Subgroups — C₂×#21422158 #28500054

Order 15104.

294 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837804	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837223	C₂	1	1	yes	maximal, characteristic
3	#21422158	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273719	C₄	1	2	yes	—
6	C₂×#14273667	—	2	1	no	—
7	#16837804	—	2	1	no	—
8	#16837804	—	2	1	no	—
9	#16837804	C₄	1	2	yes	—
10	#16837804	C₂²	1	2	yes	—
11	C₂×#14273661	—	2	1	no	—
12	C₂×#14273659	—	2	1	no	—
13	#16837223	—	2	1	no	—
14	#16837223	C₂²	1	2	yes	—
15	C₂×M₃₂×C₅₉	—	2	1	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#364	—	2	1	no	—
18	C₅₉×#364	C₂²	1	2	yes	—
19	#14273719	—	2	2	no	—
20	#14273719	—	4	2	no	—
21	#14273719	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×Dic₅₉⋊C₄	—	4	2	no	—
24	#14273667	—	2	2	no	—
25	#14273667	—	4	1	no	—
26	C₂×C₅₉⋊C₁₆	—	4	1	no	—
27	#14273661	—	2	2	no	—
28	#14273661	—	2	2	no	—
29	#14273659	—	2	2	no	—
30	#14273659	—	2	2	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
34	M₃₂×C₅₉	—	2	2	no	—
35	M₃₂×C₅₉	—	4	1	no	—
36	C₂×C₁₆×C₅₉	—	4	1	no	—
37	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	—	2	1	no	—
42	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₅₉	—	4	2	no	—
44	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
51	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₉	Q₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₅₉	—	2	1	no	—
57	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₄⋊Dic₅₉	—	2	4	no	—
60	C₄⋊Dic₅₉	—	4	2	no	—
61	Dic₅₉⋊C₄	—	4	4	no	—
62	Dic₅₉⋊C₄	—	4	4	no	—
63	C₁₆×C₅₉	—	4	2	no	—
64	C₅₉⋊C₁₆	—	4	2	no	—
65	C₂³×C₅₉	#100	1	1	yes	characteristic
66	C₂×C₄×C₅₉	—	2	1	no	—
67	C₂×C₄×C₅₉	—	2	1	no	—
68	C₂×C₄×C₅₉	—	4	2	no	—
69	C₂×C₄×C₅₉	#100	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
72	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
73	C₂×C₄×C₅₉	#103	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×Dic₅₉	—	4	4	no	—
76	C₂×Dic₅₉	—	8	4	no	—
77	C₈×C₅₉	#103	1	1	yes	characteristic
78	C₈×C₅₉	#103	1	1	yes	characteristic
79	C₈×C₅₉	#103	1	1	yes	characteristic
80	C₈×C₅₉	#103	1	1	yes	characteristic
81	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
84	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×#1366	—	59	1	no	maximal, Sylow
86	C₂²×C₅₉	—	2	2	no	—
87	C₂²×C₅₉	#333	1	1	yes	characteristic
88	C₂²×C₅₉	#333	1	1	yes	characteristic
89	C₂²×C₅₉	C₂×#100	1	1	yes	characteristic
90	C₄×C₅₉	—	4	1	no	—
91	C₄×C₅₉	—	4	1	no	—
92	C₄×C₅₉	#333	1	1	yes	characteristic
93	C₄×C₅₉	#333	1	1	yes	characteristic
94	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
95	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
96	Dic₅₉	—	8	4	no	—
97	C₂×#499	—	59	1	no	—
98	C₂×#364	—	59	1	no	—
99	C₂×#364	D₅₉	1	1	yes	characteristic
100	#1366	—	59	4	no	—
101	C₂×C₅₉	—	2	1	no	—
102	C₂×C₅₉	—	2	1	no	—
103	C₂×C₅₉	#1366	1	2	yes	complemented
104	C₂×C₅₉	C₂×#333	1	1	yes	characteristic
105	C₂×#125	—	59	2	no	—
106	C₂×M₃₂	—	2	1	no	—
107	C₂×M₃₂	—	118	1	no	—
108	#499	—	59	2	no	—
109	#499	—	59	2	no	—
110	#499	—	118	1	no	—
111	#499	—	118	1	no	—
112	C₂×#107	—	118	1	no	—
113	C₂×#106	—	118	1	no	—
114	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
115	#364	—	2	1	no	—
116	#364	—	59	2	no	—
117	#364	—	118	1	no	—
118	#364	C₂×D₅₉	1	2	yes	—
119	C₅₉	C₂×#1366	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
121	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
122	#125	—	59	4	no	—
123	#125	—	118	2	no	—
124	#125	—	236	2	no	—
125	C₂×C₄⋊C₄	—	118	2	no	—
126	C₂×C₄⋊C₄	—	236	2	no	—
127	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
128	M₃₂	—	2	2	no	—
129	M₃₂	—	4	1	no	—
130	M₃₂	—	118	2	no	—
131	M₃₂	—	236	1	no	—
132	C₂×C₁₆	—	4	1	no	—
133	C₂×C₁₆	—	236	1	no	—
134	#107	—	118	2	no	—
135	#107	—	118	2	no	—
136	#106	—	118	2	no	—
137	#106	—	118	2	no	—
138	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
139	C₄×C₈	D₂₃₆	1	1	yes	characteristic
140	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
141	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	236	2	no	—
144	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
145	C₂×C₈	#11745660	1	1	yes	characteristic
146	C₂×C₈	#11745660	1	1	yes	characteristic
147	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
148	C₂×C₈	Dic₂₃₆	1	1	yes	characteristic
149	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
150	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	#11745670	1	1	yes	characteristic
152	C₂×C₈	#11745670	1	1	yes	characteristic
153	C₂×C₈	#11745672	1	1	yes	characteristic
154	C₂×C₈	#11745672	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
156	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
157	C₄⋊C₄	—	118	4	no	—
158	C₄⋊C₄	—	236	4	no	—
159	C₄⋊C₄	—	236	4	no	—
160	C₄⋊C₄	—	236	2	no	—
161	C₄²	—	2	1	no	—
162	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
163	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
164	C₁₆	—	4	2	no	—
165	C₁₆	—	236	2	no	—
166	C₂³	#14273643	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	472	4	no	—
171	C₂×C₄	—	236	4	no	—
172	C₂×C₄	#14273643	1	1	yes	characteristic
173	C₂×C₄	#14273650	1	1	yes	characteristic
174	C₂×C₄	#14273650	1	1	yes	characteristic
175	C₂×C₄	#14273652	1	1	yes	characteristic
176	C₂×C₄	#14273652	1	1	yes	characteristic
177	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
178	C₈	#14273650	1	1	yes	characteristic
179	C₈	#14273650	1	1	yes	characteristic
180	C₈	#14273652	1	1	yes	characteristic
181	C₈	#14273652	1	1	yes	characteristic
182	C₈	C₂×#11745670	1	1	yes	characteristic
183	C₈	C₂×#11745660	1	1	yes	characteristic
184	C₈	C₂×#11745672	1	1	yes	characteristic
185	C₈	C₂×Dic₂₃₆	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#16837150	1	1	yes	characteristic
188	C₂²	#16837150	1	1	yes	characteristic
189	C₂²	C₂×#14273643	1	1	yes	characteristic
190	C₄	—	4	1	no	—
191	C₄	—	4	1	no	—
192	C₄	—	472	4	no	—
193	C₄	#16837150	1	1	yes	characteristic
194	C₄	#16837150	1	1	yes	characteristic
195	C₄	C₂×#14273650	1	1	yes	characteristic
196	C₄	C₂×#14273652	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#21422158	1	2	yes	direct product factor
200	C₂	C₂×#16837150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837150
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	#333
prime radical	C₂³×C₅₉	#100
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	C₂×#1366
LCS	G ▷ C₄²×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #21422158	C₂	direct (N × H)
103. C₂×C₅₉	#1366	semidirect
119. C₅₉	C₂×#1366	semidirect
199. C₂	#21422158	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837804	2
C₂×#16837223	2
#21422158	2
C₂×C₅₉×#364	2
C₂×#1366	59

Subgroups — C2×#21422158 #28500054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422158 #28500054