Subgroups — C₂×#22581026 #31815085

Order 17792.

564 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×#14555768	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic
4	#22581026	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×(C₂×D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×(C₂×D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₅₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×#14555768	—	2	4	no	—
13	C₂×#14555768	C₂²	1	4	yes	complemented
14	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₃₉	—	2	4	no	—
17	C₂×C₈×D₁₃₉	C₂²	1	4	yes	complemented
18	C₄×D₅₅₆	C₄	1	4	yes	—
19	C₄×D₅₅₆	C₂²	1	4	yes	—
20	C₂×C₄⋊Dic₁₃₉	C₄	1	1	yes	characteristic
21	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	1	yes	characteristic
22	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	1	yes	characteristic
23	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	complemented
24	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	complemented
25	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	complemented
26	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
29	#97×C₁₃₉	C₂²	1	4	yes	complemented
30	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₂×D₅₅₆	—	2	4	no	—
33	C₂×D₅₅₆	C₂×C₄	1	4	yes	—
34	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₁₃₉	—	2	2	no	—
37	C₂×C₄×D₁₃₉	—	2	2	no	—
38	C₂×C₄×D₁₃₉	—	2	2	no	—
39	C₂×C₄×D₁₃₉	—	2	2	no	—
40	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
43	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
44	#14555768	—	2	8	no	—
45	C₂×C₁₃₉⋊C₈	—	2	2	no	—
46	C₂×C₁₃₉⋊C₈	—	2	2	no	—
47	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
49	C₈×D₁₃₉	—	2	8	no	—
50	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
51	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
52	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
53	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃₉	—	2	2	no	—
57	C₂×C₈×C₁₃₉	D₄	1	4	yes	complemented
58	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
60	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
61	C₄²×C₁₃₉	C₂³	1	2	yes	—
62	C₂²×D₁₃₉	—	2	2	no	—
63	C₂²×D₁₃₉	—	2	2	no	—
64	C₂²×D₁₃₉	—	2	2	no	—
65	C₂²×D₁₃₉	—	2	2	no	—
66	C₂²×D₁₃₉	—	2	2	no	—
67	C₂²×D₁₃₉	M₁₆	1	4	yes	complemented
68	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
70	D₅₅₆	—	2	8	no	—
71	C₂×Dic₁₃₉	—	2	2	no	—
72	C₂×Dic₁₃₉	—	2	2	no	—
73	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
75	C₄×D₁₃₉	—	2	4	no	—
76	C₄×D₁₃₉	—	2	4	no	—
77	C₄×D₁₃₉	—	2	2	no	—
78	C₄×D₁₃₉	—	2	2	no	—
79	C₄×D₁₃₉	—	2	2	no	—
80	C₄×D₁₃₉	—	2	2	no	—
81	C₁₃₉⋊C₈	—	2	2	no	—
82	C₁₃₉⋊C₈	—	2	2	no	—
83	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₉	—	2	2	no	—
85	C₂×C₄×C₁₃₉	—	2	2	no	—
86	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃₉	#41	1	2	yes	—
91	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₃₉	—	2	2	no	—
93	C₈×C₁₃₉	C₂×D₄	1	4	yes	complemented
94	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
95	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
96	C₂²×C₁₃₉	#131	1	2	yes	—
97	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₃₉	—	2	2	no	—
99	C₂×D₁₃₉	—	2	2	no	—
100	C₂×D₁₃₉	—	2	8	no	—
101	C₂×D₁₃₉	—	2	2	no	—
102	C₂×D₁₃₉	—	2	2	no	—
103	C₂×D₁₃₉	—	2	2	no	—
104	C₂×D₁₃₉	—	4	4	no	—
105	C₂×D₁₃₉	C₂×M₁₆	1	4	yes	complemented
106	C₄×C₁₃₉	—	2	2	no	—
107	C₄×C₁₃₉	—	2	2	no	—
108	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
109	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
111	Dic₁₃₉	—	2	2	no	—
112	Dic₁₃₉	—	2	2	no	—
113	C₂×C₁₃₉	#435	1	4	yes	complemented
114	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂×C₁₃₉	C₂²×M₁₆	1	1	yes	characteristic
116	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
117	D₁₃₉	—	2	4	no	—
118	D₁₃₉	—	4	2	no	—
119	C₁₃₉	C₂×#435	1	1	yes	characteristic, complemented, Sylow
120	C₂×#435	—	139	1	no	maximal, Sylow
121	C₂²×M₁₆	—	139	1	no	—
122	C₂³×C₈	—	139	1	no	—
123	C₂×C₄×D₄	—	139	1	no	—
124	#435	—	139	8	no	—
125	C₂×C₄⋊C₈	—	139	1	no	—
126	C₂×C₂²⋊C₈	—	139	1	no	—
127	C₂×C₂²⋊C₈	—	139	1	no	—
128	C₂×#97	D₁₃₉	1	1	yes	characteristic, complemented
129	C₂²×D₄	—	139	1	no	—
130	C₂³×C₄	—	139	1	no	—
131	C₂³×C₄	—	139	1	no	—
132	C₂×M₁₆	—	278	4	no	—
133	C₂×M₁₆	—	139	4	no	—
134	C₂²×C₈	—	278	4	no	—
135	C₂²×C₈	—	139	1	no	—
136	C₂²×C₈	—	139	1	no	—
137	C₂²×C₈	—	139	4	no	—
138	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
140	C₄×D₄	—	139	4	no	—
141	C₄×D₄	—	139	4	no	—
142	C₂×C₄⋊C₄	—	139	1	no	—
143	C₂×C₂²⋊C₄	—	139	1	no	—
144	C₂×C₂²⋊C₄	—	139	1	no	—
145	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
146	C₄⋊C₈	—	139	4	no	—
147	C₂²⋊C₈	—	139	4	no	—
148	C₂²⋊C₈	—	139	4	no	—
149	#97	C₂×D₁₃₉	1	4	yes	complemented
150	C₂⁴	—	139	1	no	—
151	C₂⁴	—	139	1	no	—
152	C₂×D₄	—	278	4	no	—
153	C₂×D₄	—	139	4	no	—
154	C₂²×C₄	—	278	2	no	—
155	C₂²×C₄	—	278	2	no	—
156	C₂²×C₄	—	278	2	no	—
157	C₂²×C₄	—	278	2	no	—
158	C₂²×C₄	—	139	1	no	—
159	C₂²×C₄	—	139	1	no	—
160	C₂²×C₄	—	139	2	no	—
161	C₂²×C₄	—	139	2	no	—
162	C₂²×C₄	—	139	2	no	—
163	C₂²×C₄	—	139	2	no	—
164	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
165	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
167	M₁₆	—	278	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	278	2	no	—
170	C₂×C₈	—	278	2	no	—
171	C₂×C₈	—	278	8	no	—
172	C₂×C₈	—	139	2	no	—
173	C₂×C₈	—	139	2	no	—
174	C₂×C₈	D₅₅₆	1	4	yes	complemented
175	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
176	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
177	C₄⋊C₄	—	139	4	no	—
178	C₂²⋊C₄	—	139	4	no	—
179	C₂²⋊C₄	—	139	4	no	—
180	C₄²	C₄×D₁₃₉	1	2	yes	—
181	C₄²	C₂²×D₁₃₉	1	2	yes	—
182	C₂³	—	278	2	no	—
183	C₂³	—	278	2	no	—
184	C₂³	—	278	2	no	—
185	C₂³	—	278	2	no	—
186	C₂³	—	278	2	no	—
187	C₂³	—	139	2	no	—
188	C₂³	—	139	4	no	—
189	C₂³	—	139	2	no	—
190	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
191	D₄	—	278	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	278	2	no	—
195	C₂×C₄	—	278	2	no	—
196	C₂×C₄	—	278	4	no	—
197	C₂×C₄	—	278	4	no	—
198	C₂×C₄	—	278	2	no	—
199	C₂×C₄	—	278	2	no	—
200	C₂×C₄	—	278	2	no	—
201	C₂×C₄	—	278	2	no	—
202	C₂×C₄	—	139	2	no	—
203	C₂×C₄	—	139	2	no	—
204	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
205	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
207	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
208	C₂×C₄	#14555785	1	2	yes	—
209	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	278	2	no	—
212	C₈	—	278	2	no	—
213	C₈	C₂×D₅₅₆	1	4	yes	complemented
214	C₂²	—	278	8	no	—
215	C₂²	—	278	2	no	—
216	C₂²	—	278	2	no	—
217	C₂²	—	278	2	no	—
218	C₂²	—	278	2	no	—
219	C₂²	—	278	2	no	—
220	C₂²	—	556	4	no	—
221	C₂²	—	139	4	no	—
222	C₂²	C₄×D₅₅₆	1	2	yes	—
223	C₂²	M₁₆×D₁₃₉	1	2	yes	—
224	C₂²	#17984850	1	2	yes	—
225	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	278	2	no	—
229	C₄	—	278	2	no	—
230	C₄	C₄×D₅₅₆	1	2	yes	—
231	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
232	C₄	C₂×#14555785	1	1	yes	characteristic
233	C₂	—	278	4	no	—
234	C₂	—	556	2	no	—
235	C₂	#22581026	1	4	yes	direct product factor
236	C₂	C₂×C₄×D₅₅₆	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₁₃₉	1	1	yes	characteristic
238	C₂	C₂×#17984850	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#97×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂²×C₄
prime radical	C₂²×D₅₅₆	C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	C₂×#435
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555768	C₂	semidirect
2. C₂²×C₈×D₁₃₉	C₂	semidirect
4. #22581026	C₂	direct (N × H)
5. C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	semidirect
6. C₂×(C₂×D₁₃₉)⋊C₈	C₂	semidirect
7. C₂×(C₂×D₁₃₉)⋊C₈	C₂	semidirect
8. C₂×#97×C₁₃₉	C₂	semidirect
13. C₂×#14555768	C₂²	semidirect
17. C₂×C₈×D₁₃₉	C₂²	semidirect
23. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
24. (C₂×D₁₃₉)⋊C₈	C₂²	semidirect
25. (C₂×D₁₃₉)⋊C₈	C₂²	semidirect
29. #97×C₁₃₉	C₂²	semidirect
57. C₂×C₈×C₁₃₉	D₄	semidirect
67. C₂²×D₁₃₉	M₁₆	semidirect
93. C₈×C₁₃₉	C₂×D₄	semidirect
105. C₂×D₁₃₉	C₂×M₁₆	semidirect
113. C₂×C₁₃₉	#435	semidirect
119. C₁₃₉	C₂×#435	semidirect
128. C₂×#97	D₁₃₉	semidirect
149. #97	C₂×D₁₃₉	semidirect
174. C₂×C₈	D₅₅₆	semidirect
213. C₈	C₂×D₅₅₆	semidirect
235. C₂	#22581026	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×#14555768	2
C₂²×C₈×D₁₃₉	2
C₂×C₄×D₅₅₆	2
#22581026	2
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×(C₂×D₁₃₉)⋊C₈	2
C₂×#97×C₁₃₉	2
C₂×#435	139

Subgroups — C2×#22581026 #31815085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581026 #31815085