Subgroups — C₂×#21425973 #28741993

Order 15104.

798 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₉₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₇₂	C₂	1	2	yes	maximal, complemented
3	#21425973	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837230	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837204	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄₇₂	C₂²	1	2	yes	—
8	C₂×C₄×D₂₃₆	—	2	2	no	—
9	C₂×D₉₄₄	C₄	1	8	yes	complemented
10	C₂×D₉₄₄	C₂²	1	4	yes	—
11	C₄×D₄₇₂	—	2	4	no	—
12	C₄×D₄₇₂	C₂²	1	8	yes	complemented
13	C₂×#14273661	C₂²	1	2	yes	—
14	C₂×#14273649	—	2	2	no	—
15	#16837230	C₂²	1	4	yes	complemented
16	#16837204	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₅₉×#346	C₂²	1	4	yes	complemented
21	C₂²×D₂₃₆	—	2	2	no	—
22	C₂²×C₄×D₅₉	—	4	2	no	—
23	C₂×D₄₇₂	—	2	2	no	—
24	C₂×D₄₇₂	—	2	4	no	—
25	C₂×D₄₇₂	C₂×C₄	1	8	yes	complemented
26	C₂×D₄₇₂	C₂³	1	4	yes	—
27	C₄×D₂₃₆	—	2	8	no	—
28	C₄×D₂₃₆	—	4	4	no	—
29	C₂×C₄⋊Dic₅₉	—	2	2	no	—
30	C₂×(C₂×D₅₉)⋊C₄	—	4	2	no	—
31	D₉₄₄	C₂×C₄	1	16	yes	complemented
32	#14273661	—	2	2	no	—
33	#14273661	C₂³	1	4	yes	—
34	#14273649	—	2	8	no	—
35	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₉	—	2	2	no	—
39	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
41	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
42	C₄×C₈×C₅₉	D₄	1	2	yes	—
43	C₄×C₈×C₅₉	C₂³	1	2	yes	—
44	C₂³×D₅₉	—	4	2	no	—
45	C₂²×C₄×C₅₉	D₈	1	2	yes	—
46	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₉	—	4	2	no	—
48	C₂×D₂₃₆	—	2	8	no	—
49	C₂×D₂₃₆	—	2	4	no	—
50	C₂×D₂₃₆	—	4	2	no	—
51	C₂×D₂₃₆	—	4	4	no	—
52	C₂×C₄×D₅₉	—	4	8	no	—
53	C₂×C₄×D₅₉	—	8	8	no	—
54	C₂×C₈×C₅₉	—	2	2	no	—
55	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	#41	1	2	yes	—
59	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
60	C₄²×C₅₉	D₈	1	2	yes	—
61	C₄²×C₅₉	C₂×D₄	1	2	yes	—
62	(C₂×D₅₉)⋊C₄	—	4	8	no	—
63	C₄⋊Dic₅₉	—	2	4	no	—
64	C₄⋊Dic₅₉	—	4	2	no	—
65	D₄₇₂	—	2	8	no	—
66	D₄₇₂	—	2	4	no	—
67	D₄₇₂	C₂²×C₄	1	8	yes	complemented
68	C₁₆×C₅₉	—	2	2	no	—
69	C₁₆×C₅₉	C₂²×C₄	1	4	yes	complemented
70	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₅₉	—	4	4	no	—
72	C₂²×D₅₉	—	4	8	no	—
73	C₂²×D₅₉	—	8	4	no	—
74	C₂²×D₅₉	—	8	4	no	—
75	C₂×C₄×C₅₉	—	2	4	no	—
76	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#135	1	2	yes	—
80	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₅₉	—	4	4	no	—
82	C₂×Dic₅₉	—	8	4	no	—
83	D₂₃₆	—	2	8	no	—
84	D₂₃₆	—	4	4	no	—
85	D₂₃₆	—	4	8	no	—
86	C₄×D₅₉	—	8	16	no	—
87	C₈×C₅₉	—	2	2	no	—
88	C₈×C₅₉	C₄×D₄	1	2	yes	—
89	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
91	C₂×#2177	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	C₄×D₈	1	2	yes	—
93	C₂²×C₅₉	#509	1	4	yes	—
94	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
95	C₂×D₅₉	—	4	8	no	—
96	C₂×D₅₉	—	8	4	no	—
97	C₂×D₅₉	—	8	4	no	—
98	C₂×D₅₉	—	8	16	no	—
99	C₄×C₅₉	—	2	4	no	—
100	C₄×C₅₉	C₄×D₈	1	2	yes	—
101	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
102	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
103	Dic₅₉	—	8	4	no	—
104	C₂²×D₁₆	—	59	1	no	—
105	C₂×C₄×D₈	—	59	2	no	—
106	#2177	—	59	8	no	—
107	C₂×#367	—	59	1	no	—
108	C₂×#357	—	59	2	no	—
109	C₂×#346	D₅₉	1	1	yes	characteristic, complemented
110	C₂×C₅₉	#2177	1	4	yes	complemented
111	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
112	C₂×C₅₉	C₂×#509	1	2	yes	—
113	D₅₉	—	8	8	no	—
114	C₂²×D₈	—	59	2	no	—
115	C₂×C₄×D₄	—	118	2	no	—
116	C₂×D₁₆	—	59	8	no	—
117	C₂×D₁₆	—	59	4	no	—
118	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
119	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
120	C₄×D₈	—	59	8	no	—
121	C₄×D₈	—	118	4	no	—
122	C₂×#107	—	59	2	no	—
123	C₂×#102	—	118	2	no	—
124	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
125	#367	—	59	4	no	—
126	#357	—	59	8	no	—
127	#346	C₂×D₅₉	1	4	yes	complemented
128	C₅₉	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
129	C₂²×D₄	—	118	2	no	—
130	C₂³×C₄	—	236	2	no	—
131	C₂×D₈	—	59	8	no	—
132	C₂×D₈	—	59	4	no	—
133	C₂×D₈	—	118	2	no	—
134	C₂×D₈	—	118	4	no	—
135	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
137	C₄×D₄	—	118	8	no	—
138	C₄×D₄	—	236	4	no	—
139	C₂×C₄⋊C₄	—	118	2	no	—
140	C₂×C₂²⋊C₄	—	236	2	no	—
141	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
142	D₁₆	—	59	16	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	C₄×D₅₉	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
146	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
147	#107	—	59	4	no	—
148	#107	—	118	2	no	—
149	#102	—	118	8	no	—
150	C₄×C₈	D₂₃₆	1	2	yes	—
151	C₄×C₈	C₂²×D₅₉	1	2	yes	—
152	C₂⁴	—	236	2	no	—
153	C₂×D₄	—	118	8	no	—
154	C₂×D₄	—	118	4	no	—
155	C₂×D₄	—	236	2	no	—
156	C₂×D₄	—	236	4	no	—
157	C₂²×C₄	—	472	8	no	—
158	C₂²×C₄	—	236	2	no	—
159	C₂²×C₄	—	236	8	no	—
160	C₂²×C₄	D₄₇₂	1	2	yes	—
161	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
162	D₈	—	59	8	no	—
163	D₈	—	118	8	no	—
164	D₈	—	118	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
167	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
168	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
169	C₂×C₈	#11745685	1	2	yes	—
170	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
171	C₄⋊C₄	—	118	4	no	—
172	C₄⋊C₄	—	236	2	no	—
173	C₂²⋊C₄	—	236	8	no	—
174	C₄²	D₄₇₂	1	2	yes	—
175	C₄²	C₂×D₂₃₆	1	2	yes	—
176	C₁₆	—	2	2	no	—
177	C₁₆	C₂×C₄×D₅₉	1	4	yes	complemented
178	C₂³	—	472	4	no	—
179	C₂³	—	472	4	no	—
180	C₂³	—	236	4	no	—
181	C₂³	—	236	8	no	—
182	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
183	D₄	—	118	8	no	—
184	D₄	—	236	4	no	—
185	D₄	—	236	8	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	472	16	no	—
188	C₂×C₄	—	472	4	no	—
189	C₂×C₄	—	236	4	no	—
190	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
191	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
192	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
193	C₂×C₄	#14273749	1	2	yes	—
194	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	C₄×D₂₃₆	1	2	yes	—
197	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
198	C₈	C₂×#11745685	1	1	yes	characteristic
199	C₂²	—	472	4	no	—
200	C₂²	—	472	4	no	—
201	C₂²	—	472	16	no	—
202	C₂²	—	236	8	no	—
203	C₂²	C₄×D₄₇₂	1	2	yes	—
204	C₂²	#16837840	1	4	yes	—
205	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	472	4	no	—
208	C₄	C₄×D₄₇₂	1	2	yes	—
209	C₄	C₂×C₄×D₂₃₆	1	1	yes	characteristic
210	C₄	C₂×#14273749	1	1	yes	characteristic
211	C₂	—	472	8	no	—
212	C₂	#21425973	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₄₇₂	1	1	yes	characteristic
214	C₂	C₂×#16837840	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₉₄₄	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	C₂×#2177
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₄₇₂	C₂	semidirect
3. #21425973	C₂	direct (N × H)
4. C₂×#16837230	C₂	semidirect
5. C₂×#16837204	C₂	semidirect
6. C₂×C₅₉×#346	C₂	semidirect
9. C₂×D₉₄₄	C₄	semidirect
12. C₄×D₄₇₂	C₂²	semidirect
15. #16837230	C₂²	semidirect
16. #16837204	C₂²	semidirect
20. C₅₉×#346	C₂²	semidirect
25. C₂×D₄₇₂	C₂×C₄	semidirect
31. D₉₄₄	C₂×C₄	semidirect
39. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
67. D₄₇₂	C₂²×C₄	semidirect
69. C₁₆×C₅₉	C₂²×C₄	semidirect
109. C₂×#346	D₅₉	semidirect
110. C₂×C₅₉	#2177	semidirect
127. #346	C₂×D₅₉	semidirect
128. C₅₉	C₂×#2177	semidirect
144. C₂×C₁₆	C₄×D₅₉	semidirect
177. C₁₆	C₂×C₄×D₅₉	semidirect
212. C₂	#21425973	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₉₄₄	2
C₂×C₄×D₄₇₂	2
#21425973	2
C₂×#16837230	2
C₂×#16837204	2
C₂×C₅₉×#346	2
C₂×#2177	59

Subgroups — C2×#21425973 #28741993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425973 #28741993