Subgroups — C₂×#21423006 #28558454

Order 15104.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837532	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21423006	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837134	C₂	1	2	yes	maximal
6	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
9	#16837532	—	2	4	no	—
10	#16837532	C₂²	1	8	yes	—
11	C₄×Dic₂₃₆	C₄	1	4	yes	—
12	C₄×Dic₂₃₆	C₂²	1	4	yes	—
13	C₂×#14273661	C₄	1	1	yes	characteristic
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
15	C₂×Dic₅₉⋊C₈	—	2	2	no	—
16	C₂×#14273652	C₄	1	2	yes	—
17	C₂×#14273637	—	2	2	no	—
18	#16837153	C₂²	1	4	yes	—
19	#16837134	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#322	C₂²	1	4	yes	—
24	C₂²×Dic₁₁₈	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₃₆	—	2	4	no	—
26	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
27	C₂²×C₅₉⋊C₈	—	2	2	no	—
28	C₄×Dic₁₁₈	—	2	2	no	—
29	C₄×Dic₁₁₈	—	2	2	no	—
30	C₄×Dic₁₁₈	C₂×C₄	1	4	yes	—
31	C₄×Dic₁₁₈	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
33	C₂×Dic₅₉⋊C₄	—	2	2	no	—
34	C₂×C₄×Dic₅₉	—	2	2	no	—
35	#14273661	C₂×C₄	1	4	yes	—
36	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
38	Dic₅₉⋊C₈	—	2	8	no	—
39	#14273652	C₂×C₄	1	8	yes	—
40	#14273637	—	2	8	no	—
41	C₂²×C₈×C₅₉	—	2	1	no	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₉	D₄	1	2	yes	—
45	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₉	—	2	1	no	—
47	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	2	yes	—
49	C₄×C₈×C₅₉	C₂³	1	2	yes	—
50	C₄×C₈×C₅₉	C₂³	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	2	no	—
56	C₂²×Dic₅₉	—	2	2	no	—
57	C₂×Dic₁₁₈	—	2	4	no	—
58	C₂×Dic₁₁₈	—	2	4	no	—
59	C₂×Dic₁₁₈	—	2	2	no	—
60	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	—	2	4	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	—	4	2	no	—
65	C₂×C₈×C₅₉	M₁₆	1	4	yes	—
66	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
69	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₉	#41	1	1	yes	characteristic
72	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
73	C₄⋊Dic₅₉	—	2	2	no	—
74	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
75	Dic₅₉⋊C₄	—	2	4	no	—
76	Dic₅₉⋊C₄	—	2	4	no	—
77	C₄×Dic₅₉	—	2	4	no	—
78	C₄×Dic₅₉	—	2	4	no	—
79	C₂×C₅₉⋊C₈	—	2	4	no	—
80	C₂×C₅₉⋊C₈	—	4	4	no	—
81	Dic₂₃₆	—	2	8	no	—
82	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₅₉	#131	1	2	yes	—
89	C₂×C₄×C₅₉	#137	1	4	yes	—
90	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
93	C₂×Dic₅₉	—	2	4	no	—
94	C₂×Dic₅₉	—	2	4	no	—
95	C₂×Dic₅₉	—	4	4	no	—
96	C₂×Dic₅₉	—	4	4	no	—
97	Dic₁₁₈	—	2	4	no	—
98	Dic₁₁₈	—	2	4	no	—
99	Dic₁₁₈	—	4	4	no	—
100	C₈×C₅₉	—	2	2	no	—
101	C₈×C₅₉	—	2	4	no	—
102	C₈×C₅₉	—	4	2	no	—
103	C₈×C₅₉	C₂×M₁₆	1	4	yes	—
104	C₅₉⋊C₈	—	4	4	no	—
105	C₂×#1593	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	#436	1	2	yes	—
107	C₂²×C₅₉	#441	1	2	yes	—
108	C₂²×C₅₉	#444	1	2	yes	—
109	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₅₉	—	2	1	no	—
111	C₄×C₅₉	—	2	1	no	—
112	C₄×C₅₉	#436	1	2	yes	—
113	C₄×C₅₉	#441	1	2	yes	—
114	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
115	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
116	C₄×C₅₉	C₂×#137	1	2	yes	—
117	Dic₅₉	—	4	4	no	—
118	Dic₅₉	—	4	4	no	—
119	C₂×#446	—	59	2	no	—
120	C₂×C₄×Q₁₆	—	59	1	no	—
121	#1593	—	59	8	no	—
122	C₂×#334	—	59	1	no	—
123	C₂×Q₈⋊C₈	—	59	2	no	—
124	C₂×#322	D₅₉	1	1	yes	characteristic
125	C₂×C₅₉	#1593	1	4	yes	complemented
126	C₂×C₅₉	C₂×#436	1	1	yes	characteristic
127	C₂×C₅₉	C₂×#441	1	1	yes	characteristic
128	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	—	59	1	no	—
130	C₂×C₄×Q₈	—	59	2	no	—
131	#446	—	59	8	no	—
132	#446	—	118	4	no	—
133	C₄×Q₁₆	—	59	4	no	—
134	C₄×Q₁₆	—	59	4	no	—
135	C₂×#107	—	59	1	no	—
136	C₂×C₄⋊C₈	—	59	2	no	—
137	C₂×C₄⋊C₈	—	118	2	no	—
138	C₂×#103	—	59	2	no	—
139	C₂×#97	—	118	2	no	—
140	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
143	#334	—	59	4	no	—
144	Q₈⋊C₈	—	59	8	no	—
145	#322	C₂×D₅₉	1	4	yes	—
146	C₅₉	C₂×#1593	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	59	2	no	—
148	C₂×Q₁₆	—	59	4	no	—
149	C₂×Q₁₆	—	118	4	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	118	2	no	—
152	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
154	C₂²×C₈	D₂₃₆	1	2	yes	—
155	C₄×Q₈	—	59	4	no	—
156	C₄×Q₈	—	59	4	no	—
157	C₄×Q₈	—	118	2	no	—
158	C₄×Q₈	—	118	2	no	—
159	C₂×C₄⋊C₄	—	59	2	no	—
160	C₂×C₄⋊C₄	—	118	2	no	—
161	C₂×C₄²	—	118	2	no	—
162	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
163	#107	—	59	4	no	—
164	C₄⋊C₈	—	59	4	no	—
165	C₄⋊C₈	—	118	8	no	—
166	C₄⋊C₈	—	118	2	no	—
167	#103	—	59	8	no	—
168	#97	—	118	8	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₄×D₅₉	1	2	yes	—
171	C₄×C₈	D₂₃₆	1	2	yes	—
172	C₄×C₈	C₂²×D₅₉	1	2	yes	—
173	C₄×C₈	C₂²×D₅₉	1	2	yes	—
174	C₄×C₈	C₂²×D₅₉	1	2	yes	—
175	C₂×Q₈	—	59	4	no	—
176	C₂×Q₈	—	118	2	no	—
177	C₂×Q₈	—	118	4	no	—
178	C₂×Q₈	—	118	4	no	—
179	C₂²×C₄	—	118	2	no	—
180	C₂²×C₄	—	118	2	no	—
181	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
183	C₂²×C₄	#11745685	1	1	yes	characteristic
184	Q₁₆	—	118	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	118	4	no	—
190	C₂×C₈	—	236	4	no	—
191	C₂×C₈	#11745659	1	4	yes	—
192	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
194	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
195	C₄⋊C₄	—	59	4	no	—
196	C₄⋊C₄	—	118	2	no	—
197	C₄⋊C₄	—	118	4	no	—
198	C₄⋊C₄	—	118	4	no	—
199	C₄²	—	118	4	no	—
200	C₄²	—	118	4	no	—
201	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
202	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
203	C₄²	#11745685	1	1	yes	characteristic
204	C₄²	C₂³×D₅₉	1	1	yes	characteristic
205	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
206	Q₈	—	118	4	no	—
207	Q₈	—	118	4	no	—
208	Q₈	—	236	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	118	4	no	—
211	C₂×C₄	—	118	4	no	—
212	C₂×C₄	—	236	4	no	—
213	C₂×C₄	—	236	4	no	—
214	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
216	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
217	C₂×C₄	C₂×#11745659	1	2	yes	—
218	C₂×C₄	#14273733	1	2	yes	—
219	C₂×C₄	#14273760	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
222	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	4	no	—
225	C₈	—	4	2	no	—
226	C₈	—	236	4	no	—
227	C₈	C₂×#11745659	1	4	yes	—
228	C₂²	#16837481	1	2	yes	—
229	C₂²	#16837510	1	2	yes	—
230	C₂²	#16837525	1	2	yes	—
231	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	236	4	no	—
235	C₄	—	236	4	no	—
236	C₄	#16837481	1	2	yes	—
237	C₄	#16837510	1	2	yes	—
238	C₄	C₂²×#11745659	1	1	yes	characteristic
239	C₄	C₂×#14273733	1	1	yes	characteristic
240	C₄	C₂×#14273760	1	2	yes	—
241	C₂	#21423006	1	4	yes	direct product factor
242	C₂	C₂×#16837481	1	1	yes	characteristic
243	C₂	C₂×#16837510	1	1	yes	characteristic
244	C₂	C₂×#16837525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	C₂×#1593
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #21423006	C₂	direct (N × H)
125. C₂×C₅₉	#1593	semidirect
146. C₅₉	C₂×#1593	semidirect
241. C₂	#21423006	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×#16837532	2
C₂×C₄×Dic₂₃₆	2
#21423006	2
C₂×#16837153	2
C₂×#16837134	2
C₂×C₅₉×#322	2
C₂×#1593	59

Subgroups — C2×#21423006 #28558454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423006 #28558454