Subgroups — C₂×#21426357 #28744088

Order 15104.

854 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14273672	C₂	1	2	yes	maximal, complemented
3	C₂²×D₉₄₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837786	C₂	1	2	yes	maximal, complemented
5	#21426357	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
8	C₂²×#11745660	—	2	2	no	—
9	C₂²×D₄₇₂	C₂²	1	2	yes	—
10	C₂×#14273715	—	2	2	no	—
11	C₂×Dic₄₇₂	—	2	4	no	—
12	C₂×Dic₄₇₂	C₂²	1	4	yes	complemented
13	C₂×#14273672	—	2	8	no	—
14	C₂×#14273672	C₂²	1	8	yes	complemented
15	C₂×D₉₄₄	—	2	4	no	—
16	C₂×D₉₄₄	C₂²	1	4	yes	—
17	#16837786	—	2	4	no	—
18	#16837786	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₉	C₂²	1	4	yes	complemented
23	C₂²×Dic₁₁₈	—	2	2	no	—
24	C₂²×D₂₃₆	—	2	2	no	—
25	C₂×Dic₂₃₆	—	2	4	no	—
26	C₂×Dic₂₃₆	—	2	4	no	—
27	C₂×Dic₂₃₆	—	2	2	no	—
28	C₂×Dic₂₃₆	C₂³	1	4	yes	—
29	C₂×#11745660	—	2	8	no	—
30	C₂×#11745660	—	4	8	no	—
31	C₂×D₄₇₂	—	2	4	no	—
32	C₂×D₄₇₂	—	2	4	no	—
33	C₂×D₄₇₂	—	2	2	no	—
34	C₂×D₄₇₂	C₂³	1	4	yes	—
35	#14273715	—	2	8	no	—
36	#14273715	—	4	4	no	—
37	C₂×(C₂×D₅₉)⋊C₄	—	4	4	no	—
38	Dic₄₇₂	—	2	8	no	—
39	#14273672	—	2	16	no	—
40	D₉₄₄	—	2	8	no	—
41	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
45	C₂×C₁₆×C₅₉	D₄	1	4	yes	—
46	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	2	yes	—
49	C₄×C₈×C₅₉	C₂³	1	2	yes	—
50	C₂³×D₅₉	—	4	2	no	—
51	C₂²×C₄×C₅₉	D₈	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₉	—	4	2	no	—
54	C₂×D₂₃₆	—	2	4	no	—
55	C₂×D₂₃₆	—	4	4	no	—
56	C₂×D₂₃₆	—	4	4	no	—
57	C₂×D₂₃₆	—	4	2	no	—
58	C₂×Dic₁₁₈	—	2	4	no	—
59	C₂×Dic₁₁₈	—	4	4	no	—
60	C₂×Dic₁₁₈	—	4	4	no	—
61	C₂×Dic₁₁₈	—	4	2	no	—
62	C₂×C₈×C₅₉	D₈	1	4	yes	—
63	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₄²×C₅₉	D₈	1	2	yes	—
69	C₄²×C₅₉	C₂×D₄	1	2	yes	—
70	(C₂×D₅₉)⋊C₄	—	4	16	no	—
71	Dic₂₃₆	—	2	4	no	—
72	Dic₂₃₆	—	2	4	no	—
73	Dic₂₃₆	—	4	4	no	—
74	D₄₇₂	—	2	4	no	—
75	D₄₇₂	—	2	4	no	—
76	D₄₇₂	—	4	4	no	—
77	#11745660	—	4	16	no	—
78	C₁₆×C₅₉	C₂×D₄	1	4	yes	—
79	C₁₆×C₅₉	C₂×D₄	1	4	yes	—
80	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₅₉	—	4	4	no	—
82	C₂²×D₅₉	—	8	4	no	—
83	C₂²×D₅₉	—	8	8	no	—
84	C₂×C₄×C₅₉	—	2	4	no	—
85	C₂×C₄×C₅₉	#127	1	2	yes	—
86	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
87	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
90	C₂×Dic₅₉	—	4	4	no	—
91	C₂×Dic₅₉	—	8	4	no	—
92	D₂₃₆	—	4	4	no	—
93	D₂₃₆	—	4	4	no	—
94	D₂₃₆	—	8	4	no	—
95	Dic₁₁₈	—	4	4	no	—
96	Dic₁₁₈	—	4	4	no	—
97	Dic₁₁₈	—	8	4	no	—
98	C₈×C₅₉	#127	1	2	yes	—
99	C₈×C₅₉	C₂×D₈	1	4	yes	—
100	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
102	C₂×#2249	—	59	1	no	maximal, Sylow
103	C₂²×C₅₉	#493	1	2	yes	—
104	C₂²×C₅₉	#508	1	4	yes	—
105	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
106	C₂×D₅₉	—	8	4	no	—
107	C₂×D₅₉	—	8	8	no	—
108	C₂×D₅₉	—	16	8	no	—
109	C₄×C₅₉	—	2	4	no	—
110	C₄×C₅₉	#493	1	2	yes	—
111	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
112	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
113	Dic₅₉	—	8	4	no	—
114	C₂²×Q₃₂	—	59	1	no	—
115	C₂²×SD₃₂	—	59	2	no	—
116	C₂²×D₁₆	—	59	1	no	—
117	C₂×#495	—	59	2	no	—
118	#2249	—	59	8	no	—
119	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
120	C₂×C₅₉	#2249	1	4	yes	complemented
121	C₂×C₅₉	C₂×#493	1	1	yes	characteristic
122	C₂×C₅₉	C₂×#508	1	2	yes	—
123	D₅₉	—	16	4	no	—
124	C₂²×Q₁₆	—	59	2	no	—
125	C₂²×SD₁₆	—	118	2	no	—
126	C₂²×D₈	—	59	2	no	—
127	C₂×#124	—	118	2	no	—
128	C₂×Q₃₂	—	59	4	no	—
129	C₂×Q₃₂	—	118	4	no	—
130	C₂×SD₃₂	—	59	8	no	—
131	C₂×SD₃₂	—	118	8	no	—
132	C₂×D₁₆	—	59	4	no	—
133	C₂×D₁₆	—	118	4	no	—
134	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
135	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
136	#495	—	59	8	no	—
137	#495	—	118	4	no	—
138	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₅₉	1	4	yes	complemented
140	C₅₉	C₂×#2249	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	118	2	no	—
142	C₂²×D₄	—	118	2	no	—
143	C₂×Q₁₆	—	59	4	no	—
144	C₂×Q₁₆	—	118	4	no	—
145	C₂×Q₁₆	—	118	4	no	—
146	C₂×Q₁₆	—	118	2	no	—
147	C₂×SD₁₆	—	118	8	no	—
148	C₂×SD₁₆	—	236	8	no	—
149	C₂×D₈	—	59	4	no	—
150	C₂×D₈	—	118	4	no	—
151	C₂×D₈	—	118	4	no	—
152	C₂×D₈	—	118	2	no	—
153	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
154	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
155	#124	—	118	8	no	—
156	#124	—	236	4	no	—
157	C₂×C₂²⋊C₄	—	236	4	no	—
158	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
159	Q₃₂	—	118	8	no	—
160	SD₃₂	—	118	16	no	—
161	D₁₆	—	118	8	no	—
162	C₂×C₁₆	D₂₃₆	1	4	yes	—
163	C₂×C₁₆	D₂₃₆	1	4	yes	—
164	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
165	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
166	C₄×C₈	D₂₃₆	1	2	yes	—
167	C₄×C₈	C₂²×D₅₉	1	2	yes	—
168	C₂⁴	—	236	2	no	—
169	C₂×Q₈	—	118	4	no	—
170	C₂×Q₈	—	236	4	no	—
171	C₂×Q₈	—	236	4	no	—
172	C₂×Q₈	—	236	2	no	—
173	C₂×D₄	—	118	4	no	—
174	C₂×D₄	—	236	4	no	—
175	C₂×D₄	—	236	2	no	—
176	C₂×D₄	—	236	4	no	—
177	C₂²×C₄	—	236	2	no	—
178	C₂²×C₄	D₄₇₂	1	2	yes	—
179	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
180	Q₁₆	—	118	4	no	—
181	Q₁₆	—	118	4	no	—
182	Q₁₆	—	236	4	no	—
183	SD₁₆	—	236	16	no	—
184	D₈	—	118	4	no	—
185	D₈	—	118	4	no	—
186	D₈	—	236	4	no	—
187	C₂×C₈	D₄₇₂	1	4	yes	—
188	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
189	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
190	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
191	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
192	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
193	C₂²⋊C₄	—	236	16	no	—
194	C₄²	D₄₇₂	1	2	yes	—
195	C₄²	C₂×D₂₃₆	1	2	yes	—
196	C₁₆	C₂×D₂₃₆	1	4	yes	—
197	C₁₆	C₂×D₂₃₆	1	4	yes	—
198	C₂³	—	472	4	no	—
199	C₂³	—	472	8	no	—
200	C₂³	—	236	4	no	—
201	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
202	Q₈	—	472	4	no	—
203	Q₈	—	236	4	no	—
204	Q₈	—	236	4	no	—
205	D₄	—	472	4	no	—
206	D₄	—	236	4	no	—
207	D₄	—	236	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	472	4	no	—
210	C₂×C₄	—	236	4	no	—
211	C₂×C₄	#14273724	1	2	yes	—
212	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
213	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
214	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
216	C₈	#14273724	1	2	yes	—
217	C₈	C₂×D₄₇₂	1	4	yes	—
218	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
219	C₈	C₂²×D₂₃₆	1	1	yes	characteristic
220	C₂²	—	944	8	no	—
221	C₂²	—	472	8	no	—
222	C₂²	—	472	4	no	—
223	C₂²	#16837779	1	2	yes	—
224	C₂²	#16837834	1	4	yes	—
225	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	472	4	no	—
228	C₄	#16837779	1	2	yes	—
229	C₄	C₂×#14273724	1	1	yes	characteristic
230	C₄	C₂²×D₄₇₂	1	1	yes	characteristic
231	C₂	—	944	4	no	—
232	C₂	#21426357	1	4	yes	direct product factor
233	C₂	C₂×#16837779	1	1	yes	characteristic
234	C₂	C₂×#16837834	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₉₄₄	C₂
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#2249
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₇₂	C₂	semidirect
2. C₂²×#14273672	C₂	semidirect
4. C₂×#16837786	C₂	semidirect
5. #21426357	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
12. C₂×Dic₄₇₂	C₂²	semidirect
14. C₂×#14273672	C₂²	semidirect
18. #16837786	C₂²	semidirect
22. C₄×C₁₆×C₅₉	C₂²	semidirect
119. C₂×C₄×C₁₆	D₅₉	semidirect
120. C₂×C₅₉	#2249	semidirect
139. C₄×C₁₆	C₂×D₅₉	semidirect
140. C₅₉	C₂×#2249	semidirect
232. C₂	#21426357	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×Dic₄₇₂	2
C₂²×#14273672	2
C₂²×D₉₄₄	2
C₂×#16837786	2
#21426357	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#2249	59

Subgroups — C2×#21426357 #28744088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426357 #28744088