Subgroups — #21433172

Order 7552.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837480	C₂	1	2	yes	maximal, complemented
2	#16837474	C₂	1	4	yes	maximal, complemented
3	D₄×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	#16837459	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837455	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837369	C₂	1	4	yes	maximal, complemented
7	C₅₉×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	4	yes	—
9	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	—
10	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
12	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
15	C₂²⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
16	#14273638	C₂²	1	2	yes	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#41×C₅₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
21	#127×C₅₉	C₄	1	1	yes	characteristic
22	#124×C₅₉	C₄	1	2	yes	—
23	#121×C₅₉	C₄	1	2	yes	—
24	#121×C₅₉	C₄	1	2	yes	—
25	C₄×D₄×C₅₉	C₂²	1	2	yes	—
26	C₄×D₄×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
28	#41×C₅₉	—	4	4	no	—
29	C₂×Q₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₅₉	C₂×C₄	1	4	yes	—
33	C₂×D₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₅₉	C₂³	1	4	yes	—
38	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
39	C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
40	C₂²⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
42	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₂³	1	1	yes	characteristic
45	C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	#11745664	—	4	4	no	—
47	C₂×C₅₉⋊C₈	—	4	4	no	—
48	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
49	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
50	C₂×C₅₉⋊C₈	C₂³	1	4	yes	—
51	C₂³×C₅₉	C₂²×C₄	1	4	yes	—
52	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₅₉	—	4	2	no	—
54	D₄×C₅₉	—	2	4	no	—
55	D₄×C₅₉	—	4	2	no	—
56	D₄×C₅₉	—	4	2	no	—
57	D₄×C₅₉	—	4	4	no	—
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	2	4	no	—
60	C₂×C₄×C₅₉	—	2	4	no	—
61	C₂×C₄×C₅₉	—	2	2	no	—
62	C₂×C₄×C₅₉	—	4	1	no	—
63	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
70	C₅₉⋊C₈	—	2	2	no	—
71	C₅₉⋊C₈	—	2	4	no	—
72	C₅₉⋊C₈	—	2	2	no	—
73	C₂²×C₅₉	—	2	1	no	—
74	C₂²×C₅₉	—	2	4	no	—
75	C₂²×C₅₉	—	2	1	no	—
76	C₂²×C₅₉	—	2	4	no	—
77	C₂²×C₅₉	—	2	1	no	—
78	C₂²×C₅₉	—	2	4	no	—
79	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₅₉	—	2	1	no	—
81	C₄×C₅₉	—	2	4	no	—
82	C₄×C₅₉	—	2	1	no	—
83	C₄×C₅₉	—	2	1	no	—
84	C₄×C₅₉	—	2	1	no	—
85	C₄×C₅₉	#131	1	2	yes	—
86	C₄×C₅₉	H(5,𝔽₂)	1	2	yes	—
87	#2981	—	59	1	no	maximal, Sylow
88	C₂×C₅₉	—	4	1	no	—
89	C₂×C₅₉	—	4	4	no	—
90	C₂×C₅₉	#518	1	1	yes	characteristic
91	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
92	C₂×C₅₉	#568	1	1	yes	characteristic
93	#532	D₅₉	1	1	yes	characteristic
94	#436	—	59	2	no	—
95	#435	—	59	4	no	—
96	C₈×D₄	—	59	2	no	—
97	#432	—	59	1	no	—
98	#431	—	59	1	no	—
99	#408	—	59	4	no	—
100	C₅₉	#2981	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	C₂×D₅₉	1	2	yes	—
102	C₂×M₁₆	—	59	4	no	—
103	C₂²×C₈	—	59	4	no	—
104	C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
105	#127	Dic₅₉	1	1	yes	characteristic
106	#124	Dic₅₉	1	2	yes	—
107	#121	Dic₅₉	1	2	yes	—
108	#121	Dic₅₉	1	2	yes	—
109	C₄×D₄	C₂×D₅₉	1	2	yes	—
110	C₄×D₄	C₂×D₅₉	1	2	yes	—
111	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
112	C₄⋊C₈	—	59	2	no	—
113	C₄⋊C₈	—	59	2	no	—
114	C₂²⋊C₈	—	59	2	no	—
115	C₂²⋊C₈	—	59	2	no	—
116	C₂²⋊C₈	—	59	4	no	—
117	C₂²⋊C₈	—	59	4	no	—
118	#97	—	59	2	no	—
119	C₄×C₈	—	59	1	no	—
120	C₄×C₈	—	59	1	no	—
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×Dic₅₉	1	2	yes	—
123	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
124	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
125	C₂×D₄	C₂×Dic₅₉	1	4	yes	—
126	C₂×D₄	C₂×Dic₅₉	1	2	yes	—
127	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₅₉	1	4	yes	—
131	M₁₆	—	236	4	no	—
132	C₂×C₈	—	59	2	no	—
133	C₂×C₈	—	59	2	no	—
134	C₂×C₈	—	59	4	no	—
135	C₂×C₈	—	236	4	no	—
136	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₅₉	1	4	yes	—
139	C₂²⋊C₄	C₂×Dic₅₉	1	4	yes	—
140	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
141	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
142	C₄²	C₂²×D₅₉	1	1	yes	characteristic
143	C₄²	C₂²×D₅₉	1	1	yes	characteristic
144	C₂³	C₂²×Dic₅₉	1	4	yes	—
145	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	2	no	—
150	D₄	—	4	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₅₉	1	4	yes	—
161	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
163	C₈	—	118	2	no	—
164	C₈	—	118	4	no	—
165	C₈	—	118	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₅₉	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	4	no	—
178	C₄	#14273734	1	2	yes	—
179	C₄	#14273772	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#16837877	1	1	yes	characteristic
183	C₂	C₂×#14273734	1	1	yes	characteristic
184	C₂	#16838169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₄
Fitting	C₅₉×#532	C₂
hypercenter	#532	D₅₉
hypocenter	C₅₉	#2981
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #16837480	C₂	semidirect
2. #16837474	C₂	semidirect
3. D₄×C₅₉⋊C₈	C₂	semidirect
4. #16837459	C₂	semidirect
5. #16837455	C₂	semidirect
6. #16837369	C₂	semidirect
100. C₅₉	#2981	semidirect