Subgroups — C₁₃₉×#9621 #48004217

Order 35584.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1948×C₁₃₉	C₂	1	2	yes	maximal
2	#1845×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1770×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1296×C₁₃₉	C₂	1	2	yes	maximal
6	C₂×#107×C₁₃₉	—	2	1	no	—
7	C₂×#107×C₁₃₉	—	2	1	no	—
8	C₂×#106×C₁₃₉	—	2	2	no	—
9	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#97×C₁₃₉	—	2	1	no	—
12	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#384×C₁₃₉	C₂²	1	2	yes	—
14	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	#340×C₁₃₉	—	2	2	no	—
16	#336×C₁₃₉	—	2	1	no	—
17	#334×C₁₃₉	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃₉	—	2	1	no	—
19	C₂²×C₈×C₁₃₉	—	2	2	no	—
20	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃₉	—	2	1	no	—
28	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
29	#107×C₁₃₉	—	4	2	no	—
30	#107×C₁₃₉	—	4	2	no	—
31	#106×C₁₃₉	—	4	4	no	—
32	C₄⋊C₈×C₁₃₉	—	2	2	no	—
33	C₄⋊C₈×C₁₃₉	—	2	1	no	—
34	C₄⋊C₈×C₁₃₉	C₂×C₄	1	4	yes	—
35	C₄⋊C₈×C₁₃₉	Q₈	1	2	yes	—
36	#97×C₁₃₉	—	4	2	no	—
37	C₄×C₈×C₁₃₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₃₉	Q₈	1	2	yes	—
39	#95×C₁₃₉	—	2	1	no	—
40	C₂²×C₄×C₁₃₉	—	2	1	no	—
41	C₂²×C₄×C₁₃₉	—	2	1	no	—
42	C₂²×C₄×C₁₃₉	—	4	2	no	—
43	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
46	C₂×C₈×C₁₃₉	—	2	1	no	—
47	C₂×C₈×C₁₃₉	—	2	1	no	—
48	C₂×C₈×C₁₃₉	—	2	4	no	—
49	C₂×C₈×C₁₃₉	—	2	2	no	—
50	C₂×C₈×C₁₃₉	—	4	1	no	—
51	C₂×C₈×C₁₃₉	—	4	2	no	—
52	C₂×C₈×C₁₃₉	—	4	1	no	—
53	C₂×C₈×C₁₃₉	—	4	2	no	—
54	C₂×C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃₉	SD₁₆	1	2	yes	—
57	C₂×C₈×C₁₃₉	SD₁₆	1	2	yes	—
58	C₄⋊C₄×C₁₃₉	—	2	2	no	—
59	C₄⋊C₄×C₁₃₉	—	2	2	no	—
60	C₄⋊C₄×C₁₃₉	—	4	2	no	—
61	C₄⋊C₄×C₁₃₉	—	4	2	no	—
62	C₄⋊C₄×C₁₃₉	—	4	2	no	—
63	C₄⋊C₄×C₁₃₉	—	4	2	no	—
64	C₄⋊C₄×C₁₃₉	—	8	2	no	—
65	C₄²×C₁₃₉	—	4	1	no	—
66	C₄²×C₁₃₉	—	4	1	no	—
67	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₃₉	#41	1	1	yes	characteristic
71	C₂³×C₁₃₉	#121	1	1	yes	characteristic
72	C₂×C₄×C₁₃₉	—	2	1	no	—
73	C₂×C₄×C₁₃₉	—	2	1	no	—
74	C₂×C₄×C₁₃₉	—	4	1	no	—
75	C₂×C₄×C₁₃₉	—	4	1	no	—
76	C₂×C₄×C₁₃₉	—	4	1	no	—
77	C₂×C₄×C₁₃₉	—	4	1	no	—
78	C₂×C₄×C₁₃₉	—	4	1	no	—
79	C₂×C₄×C₁₃₉	—	4	1	no	—
80	C₂×C₄×C₁₃₉	—	8	2	no	—
81	C₂×C₄×C₁₃₉	—	8	2	no	—
82	C₂×C₄×C₁₃₉	—	8	2	no	—
83	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₉	#125	1	1	yes	characteristic
88	C₂×C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₁₃₉	#137	1	1	yes	characteristic
94	C₂×C₄×C₁₃₉	#137	1	2	yes	—
95	C₈×C₁₃₉	—	2	2	no	—
96	C₈×C₁₃₉	—	4	4	no	—
97	C₈×C₁₃₉	—	4	2	no	—
98	C₈×C₁₃₉	#106	1	4	yes	complemented
99	C₂²×C₁₃₉	#389	1	1	yes	characteristic
100	C₂²×C₁₃₉	#460	1	1	yes	characteristic
101	C₂²×C₁₃₉	#461	1	1	yes	characteristic
102	C₂²×C₁₃₉	#469	1	1	yes	characteristic
103	C₂²×C₁₃₉	#470	1	1	yes	characteristic
104	C₂²×C₁₃₉	#472	1	1	yes	characteristic
105	C₂²×C₁₃₉	#473	1	1	yes	characteristic
106	C₄×C₁₃₉	—	2	1	no	—
107	C₄×C₁₃₉	—	2	1	no	—
108	C₄×C₁₃₉	—	8	1	no	—
109	C₄×C₁₃₉	—	8	1	no	—
110	C₄×C₁₃₉	—	16	2	no	—
111	C₄×C₁₃₉	C₂×#106	1	1	yes	characteristic
112	C₄×C₁₃₉	#428	1	1	yes	characteristic
113	C₄×C₁₃₉	#440	1	2	yes	—
114	C₄×C₁₃₉	#475	1	1	yes	characteristic
115	C₄×C₁₃₉	#476	1	1	yes	characteristic
116	C₄×C₁₃₉	#501	1	2	yes	—
117	C₂×C₁₃₉	#1688	1	1	yes	characteristic
118	C₂×C₁₃₉	#1690	1	1	yes	characteristic
119	C₂×C₁₃₉	#1693	1	1	yes	characteristic
120	C₂×C₁₃₉	#1695	1	1	yes	characteristic
121	C₂×C₁₃₉	#1944	1	1	yes	characteristic
122	C₂×C₁₃₉	#1948	1	1	yes	characteristic
123	C₂×C₁₃₉	#1952	1	1	yes	characteristic
124	#9621	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₁₃₉	#9621	1	1	yes	characteristic, direct product factor, Sylow
126	#1948	C₂×C₁₃₉	1	2	yes	—
127	#1845	C₂×C₁₃₉	1	1	yes	characteristic
128	#1770	C₂×C₁₃₉	1	1	yes	characteristic
129	C₂×#334	C₂×C₁₃₉	1	1	yes	characteristic
130	#1296	C₂×C₁₃₉	1	2	yes	—
131	C₂×#107	—	2	1	no	—
132	C₂×#107	—	2	1	no	—
133	C₂×#106	—	2	2	no	—
134	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
138	#384	C₂²×C₁₃₉	1	2	yes	—
139	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
140	#340	—	2	2	no	—
141	#336	—	2	1	no	—
142	#334	C₄×C₁₃₉	1	4	yes	complemented
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
154	#107	—	4	2	no	—
155	#107	—	4	2	no	—
156	#106	—	4	4	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	C₂×C₄×C₁₃₉	1	4	yes	—
160	C₄⋊C₈	Q₈×C₁₃₉	1	2	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	C₂×C₄×C₁₃₉	1	2	yes	—
163	C₄×C₈	Q₈×C₁₃₉	1	2	yes	—
164	#95	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
170	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	C₄⋊C₄×C₁₃₉	1	2	yes	—
180	C₂×C₈	C₄⋊C₄×C₁₃₉	1	2	yes	—
181	C₂×C₈	SD₁₆×C₁₃₉	1	2	yes	—
182	C₂×C₈	SD₁₆×C₁₃₉	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	8	2	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
193	C₄²	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
194	C₄²	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
195	C₄²	#41×C₁₃₉	1	1	yes	characteristic
196	C₂³	#121×C₁₃₉	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	8	2	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
209	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
210	C₂×C₄	C₄×Q₈×C₁₃₉	1	1	yes	characteristic
211	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
212	C₂×C₄	#125×C₁₃₉	1	1	yes	characteristic
213	C₂×C₄	C₄⋊Q₈×C₁₃₉	1	1	yes	characteristic
214	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
215	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
216	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
217	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
218	C₂×C₄	#137×C₁₃₉	1	1	yes	characteristic
219	C₂×C₄	#137×C₁₃₉	1	2	yes	—
220	C₈	—	2	2	no	—
221	C₈	—	4	4	no	—
222	C₈	—	4	2	no	—
223	C₈	#106×C₁₃₉	1	4	yes	complemented
224	C₂²	#389×C₁₃₉	1	1	yes	characteristic
225	C₂²	#460×C₁₃₉	1	1	yes	characteristic
226	C₂²	#461×C₁₃₉	1	1	yes	characteristic
227	C₂²	#469×C₁₃₉	1	1	yes	characteristic
228	C₂²	#470×C₁₃₉	1	1	yes	characteristic
229	C₂²	#472×C₁₃₉	1	1	yes	characteristic
230	C₂²	#473×C₁₃₉	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	1	no	—
234	C₄	—	8	1	no	—
235	C₄	—	16	2	no	—
236	C₄	C₂×#106×C₁₃₉	1	1	yes	characteristic
237	C₄	#428×C₁₃₉	1	1	yes	characteristic
238	C₄	#440×C₁₃₉	1	2	yes	—
239	C₄	#475×C₁₃₉	1	1	yes	characteristic
240	C₄	#476×C₁₃₉	1	1	yes	characteristic
241	C₄	#501×C₁₃₉	1	2	yes	—
242	C₂	#1688×C₁₃₉	1	1	yes	characteristic
243	C₂	#1690×C₁₃₉	1	1	yes	characteristic
244	C₂	#1693×C₁₃₉	1	1	yes	characteristic
245	C₂	#1695×C₁₃₉	1	1	yes	characteristic
246	C₂	#1944×C₁₃₉	1	1	yes	characteristic
247	C₂	#1948×C₁₃₉	1	1	yes	characteristic
248	C₂	#1952×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	#121
commutator	C₄²	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. #334×C₁₃₉	C₄	semidirect
98. C₈×C₁₃₉	#106	semidirect
124. #9621	C₁₃₉	direct (N × H)
125. C₁₃₉	#9621	direct (N × H)
142. #334	C₄×C₁₃₉	semidirect
223. C₈	#106×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#1948×C₁₃₉	2
#1845×C₁₃₉	2
#1770×C₁₃₉	2
C₂×#334×C₁₃₉	2
#1296×C₁₃₉	2
#9621	139

Subgroups — C139×#9621 #48004217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9621 #48004217