Subgroups — C₂×#16837531 #21433022

Order 7552.

400 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic
2	#16837531	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₁₈	C₄	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×Dic₁₁₈	C₄	1	4	yes	—
13	C₄×Dic₁₁₈	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅₉	C₄	1	1	yes	characteristic
15	C₂×Dic₅₉⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₅₉⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
19	Dic₅₉⋊C₈	C₂²	1	4	yes	—
20	Dic₅₉⋊C₈	C₂²	1	4	yes	—
21	#14273638	C₂²	1	4	yes	—
22	#14273637	C₂²	1	4	yes	—
23	C₈×Dic₅₉	C₂²	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₅₉	C₂²	1	4	yes	—
28	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₁₁₈	—	2	4	no	—
36	C₂×Dic₁₁₈	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₅₉	—	2	2	no	—
38	C₂×C₈×C₅₉	—	2	2	no	—
39	C₂×C₈×C₅₉	C₂³	1	2	yes	—
40	C₂×C₈×C₅₉	C₂³	1	2	yes	—
41	C₄²×C₅₉	C₂×C₄	1	2	yes	—
42	C₄²×C₅₉	C₂³	1	2	yes	—
43	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
44	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
45	Dic₅₉⋊C₄	C₂×C₄	1	4	yes	—
46	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
47	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
48	C₄×Dic₅₉	C₂³	1	2	yes	—
49	C₄×Dic₅₉	C₂³	1	2	yes	—
50	C₂×C₅₉⋊C₈	—	2	2	no	—
51	C₂×C₅₉⋊C₈	Q₈	1	4	yes	complemented
52	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
53	C₂×C₅₉⋊C₈	C₂³	1	2	yes	—
54	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	—	2	2	no	—
56	C₂×C₄×C₅₉	—	2	2	no	—
57	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₅₉	#41	1	2	yes	—
62	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₅₉	—	2	2	no	—
64	C₂×Dic₅₉	—	2	2	no	—
65	C₂×Dic₅₉	—	2	2	no	—
66	C₂×Dic₅₉	M₁₆	1	4	yes	—
67	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
68	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
69	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
70	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
71	Dic₁₁₈	—	2	8	no	—
72	C₈×C₅₉	—	2	2	no	—
73	C₈×C₅₉	—	2	2	no	—
74	C₅₉⋊C₈	—	2	2	no	—
75	C₅₉⋊C₈	C₂×Q₈	1	4	yes	complemented
76	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
77	C₂²×C₅₉	C₂×M₁₆	1	2	yes	—
78	C₂²×C₅₉	#131	1	2	yes	—
79	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₅₉	—	2	2	no	—
81	C₄×C₅₉	—	2	2	no	—
82	C₄×C₅₉	C₄×Q₈	1	2	yes	—
83	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
85	Dic₅₉	—	2	2	no	—
86	Dic₅₉	—	2	2	no	—
87	Dic₅₉	—	2	2	no	—
88	Dic₅₉	C₂×M₁₆	1	4	yes	—
89	C₂×#446	—	59	1	no	maximal, Sylow
90	C₂×C₅₉	#446	1	4	yes	complemented
91	C₂×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₂×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
93	C₂×C₅₉	C₂×#131	1	1	yes	characteristic
94	C₂×C₄×Q₈	—	59	1	no	—
95	#446	—	59	8	no	—
96	C₂×C₄⋊C₈	—	59	1	no	—
97	C₂×C₄⋊C₈	—	59	1	no	—
98	C₂×C₄⋊C₈	D₅₉	1	1	yes	characteristic
99	C₂×#97	—	59	1	no	—
100	C₂×#97	—	59	1	no	—
101	C₂×C₄×C₈	—	59	1	no	—
102	C₅₉	C₂×#446	1	1	yes	characteristic, complemented, Sylow
103	C₂²×Q₈	—	59	1	no	—
104	C₂²×C₈	—	59	1	no	—
105	C₂²×C₈	—	59	1	no	—
106	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
108	C₄×Q₈	—	59	4	no	—
109	C₄×Q₈	—	59	4	no	—
110	C₂×C₄⋊C₄	—	59	1	no	—
111	C₂×C₄⋊C₄	—	59	1	no	—
112	C₂×C₄⋊C₄	—	59	1	no	—
113	C₂×C₄²	—	59	1	no	—
114	C₂×C₄²	—	59	1	no	—
115	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
116	C₄⋊C₈	—	59	4	no	—
117	C₄⋊C₈	—	59	4	no	—
118	C₄⋊C₈	C₂×D₅₉	1	4	yes	—
119	#97	—	59	4	no	—
120	#97	—	59	4	no	—
121	C₄×C₈	—	59	4	no	—
122	C₂×Q₈	—	59	4	no	—
123	C₂×Q₈	—	118	4	no	—
124	C₂²×C₄	—	59	1	no	—
125	C₂²×C₄	—	59	1	no	—
126	C₂²×C₄	—	59	1	no	—
127	C₂²×C₄	—	59	1	no	—
128	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	59	4	no	—
134	C₂×C₈	—	59	2	no	—
135	C₂×C₈	—	59	2	no	—
136	C₂×C₈	—	118	2	no	—
137	C₂×C₈	C₂²×D₅₉	1	2	yes	—
138	C₂×C₈	C₂²×D₅₉	1	2	yes	—
139	C₄⋊C₄	—	59	4	no	—
140	C₄⋊C₄	—	59	4	no	—
141	C₄⋊C₄	—	59	4	no	—
142	C₄²	—	59	2	no	—
143	C₄²	—	59	2	no	—
144	C₄²	—	59	2	no	—
145	C₄²	—	59	2	no	—
146	C₄²	C₄×D₅₉	1	2	yes	—
147	C₄²	C₂²×D₅₉	1	2	yes	—
148	C₂³	C₂×C₄×D₅₉	1	1	yes	characteristic
149	Q₈	—	118	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	59	4	no	—
153	C₂×C₄	—	59	2	no	—
154	C₂×C₄	—	59	2	no	—
155	C₂×C₄	—	59	2	no	—
156	C₂×C₄	—	59	2	no	—
157	C₂×C₄	—	118	2	no	—
158	C₂×C₄	—	118	2	no	—
159	C₂×C₄	—	118	2	no	—
160	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
163	C₂×C₄	#11745687	1	2	yes	—
164	C₂×C₄	Q₈×D₅₉	1	2	yes	—
165	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	59	4	no	—
169	C₈	—	118	2	no	—
170	C₂²	#14273692	1	2	yes	—
171	C₂²	M₁₆×D₅₉	1	2	yes	—
172	C₂²	#14273733	1	2	yes	—
173	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	59	4	no	—
177	C₄	—	118	2	no	—
178	C₄	—	118	2	no	—
179	C₄	—	118	2	no	—
180	C₄	#14273692	1	2	yes	—
181	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
182	C₄	C₂×#11745687	1	1	yes	characteristic
183	C₂	#16837531	1	4	yes	direct product factor
184	C₂	C₂×#14273692	1	1	yes	characteristic
185	C₂	C₂×M₁₆×D₅₉	1	1	yes	characteristic
186	C₂	C₂×#14273733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₂×C₄⋊C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₉
hypocenter	C₅₉	C₂×#446
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #16837531	C₂	direct (N × H)
51. C₂×C₅₉⋊C₈	Q₈	semidirect
75. C₅₉⋊C₈	C₂×Q₈	semidirect
90. C₂×C₅₉	#446	semidirect
102. C₅₉	C₂×#446	semidirect
183. C₂	#16837531	direct (N × H)

Subgroups — C2×#16837531 #21433022

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16837531 #21433022