Subgroups — C₄×#339×C₁₃₉ #48005296

Order 35584.

1214 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#339×C₁₃₉	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₁₃₉	C₂²	1	2	yes	—
6	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃₉	—	2	2	no	—
8	C₂²×C₄²×C₁₃₉	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₄×M₁₆×C₁₃₉	—	2	2	no	—
11	C₄×M₁₆×C₁₃₉	C₄	1	4	yes	complemented
12	C₂×#97×C₁₃₉	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃₉	—	2	1	no	—
14	C₄×C₄⋊C₄×C₁₃₉	C₄	1	4	yes	complemented
15	C₄×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
16	#376×C₁₃₉	—	2	1	no	—
17	C₄³×C₁₃₉	—	2	2	no	—
18	C₄³×C₁₃₉	—	2	1	no	—
19	C₄³×C₁₃₉	C₄	1	4	yes	complemented
20	#339×C₁₃₉	C₄	1	16	yes	direct product factor
21	C₂³×C₄×C₁₃₉	—	2	2	no	—
22	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
23	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₁₃₉	—	2	2	no	—
25	C₂×M₁₆×C₁₃₉	—	2	4	no	—
26	C₂×M₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₁₃₉	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₃₉	—	2	2	no	—
29	#117×C₁₃₉	—	2	4	no	—
30	#117×C₁₃₉	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
33	C₂×C₄²×C₁₃₉	—	2	16	no	—
34	C₂×C₄²×C₁₃₉	—	2	8	no	—
35	C₂×C₄²×C₁₃₉	—	2	2	no	—
36	C₂×C₄²×C₁₃₉	—	2	2	no	—
37	C₂×C₄²×C₁₃₉	—	2	4	no	—
38	C₂×C₄²×C₁₃₉	—	2	4	no	—
39	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₉	Q₈	1	2	yes	—
48	#97×C₁₃₉	—	2	4	no	—
49	C₄×C₈×C₁₃₉	—	2	4	no	—
50	#95×C₁₃₉	—	2	4	no	—
51	C₂⁴×C₁₃₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₉	—	2	16	no	—
53	C₂²×C₄×C₁₃₉	—	2	4	no	—
54	C₂²×C₄×C₁₃₉	—	2	8	no	—
55	C₂²×C₄×C₁₃₉	—	2	8	no	—
56	C₂²×C₄×C₁₃₉	—	2	4	no	—
57	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
69	M₁₆×C₁₃₉	—	2	4	no	—
70	M₁₆×C₁₃₉	—	2	8	no	—
71	M₁₆×C₁₃₉	C₄²	1	8	yes	complemented
72	C₂×C₈×C₁₃₉	—	2	4	no	—
73	C₂×C₈×C₁₃₉	—	2	8	no	—
74	C₄⋊C₄×C₁₃₉	—	2	4	no	—
75	C₄⋊C₄×C₁₃₉	C₄²	1	8	yes	complemented
76	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
77	C₄²×C₁₃₉	—	2	32	no	—
78	C₄²×C₁₃₉	—	2	4	no	—
79	C₄²×C₁₃₉	—	2	8	no	—
80	C₄²×C₁₃₉	—	2	8	no	—
81	C₄²×C₁₃₉	—	4	8	no	—
82	C₄²×C₁₃₉	C₄²	1	4	yes	—
83	C₄²×C₁₃₉	C₄²	1	8	yes	complemented
84	C₄²×C₁₃₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₁₃₉	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₁₃₉	C₄⋊C₄	1	2	yes	—
87	C₄²×C₁₃₉	C₄⋊C₄	1	4	yes	—
88	C₄²×C₁₃₉	C₄⋊C₄	1	2	yes	—
89	C₂³×C₁₃₉	—	2	4	no	—
90	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₃₉	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₁₃₉	—	2	16	no	—
95	C₂×C₄×C₁₃₉	—	2	32	no	—
96	C₂×C₄×C₁₃₉	—	2	8	no	—
97	C₂×C₄×C₁₃₉	—	2	4	no	—
98	C₂×C₄×C₁₃₉	—	2	8	no	—
99	C₂×C₄×C₁₃₉	—	2	8	no	—
100	C₂×C₄×C₁₃₉	—	2	8	no	—
101	C₂×C₄×C₁₃₉	—	4	8	no	—
102	C₂×C₄×C₁₃₉	#95	1	4	yes	—
103	C₂×C₄×C₁₃₉	#95	1	4	yes	—
104	C₂×C₄×C₁₃₉	#104	1	8	yes	—
105	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	#117	1	2	yes	—
113	C₂×C₄×C₁₃₉	#117	1	2	yes	—
114	C₂×C₄×C₁₃₉	#117	1	2	yes	—
115	C₂×C₄×C₁₃₉	C₄×D₄	1	4	yes	—
116	C₂×C₄×C₁₃₉	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₁₃₉	C₄×Q₈	1	4	yes	—
118	C₈×C₁₃₉	—	2	8	no	—
119	C₂²×C₁₃₉	—	2	2	no	—
120	C₂²×C₁₃₉	—	2	2	no	—
121	C₂²×C₁₃₉	—	2	8	no	—
122	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
123	C₂²×C₁₃₉	C₄×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₁₃₉	C₄×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₁₃₉	C₄×C₄⋊C₄	1	2	yes	—
126	C₂²×C₁₃₉	C₂×#104	1	4	yes	—
127	C₄×C₁₃₉	—	2	16	no	—
128	C₄×C₁₃₉	—	2	4	no	—
129	C₄×C₁₃₉	—	4	8	no	—
130	C₄×C₁₃₉	#339	1	8	yes	direct product factor
131	C₄×C₁₃₉	C₄³	1	1	yes	characteristic
132	C₄×C₁₃₉	#376	1	1	yes	characteristic
133	C₄×C₁₃₉	C₄×C₂²⋊C₄	1	2	yes	—
134	C₄×C₁₃₉	C₄×C₄⋊C₄	1	2	yes	—
135	C₄×C₁₃₉	C₄×C₄⋊C₄	1	2	yes	—
136	C₂×C₁₃₉	—	2	2	no	—
137	C₂×C₁₃₉	—	2	2	no	—
138	C₂×C₁₃₉	C₄×#95	1	1	yes	characteristic
139	C₂×C₁₃₉	C₂×#339	1	2	yes	—
140	C₂×C₁₃₉	C₄×#104	1	4	yes	—
141	C₄×#339	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₉	C₄×#339	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×C₄×M₁₆	C₂×C₁₃₉	1	1	yes	characteristic
144	C₄×#117	C₂×C₁₃₉	1	1	yes	characteristic
145	C₂×C₄³	C₂×C₁₃₉	1	1	yes	characteristic
146	C₂×#339	C₂×C₁₃₉	1	4	yes	—
147	C₂²×M₁₆	C₂²×C₁₃₉	1	2	yes	—
148	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
149	C₂²×C₄²	—	2	2	no	—
150	C₂²×C₄²	C₂²×C₁₃₉	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
152	C₄×M₁₆	—	2	2	no	—
153	C₄×M₁₆	C₄×C₁₃₉	1	4	yes	complemented
154	C₂×#97	—	2	1	no	—
155	C₂×C₄×C₈	—	2	1	no	—
156	C₄×C₄⋊C₄	C₄×C₁₃₉	1	4	yes	complemented
157	C₄×C₂²⋊C₄	—	2	2	no	—
158	#376	—	2	1	no	—
159	C₄³	—	2	2	no	—
160	C₄³	—	2	1	no	—
161	C₄³	C₄×C₁₃₉	1	4	yes	complemented
162	#339	C₄×C₁₃₉	1	16	yes	direct product factor
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	C₂×C₄×C₁₃₉	1	8	yes	—
169	C₂×M₁₆	C₂×C₄×C₁₃₉	1	4	yes	—
170	C₂²×C₈	—	2	2	no	—
171	#117	—	2	4	no	—
172	#117	C₂×C₄×C₁₃₉	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	16	no	—
176	C₂×C₄²	—	2	8	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
182	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
183	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
185	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
186	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
188	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
189	C₂×C₄²	Q₈×C₁₃₉	1	2	yes	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	—	2	4	no	—
193	C₂⁴	C₂×D₄×C₁₃₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
202	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
210	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	8	no	—
213	M₁₆	C₄²×C₁₃₉	1	8	yes	complemented
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₄²×C₁₃₉	1	8	yes	complemented
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	32	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	4	8	no	—
224	C₄²	C₄²×C₁₃₉	1	4	yes	—
225	C₄²	C₄²×C₁₃₉	1	8	yes	complemented
226	C₄²	C₂²⋊C₄×C₁₃₉	1	2	yes	—
227	C₄²	C₂²⋊C₄×C₁₃₉	1	2	yes	—
228	C₄²	C₄⋊C₄×C₁₃₉	1	2	yes	—
229	C₄²	C₄⋊C₄×C₁₃₉	1	4	yes	—
230	C₄²	C₄⋊C₄×C₁₃₉	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
233	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
234	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
235	C₂³	C₄×D₄×C₁₃₉	1	4	yes	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	#95×C₁₃₉	1	4	yes	—
245	C₂×C₄	#95×C₁₃₉	1	4	yes	—
246	C₂×C₄	#104×C₁₃₉	1	8	yes	—
247	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄²×C₁₃₉	1	2	yes	—
249	C₂×C₄	C₂×C₄²×C₁₃₉	1	2	yes	—
250	C₂×C₄	C₂×C₄²×C₁₃₉	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
254	C₂×C₄	#117×C₁₃₉	1	2	yes	—
255	C₂×C₄	#117×C₁₃₉	1	2	yes	—
256	C₂×C₄	#117×C₁₃₉	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₁₃₉	1	4	yes	—
258	C₂×C₄	C₄×D₄×C₁₃₉	1	4	yes	—
259	C₂×C₄	C₄×Q₈×C₁₃₉	1	4	yes	—
260	C₈	—	2	8	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
265	C₂²	C₄×C₂²⋊C₄×C₁₃₉	1	2	yes	—
266	C₂²	C₄×C₂²⋊C₄×C₁₃₉	1	2	yes	—
267	C₂²	C₄×C₄⋊C₄×C₁₃₉	1	2	yes	—
268	C₂²	C₂×#104×C₁₃₉	1	4	yes	—
269	C₄	—	2	16	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	#339×C₁₃₉	1	8	yes	direct product factor
273	C₄	C₄³×C₁₃₉	1	1	yes	characteristic
274	C₄	#376×C₁₃₉	1	1	yes	characteristic
275	C₄	C₄×C₂²⋊C₄×C₁₃₉	1	2	yes	—
276	C₄	C₄×C₄⋊C₄×C₁₃₉	1	2	yes	—
277	C₄	C₄×C₄⋊C₄×C₁₃₉	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	C₄×#95×C₁₃₉	1	1	yes	characteristic
281	C₂	C₂×#339×C₁₃₉	1	2	yes	—
282	C₂	C₄×#104×C₁₃₉	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₉	D₄
commutator	C₄	C₄³×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
socle	C₂³×C₁₃₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₃₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₉ ◁ C₂²×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×M₁₆×C₁₃₉	C₄	semidirect
14. C₄×C₄⋊C₄×C₁₃₉	C₄	semidirect
19. C₄³×C₁₃₉	C₄	semidirect
20. #339×C₁₃₉	C₄	direct (N × H)
71. M₁₆×C₁₃₉	C₄²	semidirect
75. C₄⋊C₄×C₁₃₉	C₄²	semidirect
83. C₄²×C₁₃₉	C₄²	semidirect
130. C₄×C₁₃₉	#339	direct (N × H)
141. C₄×#339	C₁₃₉	direct (N × H)
142. C₁₃₉	C₄×#339	direct (N × H)
153. C₄×M₁₆	C₄×C₁₃₉	semidirect
156. C₄×C₄⋊C₄	C₄×C₁₃₉	semidirect
161. C₄³	C₄×C₁₃₉	semidirect
162. #339	C₄×C₁₃₉	direct (N × H)
213. M₁₆	C₄²×C₁₃₉	semidirect
217. C₄⋊C₄	C₄²×C₁₃₉	semidirect
225. C₄²	C₄²×C₁₃₉	semidirect
272. C₄	#339×C₁₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₁₃₉	2
C₄×#117×C₁₃₉	2
C₂×C₄³×C₁₃₉	2
C₂×#339×C₁₃₉	2
C₄×#339	139

Subgroups — C4×#339×C139 #48005296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#339×C₁₃₉ #48005296