Subgroups — C₂×#21426204 #28743141

Order 15104.

682 conjugacy classes of subgroups.

The subgroup lattice has 240 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	C₂×#16837643	C₂	1	2	yes	maximal
3	#21426204	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837230	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837207	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₃₆	C₂²	1	2	yes	—
8	C₂×#14273706	—	2	2	no	—
9	C₂×Dic₄₇₂	—	2	4	no	—
10	C₂×Dic₄₇₂	C₂²	1	4	yes	complemented
11	#16837643	—	2	4	no	—
12	#16837643	C₂²	1	8	yes	—
13	C₂×#14273661	C₂²	1	2	yes	—
14	C₂×#14273652	—	2	2	no	—
15	#16837230	C₂²	1	4	yes	complemented
16	#16837207	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₅₉	—	2	4	no	—
19	C₂×M₃₂×C₅₉	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₁₈	—	2	2	no	—
23	C₂×Dic₂₃₆	—	2	4	no	—
24	C₂×Dic₂₃₆	—	2	4	no	—
25	C₂×Dic₂₃₆	—	2	2	no	—
26	C₂×Dic₂₃₆	C₂³	1	4	yes	—
27	#14273706	—	2	8	no	—
28	#14273706	—	4	4	no	—
29	C₂×C₄⋊Dic₅₉	—	2	2	no	—
30	C₂×Dic₅₉⋊C₄	—	4	2	no	—
31	C₂×C₂²⋊Dic₅₉	—	4	2	no	—
32	Dic₄₇₂	—	2	8	no	—
33	#14273661	—	2	2	no	—
34	#14273661	C₂³	1	4	yes	—
35	#14273652	—	2	8	no	—
36	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	—	2	2	no	—
38	C₂²×C₈×C₅₉	—	2	2	no	—
39	C₂²×C₈×C₅₉	D₄	1	2	yes	—
40	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
43	M₃₂×C₅₉	—	2	8	no	—
44	C₂×C₁₆×C₅₉	—	2	2	no	—
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	C₂⁴×C₅₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	—	2	2	no	—
50	C₂²×C₄×C₅₉	—	2	2	no	—
51	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	D₈	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	4	2	no	—
56	C₂²×Dic₅₉	—	4	2	no	—
57	C₂×Dic₁₁₈	—	2	4	no	—
58	C₂×Dic₁₁₈	—	4	4	no	—
59	C₂×Dic₁₁₈	—	4	4	no	—
60	C₂×Dic₁₁₈	—	4	2	no	—
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	—	2	2	no	—
65	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₉	#41	1	2	yes	—
69	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
70	C₂²⋊Dic₅₉	—	4	8	no	—
71	C₄⋊Dic₅₉	—	2	4	no	—
72	C₄⋊Dic₅₉	—	4	2	no	—
73	Dic₅₉⋊C₄	—	4	8	no	—
74	Dic₂₃₆	—	2	4	no	—
75	Dic₂₃₆	—	2	4	no	—
76	Dic₂₃₆	—	4	4	no	—
77	C₁₆×C₅₉	—	2	2	no	—
78	C₁₆×C₅₉	C₂×D₄	1	4	yes	—
79	C₂³×C₅₉	—	2	2	no	—
80	C₂³×C₅₉	—	2	4	no	—
81	C₂³×C₅₉	C₂×D₈	1	2	yes	—
82	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	—	2	2	no	—
85	C₂×C₄×C₅₉	—	2	2	no	—
86	C₂×C₄×C₅₉	—	2	4	no	—
87	C₂×C₄×C₅₉	#121	1	2	yes	—
88	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₉	#135	1	2	yes	—
91	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₉	—	4	4	no	—
93	C₂×Dic₅₉	—	4	4	no	—
94	C₂×Dic₅₉	—	8	4	no	—
95	C₂×Dic₅₉	—	8	4	no	—
96	Dic₁₁₈	—	4	4	no	—
97	Dic₁₁₈	—	4	4	no	—
98	Dic₁₁₈	—	8	4	no	—
99	C₈×C₅₉	—	2	2	no	—
100	C₈×C₅₉	#121	1	2	yes	—
101	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
103	C₂×#2219	—	59	1	no	maximal, Sylow
104	C₂²×C₅₉	—	2	4	no	—
105	C₂²×C₅₉	—	2	2	no	—
106	C₂²×C₅₉	—	4	4	no	—
107	C₂²×C₅₉	#466	1	2	yes	—
108	C₂²×C₅₉	#510	1	4	yes	—
109	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₅₉	—	2	2	no	—
111	C₄×C₅₉	#466	1	2	yes	—
112	C₄×C₅₉	C₂×#121	1	1	yes	characteristic
113	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
114	Dic₅₉	—	8	4	no	—
115	Dic₅₉	—	8	4	no	—
116	C₂²×Q₃₂	—	59	1	no	—
117	C₂²×M₃₂	D₅₉	1	1	yes	characteristic
118	C₂×#467	—	59	2	no	—
119	#2219	—	59	8	no	—
120	C₂×#367	—	59	1	no	—
121	C₂×#358	—	59	2	no	—
122	C₂×C₅₉	—	4	2	no	—
123	C₂×C₅₉	#2219	1	4	yes	complemented
124	C₂×C₅₉	C₂×#466	1	1	yes	characteristic
125	C₂×C₅₉	C₂×#510	1	2	yes	—
126	C₂²×Q₁₆	—	59	2	no	—
127	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
128	C₂×C₂²⋊Q₈	—	118	2	no	—
129	C₂×Q₃₂	—	59	4	no	—
130	C₂×Q₃₂	—	118	4	no	—
131	C₂×M₃₂	—	2	4	no	—
132	C₂×M₃₂	C₂×D₅₉	1	4	yes	—
133	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
134	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
135	#467	—	59	8	no	—
136	#467	—	118	4	no	—
137	C₂×#107	—	59	2	no	—
138	C₂×#103	—	118	2	no	—
139	#367	—	59	4	no	—
140	#358	—	59	8	no	—
141	C₅₉	C₂×#2219	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	118	2	no	—
143	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
144	C₂×Q₁₆	—	59	4	no	—
145	C₂×Q₁₆	—	118	2	no	—
146	C₂×Q₁₆	—	118	4	no	—
147	C₂×Q₁₆	—	118	4	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
151	C₂²×C₈	D₂₃₆	1	2	yes	—
152	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
153	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
154	C₂²⋊Q₈	—	118	8	no	—
155	C₂²⋊Q₈	—	236	4	no	—
156	C₂×C₄⋊C₄	—	118	2	no	—
157	C₂×C₄⋊C₄	—	236	2	no	—
158	C₂×C₂²⋊C₄	—	236	2	no	—
159	Q₃₂	—	118	8	no	—
160	M₃₂	—	2	8	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	C₂²⋊D₅₉	1	4	yes	—
163	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
165	#107	—	59	4	no	—
166	#107	—	118	2	no	—
167	#103	—	118	8	no	—
168	C₂⁴	D₄₇₂	1	1	yes	characteristic
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	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	236	2	no	—
176	C₂²×C₄	—	236	2	no	—
177	C₂²×C₄	D₄₇₂	1	2	yes	—
178	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
180	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
181	Q₁₆	—	118	4	no	—
182	Q₁₆	—	118	4	no	—
183	Q₁₆	—	236	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
189	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
190	C₂×C₈	#11745685	1	2	yes	—
191	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
192	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
193	C₄⋊C₄	—	118	4	no	—
194	C₄⋊C₄	—	236	2	no	—
195	C₄⋊C₄	—	236	8	no	—
196	C₂²⋊C₄	—	236	8	no	—
197	C₁₆	—	2	2	no	—
198	C₁₆	C₂×C₂²⋊D₅₉	1	4	yes	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	C₂×D₄₇₂	1	2	yes	—
202	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
203	Q₈	—	472	4	no	—
204	Q₈	—	236	4	no	—
205	Q₈	—	236	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	472	4	no	—
211	C₂×C₄	—	472	4	no	—
212	C₂×C₄	—	236	4	no	—
213	C₂×C₄	—	236	4	no	—
214	C₂×C₄	#14273701	1	2	yes	—
215	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
216	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
217	C₂×C₄	#14273749	1	2	yes	—
218	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	#14273701	1	2	yes	—
221	C₈	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
222	C₈	C₂×#11745685	1	1	yes	characteristic
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#16837638	1	2	yes	—
227	C₂²	#16837845	1	4	yes	—
228	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	472	4	no	—
231	C₄	—	472	4	no	—
232	C₄	#16837638	1	2	yes	—
233	C₄	C₂×#14273701	1	1	yes	characteristic
234	C₄	C₂×#14273749	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	#21426204	1	4	yes	direct product factor
237	C₂	C₂×#16837638	1	1	yes	characteristic
238	C₂	C₂×#16837845	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂²×M₃₂×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂⁴×C₅₉	D₈
hypercenter	C₂²×M₃₂	D₅₉
hypocenter	C₅₉	C₂×#2219
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₄₇₂	C₂	semidirect
3. #21426204	C₂	direct (N × H)
4. C₂×#16837230	C₂	semidirect
5. C₂×#16837207	C₂	semidirect
10. C₂×Dic₄₇₂	C₂²	semidirect
15. #16837230	C₂²	semidirect
16. #16837207	C₂²	semidirect
123. C₂×C₅₉	#2219	semidirect
141. C₅₉	C₂×#2219	semidirect
236. C₂	#21426204	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
C₂×#16837643	2
#21426204	2
C₂×#16837230	2
C₂×#16837207	2
C₂²×M₃₂×C₅₉	2
C₂×#2219	59

Subgroups — C2×#21426204 #28743141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21426204 #28743141