Subgroups — C₁₃₉×#33382 #48027978

Order 35584.

1124 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#504×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2913×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#2141×C₁₃₉	C₂	1	3	yes	maximal, complemented
5	#2139×C₁₃₉	C₂	1	3	yes	maximal, complemented
6	#2138×C₁₃₉	C₂	1	3	yes	maximal, complemented
7	C₂×#358×C₁₃₉	C₂	1	3	yes	maximal, complemented
8	C₂×2₋¹⁺⁴×C₁₃₉	—	2	1	no	—
9	#578×C₁₃₉	—	2	4	no	—
10	#578×C₁₃₉	C₄	1	8	yes	complemented
11	C₂×#137×C₁₃₉	—	2	3	no	—
12	C₂×#135×C₁₃₉	C₂²	1	3	yes	—
13	C₂²×Q₁₆×C₁₃₉	C₂²	1	3	yes	—
14	C₂×#131×C₁₃₉	C₂²	1	1	yes	characteristic
15	#520×C₁₃₉	—	2	1	no	—
16	#504×C₁₃₉	—	2	4	no	—
17	C₂×M₃₂×C₁₃₉	C₂²	1	3	yes	—
18	C₂²×C₁₆×C₁₃₉	C₂²	1	3	yes	—
19	#428×C₁₃₉	—	2	3	no	—
20	#427×C₁₃₉	C₂²	1	3	yes	—
21	C₂×#107×C₁₃₉	C₂²	1	3	yes	—
22	#358×C₁₃₉	C₂²	1	3	yes	—
23	#358×C₁₃₉	C₂²	1	6	yes	—
24	#358×C₁₃₉	C₂²	1	3	yes	—
25	#357×C₁₃₉	C₂²	1	1	yes	characteristic
26	#357×C₁₃₉	C₂²	1	3	yes	—
27	2₋¹⁺⁴×C₁₃₉	—	2	8	no	—
28	2₋¹⁺⁴×C₁₃₉	—	4	4	no	—
29	C₂×#41×C₁₃₉	—	2	3	no	—
30	C₂×#41×C₁₃₉	—	4	3	no	—
31	C₂×#41×C₁₃₉	D₄	1	1	yes	characteristic
32	C₂²×Q₈×C₁₃₉	—	2	3	no	—
33	C₂²×Q₈×C₁₃₉	—	4	1	no	—
34	#137×C₁₃₉	—	2	24	no	—
35	#135×C₁₃₉	—	2	6	no	—
36	#135×C₁₃₉	C₂×C₄	1	12	yes	complemented
37	C₂×Q₁₆×C₁₃₉	—	2	3	no	—
38	C₂×Q₁₆×C₁₃₉	—	2	6	no	—
39	C₂×Q₁₆×C₁₃₉	C₂×C₄	1	12	yes	complemented
40	C₂×Q₁₆×C₁₃₉	C₂³	1	3	yes	—
41	C₂×SD₁₆×C₁₃₉	—	2	3	no	—
42	C₂×D₈×C₁₃₉	C₂³	1	1	yes	characteristic
43	#131×C₁₃₉	C₂×C₄	1	4	yes	—
44	#131×C₁₃₉	D₄	1	4	yes	—
45	C₂×M₁₆×C₁₃₉	D₄	1	3	yes	—
46	C₂²×C₈×C₁₃₉	C₂³	1	3	yes	—
47	C₄×Q₈×C₁₃₉	—	4	1	no	—
48	C₄×D₄×C₁₃₉	—	4	3	no	—
49	#117×C₁₃₉	—	2	3	no	—
50	C₂×C₄⋊C₄×C₁₃₉	—	2	3	no	—
51	M₃₂×C₁₃₉	—	2	6	no	—
52	C₂×C₁₆×C₁₃₉	—	2	6	no	—
53	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₁₃₉	C₂³	1	3	yes	—
55	#107×C₁₃₉	—	2	3	no	—
56	#107×C₁₃₉	C₂³	1	3	yes	—
57	#107×C₁₃₉	C₂³	1	1	yes	characteristic
58	#106×C₁₃₉	—	2	3	no	—
59	#41×C₁₃₉	—	2	12	no	—
60	#41×C₁₃₉	—	4	24	no	—
61	#41×C₁₃₉	—	4	6	no	—
62	#41×C₁₃₉	C₂²⋊C₄	1	4	yes	—
63	#41×C₁₃₉	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₁₃₉	—	2	12	no	—
65	C₂×Q₈×C₁₃₉	—	2	3	no	—
66	C₂×Q₈×C₁₃₉	—	4	8	no	—
67	C₂×Q₈×C₁₃₉	—	4	3	no	—
68	C₂×Q₈×C₁₃₉	—	4	6	no	—
69	C₂×Q₈×C₁₃₉	D₈	1	1	yes	characteristic
70	C₂×D₄×C₁₃₉	—	2	1	no	—
71	C₂×D₄×C₁₃₉	—	4	3	no	—
72	C₂×D₄×C₁₃₉	D₈	1	3	yes	—
73	C₂²×C₄×C₁₃₉	—	4	3	no	—
74	C₂²×C₄×C₁₃₉	—	4	3	no	—
75	C₂²×C₄×C₁₃₉	—	4	3	no	—
76	C₂²×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
77	Q₁₆×C₁₃₉	—	2	12	no	—
78	Q₁₆×C₁₃₉	—	2	3	no	—
79	Q₁₆×C₁₃₉	C₂²×C₄	1	6	yes	—
80	SD₁₆×C₁₃₉	—	2	12	no	—
81	D₈×C₁₃₉	—	2	1	no	—
82	D₈×C₁₃₉	C₂²×C₄	1	2	yes	—
83	M₁₆×C₁₃₉	C₂²⋊C₄	1	12	yes	—
84	C₂×C₈×C₁₃₉	C₂²×C₄	1	6	yes	—
85	C₂×C₈×C₁₃₉	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₁₃₉	C₂×D₄	1	6	yes	—
87	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₁₃₉	—	2	1	no	—
89	C₄⋊C₄×C₁₃₉	—	2	3	no	—
90	C₄⋊C₄×C₁₃₉	—	4	3	no	—
91	C₂²⋊C₄×C₁₃₉	—	4	3	no	—
92	C₄²×C₁₃₉	—	4	3	no	—
93	C₁₆×C₁₃₉	—	2	1	no	—
94	C₁₆×C₁₃₉	—	2	3	no	—
95	C₂³×C₁₃₉	—	4	1	no	—
96	C₂³×C₁₃₉	C₂×D₈	1	3	yes	—
97	Q₈×C₁₃₉	—	2	6	no	—
98	Q₈×C₁₃₉	—	4	3	no	—
99	Q₈×C₁₃₉	—	4	12	no	—
100	Q₈×C₁₃₉	#102	1	4	yes	—
101	D₄×C₁₃₉	—	2	2	no	—
102	D₄×C₁₃₉	—	4	12	no	—
103	D₄×C₁₃₉	—	4	1	no	—
104	D₄×C₁₃₉	#102	1	12	yes	—
105	C₂×C₄×C₁₃₉	—	4	3	no	—
106	C₂×C₄×C₁₃₉	—	4	12	no	—
107	C₂×C₄×C₁₃₉	—	4	12	no	—
108	C₂×C₄×C₁₃₉	—	4	1	no	—
109	C₂×C₄×C₁₃₉	—	4	3	no	—
110	C₂×C₄×C₁₃₉	—	16	3	no	—
111	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
112	C₂×C₄×C₁₃₉	C₂×D₈	1	3	yes	—
113	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	6	yes	—
114	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₃₉	C₂×C₂²⋊C₄	1	6	yes	—
116	C₈×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²×C₁₃₉	—	2	3	no	—
119	C₂²×C₁₃₉	—	2	3	no	—
120	C₂²×C₁₃₉	—	4	2	no	—
121	C₂²×C₁₃₉	—	8	1	no	—
122	C₂²×C₁₃₉	—	8	1	no	—
123	C₂²×C₁₃₉	C₂×#102	1	6	yes	—
124	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₃₉	—	4	6	no	—
126	C₄×C₁₃₉	—	8	3	no	—
127	C₄×C₁₃₉	—	8	1	no	—
128	C₄×C₁₃₉	C₂×#102	1	6	yes	—
129	C₄×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₁₃₉	C₂²×SD₁₆	1	1	yes	characteristic
131	C₂×C₁₃₉	—	2	6	no	—
132	C₂×C₁₃₉	—	8	2	no	—
133	C₂×C₁₃₉	C₂²×#102	1	1	yes	characteristic
134	C₂×C₁₃₉	#3416	1	1	yes	characteristic
135	C₂×C₁₃₉	#3417	1	1	yes	characteristic
136	#33382	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₉	#33382	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#578	C₂×C₁₃₉	1	1	yes	characteristic
139	C₂×#504	C₂×C₁₃₉	1	1	yes	characteristic, complemented
140	#2913	C₂×C₁₃₉	1	1	yes	characteristic, complemented
141	#2141	C₂×C₁₃₉	1	3	yes	complemented
142	#2139	C₂×C₁₃₉	1	3	yes	complemented
143	#2138	C₂×C₁₃₉	1	3	yes	complemented
144	C₂×#358	C₂×C₁₃₉	1	3	yes	complemented
145	C₂×2₋¹⁺⁴	—	2	1	no	—
146	#578	—	2	4	no	—
147	#578	C₄×C₁₃₉	1	8	yes	complemented
148	C₂×#137	—	2	3	no	—
149	C₂×#135	C₂²×C₁₃₉	1	3	yes	—
150	C₂²×Q₁₆	C₂²×C₁₃₉	1	3	yes	—
151	C₂×#131	C₂²×C₁₃₉	1	1	yes	characteristic
152	#520	—	2	1	no	—
153	#504	—	2	4	no	—
154	C₂×M₃₂	C₂²×C₁₃₉	1	3	yes	—
155	C₂²×C₁₆	C₂²×C₁₃₉	1	3	yes	—
156	#428	—	2	3	no	—
157	#427	C₂²×C₁₃₉	1	3	yes	—
158	C₂×#107	C₂²×C₁₃₉	1	3	yes	—
159	#358	C₂²×C₁₃₉	1	3	yes	—
160	#358	C₂²×C₁₃₉	1	6	yes	—
161	#358	C₂²×C₁₃₉	1	3	yes	—
162	#357	C₂²×C₁₃₉	1	1	yes	characteristic
163	#357	C₂²×C₁₃₉	1	3	yes	—
164	2₋¹⁺⁴	—	2	8	no	—
165	2₋¹⁺⁴	—	4	4	no	—
166	C₂×#41	—	2	3	no	—
167	C₂×#41	—	4	3	no	—
168	C₂×#41	D₄×C₁₃₉	1	1	yes	characteristic
169	C₂²×Q₈	—	2	3	no	—
170	C₂²×Q₈	—	4	1	no	—
171	#137	—	2	24	no	—
172	#135	—	2	6	no	—
173	#135	C₂×C₄×C₁₃₉	1	12	yes	complemented
174	C₂×Q₁₆	—	2	3	no	—
175	C₂×Q₁₆	—	2	6	no	—
176	C₂×Q₁₆	C₂×C₄×C₁₃₉	1	12	yes	complemented
177	C₂×Q₁₆	C₂³×C₁₃₉	1	3	yes	—
178	C₂×SD₁₆	—	2	3	no	—
179	C₂×D₈	C₂³×C₁₃₉	1	1	yes	characteristic
180	#131	C₂×C₄×C₁₃₉	1	4	yes	—
181	#131	D₄×C₁₃₉	1	4	yes	—
182	C₂×M₁₆	D₄×C₁₃₉	1	3	yes	—
183	C₂²×C₈	C₂³×C₁₃₉	1	3	yes	—
184	C₄×Q₈	—	4	1	no	—
185	C₄×D₄	—	4	3	no	—
186	#117	—	2	3	no	—
187	C₂×C₄⋊C₄	—	2	3	no	—
188	M₃₂	—	2	6	no	—
189	C₂×C₁₆	—	2	6	no	—
190	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
191	C₂×C₁₆	C₂³×C₁₃₉	1	3	yes	—
192	#107	—	2	3	no	—
193	#107	C₂³×C₁₃₉	1	3	yes	—
194	#107	C₂³×C₁₃₉	1	1	yes	characteristic
195	#106	—	2	3	no	—
196	#41	—	2	12	no	—
197	#41	—	4	24	no	—
198	#41	—	4	6	no	—
199	#41	C₂²⋊C₄×C₁₃₉	1	4	yes	—
200	#41	SD₁₆×C₁₃₉	1	4	yes	—
201	C₂×Q₈	—	2	12	no	—
202	C₂×Q₈	—	2	3	no	—
203	C₂×Q₈	—	4	8	no	—
204	C₂×Q₈	—	4	3	no	—
205	C₂×Q₈	—	4	6	no	—
206	C₂×Q₈	D₈×C₁₃₉	1	1	yes	characteristic
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	4	3	no	—
209	C₂×D₄	D₈×C₁₃₉	1	3	yes	—
210	C₂²×C₄	—	4	3	no	—
211	C₂²×C₄	—	4	3	no	—
212	C₂²×C₄	—	4	3	no	—
213	C₂²×C₄	C₂×D₄×C₁₃₉	1	3	yes	—
214	Q₁₆	—	2	12	no	—
215	Q₁₆	—	2	3	no	—
216	Q₁₆	C₂²×C₄×C₁₃₉	1	6	yes	—
217	SD₁₆	—	2	12	no	—
218	D₈	—	2	1	no	—
219	D₈	C₂²×C₄×C₁₃₉	1	2	yes	—
220	M₁₆	C₂²⋊C₄×C₁₃₉	1	12	yes	—
221	C₂×C₈	C₂²×C₄×C₁₃₉	1	6	yes	—
222	C₂×C₈	C₂×D₄×C₁₃₉	1	3	yes	—
223	C₂×C₈	C₂×D₄×C₁₃₉	1	6	yes	—
224	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	3	no	—
227	C₄⋊C₄	—	4	3	no	—
228	C₂²⋊C₄	—	4	3	no	—
229	C₄²	—	4	3	no	—
230	C₁₆	—	2	1	no	—
231	C₁₆	—	2	3	no	—
232	C₂³	—	4	1	no	—
233	C₂³	C₂×D₈×C₁₃₉	1	3	yes	—
234	Q₈	—	2	6	no	—
235	Q₈	—	4	12	no	—
236	Q₈	—	4	3	no	—
237	Q₈	#102×C₁₃₉	1	4	yes	—
238	D₄	—	2	2	no	—
239	D₄	—	4	12	no	—
240	D₄	—	4	1	no	—
241	D₄	#102×C₁₃₉	1	12	yes	—
242	C₂×C₄	—	4	12	no	—
243	C₂×C₄	—	4	3	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	3	no	—
247	C₂×C₄	—	16	3	no	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
249	C₂×C₄	C₂×D₈×C₁₃₉	1	3	yes	—
250	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	6	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
252	C₈	C₂×C₂²⋊C₄×C₁₃₉	1	6	yes	—
253	C₈	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
254	C₈	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
255	C₂²	—	2	3	no	—
256	C₂²	—	2	3	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	8	1	no	—
259	C₂²	—	8	1	no	—
260	C₂²	C₂×#102×C₁₃₉	1	6	yes	—
261	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
262	C₄	—	4	6	no	—
263	C₄	—	8	3	no	—
264	C₄	—	8	1	no	—
265	C₄	C₂×#102×C₁₃₉	1	6	yes	—
266	C₄	C₂²×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
267	C₄	C₂²×SD₁₆×C₁₃₉	1	1	yes	characteristic
268	C₂	—	2	6	no	—
269	C₂	—	8	2	no	—
270	C₂	C₂²×#102×C₁₃₉	1	1	yes	characteristic
271	C₂	#3416×C₁₃₉	1	1	yes	characteristic
272	C₂	#3417×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
prime radical	C₂×#578×C₁₃₉	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂×#41×C₁₃₉ ◁ C₂×#131×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#504×C₁₃₉	C₂	semidirect
3. #2913×C₁₃₉	C₂	semidirect
4. #2141×C₁₃₉	C₂	semidirect
5. #2139×C₁₃₉	C₂	semidirect
6. #2138×C₁₃₉	C₂	semidirect
7. C₂×#358×C₁₃₉	C₂	semidirect
10. #578×C₁₃₉	C₄	semidirect
36. #135×C₁₃₉	C₂×C₄	semidirect
39. C₂×Q₁₆×C₁₃₉	C₂×C₄	semidirect
136. #33382	C₁₃₉	direct (N × H)
137. C₁₃₉	#33382	direct (N × H)
139. C₂×#504	C₂×C₁₃₉	semidirect
140. #2913	C₂×C₁₃₉	semidirect
141. #2141	C₂×C₁₃₉	semidirect
142. #2139	C₂×C₁₃₉	semidirect
143. #2138	C₂×C₁₃₉	semidirect
144. C₂×#358	C₂×C₁₃₉	semidirect
147. #578	C₄×C₁₃₉	semidirect
173. #135	C₂×C₄×C₁₃₉	semidirect
176. C₂×Q₁₆	C₂×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×#578×C₁₃₉	2
C₂×#504×C₁₃₉	2
#2913×C₁₃₉	2
#2141×C₁₃₉	2
#2139×C₁₃₉	2
#2138×C₁₃₉	2
C₂×#358×C₁₃₉	2
#33382	139

Subgroups — C139×#33382 #48027978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33382 #48027978