Subgroups — #3381×C₁₃₉ #31804007

Order 17792.

338 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	#557×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#499×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#498×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#479×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	#478×C₁₃₉	C₂	1	2	yes	maximal
7	#477×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#474×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
11	C₄×SD₁₆×C₁₃₉	C₂	1	2	yes	maximal
12	C₂×SD₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
14	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	#125×C₁₃₉	—	2	2	no	—
17	#125×C₁₃₉	C₂²	1	2	yes	—
18	#125×C₁₃₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₃₉	—	2	1	no	—
20	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
21	C₄×Q₈×C₁₃₉	—	2	1	no	—
22	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
28	#107×C₁₃₉	C₂²	1	2	yes	—
29	#107×C₁₃₉	C₂²	1	2	yes	—
30	#106×C₁₃₉	C₂²	1	2	yes	—
31	#106×C₁₃₉	C₂²	1	1	yes	characteristic
32	#106×C₁₃₉	C₂²	1	2	yes	—
33	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
35	#103×C₁₃₉	C₂²	1	2	yes	—
36	#103×C₁₃₉	C₂²	1	1	yes	characteristic
37	#102×C₁₃₉	C₂²	1	2	yes	—
38	#102×C₁₃₉	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃₉	—	2	1	no	—
42	C₂×Q₈×C₁₃₉	—	2	1	no	—
43	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	—	2	1	no	—
47	C₂²×C₄×C₁₃₉	—	2	2	no	—
48	SD₁₆×C₁₃₉	Q₈	1	4	yes	complemented
49	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₃₉	—	2	2	no	—
53	C₄⋊C₄×C₁₃₉	—	2	1	no	—
54	C₄⋊C₄×C₁₃₉	—	2	2	no	—
55	C₄⋊C₄×C₁₃₉	—	2	1	no	—
56	C₄⋊C₄×C₁₃₉	—	2	2	no	—
57	C₄⋊C₄×C₁₃₉	—	2	2	no	—
58	C₄⋊C₄×C₁₃₉	—	2	1	no	—
59	C₄⋊C₄×C₁₃₉	—	2	2	no	—
60	C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
62	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
67	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
68	C₄²×C₁₃₉	—	2	2	no	—
69	C₄²×C₁₃₉	—	2	1	no	—
70	C₄²×C₁₃₉	C₂³	1	2	yes	—
71	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
72	C₂³×C₁₃₉	—	2	1	no	—
73	Q₈×C₁₃₉	—	2	2	no	—
74	Q₈×C₁₃₉	—	2	1	no	—
75	Q₈×C₁₃₉	—	4	2	no	—
76	Q₈×C₁₃₉	—	8	1	no	—
77	Q₈×C₁₃₉	C₂×Q₈	1	2	yes	—
78	D₄×C₁₃₉	—	2	1	no	—
79	D₄×C₁₃₉	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₁₃₉	—	2	1	no	—
81	C₂×C₄×C₁₃₉	—	2	1	no	—
82	C₂×C₄×C₁₃₉	—	2	2	no	—
83	C₂×C₄×C₁₃₉	—	2	1	no	—
84	C₂×C₄×C₁₃₉	—	2	2	no	—
85	C₂×C₄×C₁₃₉	—	4	2	no	—
86	C₂×C₄×C₁₃₉	—	8	2	no	—
87	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₃₉	—	2	1	no	—
93	C₈×C₁₃₉	—	2	2	no	—
94	C₈×C₁₃₉	C₂×Q₈	1	2	yes	—
95	C₂²×C₁₃₉	—	2	2	no	—
96	C₂²×C₁₃₉	—	4	1	no	—
97	C₂²×C₁₃₉	—	4	1	no	—
98	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₁₃₉	—	2	2	no	—
100	C₄×C₁₃₉	—	2	2	no	—
101	C₄×C₁₃₉	—	2	2	no	—
102	C₄×C₁₃₉	—	2	1	no	—
103	C₄×C₁₃₉	—	4	1	no	—
104	C₄×C₁₃₉	—	4	2	no	—
105	C₄×C₁₃₉	—	4	1	no	—
106	C₄×C₁₃₉	—	4	2	no	—
107	C₄×C₁₃₉	#135	1	2	yes	—
108	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
109	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×C₁₃₉	—	4	2	no	—
111	C₂×C₁₃₉	D₄×Q₈	1	1	yes	characteristic
112	C₂×C₁₃₉	C₂×#135	1	1	yes	characteristic
113	C₂×C₁₃₉	#577	1	1	yes	characteristic
114	C₁₃₉	#3381	1	1	yes	characteristic, direct product factor, Sylow
115	#3381	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	#557	C₂×C₁₃₉	1	1	yes	characteristic, complemented
117	#554	C₂×C₁₃₉	1	1	yes	characteristic
118	#499	C₂×C₁₃₉	1	2	yes	complemented
119	#498	C₂×C₁₃₉	1	1	yes	characteristic, complemented
120	#479	C₂×C₁₃₉	1	2	yes	complemented
121	#478	C₂×C₁₃₉	1	2	yes	—
122	#477	C₂×C₁₃₉	1	1	yes	characteristic, complemented
123	#474	C₂×C₁₃₉	1	1	yes	characteristic
124	C₈×Q₈	C₂×C₁₃₉	1	1	yes	characteristic, complemented
125	C₄×SD₁₆	C₂×C₁₃₉	1	1	yes	characteristic
126	C₄×SD₁₆	C₂×C₁₃₉	1	2	yes	—
127	C₂×SD₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
128	C₄⋊Q₈	—	2	1	no	—
129	C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
130	C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
131	#125	—	2	2	no	—
132	#125	C₂²×C₁₃₉	1	2	yes	—
133	#125	C₂²×C₁₃₉	1	2	yes	—
134	C₂²⋊Q₈	—	2	1	no	—
135	C₂²⋊Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
138	C₄×Q₈	C₂²×C₁₃₉	1	2	yes	—
139	C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
140	C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₁₃₉	1	2	yes	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	#107	C₂²×C₁₃₉	1	2	yes	—
144	#107	C₂²×C₁₃₉	1	2	yes	—
145	#106	C₂²×C₁₃₉	1	2	yes	—
146	#106	C₂²×C₁₃₉	1	1	yes	characteristic
147	#106	C₂²×C₁₃₉	1	2	yes	—
148	C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
150	#103	C₂²×C₁₃₉	1	2	yes	—
151	#103	C₂²×C₁₃₉	1	1	yes	characteristic
152	#102	C₂²×C₁₃₉	1	2	yes	—
153	#102	C₂²×C₁₃₉	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	D₄×C₁₃₉	1	1	yes	characteristic
159	C₂×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
160	C₂×D₄	C₂³×C₁₃₉	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	2	no	—
163	SD₁₆	Q₈×C₁₃₉	1	4	yes	complemented
164	C₂×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₁₃₉	1	2	yes	—
166	C₂×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
177	C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
178	C₄⋊C₄	C₂³×C₁₃₉	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
180	C₄⋊C₄	C₂³×C₁₃₉	1	1	yes	characteristic
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	C₂³×C₁₃₉	1	2	yes	—
186	C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
187	C₂³	—	2	1	no	—
188	Q₈	—	2	2	no	—
189	Q₈	—	2	1	no	—
190	Q₈	—	4	2	no	—
191	Q₈	—	8	1	no	—
192	Q₈	C₂×Q₈×C₁₃₉	1	2	yes	—
193	D₄	—	2	1	no	—
194	D₄	C₂×Q₈×C₁₃₉	1	2	yes	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
206	C₂×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	2	no	—
209	C₈	C₂×Q₈×C₁₃₉	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	#135×C₁₃₉	1	2	yes	—
223	C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
224	C₄	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
225	C₂	—	4	2	no	—
226	C₂	D₄×Q₈×C₁₃₉	1	1	yes	characteristic
227	C₂	C₂×#135×C₁₃₉	1	1	yes	characteristic
228	C₂	#577×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₉
prime radical	C₂×D₄×C₁₃₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₄×Q₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #557×C₁₃₉	C₂	semidirect
3. #499×C₁₃₉	C₂	semidirect
4. #498×C₁₃₉	C₂	semidirect
5. #479×C₁₃₉	C₂	semidirect
7. #477×C₁₃₉	C₂	semidirect
9. C₈×Q₈×C₁₃₉	C₂	semidirect
48. SD₁₆×C₁₃₉	Q₈	semidirect
114. C₁₃₉	#3381	direct (N × H)
115. #3381	C₁₃₉	direct (N × H)
116. #557	C₂×C₁₃₉	semidirect
118. #499	C₂×C₁₃₉	semidirect
119. #498	C₂×C₁₃₉	semidirect
120. #479	C₂×C₁₃₉	semidirect
122. #477	C₂×C₁₃₉	semidirect
124. C₈×Q₈	C₂×C₁₃₉	semidirect
163. SD₁₆	Q₈×C₁₃₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#557×C₁₃₉	2
#554×C₁₃₉	2
#499×C₁₃₉	2
#498×C₁₃₉	2
#479×C₁₃₉	2
#478×C₁₃₉	2
#477×C₁₃₉	2
#474×C₁₃₉	2
C₈×Q₈×C₁₃₉	2
C₄×SD₁₆×C₁₃₉	2
#3381	139

Subgroups — #3381×C139 #31804007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3381×C₁₃₉ #31804007