Subgroups — C₅²×#547 #13107405

Order 1600.

1168 conjugacy classes of subgroups.

The subgroup lattice has 180 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	2	yes	maximal, complemented
2	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#123	C₂	1	4	yes	maximal, complemented
4	C₅²×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₅²×#121	C₂	1	2	yes	maximal, complemented
6	C₄×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅²×#41	—	2	8	no	—
10	C₂×C₅²×Q₈	C₂²	1	2	yes	—
11	C₂×C₅²×D₄	C₂²	1	4	yes	—
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅²	C₂²	1	4	yes	—
15	C₂²×C₄×C₅²	C₂²	1	2	yes	—
16	C₂²×C₄×C₅²	C₂²	1	2	yes	—
17	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
19	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
20	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
22	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₅²×C₂²⋊C₄	C₂²	1	4	yes	—
25	C₄²×C₅²	C₂²	1	1	yes	characteristic
26	C₅×#547	C₅	1	6	yes	maximal, direct product factor
27	C₂³×C₅²	C₂³	1	2	yes	—
28	C₂³×C₅²	C₂³	1	2	yes	—
29	C₅²×Q₈	—	2	4	no	—
30	C₅²×D₄	—	2	8	no	—
31	C₅²×D₄	—	2	2	no	—
32	C₅²×D₄	D₄	1	4	yes	complemented
33	C₂×C₄×C₅²	—	2	8	no	—
34	C₂×C₄×C₅²	—	2	4	no	—
35	C₂×C₄×C₅²	—	2	4	no	—
36	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅²	C₂³	1	2	yes	—
38	C₂×C₄×C₅²	C₂³	1	2	yes	—
39	C₂×C₄×C₅²	C₂³	1	4	yes	—
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂×C₅×#41	C₂×C₅	1	12	yes	complemented
43	C₅×C₄⋊Q₈	C₂×C₅	1	6	yes	complemented
44	C₅×#123	C₂×C₅	1	24	yes	complemented
45	C₅×C₂²⋊Q₈	C₂×C₅	1	12	yes	complemented
46	C₅×#121	C₂×C₅	1	12	yes	complemented
47	C₄×C₅×D₄	C₂×C₅	1	6	yes	complemented
48	C₄×C₅×D₄	C₂×C₅	1	6	yes	complemented
49	C₂×C₅×C₄⋊C₄	C₂×C₅	1	12	yes	complemented
50	C₂²×C₅²	—	2	2	no	—
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	—	2	2	no	—
53	C₂²×C₅²	—	2	2	no	—
54	C₂²×C₅²	—	2	2	no	—
55	C₂²×C₅²	C₂×D₄	1	4	yes	—
56	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
57	C₄×C₅²	—	2	2	no	—
58	C₄×C₅²	—	2	4	no	—
59	C₄×C₅²	—	2	2	no	—
60	C₄×C₅²	—	2	1	no	—
61	C₄×C₅²	C₂×D₄	1	2	yes	—
62	C₄×C₅²	#41	1	2	yes	complemented
63	C₅×#41	—	2	48	no	—
64	C₂×C₅×Q₈	C₂²×C₅	1	12	yes	—
65	C₂×C₅×D₄	C₂²×C₅	1	6	yes	—
66	C₂×C₅×D₄	C₂²×C₅	1	24	yes	—
67	C₂×C₅×D₄	C₂²×C₅	1	6	yes	—
68	C₂²×C₄×C₅	C₂²×C₅	1	12	yes	—
69	C₂²×C₄×C₅	C₂²×C₅	1	24	yes	—
70	C₂²×C₄×C₅	C₂²×C₅	1	12	yes	—
71	C₅×C₄⋊C₄	C₂²×C₅	1	6	yes	—
72	C₅×C₄⋊C₄	C₂²×C₅	1	12	yes	—
73	C₅×C₄⋊C₄	C₂²×C₅	1	24	yes	—
74	C₅×C₄⋊C₄	C₂²×C₅	1	6	yes	—
75	C₅×C₄⋊C₄	C₂²×C₅	1	12	yes	—
76	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
77	C₅×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
78	C₅×C₂²⋊C₄	C₂²×C₅	1	24	yes	—
79	C₄²×C₅	C₂²×C₅	1	6	yes	—
80	#547	C₅²	1	1	yes	characteristic, direct product factor, Sylow
81	C₂×C₅²	—	2	4	no	—
82	C₂×C₅²	—	4	2	no	—
83	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₅²	C₂×#41	1	1	yes	characteristic
85	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
86	C₂³×C₅	C₂³×C₅	1	12	yes	—
87	C₂³×C₅	C₂³×C₅	1	12	yes	—
88	C₅×Q₈	—	2	24	no	—
89	C₅×D₄	—	2	48	no	—
90	C₅×D₄	—	2	12	no	—
91	C₅×D₄	C₅×D₄	1	24	yes	complemented
92	C₂×C₄×C₅	—	2	48	no	—
93	C₂×C₄×C₅	—	2	24	no	—
94	C₂×C₄×C₅	—	2	24	no	—
95	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
96	C₂×C₄×C₅	C₂³×C₅	1	12	yes	—
97	C₂×C₄×C₅	C₂³×C₅	1	12	yes	—
98	C₂×C₄×C₅	C₂³×C₅	1	24	yes	—
99	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
100	C₂×C₄×C₅	C₂³×C₅	1	6	yes	—
101	C₂×#41	C₂×C₅²	1	2	yes	complemented
102	C₄⋊Q₈	C₂×C₅²	1	1	yes	characteristic, complemented
103	#123	C₂×C₅²	1	4	yes	complemented
104	C₂²⋊Q₈	C₂×C₅²	1	2	yes	complemented
105	#121	C₂×C₅²	1	2	yes	complemented
106	C₄×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
107	C₄×D₄	C₂×C₅²	1	1	yes	characteristic, complemented
108	C₂×C₄⋊C₄	C₂×C₅²	1	2	yes	complemented
109	C₅²	#547	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×C₅	—	2	12	no	—
111	C₂²×C₅	—	2	12	no	—
112	C₂²×C₅	—	2	12	no	—
113	C₂²×C₅	—	2	12	no	—
114	C₂²×C₅	—	2	12	no	—
115	C₂²×C₅	C₂×C₅×D₄	1	24	yes	—
116	C₂²×C₅	C₂⁴×C₅	1	6	yes	—
117	C₄×C₅	—	2	12	no	—
118	C₄×C₅	—	2	24	no	—
119	C₄×C₅	—	2	12	no	—
120	C₄×C₅	—	2	6	no	—
121	C₄×C₅	C₂×C₅×D₄	1	12	yes	—
122	C₄×C₅	C₅×#41	1	12	yes	complemented
123	#41	—	2	8	no	—
124	C₂×Q₈	C₂²×C₅²	1	2	yes	—
125	C₂×D₄	C₂²×C₅²	1	1	yes	characteristic
126	C₂×D₄	C₂²×C₅²	1	4	yes	—
127	C₂×D₄	C₂²×C₅²	1	1	yes	characteristic
128	C₂²×C₄	C₂²×C₅²	1	2	yes	—
129	C₂²×C₄	C₂²×C₅²	1	4	yes	—
130	C₂²×C₄	C₂²×C₅²	1	2	yes	—
131	C₄⋊C₄	C₂²×C₅²	1	1	yes	characteristic
132	C₄⋊C₄	C₂²×C₅²	1	2	yes	—
133	C₄⋊C₄	C₂²×C₅²	1	4	yes	—
134	C₄⋊C₄	C₂²×C₅²	1	1	yes	characteristic
135	C₄⋊C₄	C₂²×C₅²	1	2	yes	—
136	C₂²⋊C₄	C₂²×C₅²	1	2	yes	—
137	C₂²⋊C₄	C₂²×C₅²	1	2	yes	—
138	C₂²⋊C₄	C₂²×C₅²	1	4	yes	—
139	C₄²	C₂²×C₅²	1	1	yes	characteristic
140	C₂×C₅	—	2	24	no	—
141	C₂×C₅	—	4	12	no	—
142	C₂×C₅	C₂²×C₅×D₄	1	6	yes	—
143	C₂×C₅	C₂×C₅×#41	1	6	yes	—
144	C₂×C₅	C₅×2₋¹⁺⁴	1	6	yes	—
145	C₂³	C₂³×C₅²	1	2	yes	—
146	C₂³	C₂³×C₅²	1	2	yes	—
147	Q₈	—	2	4	no	—
148	D₄	—	2	8	no	—
149	D₄	—	2	2	no	—
150	D₄	C₅²×D₄	1	4	yes	complemented
151	C₂×C₄	—	2	8	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
155	C₂×C₄	C₂³×C₅²	1	2	yes	—
156	C₂×C₄	C₂³×C₅²	1	2	yes	—
157	C₂×C₄	C₂³×C₅²	1	4	yes	—
158	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
159	C₂×C₄	C₂³×C₅²	1	1	yes	characteristic
160	C₅	C₅×#547	1	6	yes	direct product factor
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	2	no	—
165	C₂²	—	2	2	no	—
166	C₂²	C₂×C₅²×D₄	1	4	yes	—
167	C₂²	C₂⁴×C₅²	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	1	no	—
172	C₄	C₂×C₅²×D₄	1	2	yes	—
173	C₄	C₅²×#41	1	2	yes	complemented
174	C₂	—	2	4	no	—
175	C₂	—	4	2	no	—
176	C₂	C₂²×C₅²×D₄	1	1	yes	characteristic
177	C₂	C₂×C₅²×#41	1	1	yes	characteristic
178	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. C₅²×C₄⋊Q₈	C₂	semidirect
3. C₅²×#123	C₂	semidirect
4. C₅²×C₂²⋊Q₈	C₂	semidirect
5. C₅²×#121	C₂	semidirect
6. C₄×C₅²×D₄	C₂	semidirect
7. C₄×C₅²×D₄	C₂	semidirect
8. C₂×C₅²×C₄⋊C₄	C₂	semidirect
26. C₅×#547	C₅	direct (N × H)
32. C₅²×D₄	D₄	semidirect
42. C₂×C₅×#41	C₂×C₅	semidirect
43. C₅×C₄⋊Q₈	C₂×C₅	semidirect
44. C₅×#123	C₂×C₅	semidirect
45. C₅×C₂²⋊Q₈	C₂×C₅	semidirect
46. C₅×#121	C₂×C₅	semidirect
47. C₄×C₅×D₄	C₂×C₅	semidirect
48. C₄×C₅×D₄	C₂×C₅	semidirect
49. C₂×C₅×C₄⋊C₄	C₂×C₅	semidirect
62. C₄×C₅²	#41	semidirect
80. #547	C₅²	direct (N × H)
91. C₅×D₄	C₅×D₄	semidirect
101. C₂×#41	C₂×C₅²	semidirect
102. C₄⋊Q₈	C₂×C₅²	semidirect
103. #123	C₂×C₅²	semidirect
104. C₂²⋊Q₈	C₂×C₅²	semidirect
105. #121	C₂×C₅²	semidirect
106. C₄×D₄	C₂×C₅²	semidirect
107. C₄×D₄	C₂×C₅²	semidirect
108. C₂×C₄⋊C₄	C₂×C₅²	semidirect
109. C₅²	#547	direct (N × H)
122. C₄×C₅	C₅×#41	semidirect
150. D₄	C₅²×D₄	semidirect
160. C₅	C₅×#547	direct (N × H)
173. C₄	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
C₅²×C₄⋊Q₈	2
C₅²×#123	2
C₅²×C₂²⋊Q₈	2
C₅²×#121	2
C₄×C₅²×D₄	2
C₂×C₅²×C₄⋊C₄	2
C₅×#547	5

Subgroups — C52×#547 #13107405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×#547 #13107405