Subgroups — C₂×#21422061 #28498425

Order 15104.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	#21422061	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₁₈	—	2	1	no	—
7	C₄×Dic₂₃₆	—	2	1	no	—
8	C₄×Dic₂₃₆	—	2	1	no	—
9	C₄×Dic₂₃₆	C₄	1	2	yes	—
10	C₄×Dic₂₃₆	C₂²	1	2	yes	—
11	C₂×#14273661	C₄	1	1	yes	characteristic
12	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
13	C₂×#14273652	—	2	1	no	—
14	#16837155	—	2	1	no	—
15	#16837155	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₉	—	2	1	no	—
19	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
20	C₂²×Dic₁₁₈	—	2	1	no	—
21	C₂×Dic₂₃₆	—	2	1	no	—
22	C₂×Dic₂₃₆	—	2	2	no	—
23	C₂×Dic₂₃₆	C₈	1	4	yes	complemented
24	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
25	C₂²×C₅₉⋊C₈	—	4	1	no	—
26	C₄×Dic₁₁₈	—	2	2	no	—
27	C₄×Dic₁₁₈	—	2	2	no	—
28	C₄×Dic₁₁₈	—	4	1	no	—
29	C₄×Dic₁₁₈	—	4	1	no	—
30	C₂×C₄⋊Dic₅₉	—	2	1	no	—
31	C₂×Dic₅₉⋊C₄	—	4	1	no	—
32	C₂×C₄×Dic₅₉	—	4	1	no	—
33	#14273661	—	2	1	no	—
34	#14273661	C₂×C₄	1	2	yes	—
35	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
36	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
37	#14273652	—	2	2	no	—
38	#14273652	—	2	2	no	—
39	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₉	—	2	4	no	—
43	C₂×C₁₆×C₅₉	—	2	2	no	—
44	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
51	C₂²×Dic₅₉	—	4	1	no	—
52	C₂²×Dic₅₉	—	4	1	no	—
53	C₂×Dic₁₁₈	—	2	4	no	—
54	C₂×Dic₁₁₈	—	2	2	no	—
55	C₂×Dic₁₁₈	—	4	1	no	—
56	C₂×Dic₁₁₈	—	4	2	no	—
57	C₂×C₈×C₅₉	—	2	2	no	—
58	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
62	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₉	D₈	1	1	yes	characteristic
66	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
67	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊Dic₅₉	—	2	2	no	—
69	C₄⋊Dic₅₉	—	4	1	no	—
70	Dic₅₉⋊C₄	—	4	2	no	—
71	Dic₅₉⋊C₄	—	4	2	no	—
72	C₄×Dic₅₉	—	4	2	no	—
73	C₄×Dic₅₉	—	4	2	no	—
74	C₂×C₅₉⋊C₈	—	4	2	no	—
75	C₂×C₅₉⋊C₈	—	8	2	no	—
76	Dic₂₃₆	—	2	4	no	—
77	Dic₂₃₆	—	2	2	no	—
78	Dic₂₃₆	C₂×C₈	1	4	yes	complemented
79	C₁₆×C₅₉	—	2	4	no	—
80	C₂³×C₅₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₉	—	2	2	no	—
82	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₉	#104	1	2	yes	—
87	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
88	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₅₉	—	4	4	no	—
93	C₂×Dic₅₉	—	4	2	no	—
94	C₂×Dic₅₉	—	4	2	no	—
95	C₂×Dic₅₉	—	8	2	no	—
96	Dic₁₁₈	—	2	4	no	—
97	Dic₁₁₈	—	4	4	no	—
98	Dic₁₁₈	—	4	2	no	—
99	C₈×C₅₉	—	2	1	no	—
100	C₈×C₅₉	—	2	1	no	—
101	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
102	C₈×C₅₉	C₂²×C₈	1	1	yes	characteristic
103	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
104	C₅₉⋊C₈	—	8	2	no	—
105	C₂×#1332	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	D₄⋊C₈	1	2	yes	—
107	C₂²×C₅₉	#358	1	2	yes	—
108	C₂²×C₅₉	#359	1	2	yes	—
109	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	D₄⋊C₈	1	2	yes	—
113	C₄×C₅₉	#358	1	2	yes	—
114	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
116	C₄×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
117	C₄×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
118	Dic₅₉	—	4	4	no	—
119	Dic₅₉	—	8	2	no	—
120	C₂×C₄×Q₁₆	—	59	1	no	—
121	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
122	C₂×#335	—	59	1	no	—
123	#1332	—	59	4	no	—
124	C₂×C₅₉	#1332	1	4	yes	complemented
125	C₂×C₅₉	C₂×D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
127	C₂×C₅₉	C₂×#359	1	1	yes	characteristic
128	C₂²×Q₁₆	—	59	1	no	—
129	C₂×C₄×Q₈	—	118	1	no	—
130	C₂²×C₁₆	—	2	1	no	—
131	C₄×Q₁₆	—	59	2	no	—
132	C₄×Q₁₆	—	59	2	no	—
133	C₄×Q₁₆	—	118	1	no	—
134	C₄×Q₁₆	—	118	1	no	—
135	C₂×#107	—	59	1	no	—
136	C₂×C₄⋊C₈	—	118	1	no	—
137	C₂×#103	—	118	1	no	—
138	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
139	C₄×C₁₆	—	2	1	no	—
140	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
141	#335	—	59	2	no	—
142	#335	—	118	1	no	—
143	C₅₉	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	118	1	no	—
145	C₂×Q₁₆	—	59	4	no	—
146	C₂×Q₁₆	—	59	2	no	—
147	C₂×Q₁₆	—	118	2	no	—
148	C₂×Q₁₆	—	118	1	no	—
149	C₂²×C₈	—	236	1	no	—
150	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
152	C₄×Q₈	—	118	2	no	—
153	C₄×Q₈	—	118	2	no	—
154	C₄×Q₈	—	236	1	no	—
155	C₄×Q₈	—	236	1	no	—
156	C₂×C₄⋊C₄	—	118	1	no	—
157	C₂×C₄⋊C₄	—	236	1	no	—
158	C₂×C₄²	—	236	1	no	—
159	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
160	C₂×C₁₆	—	2	4	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	59	2	no	—
163	#107	—	118	1	no	—
164	C₄⋊C₈	—	118	2	no	—
165	C₄⋊C₈	—	236	1	no	—
166	#103	—	118	2	no	—
167	#103	—	118	2	no	—
168	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
169	C₄×C₈	D₂₃₆	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	C₂×Q₈	—	118	4	no	—
173	C₂×Q₈	—	118	2	no	—
174	C₂×Q₈	—	236	1	no	—
175	C₂×Q₈	—	236	2	no	—
176	C₂²×C₄	—	236	1	no	—
177	C₂²×C₄	—	236	1	no	—
178	C₂²×C₄	#11745660	1	1	yes	characteristic
179	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
180	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
181	Q₁₆	—	59	4	no	—
182	Q₁₆	—	118	4	no	—
183	Q₁₆	—	118	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	472	2	no	—
186	C₂×C₈	—	236	2	no	—
187	C₂×C₈	C₈×D₅₉	1	2	yes	—
188	C₂×C₈	#11745659	1	2	yes	—
189	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
190	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
193	C₄⋊C₄	—	118	2	no	—
194	C₄⋊C₄	—	236	1	no	—
195	C₄⋊C₄	—	236	2	no	—
196	C₄⋊C₄	—	236	2	no	—
197	C₄²	—	236	2	no	—
198	C₄²	—	236	2	no	—
199	C₄²	#11745660	1	1	yes	characteristic
200	C₄²	D₄₇₂	1	1	yes	characteristic
201	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
202	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
203	C₁₆	—	2	4	no	—
204	C₂³	#14273649	1	1	yes	characteristic
205	Q₈	—	118	4	no	—
206	Q₈	—	236	4	no	—
207	Q₈	—	236	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	472	2	no	—
210	C₂×C₄	—	236	2	no	—
211	C₂×C₄	—	236	4	no	—
212	C₂×C₄	—	236	2	no	—
213	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
214	C₂×C₄	#14273649	1	1	yes	characteristic
215	C₂×C₄	#14273649	1	1	yes	characteristic
216	C₂×C₄	#14273649	1	1	yes	characteristic
217	C₂×C₄	#14273655	1	2	yes	—
218	C₂×C₄	#14273672	1	2	yes	—
219	C₂×C₄	Dic₄₇₂	1	2	yes	—
220	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
222	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	472	2	no	—
226	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
227	C₈	C₂×C₈×D₅₉	1	1	yes	characteristic
228	C₈	C₂×#11745659	1	1	yes	characteristic
229	C₂²	#16837131	1	2	yes	—
230	C₂²	#16837207	1	2	yes	—
231	C₂²	#16837210	1	2	yes	—
232	C₂²	C₂×#14273649	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	472	2	no	—
236	C₄	—	236	4	no	—
237	C₄	#16837131	1	2	yes	—
238	C₄	#16837207	1	2	yes	—
239	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
240	C₄	C₂×#14273655	1	1	yes	characteristic
241	C₄	C₂×#14273672	1	1	yes	characteristic
242	C₄	C₂×Dic₄₇₂	1	1	yes	characteristic
243	C₂	#21422061	1	4	yes	direct product factor
244	C₂	C₂×#16837131	1	1	yes	characteristic
245	C₂	C₂×#16837207	1	1	yes	characteristic
246	C₂	C₂×#16837210	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₂×#1332
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #21422061	C₂	direct (N × H)
23. C₂×Dic₂₃₆	C₈	semidirect
78. Dic₂₃₆	C₂×C₈	semidirect
124. C₂×C₅₉	#1332	semidirect
143. C₅₉	C₂×#1332	semidirect
243. C₂	#21422061	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×C₄×Dic₂₃₆	2
C₂×#16837155	2
#21422061	2
C₂×C₄×C₁₆×C₅₉	2
C₂×#1332	59

Subgroups — C2×#21422061 #28498425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422061 #28498425