Subgroups — C₁₃₉×#14165 #48008761

Order 35584.

1646 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#381×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂²⋊C₄×C₁₃₉	C₂	1	2	yes	maximal
3	#1439×C₁₃₉	C₂	1	4	yes	maximal
4	#1436×C₁₃₉	C₂	1	8	yes	maximal, complemented
5	C₂⁴×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
7	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	4	yes	—
8	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	4	yes	—
10	#381×C₁₃₉	C₄	1	8	yes	complemented
11	C₄×C₂²⋊C₄×C₁₃₉	—	2	8	no	—
12	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
13	C₂×#95×C₁₃₉	C₂²	1	8	yes	—
14	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
15	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
16	C₂⁵×C₁₃₉	C₂³	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₉	—	4	2	no	—
18	C₂³×C₄×C₁₃₉	—	4	2	no	—
19	C₂³×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
20	C₂³×C₄×C₁₃₉	D₄	1	4	yes	—
21	C₂³×C₄×C₁₃₉	Q₈	1	4	yes	—
22	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₃₉	C₂³	1	8	yes	—
24	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₃₉	—	2	16	no	—
26	C₂×C₄⋊C₄×C₁₃₉	—	4	4	no	—
27	C₂×C₂²⋊C₄×C₁₃₉	—	2	8	no	—
28	C₂×C₂²⋊C₄×C₁₃₉	—	2	16	no	—
29	C₂×C₂²⋊C₄×C₁₃₉	—	4	2	no	—
30	C₂×C₂²⋊C₄×C₁₃₉	—	4	2	no	—
31	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₁₃₉	—	2	8	no	—
33	C₂×C₄²×C₁₃₉	—	4	4	no	—
34	C₂×C₄²×C₁₃₉	—	4	4	no	—
35	#95×C₁₃₉	—	2	8	no	—
36	#95×C₁₃₉	—	2	16	no	—
37	#95×C₁₃₉	—	4	4	no	—
38	#95×C₁₃₉	C₂×C₄	1	16	yes	—
39	C₂⁴×C₁₃₉	—	2	8	no	—
40	C₂⁴×C₁₃₉	—	2	2	no	—
41	C₂⁴×C₁₃₉	—	2	2	no	—
42	C₂⁴×C₁₃₉	C₂²×C₄	1	2	yes	—
43	C₂⁴×C₁₃₉	C₂×D₄	1	2	yes	—
44	C₂⁴×C₁₃₉	C₂×Q₈	1	2	yes	—
45	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	—	2	4	no	—
47	C₂²×C₄×C₁₃₉	—	2	4	no	—
48	C₂²×C₄×C₁₃₉	—	2	8	no	—
49	C₂²×C₄×C₁₃₉	—	2	8	no	—
50	C₂²×C₄×C₁₃₉	—	2	8	no	—
51	C₂²×C₄×C₁₃₉	—	2	8	no	—
52	C₂²×C₄×C₁₃₉	—	2	8	no	—
53	C₂²×C₄×C₁₃₉	—	2	8	no	—
54	C₂²×C₄×C₁₃₉	—	2	8	no	—
55	C₂²×C₄×C₁₃₉	—	2	4	no	—
56	C₂²×C₄×C₁₃₉	—	2	8	no	—
57	C₂²×C₄×C₁₃₉	—	2	4	no	—
58	C₂²×C₄×C₁₃₉	—	4	8	no	—
59	C₂²×C₄×C₁₃₉	—	4	8	no	—
60	C₂²×C₄×C₁₃₉	—	4	4	no	—
61	C₂²×C₄×C₁₃₉	—	4	4	no	—
62	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
67	C₂²×C₄×C₁₃₉	#41	1	16	yes	—
68	C₄⋊C₄×C₁₃₉	—	4	16	no	—
69	C₂²⋊C₄×C₁₃₉	—	2	32	no	—
70	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
71	C₄²×C₁₃₉	—	4	16	no	—
72	C₂³×C₁₃₉	—	2	8	no	—
73	C₂³×C₁₃₉	—	2	4	no	—
74	C₂³×C₁₃₉	—	2	8	no	—
75	C₂³×C₁₃₉	—	2	8	no	—
76	C₂³×C₁₃₉	—	2	4	no	—
77	C₂³×C₁₃₉	—	2	2	no	—
78	C₂³×C₁₃₉	—	2	2	no	—
79	C₂³×C₁₃₉	—	4	4	no	—
80	C₂³×C₁₃₉	—	4	8	no	—
81	C₂³×C₁₃₉	—	4	4	no	—
82	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
83	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
84	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
85	C₂³×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂³×C₁₃₉	C₂×#41	1	8	yes	—
87	C₂³×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
88	C₂³×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
89	C₂×C₄×C₁₃₉	—	2	8	no	—
90	C₂×C₄×C₁₃₉	—	2	8	no	—
91	C₂×C₄×C₁₃₉	—	2	4	no	—
92	C₂×C₄×C₁₃₉	—	2	4	no	—
93	C₂×C₄×C₁₃₉	—	2	16	no	—
94	C₂×C₄×C₁₃₉	—	2	8	no	—
95	C₂×C₄×C₁₃₉	—	2	16	no	—
96	C₂×C₄×C₁₃₉	—	2	16	no	—
97	C₂×C₄×C₁₃₉	—	2	16	no	—
98	C₂×C₄×C₁₃₉	—	4	16	no	—
99	C₂×C₄×C₁₃₉	—	4	8	no	—
100	C₂×C₄×C₁₃₉	—	4	8	no	—
101	C₂×C₄×C₁₃₉	—	4	4	no	—
102	C₂×C₄×C₁₃₉	—	4	8	no	—
103	C₂×C₄×C₁₃₉	—	4	8	no	—
104	C₂×C₄×C₁₃₉	—	4	8	no	—
105	C₂×C₄×C₁₃₉	—	4	4	no	—
106	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	—
107	C₂²×C₁₃₉	—	2	4	no	—
108	C₂²×C₁₃₉	—	2	8	no	—
109	C₂²×C₁₃₉	—	4	8	no	—
110	C₂²×C₁₃₉	—	4	4	no	—
111	C₂²×C₁₃₉	—	4	4	no	—
112	C₂²×C₁₃₉	—	4	8	no	—
113	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₁₃₉	C₄×#41	1	4	yes	—
115	C₂²×C₁₃₉	#518	1	2	yes	—
116	C₂²×C₁₃₉	#525	1	4	yes	—
117	C₂²×C₁₃₉	#533	1	4	yes	—
118	C₂²×C₁₃₉	#534	1	2	yes	—
119	C₂²×C₁₃₉	#543	1	1	yes	characteristic
120	C₂²×C₁₃₉	#543	1	1	yes	characteristic
121	C₂²×C₁₃₉	#551	1	4	yes	—
122	C₂²×C₁₃₉	#553	1	8	yes	—
123	C₂²×C₁₃₉	#559	1	4	yes	—
124	C₄×C₁₃₉	—	2	8	no	—
125	C₄×C₁₃₉	—	4	16	no	—
126	C₄×C₁₃₉	—	4	8	no	—
127	C₂×C₁₃₉	—	4	4	no	—
128	C₂×C₁₃₉	#2296	1	2	yes	—
129	C₂×C₁₃₉	#2315	1	1	yes	characteristic
130	C₂×C₁₃₉	#2374	1	4	yes	—
131	C₂×C₁₃₉	#2600	1	4	yes	—
132	C₂×C₁₃₉	#2607	1	2	yes	—
133	C₂×C₁₃₉	#2608	1	2	yes	—
134	#14165	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₁₃₉	#14165	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×#381	C₂×C₁₃₉	1	1	yes	characteristic
137	C₂×C₄×C₂²⋊C₄	C₂×C₁₃₉	1	2	yes	—
138	#1439	C₂×C₁₃₉	1	4	yes	—
139	#1436	C₂×C₁₃₉	1	8	yes	complemented
140	C₂⁴×C₄	C₂²×C₁₃₉	1	1	yes	characteristic
141	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
142	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	4	yes	—
143	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	2	yes	—
144	C₂²×C₄²	C₂²×C₁₃₉	1	4	yes	—
145	#381	C₄×C₁₃₉	1	8	yes	complemented
146	C₄×C₂²⋊C₄	—	2	8	no	—
147	C₂×#95	C₂²×C₁₃₉	1	8	yes	—
148	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
149	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
150	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
151	C₂⁵	C₂³×C₁₃₉	1	1	yes	characteristic
152	C₂³×C₄	—	4	2	no	—
153	C₂³×C₄	—	4	2	no	—
154	C₂³×C₄	C₂×C₄×C₁₃₉	1	4	yes	—
155	C₂³×C₄	D₄×C₁₃₉	1	4	yes	—
156	C₂³×C₄	Q₈×C₁₃₉	1	4	yes	—
157	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
158	C₂³×C₄	C₂³×C₁₃₉	1	8	yes	—
159	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
160	C₂×C₄⋊C₄	—	2	16	no	—
161	C₂×C₄⋊C₄	—	4	4	no	—
162	C₂×C₂²⋊C₄	—	2	8	no	—
163	C₂×C₂²⋊C₄	—	2	16	no	—
164	C₂×C₂²⋊C₄	—	4	2	no	—
165	C₂×C₂²⋊C₄	—	4	2	no	—
166	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	8	yes	—
167	C₂×C₄²	—	2	8	no	—
168	C₂×C₄²	—	4	4	no	—
169	C₂×C₄²	—	4	4	no	—
170	#95	—	2	8	no	—
171	#95	—	2	16	no	—
172	#95	—	4	4	no	—
173	#95	C₂×C₄×C₁₃₉	1	16	yes	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	—	2	2	no	—
176	C₂⁴	—	2	2	no	—
177	C₂⁴	C₂²×C₄×C₁₃₉	1	2	yes	—
178	C₂⁴	C₂×D₄×C₁₃₉	1	2	yes	—
179	C₂⁴	C₂×Q₈×C₁₃₉	1	2	yes	—
180	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	2	8	no	—
185	C₂²×C₄	—	2	8	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	8	no	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	16	yes	complemented
198	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
200	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
201	C₂²×C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
202	C₂²×C₄	#41×C₁₃₉	1	16	yes	—
203	C₄⋊C₄	—	4	16	no	—
204	C₂²⋊C₄	—	2	32	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₄²	—	4	16	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	8	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	4	8	no	—
216	C₂³	—	4	4	no	—
217	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
218	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
219	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
220	C₂³	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
221	C₂³	C₂×#41×C₁₃₉	1	8	yes	—
222	C₂³	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
223	C₂³	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	4	16	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	8	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	4	4	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	4	8	no	—
248	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
249	C₂²	C₄×#41×C₁₃₉	1	4	yes	—
250	C₂²	#518×C₁₃₉	1	2	yes	—
251	C₂²	#525×C₁₃₉	1	4	yes	—
252	C₂²	#533×C₁₃₉	1	4	yes	—
253	C₂²	#534×C₁₃₉	1	2	yes	—
254	C₂²	#543×C₁₃₉	1	1	yes	characteristic
255	C₂²	#543×C₁₃₉	1	1	yes	characteristic
256	C₂²	#551×C₁₃₉	1	4	yes	—
257	C₂²	#553×C₁₃₉	1	8	yes	—
258	C₂²	#559×C₁₃₉	1	4	yes	—
259	C₄	—	2	8	no	—
260	C₄	—	4	16	no	—
261	C₄	—	4	8	no	—
262	C₂	—	4	4	no	—
263	C₂	#2296×C₁₃₉	1	2	yes	—
264	C₂	#2315×C₁₃₉	1	1	yes	characteristic
265	C₂	#2374×C₁₃₉	1	4	yes	—
266	C₂	#2600×C₁₃₉	1	4	yes	—
267	C₂	#2607×C₁₃₉	1	2	yes	—
268	C₂	#2608×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₉
prime radical	C₂⁵×C₁₃₉	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. #1436×C₁₃₉	C₂	semidirect
10. #381×C₁₃₉	C₄	semidirect
62. C₂²×C₄×C₁₃₉	C₄⋊C₄	semidirect
134. #14165	C₁₃₉	direct (N × H)
135. C₁₃₉	#14165	direct (N × H)
139. #1436	C₂×C₁₃₉	semidirect
145. #381	C₄×C₁₃₉	semidirect
197. C₂²×C₄	C₄⋊C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#381×C₁₃₉	2
C₂×C₄×C₂²⋊C₄×C₁₃₉	2
#1439×C₁₃₉	2
#1436×C₁₃₉	2
#14165	139

Subgroups — C139×#14165 #48008761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#14165 #48008761