Subgroups — C₁₃₉×#36179 #48030775

Order 35584.

3618 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	C₂×#547×C₁₃₉	C₂	1	6	yes	maximal, complemented
2	#2868×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2856×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#2542×C₁₃₉	C₂	1	6	yes	maximal, complemented
5	#2501×C₁₃₉	C₂	1	12	yes	maximal, complemented
6	C₄×#127×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#384×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₄²×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₃₉	C₂²	1	4	yes	—
10	#547×C₁₃₉	—	2	48	no	—
11	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	6	yes	—
12	C₂×#127×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×#123×C₁₃₉	C₂²	1	24	yes	—
14	C₂×C₂²⋊Q₈×C₁₃₉	C₂²	1	12	yes	—
15	C₂×#121×C₁₃₉	C₂²	1	12	yes	—
16	C₂×C₄×D₄×C₁₃₉	C₂²	1	6	yes	—
17	C₂×C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
18	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	12	yes	—
19	C₂²×C₄²×C₁₃₉	C₂²	1	2	yes	—
20	#399×C₁₃₉	C₂²	1	24	yes	—
21	#390×C₁₃₉	C₂²	1	12	yes	—
22	#390×C₁₃₉	C₂²	1	6	yes	—
23	#384×C₁₃₉	C₂²	1	1	yes	characteristic
24	#384×C₁₃₉	C₂²	1	12	yes	—
25	#384×C₁₃₉	C₂²	1	2	yes	—
26	#384×C₁₃₉	C₂²	1	6	yes	—
27	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
28	C₄×C₂²⋊C₄×C₁₃₉	C₂²	1	6	yes	—
29	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×#41×C₁₃₉	—	2	48	no	—
31	C₂²×Q₈×C₁₃₉	C₂³	1	4	yes	—
32	C₂²×D₄×C₁₃₉	C₂³	1	6	yes	—
33	C₂²×D₄×C₁₃₉	C₂³	1	8	yes	—
34	C₂²×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃₉	C₂³	1	8	yes	—
36	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₃₉	C₂³	1	6	yes	—
38	C₄⋊Q₈×C₁₃₉	—	2	24	no	—
39	#127×C₁₃₉	—	2	4	no	—
40	#123×C₁₃₉	—	2	96	no	—
41	C₂²⋊Q₈×C₁₃₉	—	2	48	no	—
42	#121×C₁₃₉	—	2	48	no	—
43	C₄×D₄×C₁₃₉	—	2	24	no	—
44	C₄×D₄×C₁₃₉	D₄	1	24	yes	complemented
45	C₂×C₄⋊C₄×C₁₃₉	—	2	48	no	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	24	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	12	yes	—
48	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
49	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	12	yes	—
50	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
51	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	12	yes	—
52	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	24	yes	—
53	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
54	C₂×C₄²×C₁₃₉	—	2	8	no	—
55	C₂×C₄²×C₁₃₉	C₂³	1	6	yes	—
56	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
57	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
58	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
59	#95×C₁₃₉	C₂³	1	6	yes	—
60	C₂⁴×C₁₃₉	C₂⁴	1	4	yes	—
61	C₂⁴×C₁₃₉	C₂⁴	1	2	yes	—
62	#41×C₁₃₉	—	4	64	no	—
63	C₂×Q₈×C₁₃₉	—	2	24	no	—
64	C₂×D₄×C₁₃₉	—	2	48	no	—
65	C₂×D₄×C₁₃₉	—	2	12	no	—
66	C₂×D₄×C₁₃₉	—	2	12	no	—
67	C₂×D₄×C₁₃₉	—	2	12	no	—
68	C₂×D₄×C₁₃₉	C₂×D₄	1	12	yes	—
69	C₂²×C₄×C₁₃₉	—	2	12	no	—
70	C₂²×C₄×C₁₃₉	—	2	12	no	—
71	C₂²×C₄×C₁₃₉	—	2	48	no	—
72	C₂²×C₄×C₁₃₉	—	2	24	no	—
73	C₂²×C₄×C₁₃₉	C₂×D₄	1	24	yes	—
74	C₂²×C₄×C₁₃₉	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₁₃₉	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
77	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
80	C₄⋊C₄×C₁₃₉	—	2	12	no	—
81	C₄⋊C₄×C₁₃₉	—	2	24	no	—
82	C₄⋊C₄×C₁₃₉	—	2	48	no	—
83	C₄⋊C₄×C₁₃₉	—	2	24	no	—
84	C₄⋊C₄×C₁₃₉	C₂×D₄	1	12	yes	—
85	C₂²⋊C₄×C₁₃₉	—	2	24	no	—
86	C₂²⋊C₄×C₁₃₉	—	2	48	no	—
87	C₂²⋊C₄×C₁₃₉	C₂×D₄	1	24	yes	—
88	C₄²×C₁₃₉	—	2	12	no	—
89	C₄²×C₁₃₉	—	2	6	no	—
90	C₄²×C₁₃₉	C₂×D₄	1	12	yes	—
91	C₄²×C₁₃₉	#41	1	4	yes	complemented
92	C₂³×C₁₃₉	—	2	2	no	—
93	C₂³×C₁₃₉	—	2	6	no	—
94	C₂³×C₁₃₉	—	2	12	no	—
95	C₂³×C₁₃₉	—	2	4	no	—
96	C₂³×C₁₃₉	—	2	12	no	—
97	C₂³×C₁₃₉	C₂²×D₄	1	12	yes	—
98	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
99	Q₈×C₁₃₉	—	4	16	no	—
100	D₄×C₁₃₉	—	4	32	no	—
101	D₄×C₁₃₉	—	4	24	no	—
102	D₄×C₁₃₉	#127	1	16	yes	complemented
103	C₂×C₄×C₁₃₉	—	2	6	no	—
104	C₂×C₄×C₁₃₉	—	2	6	no	—
105	C₂×C₄×C₁₃₉	—	2	6	no	—
106	C₂×C₄×C₁₃₉	—	2	48	no	—
107	C₂×C₄×C₁₃₉	—	2	12	no	—
108	C₂×C₄×C₁₃₉	—	2	24	no	—
109	C₂×C₄×C₁₃₉	—	4	32	no	—
110	C₂×C₄×C₁₃₉	—	4	16	no	—
111	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
112	C₂×C₄×C₁₃₉	C₂²×D₄	1	12	yes	—
113	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
114	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
115	C₂×C₄×C₁₃₉	C₂×#41	1	6	yes	—
116	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	6	yes	—
117	C₂²×C₁₃₉	—	2	12	no	—
118	C₂²×C₁₃₉	—	2	12	no	—
119	C₂²×C₁₃₉	—	4	4	no	—
120	C₂²×C₁₃₉	—	4	12	no	—
121	C₂²×C₁₃₉	—	4	12	no	—
122	C₂²×C₁₃₉	C₂×#127	1	8	yes	—
123	C₂²×C₁₃₉	C₂³×D₄	1	3	yes	—
124	C₂²×C₁₃₉	C₂²×#41	1	1	yes	characteristic
125	C₂²×C₁₃₉	C₂×2₋¹⁺⁴	1	3	yes	—
126	C₄×C₁₃₉	—	2	12	no	—
127	C₄×C₁₃₉	—	2	6	no	—
128	C₄×C₁₃₉	—	4	4	no	—
129	C₄×C₁₃₉	—	4	8	no	—
130	C₄×C₁₃₉	C₂×#127	1	4	yes	—
131	C₄×C₁₃₉	#547	1	12	yes	complemented
132	C₂×C₁₃₉	—	2	8	no	—
133	C₂×C₁₃₉	—	8	4	no	—
134	C₂×C₁₃₉	C₂²×#127	1	1	yes	characteristic
135	C₂×C₁₃₉	C₂×#547	1	3	yes	—
136	C₂×C₁₃₉	#3499	1	3	yes	—
137	#36179	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₁₃₉	#36179	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#547	C₂×C₁₃₉	1	6	yes	complemented
140	#2868	C₂×C₁₃₉	1	1	yes	characteristic, complemented
141	#2856	C₂×C₁₃₉	1	2	yes	complemented
142	#2542	C₂×C₁₃₉	1	6	yes	complemented
143	#2501	C₂×C₁₃₉	1	12	yes	complemented
144	C₄×#127	C₂×C₁₃₉	1	1	yes	characteristic, complemented
145	C₂×#384	C₂×C₁₃₉	1	2	yes	complemented
146	C₄²×D₄	C₂×C₁₃₉	1	1	yes	characteristic, complemented
147	C₂²×#41	C₂²×C₁₃₉	1	4	yes	—
148	#547	—	2	48	no	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	6	yes	—
150	C₂×#127	C₂²×C₁₃₉	1	1	yes	characteristic
151	C₂×#123	C₂²×C₁₃₉	1	24	yes	—
152	C₂×C₂²⋊Q₈	C₂²×C₁₃₉	1	12	yes	—
153	C₂×#121	C₂²×C₁₃₉	1	12	yes	—
154	C₂×C₄×D₄	C₂²×C₁₃₉	1	6	yes	—
155	C₂×C₄×D₄	C₂²×C₁₃₉	1	3	yes	—
156	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	12	yes	—
157	C₂²×C₄²	C₂²×C₁₃₉	1	2	yes	—
158	#399	C₂²×C₁₃₉	1	24	yes	—
159	#390	C₂²×C₁₃₉	1	6	yes	—
160	#390	C₂²×C₁₃₉	1	12	yes	—
161	#384	C₂²×C₁₃₉	1	1	yes	characteristic
162	#384	C₂²×C₁₃₉	1	2	yes	—
163	#384	C₂²×C₁₃₉	1	6	yes	—
164	#384	C₂²×C₁₃₉	1	12	yes	—
165	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	3	yes	—
166	C₄×C₂²⋊C₄	C₂²×C₁₃₉	1	6	yes	—
167	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
168	C₂×#41	—	2	48	no	—
169	C₂²×Q₈	C₂³×C₁₃₉	1	4	yes	—
170	C₂²×D₄	C₂³×C₁₃₉	1	6	yes	—
171	C₂²×D₄	C₂³×C₁₃₉	1	8	yes	—
172	C₂²×D₄	C₂³×C₁₃₉	1	1	yes	characteristic
173	C₂³×C₄	C₂³×C₁₃₉	1	8	yes	—
174	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
175	C₂³×C₄	C₂³×C₁₃₉	1	6	yes	—
176	C₄⋊Q₈	—	2	24	no	—
177	#127	—	2	4	no	—
178	#123	—	2	96	no	—
179	C₂²⋊Q₈	—	2	48	no	—
180	#121	—	2	48	no	—
181	C₄×D₄	—	2	24	no	—
182	C₄×D₄	D₄×C₁₃₉	1	24	yes	complemented
183	C₂×C₄⋊C₄	—	2	48	no	—
184	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	12	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	24	yes	—
187	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	12	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	3	yes	—
189	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	12	yes	—
190	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	24	yes	—
191	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	6	yes	—
192	C₂×C₄²	—	2	8	no	—
193	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
194	C₂×C₄²	C₂³×C₁₃₉	1	6	yes	—
195	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
196	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
197	#95	C₂³×C₁₃₉	1	6	yes	—
198	C₂⁴	C₂⁴×C₁₃₉	1	4	yes	—
199	C₂⁴	C₂⁴×C₁₃₉	1	2	yes	—
200	#41	—	4	64	no	—
201	C₂×Q₈	—	2	24	no	—
202	C₂×D₄	—	2	48	no	—
203	C₂×D₄	—	2	12	no	—
204	C₂×D₄	—	2	12	no	—
205	C₂×D₄	—	2	12	no	—
206	C₂×D₄	C₂×D₄×C₁₃₉	1	12	yes	—
207	C₂²×C₄	—	2	12	no	—
208	C₂²×C₄	—	2	12	no	—
209	C₂²×C₄	—	2	48	no	—
210	C₂²×C₄	—	2	24	no	—
211	C₂²×C₄	C₂×D₄×C₁₃₉	1	24	yes	—
212	C₂²×C₄	C₂⁴×C₁₃₉	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₁₃₉	1	8	yes	—
214	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
215	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
216	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
218	C₄⋊C₄	—	2	12	no	—
219	C₄⋊C₄	—	2	24	no	—
220	C₄⋊C₄	—	2	48	no	—
221	C₄⋊C₄	—	2	24	no	—
222	C₄⋊C₄	C₂×D₄×C₁₃₉	1	12	yes	—
223	C₂²⋊C₄	—	2	24	no	—
224	C₂²⋊C₄	—	2	48	no	—
225	C₂²⋊C₄	C₂×D₄×C₁₃₉	1	24	yes	—
226	C₄²	—	2	12	no	—
227	C₄²	—	2	6	no	—
228	C₄²	C₂×D₄×C₁₃₉	1	12	yes	—
229	C₄²	#41×C₁₃₉	1	4	yes	complemented
230	C₂³	—	2	2	no	—
231	C₂³	—	2	6	no	—
232	C₂³	—	2	12	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	12	no	—
235	C₂³	C₂²×D₄×C₁₃₉	1	12	yes	—
236	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
237	Q₈	—	4	16	no	—
238	D₄	—	4	32	no	—
239	D₄	—	4	24	no	—
240	D₄	#127×C₁₃₉	1	16	yes	complemented
241	C₂×C₄	—	2	6	no	—
242	C₂×C₄	—	2	6	no	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	48	no	—
245	C₂×C₄	—	2	12	no	—
246	C₂×C₄	—	2	24	no	—
247	C₂×C₄	—	4	32	no	—
248	C₂×C₄	—	4	16	no	—
249	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
250	C₂×C₄	C₂²×D₄×C₁₃₉	1	12	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
252	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
253	C₂×C₄	C₂×#41×C₁₃₉	1	6	yes	—
254	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	6	yes	—
255	C₂²	—	2	12	no	—
256	C₂²	—	2	12	no	—
257	C₂²	—	4	4	no	—
258	C₂²	—	4	12	no	—
259	C₂²	—	4	12	no	—
260	C₂²	C₂×#127×C₁₃₉	1	8	yes	—
261	C₂²	C₂³×D₄×C₁₃₉	1	3	yes	—
262	C₂²	C₂²×#41×C₁₃₉	1	1	yes	characteristic
263	C₂²	C₂×2₋¹⁺⁴×C₁₃₉	1	3	yes	—
264	C₄	—	2	12	no	—
265	C₄	—	2	6	no	—
266	C₄	—	4	4	no	—
267	C₄	—	4	8	no	—
268	C₄	C₂×#127×C₁₃₉	1	4	yes	—
269	C₄	#547×C₁₃₉	1	12	yes	complemented
270	C₂	—	2	8	no	—
271	C₂	—	8	4	no	—
272	C₂	C₂²×#127×C₁₃₉	1	1	yes	characteristic
273	C₂	C₂×#547×C₁₃₉	1	3	yes	—
274	C₂	#3499×C₁₃₉	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#547×C₁₃₉	C₂	semidirect
2. #2868×C₁₃₉	C₂	semidirect
3. #2856×C₁₃₉	C₂	semidirect
4. #2542×C₁₃₉	C₂	semidirect
5. #2501×C₁₃₉	C₂	semidirect
6. C₄×#127×C₁₃₉	C₂	semidirect
7. C₂×#384×C₁₃₉	C₂	semidirect
8. C₄²×D₄×C₁₃₉	C₂	semidirect
44. C₄×D₄×C₁₃₉	D₄	semidirect
91. C₄²×C₁₃₉	#41	semidirect
102. D₄×C₁₃₉	#127	semidirect
131. C₄×C₁₃₉	#547	semidirect
137. #36179	C₁₃₉	direct (N × H)
138. C₁₃₉	#36179	direct (N × H)
139. C₂×#547	C₂×C₁₃₉	semidirect
140. #2868	C₂×C₁₃₉	semidirect
141. #2856	C₂×C₁₃₉	semidirect
142. #2542	C₂×C₁₃₉	semidirect
143. #2501	C₂×C₁₃₉	semidirect
144. C₄×#127	C₂×C₁₃₉	semidirect
145. C₂×#384	C₂×C₁₃₉	semidirect
146. C₄²×D₄	C₂×C₁₃₉	semidirect
182. C₄×D₄	D₄×C₁₃₉	semidirect
229. C₄²	#41×C₁₃₉	semidirect
240. D₄	#127×C₁₃₉	semidirect
269. C₄	#547×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#547×C₁₃₉	2
#2868×C₁₃₉	2
#2856×C₁₃₉	2
#2542×C₁₃₉	2
#2501×C₁₃₉	2
C₄×#127×C₁₃₉	2
C₂×#384×C₁₃₉	2
C₄²×D₄×C₁₃₉	2
#36179	139

Subgroups — C139×#36179 #48030775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#36179 #48030775