Subgroups — C₁₃₉×#36082 #48030678

Order 35584.

5432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#548×C₁₃₉	C₂	1	3	yes	maximal, complemented
3	#2616×C₁₃₉	C₂	1	3	yes	maximal, complemented
4	#2435×C₁₃₉	C₂	1	9	yes	maximal, complemented
5	#2408×C₁₃₉	C₂	1	9	yes	maximal, complemented
6	#2388×C₁₃₉	C₂	1	3	yes	maximal, complemented
7	#2341×C₁₃₉	C₂	1	3	yes	maximal, complemented
8	C₂×H(5,𝔽₂)×C₁₃₉	—	2	24	no	—
9	C₂²×#41×C₁₃₉	C₂²	1	3	yes	—
10	C₂²×#41×C₁₃₉	C₂²	1	3	yes	—
11	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
12	C₂³×D₄×C₁₃₉	C₂²	1	9	yes	—
13	#548×C₁₃₉	—	2	24	no	—
14	C₂×#124×C₁₃₉	C₂²	1	9	yes	complemented
15	C₂×C₂²⋊Q₈×C₁₃₉	C₂²	1	9	yes	complemented
16	C₂×#121×C₁₃₉	C₂²	1	9	yes	complemented
17	C₂×#120×C₁₃₉	C₂²	1	6	yes	complemented
18	C₂×C₄×Q₈×C₁₃₉	C₂²	1	3	yes	complemented
19	C₂×C₄×D₄×C₁₃₉	C₂²	1	9	yes	complemented
20	#395×C₁₃₉	C₂²	1	3	yes	complemented
21	#394×C₁₃₉	C₂²	1	18	yes	complemented
22	#394×C₁₃₉	C₂²	1	9	yes	complemented
23	#393×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
24	#392×C₁₃₉	C₂²	1	9	yes	complemented
25	#390×C₁₃₉	C₂²	1	18	yes	complemented
26	#380×C₁₃₉	C₂²	1	9	yes	complemented
27	#379×C₁₃₉	C₂²	1	18	yes	complemented
28	C₄×C₂²⋊C₄×C₁₃₉	C₂²	1	9	yes	complemented
29	C₂⁵×C₁₃₉	C₂³	1	6	yes	—
30	H(5,𝔽₂)×C₁₃₉	—	2	128	no	—
31	C₂×#41×C₁₃₉	—	2	24	no	—
32	C₂×#41×C₁₃₉	—	2	36	no	—
33	C₂×#41×C₁₃₉	D₄	1	24	yes	complemented
34	C₂²×Q₈×C₁₃₉	—	2	12	no	—
35	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃₉	C₂³	1	3	yes	—
37	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₁₃₉	—	2	72	no	—
39	C₂²×D₄×C₁₃₉	—	2	36	no	—
40	C₂²×D₄×C₁₃₉	C₂³	1	9	yes	—
41	C₂²×D₄×C₁₃₉	C₂³	1	9	yes	—
42	C₂²×D₄×C₁₃₉	C₂³	1	9	yes	—
43	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
44	C₂³×C₄×C₁₃₉	C₂³	1	9	yes	—
45	#124×C₁₃₉	—	2	36	no	—
46	C₂²⋊Q₈×C₁₃₉	—	2	36	no	—
47	#121×C₁₃₉	—	2	36	no	—
48	#120×C₁₃₉	—	4	12	no	—
49	C₄×Q₈×C₁₃₉	—	2	12	no	—
50	C₄×D₄×C₁₃₉	—	2	36	no	—
51	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	9	yes	—
52	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	9	yes	—
53	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	3	yes	—
54	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	9	yes	—
55	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	18	yes	—
56	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	18	yes	—
57	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	18	yes	—
58	C₂×C₄²×C₁₃₉	C₂³	1	9	yes	—
59	#95×C₁₃₉	C₂³	1	9	yes	—
60	#95×C₁₃₉	C₂³	1	3	yes	—
61	C₂⁴×C₁₃₉	—	2	36	no	—
62	C₂⁴×C₁₃₉	—	4	6	no	—
63	C₂⁴×C₁₃₉	—	4	18	no	—
64	C₂⁴×C₁₃₉	C₂⁴	1	9	yes	—
65	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	#41×C₁₃₉	—	2	96	no	—
67	#41×C₁₃₉	—	2	96	no	—
68	C₂×Q₈×C₁₃₉	—	2	6	no	—
69	C₂×Q₈×C₁₃₉	—	2	18	no	—
70	C₂×Q₈×C₁₃₉	—	4	16	no	—
71	C₂×Q₈×C₁₃₉	C₂×D₄	1	12	yes	—
72	C₂×D₄×C₁₃₉	—	2	288	no	—
73	C₂×D₄×C₁₃₉	—	2	36	no	—
74	C₂×D₄×C₁₃₉	—	2	18	no	—
75	C₂×D₄×C₁₃₉	—	2	18	no	—
76	C₂×D₄×C₁₃₉	—	2	18	no	—
77	C₂×D₄×C₁₃₉	—	2	36	no	—
78	C₂×D₄×C₁₃₉	—	4	9	no	—
79	C₂×D₄×C₁₃₉	C₂×D₄	1	36	yes	—
80	C₂²×C₄×C₁₃₉	—	2	36	no	—
81	C₂²×C₄×C₁₃₉	—	2	18	no	—
82	C₂²×C₄×C₁₃₉	C₂×D₄	1	36	yes	—
83	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
84	C₂²×C₄×C₁₃₉	C₂⁴	1	9	yes	—
85	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
86	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
87	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
88	C₄⋊C₄×C₁₃₉	—	2	18	no	—
89	C₄⋊C₄×C₁₃₉	—	2	18	no	—
90	C₂²⋊C₄×C₁₃₉	—	2	18	no	—
91	C₂²⋊C₄×C₁₃₉	—	2	36	no	—
92	C₂²⋊C₄×C₁₃₉	—	2	6	no	—
93	C₂²⋊C₄×C₁₃₉	—	4	18	no	—
94	C₂²⋊C₄×C₁₃₉	—	4	18	no	—
95	C₄²×C₁₃₉	—	2	18	no	—
96	C₂³×C₁₃₉	—	2	48	no	—
97	C₂³×C₁₃₉	—	2	3	no	—
98	C₂³×C₁₃₉	—	2	18	no	—
99	C₂³×C₁₃₉	—	2	9	no	—
100	C₂³×C₁₃₉	—	2	18	no	—
101	C₂³×C₁₃₉	—	2	3	no	—
102	C₂³×C₁₃₉	—	2	9	no	—
103	C₂³×C₁₃₉	—	8	36	no	—
104	C₂³×C₁₃₉	—	8	36	no	—
105	C₂³×C₁₃₉	C₂²×D₄	1	18	yes	—
106	C₂³×C₁₃₉	H(5,𝔽₂)	1	2	yes	complemented
107	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
108	Q₈×C₁₃₉	—	2	8	no	—
109	Q₈×C₁₃₉	—	4	12	no	—
110	Q₈×C₁₃₉	#120	1	16	yes	complemented
111	D₄×C₁₃₉	—	2	72	no	—
112	D₄×C₁₃₉	—	2	72	no	—
113	D₄×C₁₃₉	—	4	36	no	—
114	C₂×C₄×C₁₃₉	—	2	6	no	—
115	C₂×C₄×C₁₃₉	—	2	72	no	—
116	C₂×C₄×C₁₃₉	—	2	3	no	—
117	C₂×C₄×C₁₃₉	—	2	9	no	—
118	C₂×C₄×C₁₃₉	—	2	9	no	—
119	C₂×C₄×C₁₃₉	—	2	9	no	—
120	C₂×C₄×C₁₃₉	—	2	6	no	—
121	C₂×C₄×C₁₃₉	—	2	9	no	—
122	C₂×C₄×C₁₃₉	—	4	12	no	—
123	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₁₃₉	C₂²×D₄	1	18	yes	—
125	C₂²×C₁₃₉	—	2	18	no	—
126	C₂²×C₁₃₉	—	2	6	no	—
127	C₂²×C₁₃₉	—	2	18	no	—
128	C₂²×C₁₃₉	—	2	18	no	—
129	C₂²×C₁₃₉	—	4	1	no	—
130	C₂²×C₁₃₉	—	4	18	no	—
131	C₂²×C₁₃₉	—	4	3	no	—
132	C₂²×C₁₃₉	—	4	18	no	—
133	C₂²×C₁₃₉	—	8	48	no	—
134	C₂²×C₁₃₉	C₂³×D₄	1	3	yes	—
135	C₂²×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₁₃₉	#585	1	3	yes	—
137	C₄×C₁₃₉	—	2	6	no	—
138	C₄×C₁₃₉	—	4	3	no	—
139	C₄×C₁₃₉	—	4	9	no	—
140	C₄×C₁₃₉	—	4	3	no	—
141	C₄×C₁₃₉	C₂×#120	1	12	yes	—
142	C₂×C₁₃₉	—	4	2	no	—
143	C₂×C₁₃₉	—	4	18	no	—
144	C₂×C₁₃₉	C₂²×#120	1	1	yes	characteristic
145	C₂×C₁₃₉	#3488	1	3	yes	—
146	C₂×C₁₃₉	#3500	1	3	yes	—
147	#36082	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₁₃₉	#36082	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×H(5,𝔽₂)	C₂×C₁₃₉	1	1	yes	characteristic, complemented
150	C₂×#548	C₂×C₁₃₉	1	3	yes	complemented
151	#2616	C₂×C₁₃₉	1	3	yes	complemented
152	#2435	C₂×C₁₃₉	1	9	yes	complemented
153	#2408	C₂×C₁₃₉	1	9	yes	complemented
154	#2388	C₂×C₁₃₉	1	3	yes	complemented
155	#2341	C₂×C₁₃₉	1	3	yes	complemented
156	C₂×H(5,𝔽₂)	—	2	24	no	—
157	C₂²×#41	C₂²×C₁₃₉	1	3	yes	—
158	C₂²×#41	C₂²×C₁₃₉	1	3	yes	—
159	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic, complemented
160	C₂³×D₄	C₂²×C₁₃₉	1	9	yes	—
161	#548	—	2	24	no	—
162	C₂×#124	C₂²×C₁₃₉	1	9	yes	complemented
163	C₂×C₂²⋊Q₈	C₂²×C₁₃₉	1	9	yes	complemented
164	C₂×#121	C₂²×C₁₃₉	1	9	yes	complemented
165	C₂×#120	C₂²×C₁₃₉	1	6	yes	complemented
166	C₂×C₄×Q₈	C₂²×C₁₃₉	1	3	yes	complemented
167	C₂×C₄×D₄	C₂²×C₁₃₉	1	9	yes	complemented
168	#395	C₂²×C₁₃₉	1	3	yes	complemented
169	#394	C₂²×C₁₃₉	1	18	yes	complemented
170	#394	C₂²×C₁₃₉	1	9	yes	complemented
171	#393	C₂²×C₁₃₉	1	1	yes	characteristic, complemented
172	#392	C₂²×C₁₃₉	1	9	yes	complemented
173	#390	C₂²×C₁₃₉	1	18	yes	complemented
174	#380	C₂²×C₁₃₉	1	9	yes	complemented
175	#379	C₂²×C₁₃₉	1	18	yes	complemented
176	C₄×C₂²⋊C₄	C₂²×C₁₃₉	1	9	yes	complemented
177	C₂⁵	C₂³×C₁₃₉	1	6	yes	—
178	H(5,𝔽₂)	—	2	128	no	—
179	C₂×#41	—	2	24	no	—
180	C₂×#41	—	2	36	no	—
181	C₂×#41	D₄×C₁₃₉	1	24	yes	complemented
182	C₂²×Q₈	—	2	12	no	—
183	C₂²×Q₈	C₂³×C₁₃₉	1	3	yes	—
184	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
185	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
186	C₂²×D₄	—	2	72	no	—
187	C₂²×D₄	—	2	36	no	—
188	C₂²×D₄	C₂³×C₁₃₉	1	9	yes	—
189	C₂²×D₄	C₂³×C₁₃₉	1	9	yes	—
190	C₂²×D₄	C₂³×C₁₃₉	1	9	yes	—
191	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
192	C₂³×C₄	C₂³×C₁₃₉	1	9	yes	—
193	#124	—	2	36	no	—
194	C₂²⋊Q₈	—	2	36	no	—
195	#121	—	2	36	no	—
196	#120	—	4	12	no	—
197	C₄×Q₈	—	2	12	no	—
198	C₄×D₄	—	2	36	no	—
199	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	9	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	9	yes	—
201	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	3	yes	—
202	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	9	yes	—
203	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	18	yes	—
204	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	18	yes	—
205	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	18	yes	—
206	C₂×C₄²	C₂³×C₁₃₉	1	9	yes	—
207	#95	C₂³×C₁₃₉	1	3	yes	—
208	#95	C₂³×C₁₃₉	1	9	yes	—
209	C₂⁴	—	2	36	no	—
210	C₂⁴	—	4	6	no	—
211	C₂⁴	—	4	18	no	—
212	C₂⁴	C₂⁴×C₁₃₉	1	9	yes	—
213	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
214	#41	—	2	96	no	—
215	#41	—	2	96	no	—
216	C₂×Q₈	—	2	6	no	—
217	C₂×Q₈	—	2	18	no	—
218	C₂×Q₈	—	4	16	no	—
219	C₂×Q₈	C₂×D₄×C₁₃₉	1	12	yes	—
220	C₂×D₄	—	2	288	no	—
221	C₂×D₄	—	2	36	no	—
222	C₂×D₄	—	2	18	no	—
223	C₂×D₄	—	2	18	no	—
224	C₂×D₄	—	2	18	no	—
225	C₂×D₄	—	2	36	no	—
226	C₂×D₄	—	4	9	no	—
227	C₂×D₄	C₂×D₄×C₁₃₉	1	36	yes	—
228	C₂²×C₄	—	2	36	no	—
229	C₂²×C₄	—	2	18	no	—
230	C₂²×C₄	C₂×D₄×C₁₃₉	1	36	yes	—
231	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
232	C₂²×C₄	C₂⁴×C₁₃₉	1	9	yes	—
233	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
234	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
235	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
236	C₄⋊C₄	—	2	18	no	—
237	C₄⋊C₄	—	2	18	no	—
238	C₂²⋊C₄	—	2	18	no	—
239	C₂²⋊C₄	—	2	36	no	—
240	C₂²⋊C₄	—	2	6	no	—
241	C₂²⋊C₄	—	4	18	no	—
242	C₂²⋊C₄	—	4	18	no	—
243	C₄²	—	2	18	no	—
244	C₂³	—	2	48	no	—
245	C₂³	—	2	3	no	—
246	C₂³	—	2	18	no	—
247	C₂³	—	2	9	no	—
248	C₂³	—	2	18	no	—
249	C₂³	—	2	3	no	—
250	C₂³	—	2	9	no	—
251	C₂³	—	8	36	no	—
252	C₂³	—	8	36	no	—
253	C₂³	C₂²×D₄×C₁₃₉	1	18	yes	—
254	C₂³	H(5,𝔽₂)×C₁₃₉	1	2	yes	complemented
255	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
256	Q₈	—	2	8	no	—
257	Q₈	—	4	12	no	—
258	Q₈	#120×C₁₃₉	1	16	yes	complemented
259	D₄	—	2	72	no	—
260	D₄	—	2	72	no	—
261	D₄	—	4	36	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	72	no	—
264	C₂×C₄	—	2	3	no	—
265	C₂×C₄	—	2	9	no	—
266	C₂×C₄	—	2	9	no	—
267	C₂×C₄	—	2	9	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	9	no	—
270	C₂×C₄	—	4	12	no	—
271	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃₉	1	18	yes	—
273	C₂²	—	2	18	no	—
274	C₂²	—	2	6	no	—
275	C₂²	—	2	18	no	—
276	C₂²	—	2	18	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	18	no	—
279	C₂²	—	4	18	no	—
280	C₂²	—	4	3	no	—
281	C₂²	—	8	48	no	—
282	C₂²	C₂³×D₄×C₁₃₉	1	3	yes	—
283	C₂²	C₂×H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
284	C₂²	#585×C₁₃₉	1	3	yes	—
285	C₄	—	2	6	no	—
286	C₄	—	4	3	no	—
287	C₄	—	4	9	no	—
288	C₄	—	4	3	no	—
289	C₄	C₂×#120×C₁₃₉	1	12	yes	—
290	C₂	—	4	18	no	—
291	C₂	—	4	2	no	—
292	C₂	C₂²×#120×C₁₃₉	1	1	yes	characteristic
293	C₂	#3488×C₁₃₉	1	3	yes	—
294	C₂	#3500×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. C₂²×H(5,𝔽₂)×C₁₃₉	C₂	semidirect
2. C₂×#548×C₁₃₉	C₂	semidirect
3. #2616×C₁₃₉	C₂	semidirect
4. #2435×C₁₃₉	C₂	semidirect
5. #2408×C₁₃₉	C₂	semidirect
6. #2388×C₁₃₉	C₂	semidirect
7. #2341×C₁₃₉	C₂	semidirect
11. C₂³×Q₈×C₁₃₉	C₂²	semidirect
14. C₂×#124×C₁₃₉	C₂²	semidirect
15. C₂×C₂²⋊Q₈×C₁₃₉	C₂²	semidirect
16. C₂×#121×C₁₃₉	C₂²	semidirect
17. C₂×#120×C₁₃₉	C₂²	semidirect
18. C₂×C₄×Q₈×C₁₃₉	C₂²	semidirect
19. C₂×C₄×D₄×C₁₃₉	C₂²	semidirect
20. #395×C₁₃₉	C₂²	semidirect
21. #394×C₁₃₉	C₂²	semidirect
22. #394×C₁₃₉	C₂²	semidirect
23. #393×C₁₃₉	C₂²	semidirect
24. #392×C₁₃₉	C₂²	semidirect
25. #390×C₁₃₉	C₂²	semidirect
26. #380×C₁₃₉	C₂²	semidirect
27. #379×C₁₃₉	C₂²	semidirect
28. C₄×C₂²⋊C₄×C₁₃₉	C₂²	semidirect
33. C₂×#41×C₁₃₉	D₄	semidirect
106. C₂³×C₁₃₉	H(5,𝔽₂)	semidirect
110. Q₈×C₁₃₉	#120	semidirect
147. #36082	C₁₃₉	direct (N × H)
148. C₁₃₉	#36082	direct (N × H)
149. C₂²×H(5,𝔽₂)	C₂×C₁₃₉	semidirect
150. C₂×#548	C₂×C₁₃₉	semidirect
151. #2616	C₂×C₁₃₉	semidirect
152. #2435	C₂×C₁₃₉	semidirect
153. #2408	C₂×C₁₃₉	semidirect
154. #2388	C₂×C₁₃₉	semidirect
155. #2341	C₂×C₁₃₉	semidirect
159. C₂³×Q₈	C₂²×C₁₃₉	semidirect
162. C₂×#124	C₂²×C₁₃₉	semidirect
163. C₂×C₂²⋊Q₈	C₂²×C₁₃₉	semidirect
164. C₂×#121	C₂²×C₁₃₉	semidirect
165. C₂×#120	C₂²×C₁₃₉	semidirect
166. C₂×C₄×Q₈	C₂²×C₁₃₉	semidirect
167. C₂×C₄×D₄	C₂²×C₁₃₉	semidirect
168. #395	C₂²×C₁₃₉	semidirect
169. #394	C₂²×C₁₃₉	semidirect
170. #394	C₂²×C₁₃₉	semidirect
171. #393	C₂²×C₁₃₉	semidirect
172. #392	C₂²×C₁₃₉	semidirect
173. #390	C₂²×C₁₃₉	semidirect
174. #380	C₂²×C₁₃₉	semidirect
175. #379	C₂²×C₁₃₉	semidirect
176. C₄×C₂²⋊C₄	C₂²×C₁₃₉	semidirect
181. C₂×#41	D₄×C₁₃₉	semidirect
254. C₂³	H(5,𝔽₂)×C₁₃₉	semidirect
258. Q₈	#120×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×H(5,𝔽₂)×C₁₃₉	2
C₂×#548×C₁₃₉	2
#2616×C₁₃₉	2
#2435×C₁₃₉	2
#2408×C₁₃₉	2
#2388×C₁₃₉	2
#2341×C₁₃₉	2
#36082	139

Subgroups — C139×#36082 #48030678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#36082 #48030678