Subgroups — C₂×#2263×C₁₃₉ #48028575

Order 35584.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2263×C₁₃₉	C₂	1	8	yes	maximal, direct product factor
4	C₂×SD₆₄×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₂×D₃₂×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×M₆₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₁₃₉	—	2	1	no	—
8	C₂²×D₈×C₁₃₉	—	2	1	no	—
9	#508×C₁₃₉	—	2	2	no	—
10	#508×C₁₃₉	C₂²	1	4	yes	complemented
11	C₂×Q₃₂×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
12	C₂×SD₃₂×C₁₃₉	—	2	1	no	—
13	C₂×D₁₆×C₁₃₉	—	2	1	no	—
14	C₂×D₁₆×C₁₃₉	—	2	2	no	—
15	C₂×D₁₆×C₁₃₉	C₂²	1	2	yes	—
16	C₂×D₁₆×C₁₃₉	C₂²	1	4	yes	complemented
17	C₂×D₁₆×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
19	SD₆₄×C₁₃₉	C₂²	1	8	yes	complemented
20	D₃₂×C₁₃₉	C₂²	1	8	yes	complemented
21	M₆₄×C₁₃₉	C₂²	1	4	yes	complemented
22	C₂×C₃₂×C₁₃₉	C₂²	1	2	yes	complemented
23	C₂×#41×C₁₃₉	—	4	1	no	—
24	C₂²×D₄×C₁₃₉	—	4	1	no	—
25	#135×C₁₃₉	—	2	4	no	—
26	#135×C₁₃₉	—	4	2	no	—
27	C₂×Q₁₆×C₁₃₉	—	2	1	no	—
28	C₂×SD₁₆×C₁₃₉	—	4	1	no	—
29	C₂×D₈×C₁₃₉	—	2	2	no	—
30	C₂×D₈×C₁₃₉	—	2	4	no	—
31	C₂×D₈×C₁₃₉	—	2	1	no	—
32	C₂×D₈×C₁₃₉	—	4	1	no	—
33	C₂×D₈×C₁₃₉	—	4	2	no	—
34	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
35	Q₃₂×C₁₃₉	—	2	1	no	—
36	Q₃₂×C₁₃₉	C₂³	1	2	yes	complemented
37	SD₃₂×C₁₃₉	—	2	4	no	—
38	D₁₆×C₁₃₉	—	2	2	no	—
39	D₁₆×C₁₃₉	—	2	4	no	—
40	D₁₆×C₁₃₉	—	2	1	no	—
41	D₁₆×C₁₃₉	C₂³	1	4	yes	—
42	D₁₆×C₁₃₉	C₂³	1	2	yes	complemented
43	C₂×C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₁₃₉	D₄	1	2	yes	—
45	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₁₃₉	C₂³	1	2	yes	—
47	C₃₂×C₁₃₉	C₂³	1	4	yes	complemented
48	C₂⁴×C₁₃₉	—	8	1	no	—
49	#41×C₁₃₉	—	4	4	no	—
50	#41×C₁₃₉	—	8	2	no	—
51	C₂×Q₈×C₁₃₉	—	4	1	no	—
52	C₂×D₄×C₁₃₉	—	4	4	no	—
53	C₂×D₄×C₁₃₉	—	4	2	no	—
54	C₂×D₄×C₁₃₉	—	4	1	no	—
55	C₂×D₄×C₁₃₉	—	8	1	no	—
56	C₂×D₄×C₁₃₉	—	8	2	no	—
57	C₂×D₄×C₁₃₉	—	8	1	no	—
58	C₂²×C₄×C₁₃₉	—	8	1	no	—
59	C₂²×C₄×C₁₃₉	D₈	1	1	yes	characteristic
60	Q₁₆×C₁₃₉	—	2	2	no	—
61	Q₁₆×C₁₃₉	—	4	1	no	—
62	SD₁₆×C₁₃₉	—	4	4	no	—
63	D₈×C₁₃₉	—	2	4	no	—
64	D₈×C₁₃₉	—	2	2	no	—
65	D₈×C₁₃₉	—	4	2	no	—
66	D₈×C₁₃₉	—	4	4	no	—
67	D₈×C₁₃₉	—	4	1	no	—
68	C₂×C₈×C₁₃₉	D₈	1	1	yes	characteristic
69	C₂×C₈×C₁₃₉	D₈	1	2	yes	—
70	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
72	C₁₆×C₁₃₉	C₂×D₄	1	1	yes	characteristic
73	C₁₆×C₁₃₉	C₂×D₄	1	2	yes	—
74	C₁₆×C₁₃₉	C₂⁴	1	1	yes	characteristic
75	C₂³×C₁₃₉	—	8	4	no	—
76	C₂³×C₁₃₉	—	8	1	no	—
77	C₂³×C₁₃₉	—	8	2	no	—
78	C₂³×C₁₃₉	—	16	4	no	—
79	C₂³×C₁₃₉	D₁₆	1	1	yes	characteristic
80	Q₈×C₁₃₉	—	4	2	no	—
81	Q₈×C₁₃₉	—	8	1	no	—
82	D₄×C₁₃₉	—	4	4	no	—
83	D₄×C₁₃₉	—	4	2	no	—
84	D₄×C₁₃₉	—	8	4	no	—
85	D₄×C₁₃₉	—	8	4	no	—
86	D₄×C₁₃₉	—	8	2	no	—
87	D₄×C₁₃₉	—	8	1	no	—
88	C₂×C₄×C₁₃₉	—	8	1	no	—
89	C₂×C₄×C₁₃₉	—	8	4	no	—
90	C₂×C₄×C₁₃₉	D₁₆	1	2	yes	—
91	C₂×C₄×C₁₃₉	D₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
94	C₈×C₁₃₉	C₂×D₈	1	1	yes	characteristic
95	C₈×C₁₃₉	C₂×D₈	1	2	yes	—
96	C₈×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₁₃₉	—	2	2	no	—
98	C₂²×C₁₃₉	—	8	2	no	—
99	C₂²×C₁₃₉	—	8	4	no	—
100	C₂²×C₁₃₉	—	16	8	no	—
101	C₂²×C₁₃₉	—	16	2	no	—
102	C₂²×C₁₃₉	—	16	4	no	—
103	C₂²×C₁₃₉	C₂×D₁₆	1	2	yes	—
104	C₂²×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
105	C₄×C₁₃₉	—	8	2	no	—
106	C₄×C₁₃₉	C₂×D₁₆	1	2	yes	—
107	C₄×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
108	C₄×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
109	C₂×C₁₃₉	—	2	2	no	—
110	C₂×C₁₃₉	—	16	2	no	—
111	C₂×C₁₃₉	—	16	4	no	—
112	C₂×C₁₃₉	#2263	1	2	yes	direct product factor
113	C₂×C₁₃₉	C₂²×D₁₆	1	1	yes	characteristic
114	C₂×#2263	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₁₃₉	C₂×#2263	1	1	yes	characteristic, direct product factor, Sylow
116	C₂×#508	C₂×C₁₃₉	1	1	yes	characteristic, complemented
117	C₂²×D₁₆	C₂×C₁₃₉	1	1	yes	characteristic, complemented
118	#2263	C₂×C₁₃₉	1	8	yes	direct product factor
119	C₂×SD₆₄	C₂×C₁₃₉	1	2	yes	complemented
120	C₂×D₃₂	C₂×C₁₃₉	1	2	yes	complemented
121	C₂×M₆₄	C₂×C₁₃₉	1	1	yes	characteristic, complemented
122	C₂×#135	—	2	1	no	—
123	C₂²×D₈	—	2	1	no	—
124	#508	—	2	2	no	—
125	#508	C₂²×C₁₃₉	1	4	yes	complemented
126	C₂×Q₃₂	C₂²×C₁₃₉	1	1	yes	characteristic, complemented
127	C₂×SD₃₂	—	2	1	no	—
128	C₂×D₁₆	—	2	1	no	—
129	C₂×D₁₆	—	2	2	no	—
130	C₂×D₁₆	C₂²×C₁₃₉	1	2	yes	—
131	C₂×D₁₆	C₂²×C₁₃₉	1	4	yes	complemented
132	C₂×D₁₆	C₂²×C₁₃₉	1	1	yes	characteristic, complemented
133	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
134	SD₆₄	C₂²×C₁₃₉	1	8	yes	complemented
135	D₃₂	C₂²×C₁₃₉	1	8	yes	complemented
136	M₆₄	C₂²×C₁₃₉	1	4	yes	complemented
137	C₂×C₃₂	C₂²×C₁₃₉	1	2	yes	complemented
138	C₂×#41	—	4	1	no	—
139	C₂²×D₄	—	4	1	no	—
140	#135	—	2	4	no	—
141	#135	—	4	2	no	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×SD₁₆	—	4	1	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	—	2	1	no	—
147	C₂×D₈	—	4	1	no	—
148	C₂×D₈	—	4	2	no	—
149	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
150	Q₃₂	—	2	1	no	—
151	Q₃₂	C₂³×C₁₃₉	1	2	yes	complemented
152	SD₃₂	—	2	4	no	—
153	D₁₆	—	2	2	no	—
154	D₁₆	—	2	4	no	—
155	D₁₆	—	2	1	no	—
156	D₁₆	C₂³×C₁₃₉	1	4	yes	—
157	D₁₆	C₂³×C₁₃₉	1	2	yes	complemented
158	C₂×C₁₆	D₄×C₁₃₉	1	1	yes	characteristic
159	C₂×C₁₆	D₄×C₁₃₉	1	2	yes	—
160	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₁₃₉	1	2	yes	—
162	C₃₂	C₂³×C₁₃₉	1	4	yes	complemented
163	C₂⁴	—	8	1	no	—
164	#41	—	4	4	no	—
165	#41	—	8	2	no	—
166	C₂×Q₈	—	4	1	no	—
167	C₂×D₄	—	4	2	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	1	no	—
170	C₂×D₄	—	8	1	no	—
171	C₂×D₄	—	8	1	no	—
172	C₂×D₄	—	8	2	no	—
173	C₂²×C₄	—	8	1	no	—
174	C₂²×C₄	D₈×C₁₃₉	1	1	yes	characteristic
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	4	1	no	—
177	SD₁₆	—	4	4	no	—
178	D₈	—	2	4	no	—
179	D₈	—	2	2	no	—
180	D₈	—	4	2	no	—
181	D₈	—	4	4	no	—
182	D₈	—	4	1	no	—
183	C₂×C₈	D₈×C₁₃₉	1	1	yes	characteristic
184	C₂×C₈	D₈×C₁₃₉	1	2	yes	—
185	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
187	C₁₆	C₂×D₄×C₁₃₉	1	1	yes	characteristic
188	C₁₆	C₂×D₄×C₁₃₉	1	2	yes	—
189	C₁₆	C₂⁴×C₁₃₉	1	1	yes	characteristic
190	C₂³	—	8	1	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	16	4	no	—
194	C₂³	D₁₆×C₁₃₉	1	1	yes	characteristic
195	Q₈	—	4	2	no	—
196	Q₈	—	8	1	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	8	4	no	—
205	C₂×C₄	D₁₆×C₁₃₉	1	1	yes	characteristic
206	C₂×C₄	D₁₆×C₁₃₉	1	2	yes	—
207	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
209	C₈	C₂×D₈×C₁₃₉	1	1	yes	characteristic
210	C₈	C₂×D₈×C₁₃₉	1	2	yes	—
211	C₈	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	8	2	no	—
214	C₂²	—	8	4	no	—
215	C₂²	—	16	8	no	—
216	C₂²	—	16	4	no	—
217	C₂²	—	16	2	no	—
218	C₂²	C₂×D₁₆×C₁₃₉	1	2	yes	—
219	C₂²	C₂×D₁₆×C₁₃₉	1	1	yes	characteristic
220	C₄	—	8	2	no	—
221	C₄	C₂×D₁₆×C₁₃₉	1	2	yes	—
222	C₄	C₂×D₁₆×C₁₃₉	1	1	yes	characteristic
223	C₄	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	16	4	no	—
226	C₂	—	16	2	no	—
227	C₂	#2263×C₁₃₉	1	2	yes	direct product factor
228	C₂	C₂²×D₁₆×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₁₃₉
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×C₈×C₁₃₉ ◁ C₂²×C₁₆×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₁₃₉	C₂	semidirect
2. C₂²×D₁₆×C₁₃₉	C₂	semidirect
3. #2263×C₁₃₉	C₂	direct (N × H)
4. C₂×SD₆₄×C₁₃₉	C₂	semidirect
5. C₂×D₃₂×C₁₃₉	C₂	semidirect
6. C₂×M₆₄×C₁₃₉	C₂	semidirect
10. #508×C₁₃₉	C₂²	semidirect
11. C₂×Q₃₂×C₁₃₉	C₂²	semidirect
16. C₂×D₁₆×C₁₃₉	C₂²	semidirect
17. C₂×D₁₆×C₁₃₉	C₂²	semidirect
19. SD₆₄×C₁₃₉	C₂²	semidirect
20. D₃₂×C₁₃₉	C₂²	semidirect
21. M₆₄×C₁₃₉	C₂²	semidirect
22. C₂×C₃₂×C₁₃₉	C₂²	semidirect
36. Q₃₂×C₁₃₉	C₂³	semidirect
42. D₁₆×C₁₃₉	C₂³	semidirect
47. C₃₂×C₁₃₉	C₂³	semidirect
112. C₂×C₁₃₉	#2263	direct (N × H)
114. C₂×#2263	C₁₃₉	direct (N × H)
115. C₁₃₉	C₂×#2263	direct (N × H)
116. C₂×#508	C₂×C₁₃₉	semidirect
117. C₂²×D₁₆	C₂×C₁₃₉	semidirect
118. #2263	C₂×C₁₃₉	direct (N × H)
119. C₂×SD₆₄	C₂×C₁₃₉	semidirect
120. C₂×D₃₂	C₂×C₁₃₉	semidirect
121. C₂×M₆₄	C₂×C₁₃₉	semidirect
125. #508	C₂²×C₁₃₉	semidirect
126. C₂×Q₃₂	C₂²×C₁₃₉	semidirect
131. C₂×D₁₆	C₂²×C₁₃₉	semidirect
132. C₂×D₁₆	C₂²×C₁₃₉	semidirect
134. SD₆₄	C₂²×C₁₃₉	semidirect
135. D₃₂	C₂²×C₁₃₉	semidirect
136. M₆₄	C₂²×C₁₃₉	semidirect
137. C₂×C₃₂	C₂²×C₁₃₉	semidirect
151. Q₃₂	C₂³×C₁₃₉	semidirect
157. D₁₆	C₂³×C₁₃₉	semidirect
162. C₃₂	C₂³×C₁₃₉	semidirect
227. C₂	#2263×C₁₃₉	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×#508×C₁₃₉	2
C₂²×D₁₆×C₁₃₉	2
#2263×C₁₃₉	2
C₂×SD₆₄×C₁₃₉	2
C₂×D₃₂×C₁₃₉	2
C₂×M₆₄×C₁₃₉	2
C₂×#2263	139

Subgroups — C2×#2263×C139 #48028575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2263×C₁₃₉ #48028575