Subgroups — C₁₃₉×#29222 #48023818

Order 35584.

654 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2862×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2101×C₁₃₉	C₂	1	4	yes	maximal
4	#1991×C₁₃₉	C₂	1	2	yes	maximal
5	#1988×C₁₃₉	C₂	1	2	yes	maximal
6	#1987×C₁₃₉	C₂	1	2	yes	maximal
7	#1835×C₁₃₉	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×#125×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
14	#391×C₁₃₉	—	2	2	no	—
15	#391×C₁₃₉	—	2	2	no	—
16	#389×C₁₃₉	C₂²	1	2	yes	—
17	#387×C₁₃₉	—	2	2	no	—
18	#382×C₁₃₉	C₂²	1	1	yes	characteristic
19	#382×C₁₃₉	C₂²	1	2	yes	—
20	#382×C₁₃₉	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
24	#340×C₁₃₉	C₂²	1	4	yes	—
25	#340×C₁₃₉	C₂²	1	4	yes	—
26	#340×C₁₃₉	C₂²	1	4	yes	—
27	#340×C₁₃₉	C₂²	1	4	yes	—
28	#336×C₁₃₉	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃₉	—	2	1	no	—
30	C₂²×Q₈×C₁₃₉	—	2	1	no	—
31	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
32	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
33	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
34	#125×C₁₃₉	—	2	2	no	—
35	#125×C₁₃₉	Q₈	1	4	yes	complemented
36	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₉	—	2	2	no	—
45	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₁₃₉	—	2	4	no	—
51	C₄×C₈×C₁₃₉	—	2	4	no	—
52	#95×C₁₃₉	—	2	2	no	—
53	#95×C₁₃₉	—	2	2	no	—
54	#95×C₁₃₉	—	2	2	no	—
55	#95×C₁₃₉	—	2	4	no	—
56	#95×C₁₃₉	—	2	4	no	—
57	C₂×Q₈×C₁₃₉	—	4	4	no	—
58	C₂×Q₈×C₁₃₉	—	4	2	no	—
59	C₂×Q₈×C₁₃₉	—	4	2	no	—
60	C₂×Q₈×C₁₃₉	—	4	4	no	—
61	C₂²×C₄×C₁₃₉	—	2	2	no	—
62	C₂²×C₄×C₁₃₉	—	2	2	no	—
63	C₂²×C₄×C₁₃₉	—	2	2	no	—
64	C₂²×C₄×C₁₃₉	—	2	2	no	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₁₃₉	—	2	4	no	—
71	C₂×C₈×C₁₃₉	—	2	4	no	—
72	C₂×C₈×C₁₃₉	—	2	4	no	—
73	C₄⋊C₄×C₁₃₉	—	2	8	no	—
74	C₄⋊C₄×C₁₃₉	—	2	2	no	—
75	C₄⋊C₄×C₁₃₉	—	2	4	no	—
76	C₄⋊C₄×C₁₃₉	—	2	4	no	—
77	C₄⋊C₄×C₁₃₉	—	2	4	no	—
78	C₄⋊C₄×C₁₃₉	—	4	4	no	—
79	C₄⋊C₄×C₁₃₉	—	4	2	no	—
80	C₄⋊C₄×C₁₃₉	—	4	2	no	—
81	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₃₉	—	2	4	no	—
83	C₄²×C₁₃₉	—	2	2	no	—
84	C₄²×C₁₃₉	—	4	2	no	—
85	C₄²×C₁₃₉	—	4	4	no	—
86	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
87	C₄²×C₁₃₉	#41	1	2	yes	—
88	C₄²×C₁₃₉	#41	1	4	yes	—
89	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₁₃₉	—	8	4	no	—
91	Q₈×C₁₃₉	—	8	4	no	—
92	C₂×C₄×C₁₃₉	—	2	2	no	—
93	C₂×C₄×C₁₃₉	—	2	2	no	—
94	C₂×C₄×C₁₃₉	—	2	2	no	—
95	C₂×C₄×C₁₃₉	—	2	4	no	—
96	C₂×C₄×C₁₃₉	—	2	2	no	—
97	C₂×C₄×C₁₃₉	—	2	2	no	—
98	C₂×C₄×C₁₃₉	—	2	2	no	—
99	C₂×C₄×C₁₃₉	—	4	2	no	—
100	C₂×C₄×C₁₃₉	—	4	2	no	—
101	C₂×C₄×C₁₃₉	—	4	2	no	—
102	C₂×C₄×C₁₃₉	—	4	2	no	—
103	C₂×C₄×C₁₃₉	—	4	2	no	—
104	C₂×C₄×C₁₃₉	—	4	2	no	—
105	C₂×C₄×C₁₃₉	—	4	4	no	—
106	C₂×C₄×C₁₃₉	—	4	2	no	—
107	C₂×C₄×C₁₃₉	—	4	4	no	—
108	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₃₉	#124	1	2	yes	—
110	C₂×C₄×C₁₃₉	#124	1	2	yes	—
111	C₂×C₄×C₁₃₉	#135	1	4	yes	—
112	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
115	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
116	C₈×C₁₃₉	—	4	4	no	—
117	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²×C₁₃₉	C₂×#124	1	1	yes	characteristic
119	C₂²×C₁₃₉	#536	1	1	yes	characteristic
120	C₂²×C₁₃₉	C₂×#135	1	2	yes	—
121	C₂²×C₁₃₉	#575	1	2	yes	—
122	C₄×C₁₃₉	—	2	4	no	—
123	C₄×C₁₃₉	—	2	2	no	—
124	C₄×C₁₃₉	—	4	4	no	—
125	C₄×C₁₃₉	—	4	2	no	—
126	C₄×C₁₃₉	—	8	2	no	—
127	C₄×C₁₃₉	—	8	2	no	—
128	C₄×C₁₃₉	—	8	2	no	—
129	C₄×C₁₃₉	#488	1	4	yes	—
130	C₄×C₁₃₉	#556	1	2	yes	—
131	C₄×C₁₃₉	#557	1	2	yes	—
132	C₂×C₁₃₉	#2568	1	1	yes	characteristic
133	C₂×C₁₃₉	#3076	1	2	yes	—
134	C₂×C₁₃₉	C₂×#488	1	1	yes	characteristic
135	C₂×C₁₃₉	#3140	1	1	yes	characteristic
136	C₂×C₁₃₉	#3233	1	2	yes	—
137	#29222	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₉	#29222	1	1	yes	characteristic, direct product factor, Sylow
139	#2862	C₂×C₁₃₉	1	1	yes	characteristic
140	C₄×#125	C₂×C₁₃₉	1	1	yes	characteristic
141	#2101	C₂×C₁₃₉	1	4	yes	—
142	#1991	C₂×C₁₃₉	1	2	yes	—
143	#1988	C₂×C₁₃₉	1	2	yes	—
144	#1987	C₂×C₁₃₉	1	2	yes	—
145	#1835	C₂×C₁₃₉	1	2	yes	—
146	C₄×C₄⋊C₈	C₂×C₁₃₉	1	1	yes	characteristic
147	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂×#125	C₂²×C₁₃₉	1	1	yes	characteristic
150	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃₉	1	2	yes	—
152	#391	—	2	2	no	—
153	#391	—	2	2	no	—
154	#389	C₂²×C₁₃₉	1	2	yes	—
155	#387	—	2	2	no	—
156	#382	C₂²×C₁₃₉	1	1	yes	characteristic
157	#382	C₂²×C₁₃₉	1	2	yes	—
158	#382	C₂²×C₁₃₉	1	2	yes	—
159	C₄×C₄⋊C₄	—	2	2	no	—
160	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
161	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
162	#340	C₂²×C₁₃₉	1	4	yes	—
163	#340	C₂²×C₁₃₉	1	4	yes	—
164	#340	C₂²×C₁₃₉	1	4	yes	—
165	#340	C₂²×C₁₃₉	1	4	yes	—
166	#336	C₂²×C₁₃₉	1	2	yes	—
167	C₂²×Q₈	—	2	1	no	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	4	no	—
172	#125	—	2	2	no	—
173	#125	Q₈×C₁₃₉	1	4	yes	complemented
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
184	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄×C₈	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	#95	—	2	4	no	—
194	#95	—	2	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₄⋊C₄	—	2	8	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
225	C₄²	#41×C₁₃₉	1	2	yes	—
226	C₄²	#41×C₁₃₉	1	4	yes	—
227	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
228	Q₈	—	8	4	no	—
229	Q₈	—	8	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
247	C₂×C₄	#124×C₁₃₉	1	2	yes	—
248	C₂×C₄	#124×C₁₃₉	1	2	yes	—
249	C₂×C₄	#135×C₁₃₉	1	4	yes	—
250	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
253	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
256	C₂²	C₂×#124×C₁₃₉	1	1	yes	characteristic
257	C₂²	#536×C₁₃₉	1	1	yes	characteristic
258	C₂²	C₂×#135×C₁₃₉	1	2	yes	—
259	C₂²	#575×C₁₃₉	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	#488×C₁₃₉	1	4	yes	—
268	C₄	#556×C₁₃₉	1	2	yes	—
269	C₄	#557×C₁₃₉	1	2	yes	—
270	C₂	#2568×C₁₃₉	1	1	yes	characteristic
271	C₂	#3076×C₁₃₉	1	2	yes	—
272	C₂	C₂×#488×C₁₃₉	1	1	yes	characteristic
273	C₂	#3140×C₁₃₉	1	1	yes	characteristic
274	C₂	#3233×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₄³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
35. #125×C₁₃₉	Q₈	semidirect
137. #29222	C₁₃₉	direct (N × H)
138. C₁₃₉	#29222	direct (N × H)
173. #125	Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2862×C₁₃₉	2
C₄×#125×C₁₃₉	2
#2101×C₁₃₉	2
#1991×C₁₃₉	2
#1988×C₁₃₉	2
#1987×C₁₃₉	2
#1835×C₁₃₉	2
C₄×C₄⋊C₈×C₁₃₉	2
#29222	139

Subgroups — C139×#29222 #48023818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29222 #48023818