Subgroups — C₂×C₄×#14273724 #28747203

Order 15104.

5694 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273724	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₂₃₆	C₂	1	6	yes	maximal, complemented
3	C₄×#14273724	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16837304	C₂	1	6	yes	maximal, complemented
5	C₂×#16837272	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄³×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₃₆	C₂²	1	6	yes	—
8	C₂³×C₄×D₅₉	C₂²	1	4	yes	—
9	C₂×#14273724	C₄	1	16	yes	direct product factor
10	C₂×#14273724	C₂²	1	8	yes	—
11	C₂×C₄×D₂₃₆	—	2	48	no	—
12	C₂×C₄×D₂₃₆	C₂²	1	48	yes	complemented
13	C₂²×C₄⋊Dic₅₉	C₂²	1	6	yes	—
14	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	12	yes	—
15	#16837304	C₂²	1	48	yes	complemented
16	#16837272	C₂²	1	8	yes	complemented
17	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
20	C₄³×C₅₉	C₂²	1	8	yes	complemented
21	C₂⁴×D₅₉	C₂³	1	4	yes	—
22	C₂²×D₂₃₆	—	2	24	no	—
23	C₂²×D₂₃₆	—	2	24	no	—
24	C₂²×D₂₃₆	C₂×C₄	1	48	yes	complemented
25	C₂²×D₂₃₆	C₂³	1	24	yes	—
26	C₂³×Dic₅₉	C₂³	1	4	yes	—
27	C₂²×C₄×D₅₉	—	2	48	no	—
28	C₂²×C₄×D₅₉	C₂³	1	16	yes	—
29	#14273724	C₂×C₄	1	64	yes	direct product factor
30	C₄×D₂₃₆	—	2	192	no	—
31	C₂×C₄⋊Dic₅₉	—	2	24	no	—
32	C₂×C₄⋊Dic₅₉	C₂³	1	24	yes	—
33	C₂×(C₂×D₅₉)⋊C₄	—	2	48	no	—
34	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	48	yes	—
35	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
36	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	—	2	12	no	—
39	C₂×C₄²×C₅₉	C₂×C₄	1	8	yes	complemented
40	C₂×C₄²×C₅₉	D₄	1	24	yes	complemented
41	C₂×C₄²×C₅₉	C₂³	1	12	yes	—
42	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₅₉	C₂³	1	12	yes	—
44	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
45	C₂³×D₅₉	—	2	24	no	—
46	C₂³×D₅₉	—	2	24	no	—
47	C₂³×D₅₉	C₂²×C₄	1	16	yes	—
48	C₂³×D₅₉	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₅₉	—	2	12	no	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	12	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	12	yes	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	12	yes	—
54	C₂²×C₄×C₅₉	#41	1	12	yes	—
55	C₂²×C₄×C₅₉	C₂⁴	1	6	yes	—
56	C₂²×C₄×C₅₉	C₂⁴	1	6	yes	—
57	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
58	C₂²×Dic₅₉	—	2	24	no	—
59	C₂²×Dic₅₉	C₂⁴	1	8	yes	—
60	C₂×D₂₃₆	—	2	192	no	—
61	C₂×D₂₃₆	—	2	48	no	—
62	C₂×D₂₃₆	—	2	48	no	—
63	C₂×D₂₃₆	C₂²×C₄	1	96	yes	complemented
64	C₂×C₄×D₅₉	—	2	96	no	—
65	C₂×C₄×D₅₉	—	4	64	no	—
66	C₄²×C₅₉	—	2	24	no	—
67	C₄²×C₅₉	C₂²×C₄	1	16	yes	complemented
68	C₄²×C₅₉	C₂×D₄	1	48	yes	complemented
69	(C₂×D₅₉)⋊C₄	—	2	96	no	—
70	C₄⋊Dic₅₉	—	2	48	no	—
71	C₂³×C₅₉	C₂³×C₄	1	2	yes	—
72	C₂³×C₅₉	C₂²×D₄	1	6	yes	—
73	C₂³×C₅₉	C₂×#41	1	6	yes	—
74	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
75	C₂²×D₅₉	—	2	96	no	—
76	C₂²×D₅₉	—	2	24	no	—
77	C₂²×D₅₉	—	2	24	no	—
78	C₂²×D₅₉	—	4	16	no	—
79	C₂²×D₅₉	—	4	48	no	—
80	C₂²×D₅₉	C₂³×C₄	1	16	yes	—
81	C₂×C₄×C₅₉	—	2	24	no	—
82	C₂×C₄×C₅₉	—	2	12	no	—
83	C₂×C₄×C₅₉	C₄×D₄	1	48	yes	complemented
84	C₂×C₄×C₅₉	#127	1	16	yes	complemented
85	C₂×C₄×C₅₉	C₂³×C₄	1	12	yes	—
86	C₂×C₄×C₅₉	C₂²×D₄	1	12	yes	—
87	C₂×C₄×C₅₉	C₂²×D₄	1	12	yes	—
88	C₂×C₄×C₅₉	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₅₉	C₂×#41	1	12	yes	—
90	C₂×Dic₅₉	—	2	24	no	—
91	C₂×Dic₅₉	—	4	16	no	—
92	D₂₃₆	—	2	192	no	—
93	C₄×D₅₉	—	4	64	no	—
94	C₂×C₄×#127	—	59	1	no	maximal, Sylow
95	C₂²×C₅₉	C₂×C₄×D₄	1	12	yes	—
96	C₂²×C₅₉	C₂×#127	1	4	yes	—
97	C₂²×C₅₉	#532	1	12	yes	—
98	C₂²×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
99	C₂²×C₅₉	C₂³×D₄	1	3	yes	—
100	C₂²×C₅₉	C₂²×#41	1	3	yes	—
101	C₂×D₅₉	—	2	48	no	—
102	C₂×D₅₉	—	4	64	no	—
103	C₂×D₅₉	—	4	8	no	—
104	C₂×D₅₉	—	4	24	no	—
105	C₄×C₅₉	—	2	12	no	—
106	C₄×C₅₉	C₂×C₄×D₄	1	24	yes	complemented
107	C₄×C₅₉	C₂×#127	1	8	yes	complemented
108	Dic₅₉	—	4	8	no	—
109	C₂²×#127	—	59	1	no	—
110	C₂²×C₄×D₄	—	59	6	no	—
111	C₄×#127	—	59	16	no	—
112	C₂×#390	—	59	6	no	—
113	C₂×#384	—	59	1	no	—
114	C₂×C₄³	D₅₉	1	1	yes	characteristic, complemented
115	C₂×C₅₉	C₄×#127	1	8	yes	complemented
116	C₂×C₅₉	C₂²×C₄×D₄	1	3	yes	—
117	C₂×C₅₉	C₂²×#127	1	1	yes	characteristic
118	C₂×C₅₉	C₂×#532	1	3	yes	—
119	D₅₉	—	4	16	no	—
120	C₂³×D₄	—	59	6	no	—
121	C₂⁴×C₄	—	59	4	no	—
122	C₂×#127	—	59	16	no	—
123	C₂×#127	—	59	8	no	—
124	C₂×C₄×D₄	—	59	48	no	—
125	C₂×C₄×D₄	—	118	48	no	—
126	C₂²×C₄⋊C₄	—	59	6	no	—
127	C₂²×C₂²⋊C₄	—	59	12	no	—
128	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
129	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
130	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
131	#390	—	59	48	no	—
132	#384	—	59	8	no	—
133	C₄³	C₂×D₅₉	1	8	yes	complemented
134	C₅₉	C₂×C₄×#127	1	1	yes	characteristic, complemented, Sylow
135	C₂⁵	—	59	4	no	—
136	C₂²×D₄	—	59	48	no	—
137	C₂²×D₄	—	59	24	no	—
138	C₂²×D₄	—	118	24	no	—
139	C₂²×D₄	—	118	24	no	—
140	C₂³×C₄	—	59	4	no	—
141	C₂³×C₄	—	59	16	no	—
142	C₂³×C₄	—	118	48	no	—
143	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
144	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
145	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
146	#127	—	59	64	no	—
147	C₄×D₄	—	118	192	no	—
148	C₂×C₄⋊C₄	—	59	24	no	—
149	C₂×C₄⋊C₄	—	118	24	no	—
150	C₂×C₂²⋊C₄	—	59	48	no	—
151	C₂×C₂²⋊C₄	—	118	48	no	—
152	C₂×C₄²	—	2	12	no	—
153	C₂×C₄²	C₄×D₅₉	1	8	yes	complemented
154	C₂×C₄²	D₂₃₆	1	24	yes	complemented
155	C₂×C₄²	C₂²×D₅₉	1	12	yes	—
156	C₂×C₄²	C₂²×D₅₉	1	4	yes	—
157	C₂×C₄²	C₂²×D₅₉	1	12	yes	—
158	C₂⁴	—	59	16	no	—
159	C₂⁴	—	59	8	no	—
160	C₂⁴	—	118	24	no	—
161	C₂⁴	—	118	24	no	—
162	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
163	C₂×D₄	—	59	96	no	—
164	C₂×D₄	—	118	192	no	—
165	C₂×D₄	—	118	48	no	—
166	C₂×D₄	—	118	48	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	59	8	no	—
169	C₂²×C₄	—	118	24	no	—
170	C₂²×C₄	—	118	96	no	—
171	C₂²×C₄	—	236	64	no	—
172	C₂²×C₄	C₂×C₄×D₅₉	1	12	yes	—
173	C₂²×C₄	C₂×D₂₃₆	1	12	yes	—
174	C₂²×C₄	C₂×D₂₃₆	1	12	yes	—
175	C₂²×C₄	C₂×D₂₃₆	1	12	yes	—
176	C₂²×C₄	#11745685	1	12	yes	—
177	C₂²×C₄	C₂³×D₅₉	1	6	yes	—
178	C₂²×C₄	C₂³×D₅₉	1	6	yes	—
179	C₂²×C₄	C₂³×D₅₉	1	2	yes	—
180	C₄⋊C₄	—	118	48	no	—
181	C₂²⋊C₄	—	118	96	no	—
182	C₄²	—	2	24	no	—
183	C₄²	C₂×C₄×D₅₉	1	16	yes	complemented
184	C₄²	C₂×D₂₃₆	1	48	yes	complemented
185	C₂³	—	59	16	no	—
186	C₂³	—	118	24	no	—
187	C₂³	—	118	24	no	—
188	C₂³	—	118	96	no	—
189	C₂³	—	236	16	no	—
190	C₂³	—	236	48	no	—
191	C₂³	C₂²×C₄×D₅₉	1	2	yes	—
192	C₂³	C₂²×D₂₃₆	1	6	yes	—
193	C₂³	C₂×#11745685	1	6	yes	—
194	C₂³	C₂⁴×D₅₉	1	1	yes	characteristic
195	D₄	—	118	192	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	118	24	no	—
199	C₂×C₄	—	236	16	no	—
200	C₂×C₄	—	236	64	no	—
201	C₂×C₄	C₄×D₂₃₆	1	48	yes	complemented
202	C₂×C₄	#14273724	1	16	yes	direct product factor
203	C₂×C₄	C₂²×C₄×D₅₉	1	12	yes	—
204	C₂×C₄	C₂²×D₂₃₆	1	12	yes	—
205	C₂×C₄	C₂²×D₂₃₆	1	12	yes	—
206	C₂×C₄	C₂²×D₂₃₆	1	12	yes	—
207	C₂×C₄	C₂×#11745685	1	12	yes	—
208	C₂²	—	118	48	no	—
209	C₂²	—	236	8	no	—
210	C₂²	—	236	24	no	—
211	C₂²	—	236	64	no	—
212	C₂²	C₂×C₄×D₂₃₆	1	12	yes	—
213	C₂²	C₂×#14273724	1	4	yes	—
214	C₂²	#16837952	1	12	yes	—
215	C₂²	C₂³×C₄×D₅₉	1	1	yes	characteristic
216	C₂²	C₂³×D₂₃₆	1	3	yes	—
217	C₂²	C₂²×#11745685	1	3	yes	—
218	C₄	—	2	12	no	—
219	C₄	—	236	8	no	—
220	C₄	C₂×C₄×D₂₃₆	1	24	yes	complemented
221	C₄	C₂×#14273724	1	8	yes	direct product factor
222	C₂	—	236	16	no	—
223	C₂	C₄×#14273724	1	8	yes	direct product factor
224	C₂	C₂²×C₄×D₂₃₆	1	3	yes	—
225	C₂	C₂²×#14273724	1	1	yes	characteristic
226	C₂	C₂×#16837952	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₉
Fitting	C₂×C₄³×C₅₉	C₂
socle	C₂⁴×C₅₉	C₂⁴
prime radical	C₂²×#14273724	C₂
cosocle	C₂³×C₅₉	C₂⁵
hypercenter	C₂×C₄³	D₅₉
hypocenter	C₅₉	C₂×C₄×#127
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×D₂₃₆	C₂	semidirect
3. C₄×#14273724	C₂	direct (N × H)
4. C₂×#16837304	C₂	semidirect
5. C₂×#16837272	C₂	semidirect
6. C₂×C₄³×C₅₉	C₂	semidirect
9. C₂×#14273724	C₄	direct (N × H)
12. C₂×C₄×D₂₃₆	C₂²	semidirect
15. #16837304	C₂²	semidirect
16. #16837272	C₂²	semidirect
20. C₄³×C₅₉	C₂²	semidirect
24. C₂²×D₂₃₆	C₂×C₄	semidirect
29. #14273724	C₂×C₄	direct (N × H)
39. C₂×C₄²×C₅₉	C₂×C₄	semidirect
40. C₂×C₄²×C₅₉	D₄	semidirect
63. C₂×D₂₃₆	C₂²×C₄	semidirect
67. C₄²×C₅₉	C₂²×C₄	semidirect
68. C₄²×C₅₉	C₂×D₄	semidirect
83. C₂×C₄×C₅₉	C₄×D₄	semidirect
84. C₂×C₄×C₅₉	#127	semidirect
106. C₄×C₅₉	C₂×C₄×D₄	semidirect
107. C₄×C₅₉	C₂×#127	semidirect
114. C₂×C₄³	D₅₉	semidirect
115. C₂×C₅₉	C₄×#127	semidirect
133. C₄³	C₂×D₅₉	semidirect
134. C₅₉	C₂×C₄×#127	semidirect
153. C₂×C₄²	C₄×D₅₉	semidirect
154. C₂×C₄²	D₂₃₆	semidirect
183. C₄²	C₂×C₄×D₅₉	semidirect
184. C₄²	C₂×D₂₃₆	semidirect
201. C₂×C₄	C₄×D₂₃₆	semidirect
202. C₂×C₄	#14273724	direct (N × H)
220. C₄	C₂×C₄×D₂₃₆	semidirect
221. C₄	C₂×#14273724	direct (N × H)
223. C₂	C₄×#14273724	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×#14273724	2
C₂²×C₄×D₂₃₆	2
C₄×#14273724	2
C₂×#16837304	2
C₂×#16837272	2
C₂×C₄³×C₅₉	2
C₂×C₄×#127	59

Subgroups — C2×C4×#14273724 #28747203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14273724 #28747203