Subgroups — #21438591

Order 7552.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉⋊#564	C₂	1	4	yes	maximal, complemented
2	#16838131	C₂	1	4	yes	maximal, complemented
3	#16838130	C₂	1	4	yes	maximal, complemented
4	#16838127	C₂	1	4	yes	maximal, complemented
5	#16838014	C₂	1	4	yes	maximal
6	#16838012	C₂	1	8	yes	maximal
7	#16837959	C₂	1	2	yes	maximal
8	C₅₉×#533	C₂	1	1	yes	maximal, characteristic
9	#14273727	C₂²	1	2	yes	—
10	#14273727	C₂²	1	2	yes	—
11	#14273727	C₂²	1	4	yes	—
12	#14273725	C₂²	1	8	yes	—
13	#14273720	C₂²	1	16	yes	—
14	#14273719	C₂²	1	4	yes	—
15	#14273719	C₂²	1	4	yes	—
16	#14273718	C₂²	1	16	yes	—
17	#14273718	C₂²	1	8	yes	—
18	#14273717	C₂²	1	16	yes	—
19	#14273717	C₂²	1	16	yes	—
20	#14273706	C₂²	1	4	yes	—
21	#14273706	C₂²	1	4	yes	—
22	Q₈×Dic₅₉	C₂²	1	8	yes	—
23	C₄×Dic₁₁₈	C₂²	1	4	yes	—
24	C₄×Dic₁₁₈	C₂²	1	4	yes	—
25	C₄×Dic₁₁₈	C₂²	1	8	yes	—
26	#14273692	C₂²	1	8	yes	—
27	#14273686	C₂²	1	4	yes	—
28	C₄⋊Q₈×C₅₉	C₂²	1	2	yes	—
29	#125×C₅₉	C₂²	1	2	yes	—
30	C₂²⋊Q₈×C₅₉	C₂²	1	4	yes	—
31	C₄×Q₈×C₅₉	C₂²	1	4	yes	—
32	#117×C₅₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×Q₈×C₅₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
37	C₂×Dic₁₁₈	C₂³	1	4	yes	—
38	C₂×Dic₁₁₈	C₂³	1	4	yes	—
39	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
40	C₄⋊C₄×C₅₉	C₂³	1	8	yes	—
41	C₄⋊C₄×C₅₉	C₂³	1	4	yes	—
42	C₂²⋊C₄×C₅₉	C₂³	1	4	yes	—
43	C₄²×C₅₉	C₂³	1	2	yes	—
44	C₄²×C₅₉	C₂³	1	1	yes	characteristic
45	C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄²×C₅₉	C₂³	1	4	yes	—
47	C₂²⋊Dic₅₉	C₂³	1	8	yes	—
48	C₄⋊Dic₅₉	C₂³	1	4	yes	—
49	C₄⋊Dic₅₉	C₂³	1	4	yes	—
50	C₄⋊Dic₅₉	C₂³	1	4	yes	—
51	C₄⋊Dic₅₉	C₂³	1	4	yes	—
52	C₄⋊Dic₅₉	C₂³	1	16	yes	—
53	Dic₅₉⋊C₄	C₂³	1	16	yes	—
54	Dic₅₉⋊C₄	C₂³	1	8	yes	—
55	Dic₅₉⋊C₄	C₂³	1	8	yes	—
56	Dic₅₉⋊C₄	C₂³	1	16	yes	—
57	C₄×Dic₅₉	C₂³	1	8	yes	—
58	C₄×Dic₅₉	C₂³	1	16	yes	—
59	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
60	Q₈×C₅₉	—	2	8	no	—
61	C₂×C₄×C₅₉	—	2	4	no	—
62	C₂×C₄×C₅₉	—	2	2	no	—
63	C₂×C₄×C₅₉	C₂⁴	1	8	yes	—
64	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂⁴	1	2	yes	—
66	C₂×C₄×C₅₉	C₂⁴	1	2	yes	—
67	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₅₉	C₂⁴	1	8	yes	—
69	C₂×Dic₅₉	C₂⁴	1	8	yes	—
70	Dic₁₁₈	—	2	8	no	—
71	Dic₁₁₈	—	2	8	no	—
72	C₂²×C₅₉	—	2	2	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	C₂⁵	1	1	yes	characteristic
75	C₄×C₅₉	—	2	2	no	—
76	C₄×C₅₉	—	2	1	no	—
77	C₄×C₅₉	—	2	8	no	—
78	C₄×C₅₉	2₋¹⁺⁴	1	4	yes	—
79	C₄×C₅₉	2₋¹⁺⁴	1	2	yes	—
80	Dic₅₉	—	2	8	no	—
81	Dic₅₉	—	2	8	no	—
82	#3556	—	59	1	no	maximal, Sylow
83	C₂×C₅₉	—	4	1	no	—
84	C₂×C₅₉	C₂×2₋¹⁺⁴	1	2	yes	—
85	C₂×C₅₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
86	#564	—	59	4	no	—
87	#557	—	59	4	no	—
88	#557	—	59	4	no	—
89	#557	—	59	4	no	—
90	#541	—	59	8	no	—
91	#541	—	59	4	no	—
92	#533	—	59	2	no	—
93	#533	D₅₉	1	1	yes	characteristic
94	C₅₉	#3556	1	1	yes	characteristic, complemented, Sylow
95	C₄⋊Q₈	—	59	8	no	—
96	C₄⋊Q₈	—	59	2	no	—
97	C₄⋊Q₈	—	59	2	no	—
98	C₄⋊Q₈	—	59	4	no	—
99	C₄⋊Q₈	C₂×D₅₉	1	2	yes	—
100	#126	—	59	16	no	—
101	#125	—	59	8	no	—
102	#125	—	59	16	no	—
103	#125	—	59	16	no	—
104	#125	—	59	16	no	—
105	#125	—	59	4	no	—
106	#125	—	59	4	no	—
107	#125	C₂×D₅₉	1	2	yes	—
108	C₂²⋊Q₈	—	59	4	no	—
109	C₂²⋊Q₈	—	59	4	no	—
110	C₂²⋊Q₈	C₂×D₅₉	1	4	yes	—
111	C₄×Q₈	—	59	8	no	—
112	C₄×Q₈	—	59	8	no	—
113	C₄×Q₈	—	59	4	no	—
114	C₄×Q₈	—	59	4	no	—
115	C₄×Q₈	—	59	8	no	—
116	C₄×Q₈	C₂×D₅₉	1	4	yes	—
117	#117	—	59	4	no	—
118	#117	C₂×D₅₉	1	2	yes	—
119	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
120	C₂×Q₈	—	59	4	no	—
121	C₂×Q₈	—	59	4	no	—
122	C₂×Q₈	C₂²×D₅₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
125	C₄⋊C₄	—	59	4	no	—
126	C₄⋊C₄	—	59	4	no	—
127	C₄⋊C₄	—	59	4	no	—
128	C₄⋊C₄	—	59	4	no	—
129	C₄⋊C₄	—	59	8	no	—
130	C₄⋊C₄	—	59	8	no	—
131	C₄⋊C₄	—	59	16	no	—
132	C₄⋊C₄	—	59	16	no	—
133	C₄⋊C₄	—	59	16	no	—
134	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
135	C₄⋊C₄	C₂²×D₅₉	1	4	yes	—
136	C₄⋊C₄	C₂²×D₅₉	1	8	yes	—
137	C₂²⋊C₄	—	59	8	no	—
138	C₂²⋊C₄	C₂²×D₅₉	1	4	yes	—
139	C₄²	—	59	8	no	—
140	C₄²	—	59	16	no	—
141	C₄²	C₂²×D₅₉	1	1	yes	characteristic
142	C₄²	C₂²×D₅₉	1	2	yes	—
143	C₄²	C₂²×D₅₉	1	1	yes	characteristic
144	C₄²	C₂²×D₅₉	1	4	yes	—
145	C₂³	C₂³×D₅₉	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	118	8	no	—
148	Q₈	—	118	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	59	8	no	—
152	C₂×C₄	—	59	8	no	—
153	C₂×C₄	C₂³×D₅₉	1	8	yes	—
154	C₂×C₄	C₂³×D₅₉	1	2	yes	—
155	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₅₉	1	2	yes	—
157	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂⁴×D₅₉	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	1	no	—
164	C₄	—	118	8	no	—
165	C₄	—	118	8	no	—
166	C₄	#14273774	1	4	yes	—
167	C₄	#14273775	1	2	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#14273774	1	2	yes	—
170	C₂	C₂×#14273775	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅₉
commutator, socle	C₂²×C₅₉	C₂⁵
Fitting	C₅₉×#533	C₂
hypercenter	#533	D₅₉
hypocenter	C₅₉	#3556
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₅₉⋊#564	C₂	semidirect
2. #16838131	C₂	semidirect
3. #16838130	C₂	semidirect
4. #16838127	C₂	semidirect
94. C₅₉	#3556	semidirect