Subgroups — C₂×#21424237 #28588919

Order 15104.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837426	C₂	1	2	yes	maximal
2	#21424237	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16837167	C₂	1	4	yes	maximal
4	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11745664	C₂²	1	4	yes	—
6	#16837426	—	2	4	no	—
7	#16837426	C₂²	1	8	yes	—
8	C₂×C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
9	#16837167	C₂²	1	16	yes	—
10	C₂³×C₈×C₅₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
12	C₂×#97×C₅₉	C₄	1	4	yes	—
13	C₂×#97×C₅₉	C₄	1	4	yes	—
14	C₂×#97×C₅₉	C₂²	1	4	yes	—
15	C₂×#11745664	—	2	4	no	—
16	C₂×#11745664	—	2	16	no	—
17	C₂×#11745664	C₂³	1	8	yes	—
18	C₂²×C₅₉⋊C₈	—	2	4	no	—
19	C₄⋊(C₅₉⋊C₈)	—	2	16	no	—
20	C₂³×C₄×C₅₉	Q₈	1	2	yes	—
21	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	—	2	8	no	—
23	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₉	C₂³	1	4	yes	—
26	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₉	D₄	1	2	yes	—
29	C₂×C₄²×C₅₉	D₄	1	2	yes	—
30	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
31	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
32	#97×C₅₉	C₂×C₄	1	16	yes	—
33	C₂⁴×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	—	4	4	no	—
35	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
37	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
42	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₉	#41	1	8	yes	—
44	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₈×C₅₉	—	2	8	no	—
46	C₂×C₈×C₅₉	—	2	8	no	—
47	C₂×C₈×C₅₉	—	2	8	no	—
48	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
49	C₄²×C₅₉	C₄⋊C₄	1	4	yes	—
50	C₄²×C₅₉	C₄⋊C₄	1	4	yes	—
51	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
52	C₄²×C₅₉	C₂×D₄	1	4	yes	—
53	#11745664	—	2	16	no	—
54	#11745664	—	4	8	no	—
55	C₂×C₅₉⋊C₈	—	2	8	no	—
56	C₂×C₅₉⋊C₈	—	4	8	no	—
57	C₂³×C₅₉	—	2	4	no	—
58	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₅₉	#125	1	4	yes	—
61	C₂³×C₅₉	C₂²×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₅₉	—	2	8	no	—
63	C₂×C₄×C₅₉	—	4	4	no	—
64	C₂×C₄×C₅₉	—	4	4	no	—
65	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₉	#117	1	4	yes	—
69	C₂×C₄×C₅₉	#117	1	4	yes	—
70	C₂×C₄×C₅₉	#124	1	4	yes	—
71	C₂×C₄×C₅₉	#124	1	4	yes	—
72	C₂×C₄×C₅₉	#125	1	4	yes	—
73	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₄×C₅₉	C₂×#41	1	4	yes	—
76	C₈×C₅₉	—	2	8	no	—
77	C₅₉⋊C₈	—	4	8	no	—
78	C₂×#1838	—	59	1	no	maximal, Sylow
79	C₂²×C₅₉	—	2	8	no	—
80	C₂²×C₅₉	—	2	2	no	—
81	C₂²×C₅₉	—	2	2	no	—
82	C₂²×C₅₉	#382	1	2	yes	—
83	C₂²×C₅₉	#382	1	2	yes	—
84	C₂²×C₅₉	#430	1	4	yes	—
85	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₅₉	C₂×#125	1	2	yes	—
87	C₄×C₅₉	—	2	8	no	—
88	C₄×C₅₉	#382	1	2	yes	—
89	C₄×C₅₉	#382	1	2	yes	—
90	C₄×C₅₉	C₂×#117	1	2	yes	—
91	C₄×C₅₉	C₂×#124	1	2	yes	—
92	C₂×#423	—	59	2	no	—
93	C₂²×#97	D₅₉	1	1	yes	characteristic
94	#1838	—	59	8	no	—
95	C₂×#341	—	59	4	no	—
96	C₂×C₅₉	—	2	2	no	—
97	C₂×C₅₉	—	2	2	no	—
98	C₂×C₅₉	#1838	1	4	yes	complemented
99	C₂×C₅₉	C₂×#382	1	1	yes	characteristic
100	C₂×C₅₉	C₂×#430	1	2	yes	—
101	C₂²×M₁₆	—	59	4	no	—
102	C₂³×C₈	C₂×D₅₉	1	2	yes	—
103	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
104	#423	—	59	8	no	—
105	#423	—	118	4	no	—
106	C₂×C₄⋊C₈	—	118	4	no	—
107	C₂×#97	Dic₅₉	1	4	yes	—
108	C₂×#97	Dic₅₉	1	4	yes	—
109	C₂×#97	C₂×D₅₉	1	4	yes	—
110	#341	—	59	16	no	—
111	C₅₉	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
112	C₂³×C₄	Dic₁₁₈	1	2	yes	—
113	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
114	C₂×M₁₆	—	59	8	no	—
115	C₂×M₁₆	—	118	4	no	—
116	C₂×M₁₆	—	118	16	no	—
117	C₂²×C₈	—	2	8	no	—
118	C₂²×C₈	—	118	4	no	—
119	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
120	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
121	C₂²×C₈	C₂²×D₅₉	1	4	yes	—
122	C₂×C₄²	Dic₁₁₈	1	2	yes	—
123	C₂×C₄²	D₂₃₆	1	2	yes	—
124	C₂×C₄²	D₂₃₆	1	2	yes	—
125	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
126	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
127	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
128	C₄⋊C₈	—	118	16	no	—
129	#97	C₂×Dic₅₉	1	16	yes	—
130	C₂⁴	C₂×Dic₁₁₈	1	1	yes	characteristic
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
133	C₂²×C₄	C₄⋊Dic₅₉	1	4	yes	—
134	C₂²×C₄	C₂×Dic₁₁₈	1	4	yes	—
135	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
138	C₂²×C₄	#11745685	1	8	yes	—
139	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
141	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
142	M₁₆	—	118	16	no	—
143	M₁₆	—	236	8	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	2	8	no	—
147	C₂×C₈	—	118	8	no	—
148	C₂×C₈	—	236	8	no	—
149	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
150	C₄²	C₄⋊Dic₅₉	1	4	yes	—
151	C₄²	C₄⋊Dic₅₉	1	4	yes	—
152	C₄²	C₂×D₂₃₆	1	4	yes	—
153	C₄²	C₂²×Dic₅₉	1	4	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
156	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
157	C₂³	#14273719	1	4	yes	—
158	C₂³	C₂²×Dic₁₁₈	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
163	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
165	C₂×C₄	#14273686	1	4	yes	—
166	C₂×C₄	#14273686	1	4	yes	—
167	C₂×C₄	#14273715	1	4	yes	—
168	C₂×C₄	#14273715	1	4	yes	—
169	C₂×C₄	#14273719	1	4	yes	—
170	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
171	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
172	C₂×C₄	C₂×#11745685	1	4	yes	—
173	C₈	—	2	8	no	—
174	C₈	—	236	8	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	#16837268	1	2	yes	—
179	C₂²	#16837268	1	2	yes	—
180	C₂²	#16837451	1	4	yes	—
181	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
182	C₂²	C₂×#14273719	1	2	yes	—
183	C₄	—	2	8	no	—
184	C₄	#16837268	1	2	yes	—
185	C₄	#16837268	1	2	yes	—
186	C₄	C₂×#14273686	1	2	yes	—
187	C₄	C₂×#14273715	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	#21424237	1	4	yes	direct product factor
191	C₂	C₂×#16837268	1	1	yes	characteristic
192	C₂	C₂×#16837451	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₂×Q₈
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	C₂×#1838
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
2. #21424237	C₂	direct (N × H)
98. C₂×C₅₉	#1838	semidirect
111. C₅₉	C₂×#1838	semidirect
190. C₂	#21424237	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×#16837426	2
#21424237	2
C₂×#16837167	2
C₂²×#97×C₅₉	2
C₂×#1838	59

Subgroups — C2×#21424237 #28588919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21424237 #28588919