×#548	1	6	yes	direct product factor
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	3	no	—
144	C₂²	—	2	3	no	—
145	C₂²	—	2	3	no	—
146	C₂²	C₅²×#41	1	2	yes	complemented
147	C₂²	C₂⁴×C₅²	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	3	no	—
150	C₄	—	2	3	no	—
151	C₄	—	2	1	no	—
152	C₄	C₂×C₅²×D₄	1	6	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	4	3	no	—
155	C₂	C₂²×C₅²×D₄	1	1	yes	characteristic
156	C₂	C₂×C₅²×#41	1	1	yes	characteristic
157	C₂	C₅²×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×#41	C₂	semidirect
2. C₂²×C₅²×Q₈	C₂	semidirect
3. C₅²×#124	C₂	semidirect
4. C₅²×C₂²⋊Q₈	C₂	semidirect
5. C₅²×#121	C₂	semidirect
6. C₄×C₅²×Q₈	C₂	semidirect
7. C₄×C₅²×D₄	C₂	semidirect
22. C₅×#548	C₅	direct (N × H)
26. C₅²×Q₈	D₄	semidirect
36. C₂×C₅×#41	C₂×C₅	semidirect
37. C₂²×C₅×Q₈	C₂×C₅	semidirect
38. C₅×#124	C₂×C₅	semidirect
39. C₅×C₂²⋊Q₈	C₂×C₅	semidirect
40. C₅×#121	C₂×C₅	semidirect
41. C₄×C₅×Q₈	C₂×C₅	semidirect
42. C₄×C₅×D₄	C₂×C₅	semidirect
48. C₂²×C₅²	#41	semidirect
69. #548	C₅²	direct (N × H)
78. C₅×Q₈	C₅×D₄	semidirect
88. C₂×#41	C₂×C₅²	semidirect
89. C₂²×Q₈	C₂×C₅²	semidirect
90. #124	C₂×C₅²	semidirect
91. C₂²⋊Q₈	C₂×C₅²	semidirect
92. #121	C₂×C₅²	semidirect
93. C₄×Q₈	C₂×C₅²	semidirect
94. C₄×D₄	C₂×C₅²	semidirect
95. C₅²	#548	direct (N × H)
101. C₂²×C₅	C₅×#41	semidirect
130. Q₈	C₅²×D₄	semidirect
140. C₅	C₅×#548	direct (N × H)
146. C₂²	C₅²×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₂²×C₅²×Q₈	2
C₅²×#124	2
C₅²×C₂²⋊Q₈	2
C₅²×#121	2
C₄×C₅²×Q₈	2
C₄×C₅²×D₄	2
C₅×#548	5

Subgroups — C52×#548 #13107406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#548 #13107406