Subgroups — Q₈×#121×C₁₃₉ #48031137

Order 35584.

3016 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#2857×C₁₃₉	C₂	1	3	yes	maximal, complemented
5	#2546×C₁₃₉	C₂	1	3	yes	maximal, complemented
6	#2538×C₁₃₉	C₂	1	3	yes	maximal, complemented
7	#2429×C₁₃₉	C₂	1	6	yes	maximal, complemented
8	#2417×C₁₃₉	C₂	1	6	yes	maximal, complemented
9	Q₈×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	Q₈×C₂²⋊C₄×C₁₃₉	C₂	1	2	yes	maximal, complemented
11	C₄×#121×C₁₃₉	C₂	1	3	yes	maximal
12	C₂³×Q₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	D₄×Q₈×C₁₃₉	—	2	16	no	—
15	D₄×Q₈×C₁₃₉	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	3	yes	—
17	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	6	yes	—
18	C₂×C₂²⋊Q₈×C₁₃₉	C₂²	1	6	yes	—
19	C₂×C₂²⋊Q₈×C₁₃₉	C₂²	1	6	yes	—
20	C₂×C₂²⋊Q₈×C₁₃₉	C₂²	1	6	yes	—
21	C₂×#121×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₁₃₉	—	2	8	no	—
23	C₂×C₄×Q₈×C₁₃₉	C₂²	1	3	yes	—
24	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₁₃₉	C₂²	1	3	yes	—
27	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
29	C₂×C₄×D₄×C₁₃₉	C₂²	1	6	yes	—
30	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
31	C₂²×C₄²×C₁₃₉	C₂²	1	3	yes	—
32	#398×C₁₃₉	C₂²	1	12	yes	—
33	#396×C₁₃₉	C₂²	1	12	yes	—
34	#391×C₁₃₉	C₂²	1	6	yes	—
35	#391×C₁₃₉	C₂²	1	6	yes	—
36	#390×C₁₃₉	C₂²	1	3	yes	—
37	#389×C₁₃₉	C₂²	1	6	yes	—
38	#389×C₁₃₉	C₂²	1	3	yes	—
39	#389×C₁₃₉	C₂²	1	6	yes	—
40	#389×C₁₃₉	C₂²	1	6	yes	—
41	#388×C₁₃₉	C₂²	1	6	yes	—
42	#386×C₁₃₉	C₂²	1	6	yes	—
43	#384×C₁₃₉	C₂²	1	3	yes	—
44	#380×C₁₃₉	C₂²	1	6	yes	—
45	#380×C₁₃₉	C₂²	1	3	yes	—
46	#380×C₁₃₉	C₂²	1	6	yes	—
47	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
48	C₄×C₂²⋊C₄×C₁₃₉	C₂²	1	6	yes	—
49	C₂²×Q₈×C₁₃₉	—	2	4	no	—
50	C₂²×Q₈×C₁₃₉	—	2	4	no	—
51	C₂²×Q₈×C₁₃₉	—	2	8	no	—
52	C₂²×Q₈×C₁₃₉	—	2	8	no	—
53	C₂²×Q₈×C₁₃₉	—	2	8	no	—
54	C₂²×Q₈×C₁₃₉	D₄	1	8	yes	complemented
55	C₂²×Q₈×C₁₃₉	C₂³	1	6	yes	—
56	C₂²×Q₈×C₁₃₉	C₂³	1	3	yes	—
57	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₁₃₉	C₂³	1	2	yes	—
59	C₂²×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₁₃₉	C₂³	1	6	yes	—
61	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
62	C₂³×C₄×C₁₃₉	C₂³	1	3	yes	—
63	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₁₃₉	—	2	24	no	—
65	C₄⋊Q₈×C₁₃₉	—	2	12	no	—
66	C₂²⋊Q₈×C₁₃₉	—	2	24	no	—
67	C₂²⋊Q₈×C₁₃₉	—	2	24	no	—
68	C₂²⋊Q₈×C₁₃₉	—	2	24	no	—
69	#121×C₁₃₉	Q₈	1	8	yes	direct product factor
70	C₄×Q₈×C₁₃₉	—	2	4	no	—
71	C₄×Q₈×C₁₃₉	—	2	8	no	—
72	C₄×Q₈×C₁₃₉	—	2	12	no	—
73	C₄×Q₈×C₁₃₉	—	2	12	no	—
74	C₄×Q₈×C₁₃₉	D₄	1	8	yes	complemented
75	C₄×D₄×C₁₃₉	—	2	24	no	—
76	C₄×D₄×C₁₃₉	—	2	12	no	—
77	C₂×C₄⋊C₄×C₁₃₉	—	2	12	no	—
78	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
79	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	12	yes	—
80	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
81	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
82	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
83	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
84	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
85	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
86	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
87	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
88	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
90	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
91	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
92	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
93	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	2	yes	—
94	C₂×C₄²×C₁₃₉	—	2	12	no	—
95	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
96	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
97	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
98	C₂×C₄²×C₁₃₉	C₂³	1	6	yes	—
99	C₂×C₄²×C₁₃₉	C₂³	1	3	yes	—
100	#95×C₁₃₉	C₂³	1	12	yes	—
101	#95×C₁₃₉	C₂³	1	6	yes	—
102	#95×C₁₃₉	C₂³	1	6	yes	—
103	#95×C₁₃₉	C₂³	1	6	yes	—
104	C₂⁴×C₁₃₉	C₂⁴	1	2	yes	—
105	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
106	C₂×Q₈×C₁₃₉	—	2	32	no	—
107	C₂×Q₈×C₁₃₉	—	2	6	no	—
108	C₂×Q₈×C₁₃₉	—	2	12	no	—
109	C₂×Q₈×C₁₃₉	—	2	12	no	—
110	C₂×Q₈×C₁₃₉	—	2	6	no	—
111	C₂×Q₈×C₁₃₉	—	2	12	no	—
112	C₂×Q₈×C₁₃₉	—	4	32	no	—
113	C₂×Q₈×C₁₃₉	C₂×D₄	1	12	yes	—
114	C₂×Q₈×C₁₃₉	C₂×D₄	1	4	yes	—
115	C₂×Q₈×C₁₃₉	C₂×D₄	1	4	yes	—
116	C₂×Q₈×C₁₃₉	#41	1	4	yes	—
117	C₂×D₄×C₁₃₉	—	2	4	no	—
118	C₂×D₄×C₁₃₉	—	2	4	no	—
119	C₂×D₄×C₁₃₉	—	2	2	no	—
120	C₂×D₄×C₁₃₉	—	2	2	no	—
121	C₂×D₄×C₁₃₉	C₂×Q₈	1	8	yes	complemented
122	C₂×D₄×C₁₃₉	C₂×Q₈	1	4	yes	complemented
123	C₂²×C₄×C₁₃₉	—	2	2	no	—
124	C₂²×C₄×C₁₃₉	—	2	6	no	—
125	C₂²×C₄×C₁₃₉	—	2	6	no	—
126	C₂²×C₄×C₁₃₉	—	2	2	no	—
127	C₂²×C₄×C₁₃₉	—	2	6	no	—
128	C₂²×C₄×C₁₃₉	—	2	6	no	—
129	C₂²×C₄×C₁₃₉	—	2	12	no	—
130	C₂²×C₄×C₁₃₉	—	2	12	no	—
131	C₂²×C₄×C₁₃₉	—	2	12	no	—
132	C₂²×C₄×C₁₃₉	—	4	3	no	—
133	C₂²×C₄×C₁₃₉	C₂×D₄	1	12	yes	—
134	C₂²×C₄×C₁₃₉	C₂×Q₈	1	4	yes	complemented
135	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
136	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
137	C₂²×C₄×C₁₃₉	C₂⁴	1	6	yes	—
138	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
139	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
140	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
141	C₂²×C₄×C₁₃₉	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₁₃₉	C₂⁴	1	3	yes	—
143	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₁₃₉	—	2	12	no	—
145	C₄⋊C₄×C₁₃₉	—	2	6	no	—
146	C₄⋊C₄×C₁₃₉	—	2	12	no	—
147	C₄⋊C₄×C₁₃₉	—	2	6	no	—
148	C₄⋊C₄×C₁₃₉	—	2	12	no	—
149	C₄⋊C₄×C₁₃₉	—	2	24	no	—
150	C₄⋊C₄×C₁₃₉	—	2	12	no	—
151	C₄⋊C₄×C₁₃₉	—	2	6	no	—
152	C₄⋊C₄×C₁₃₉	—	4	6	no	—
153	C₄⋊C₄×C₁₃₉	C₂×D₄	1	12	yes	—
154	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	4	yes	complemented
155	C₂²⋊C₄×C₁₃₉	—	2	12	no	—
156	C₂²⋊C₄×C₁₃₉	—	2	12	no	—
157	C₂²⋊C₄×C₁₃₉	—	2	12	no	—
158	C₂²⋊C₄×C₁₃₉	—	4	6	no	—
159	C₂²⋊C₄×C₁₃₉	C₂×Q₈	1	8	yes	complemented
160	C₄²×C₁₃₉	—	2	6	no	—
161	C₄²×C₁₃₉	—	2	6	no	—
162	C₄²×C₁₃₉	—	2	6	no	—
163	C₄²×C₁₃₉	—	2	12	no	—
164	C₄²×C₁₃₉	C₂×D₄	1	12	yes	—
165	C₂³×C₁₃₉	—	2	1	no	—
166	C₂³×C₁₃₉	—	2	1	no	—
167	C₂³×C₁₃₉	—	2	1	no	—
168	C₂³×C₁₃₉	—	2	1	no	—
169	C₂³×C₁₃₉	—	2	2	no	—
170	C₂³×C₁₃₉	—	2	2	no	—
171	C₂³×C₁₃₉	—	2	2	no	—
172	C₂³×C₁₃₉	—	2	2	no	—
173	C₂³×C₁₃₉	—	2	2	no	—
174	C₂³×C₁₃₉	—	2	2	no	—
175	C₂³×C₁₃₉	C₂²×D₄	1	2	yes	—
176	C₂³×C₁₃₉	C₂²×Q₈	1	4	yes	—
177	C₂³×C₁₃₉	C₂²×Q₈	1	2	yes	—
178	C₂³×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
179	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
180	Q₈×C₁₃₉	—	2	24	no	—
181	Q₈×C₁₃₉	—	4	24	no	—
182	Q₈×C₁₃₉	#121	1	16	yes	direct product factor
183	D₄×C₁₃₉	—	2	16	no	—
184	D₄×C₁₃₉	—	2	8	no	—
185	C₂×C₄×C₁₃₉	—	2	1	no	—
186	C₂×C₄×C₁₃₉	—	2	2	no	—
187	C₂×C₄×C₁₃₉	—	2	3	no	—
188	C₂×C₄×C₁₃₉	—	2	3	no	—
189	C₂×C₄×C₁₃₉	—	2	1	no	—
190	C₂×C₄×C₁₃₉	—	2	3	no	—
191	C₂×C₄×C₁₃₉	—	2	2	no	—
192	C₂×C₄×C₁₃₉	—	2	3	no	—
193	C₂×C₄×C₁₃₉	—	2	24	no	—
194	C₂×C₄×C₁₃₉	—	2	8	no	—
195	C₂×C₄×C₁₃₉	—	2	3	no	—
196	C₂×C₄×C₁₃₉	—	2	6	no	—
197	C₂×C₄×C₁₃₉	—	2	6	no	—
198	C₂×C₄×C₁₃₉	—	2	6	no	—
199	C₂×C₄×C₁₃₉	—	2	6	no	—
200	C₂×C₄×C₁₃₉	—	2	6	no	—
201	C₂×C₄×C₁₃₉	—	2	3	no	—
202	C₂×C₄×C₁₃₉	—	2	3	no	—
203	C₂×C₄×C₁₃₉	—	2	6	no	—
204	C₂×C₄×C₁₃₉	—	4	12	no	—
205	C₂×C₄×C₁₃₉	—	4	24	no	—
206	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
207	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
208	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
209	C₂×C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
210	C₂×C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
212	C₂×C₄×C₁₃₉	C₂²×Q₈	1	4	yes	—
213	C₂×C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
214	C₂×C₄×C₁₃₉	C₂×#41	1	6	yes	—
215	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
216	C₂²×C₁₃₉	—	2	2	no	—
217	C₂²×C₁₃₉	—	2	2	no	—
218	C₂²×C₁₃₉	—	2	2	no	—
219	C₂²×C₁₃₉	—	2	2	no	—
220	C₂²×C₁₃₉	—	2	2	no	—
221	C₂²×C₁₃₉	—	2	2	no	—
222	C₂²×C₁₃₉	—	2	4	no	—
223	C₂²×C₁₃₉	—	2	4	no	—
224	C₂²×C₁₃₉	—	2	4	no	—
225	C₂²×C₁₃₉	—	4	2	no	—
226	C₂²×C₁₃₉	—	4	2	no	—
227	C₂²×C₁₃₉	—	4	2	no	—
228	C₂²×C₁₃₉	—	4	2	no	—
229	C₂²×C₁₃₉	D₄×Q₈	1	4	yes	complemented
230	C₂²×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
231	C₂²×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
232	C₂²×C₁₃₉	C₂³×Q₈	1	1	yes	characteristic
233	C₂²×C₁₃₉	C₂²×#41	1	1	yes	characteristic
234	C₂²×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂²×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
236	C₂²×C₁₃₉	#585	1	1	yes	characteristic
237	C₄×C₁₃₉	—	2	6	no	—
238	C₄×C₁₃₉	—	2	2	no	—
239	C₄×C₁₃₉	—	2	4	no	—
240	C₄×C₁₃₉	—	2	6	no	—
241	C₄×C₁₃₉	—	4	6	no	—
242	C₄×C₁₃₉	—	4	3	no	—
243	C₄×C₁₃₉	—	4	6	no	—
244	C₄×C₁₃₉	C₂×#121	1	12	yes	—
245	C₄×C₁₃₉	D₄×Q₈	1	4	yes	complemented
246	C₂×C₁₃₉	—	2	4	no	—
247	C₂×C₁₃₉	—	4	4	no	—
248	C₂×C₁₃₉	C₂²×#121	1	1	yes	characteristic
249	C₂×C₁₃₉	C₂×D₄×Q₈	1	1	yes	characteristic
250	C₂×C₁₃₉	C₂×D₄×Q₈	1	1	yes	characteristic
251	C₂×C₁₃₉	Q₈×#41	1	1	yes	characteristic
252	C₂×C₁₃₉	#3489	1	1	yes	characteristic
253	C₂×C₁₃₉	#3497	1	1	yes	characteristic
254	C₂×C₁₃₉	#3523	1	1	yes	characteristic
255	Q₈×#121	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₁₃₉	Q₈×#121	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×D₄×Q₈	C₂×C₁₃₉	1	2	yes	complemented
258	C₂×D₄×Q₈	C₂×C₁₃₉	1	1	yes	characteristic, complemented
259	C₂²×C₄×Q₈	C₂×C₁₃₉	1	1	yes	characteristic, complemented
260	#2857	C₂×C₁₃₉	1	3	yes	complemented
261	#2546	C₂×C₁₃₉	1	3	yes	complemented
262	#2538	C₂×C₁₃₉	1	3	yes	complemented
263	#2429	C₂×C₁₃₉	1	6	yes	complemented
264	#2417	C₂×C₁₃₉	1	6	yes	complemented
265	Q₈×C₄⋊C₄	C₂×C₁₃₉	1	1	yes	characteristic, complemented
266	Q₈×C₂²⋊C₄	C₂×C₁₃₉	1	2	yes	complemented
267	C₄×#121	C₂×C₁₃₉	1	3	yes	—
268	C₂³×Q₈	C₂²×C₁₃₉	1	2	yes	—
269	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
270	D₄×Q₈	—	2	16	no	—
271	D₄×Q₈	—	2	8	no	—
272	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	3	yes	—
273	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	6	yes	—
274	C₂×C₂²⋊Q₈	C₂²×C₁₃₉	1	6	yes	—
275	C₂×C₂²⋊Q₈	C₂²×C₁₃₉	1	6	yes	—
276	C₂×C₂²⋊Q₈	C₂²×C₁₃₉	1	6	yes	—
277	C₂×#121	C₂²×C₁₃₉	1	1	yes	characteristic
278	C₂×C₄×Q₈	—	2	8	no	—
279	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₁₃₉	1	2	yes	—
281	C₂×C₄×Q₈	C₂²×C₁₃₉	1	3	yes	—
282	C₂×C₄×Q₈	C₂²×C₁₃₉	1	3	yes	—
283	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
284	C₂×C₄×D₄	C₂²×C₁₃₉	1	6	yes	—
285	C₂×C₄×D₄	C₂²×C₁₃₉	1	3	yes	—
286	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	3	yes	—
287	C₂²×C₄²	C₂²×C₁₃₉	1	3	yes	—
288	#398	C₂²×C₁₃₉	1	12	yes	—
289	#396	C₂²×C₁₃₉	1	12	yes	—
290	#391	C₂²×C₁₃₉	1	6	yes	—
291	#391	C₂²×C₁₃₉	1	6	yes	—
292	#390	C₂²×C₁₃₉	1	3	yes	—
293	#389	C₂²×C₁₃₉	1	6	yes	—
294	#389	C₂²×C₁₃₉	1	6	yes	—
295	#389	C₂²×C₁₃₉	1	6	yes	—
296	#389	C₂²×C₁₃₉	1	3	yes	—
297	#388	C₂²×C₁₃₉	1	6	yes	—
298	#386	C₂²×C₁₃₉	1	6	yes	—
299	#384	C₂²×C₁₃₉	1	3	yes	—
300	#380	C₂²×C₁₃₉	1	6	yes	—
301	#380	C₂²×C₁₃₉	1	3	yes	—
302	#380	C₂²×C₁₃₉	1	6	yes	—
303	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	3	yes	—
304	C₄×C₂²⋊C₄	C₂²×C₁₃₉	1	6	yes	—
305	C₂²×Q₈	—	2	4	no	—
306	C₂²×Q₈	—	2	4	no	—
307	C₂²×Q₈	—	2	8	no	—
308	C₂²×Q₈	—	2	8	no	—
309	C₂²×Q₈	—	2	8	no	—
310	C₂²×Q₈	D₄×C₁₃₉	1	8	yes	complemented
311	C₂²×Q₈	C₂³×C₁₃₉	1	6	yes	—
312	C₂²×Q₈	C₂³×C₁₃₉	1	3	yes	—
313	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₁₃₉	1	2	yes	—
315	C₂²×D₄	C₂³×C₁₃₉	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₁₃₉	1	6	yes	—
317	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
318	C₂³×C₄	C₂³×C₁₃₉	1	3	yes	—
319	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
320	C₄⋊Q₈	—	2	24	no	—
321	C₄⋊Q₈	—	2	12	no	—
322	C₂²⋊Q₈	—	2	24	no	—
323	C₂²⋊Q₈	—	2	24	no	—
324	C₂²⋊Q₈	—	2	24	no	—
325	#121	Q₈×C₁₃₉	1	8	yes	direct product factor
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	—	2	8	no	—
328	C₄×Q₈	—	2	12	no	—
329	C₄×Q₈	—	2	12	no	—
330	C₄×Q₈	D₄×C₁₃₉	1	8	yes	complemented
331	C₄×D₄	—	2	24	no	—
332	C₄×D₄	—	2	12	no	—
333	C₂×C₄⋊C₄	—	2	12	no	—
334	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	3	yes	—
335	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
336	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	3	yes	—
337	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
338	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
339	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
340	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	12	yes	—
341	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	3	yes	—
342	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	6	yes	—
343	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	3	yes	—
344	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	6	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	6	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	6	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	6	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	2	yes	—
350	C₂×C₄²	—	2	12	no	—
351	C₂×C₄²	C₂³×C₁₃₉	1	6	yes	—
352	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
353	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
354	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
355	C₂×C₄²	C₂³×C₁₃₉	1	3	yes	—
356	#95	C₂³×C₁₃₉	1	6	yes	—
357	#95	C₂³×C₁₃₉	1	6	yes	—
358	#95	C₂³×C₁₃₉	1	6	yes	—
359	#95	C₂³×C₁₃₉	1	12	yes	—
360	C₂⁴	C₂⁴×C₁₃₉	1	2	yes	—
361	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
362	C₂×Q₈	—	2	32	no	—
363	C₂×Q₈	—	2	6	no	—
364	C₂×Q₈	—	2	12	no	—
365	C₂×Q₈	—	2	12	no	—
366	C₂×Q₈	—	2	6	no	—
367	C₂×Q₈	—	2	12	no	—
368	C₂×Q₈	—	4	32	no	—
369	C₂×Q₈	C₂×D₄×C₁₃₉	1	12	yes	—
370	C₂×Q₈	C₂×D₄×C₁₃₉	1	4	yes	—
371	C₂×Q₈	C₂×D₄×C₁₃₉	1	4	yes	—
372	C₂×Q₈	#41×C₁₃₉	1	4	yes	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	C₂×Q₈×C₁₃₉	1	8	yes	complemented
378	C₂×D₄	C₂×Q₈×C₁₃₉	1	4	yes	complemented
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	6	no	—
381	C₂²×C₄	—	2	6	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	6	no	—
384	C₂²×C₄	—	2	6	no	—
385	C₂²×C₄	—	2	12	no	—
386	C₂²×C₄	—	2	12	no	—
387	C₂²×C₄	—	2	12	no	—
388	C₂²×C₄	—	4	3	no	—
389	C₂²×C₄	C₂×D₄×C₁₃₉	1	12	yes	—
390	C₂²×C₄	C₂×Q₈×C₁₃₉	1	4	yes	complemented
391	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
392	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
393	C₂²×C₄	C₂⁴×C₁₃₉	1	6	yes	—
394	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
395	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
396	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₁₃₉	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₁₃₉	1	3	yes	—
399	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
400	C₄⋊C₄	—	2	12	no	—
401	C₄⋊C₄	—	2	6	no	—
402	C₄⋊C₄	—	2	12	no	—
403	C₄⋊C₄	—	2	6	no	—
404	C₄⋊C₄	—	2	12	no	—
405	C₄⋊C₄	—	2	24	no	—
406	C₄⋊C₄	—	2	12	no	—
407	C₄⋊C₄	—	2	6	no	—
408	C₄⋊C₄	—	4	6	no	—
409	C₄⋊C₄	C₂×D₄×C₁₃₉	1	12	yes	—
410	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	4	yes	complemented
411	C₂²⋊C₄	—	2	12	no	—
412	C₂²⋊C₄	—	2	12	no	—
413	C₂²⋊C₄	—	2	12	no	—
414	C₂²⋊C₄	—	4	6	no	—
415	C₂²⋊C₄	C₂×Q₈×C₁₃₉	1	8	yes	complemented
416	C₄²	—	2	6	no	—
417	C₄²	—	2	6	no	—
418	C₄²	—	2	6	no	—
419	C₄²	—	2	12	no	—
420	C₄²	C₂×D₄×C₁₃₉	1	12	yes	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	1	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	C₂²×D₄×C₁₃₉	1	2	yes	—
432	C₂³	C₂²×Q₈×C₁₃₉	1	4	yes	—
433	C₂³	C₂²×Q₈×C₁₃₉	1	2	yes	—
434	C₂³	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
435	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
436	Q₈	—	2	24	no	—
437	Q₈	—	4	24	no	—
438	Q₈	#121×C₁₃₉	1	16	yes	direct product factor
439	D₄	—	2	16	no	—
440	D₄	—	2	8	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	3	no	—
444	C₂×C₄	—	2	3	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	3	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	3	no	—
449	C₂×C₄	—	2	24	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	3	no	—
452	C₂×C₄	—	2	6	no	—
453	C₂×C₄	—	2	6	no	—
454	C₂×C₄	—	2	6	no	—
455	C₂×C₄	—	2	6	no	—
456	C₂×C₄	—	2	6	no	—
457	C₂×C₄	—	2	3	no	—
458	C₂×C₄	—	2	3	no	—
459	C₂×C₄	—	2	6	no	—
460	C₂×C₄	—	4	12	no	—
461	C₂×C₄	—	4	24	no	—
462	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
463	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
464	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
465	C₂×C₄	C₂²×D₄×C₁₃₉	1	6	yes	—
466	C₂×C₄	C₂²×D₄×C₁₃₉	1	2	yes	—
467	C₂×C₄	C₂²×Q₈×C₁₃₉	1	2	yes	—
468	C₂×C₄	C₂²×Q₈×C₁₃₉	1	4	yes	—
469	C₂×C₄	C₂²×Q₈×C₁₃₉	1	2	yes	—
470	C₂×C₄	C₂×#41×C₁₃₉	1	6	yes	—
471	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	2	yes	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	D₄×Q₈×C₁₃₉	1	4	yes	complemented
486	C₂²	C₂³×D₄×C₁₃₉	1	1	yes	characteristic
487	C₂²	C₂³×D₄×C₁₃₉	1	1	yes	characteristic
488	C₂²	C₂³×Q₈×C₁₃₉	1	1	yes	characteristic
489	C₂²	C₂²×#41×C₁₃₉	1	1	yes	characteristic
490	C₂²	C₂×2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
491	C₂²	C₂×2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
492	C₂²	#585×C₁₃₉	1	1	yes	characteristic
493	C₄	—	2	6	no	—
494	C₄	—	2	2	no	—
495	C₄	—	2	4	no	—
496	C₄	—	2	6	no	—
497	C₄	—	4	6	no	—
498	C₄	—	4	3	no	—
499	C₄	—	4	6	no	—
500	C₄	C₂×#121×C₁₃₉	1	12	yes	—
501	C₄	D₄×Q₈×C₁₃₉	1	4	yes	complemented
502	C₂	—	2	4	no	—
503	C₂	—	4	4	no	—
504	C₂	C₂²×#121×C₁₃₉	1	1	yes	characteristic
505	C₂	C₂×D₄×Q₈×C₁₃₉	1	1	yes	characteristic
506	C₂	C₂×D₄×Q₈×C₁₃₉	1	1	yes	characteristic
507	C₂	Q₈×#41×C₁₃₉	1	1	yes	characteristic
508	C₂	#3489×C₁₃₉	1	1	yes	characteristic
509	C₂	#3497×C₁₃₉	1	1	yes	characteristic
510	C₂	#3523×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₉
prime radical	C₂×#121×C₁₃₉	C₂²
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×D₄×Q₈×C₁₃₉	C₂	semidirect
2. C₂×D₄×Q₈×C₁₃₉	C₂	semidirect
3. C₂²×C₄×Q₈×C₁₃₉	C₂	semidirect
4. #2857×C₁₃₉	C₂	semidirect
5. #2546×C₁₃₉	C₂	semidirect
6. #2538×C₁₃₉	C₂	semidirect
7. #2429×C₁₃₉	C₂	semidirect
8. #2417×C₁₃₉	C₂	semidirect
9. Q₈×C₄⋊C₄×C₁₃₉	C₂	semidirect
10. Q₈×C₂²⋊C₄×C₁₃₉	C₂	semidirect
54. C₂²×Q₈×C₁₃₉	D₄	semidirect
69. #121×C₁₃₉	Q₈	direct (N × H)
74. C₄×Q₈×C₁₃₉	D₄	semidirect
121. C₂×D₄×C₁₃₉	C₂×Q₈	semidirect
122. C₂×D₄×C₁₃₉	C₂×Q₈	semidirect
134. C₂²×C₄×C₁₃₉	C₂×Q₈	semidirect
154. C₄⋊C₄×C₁₃₉	C₂×Q₈	semidirect
159. C₂²⋊C₄×C₁₃₉	C₂×Q₈	semidirect
182. Q₈×C₁₃₉	#121	direct (N × H)
229. C₂²×C₁₃₉	D₄×Q₈	semidirect
245. C₄×C₁₃₉	D₄×Q₈	semidirect
255. Q₈×#121	C₁₃₉	direct (N × H)
256. C₁₃₉	Q₈×#121	direct (N × H)
257. C₂×D₄×Q₈	C₂×C₁₃₉	semidirect
258. C₂×D₄×Q₈	C₂×C₁₃₉	semidirect
259. C₂²×C₄×Q₈	C₂×C₁₃₉	semidirect
260. #2857	C₂×C₁₃₉	semidirect
261. #2546	C₂×C₁₃₉	semidirect
262. #2538	C₂×C₁₃₉	semidirect
263. #2429	C₂×C₁₃₉	semidirect
264. #2417	C₂×C₁₃₉	semidirect
265. Q₈×C₄⋊C₄	C₂×C₁₃₉	semidirect
266. Q₈×C₂²⋊C₄	C₂×C₁₃₉	semidirect
310. C₂²×Q₈	D₄×C₁₃₉	semidirect
325. #121	Q₈×C₁₃₉	direct (N × H)
330. C₄×Q₈	D₄×C₁₃₉	semidirect
377. C₂×D₄	C₂×Q₈×C₁₃₉	semidirect
378. C₂×D₄	C₂×Q₈×C₁₃₉	semidirect
390. C₂²×C₄	C₂×Q₈×C₁₃₉	semidirect
410. C₄⋊C₄	C₂×Q₈×C₁₃₉	semidirect
415. C₂²⋊C₄	C₂×Q₈×C₁₃₉	semidirect
438. Q₈	#121×C₁₃₉	direct (N × H)
485. C₂²	D₄×Q₈×C₁₃₉	semidirect
501. C₄	D₄×Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₁₃₉	2
C₂²×C₄×Q₈×C₁₃₉	2
#2857×C₁₃₉	2
#2546×C₁₃₉	2
#2538×C₁₃₉	2
#2429×C₁₃₉	2
#2417×C₁₃₉	2
Q₈×C₄⋊C₄×C₁₃₉	2
Q₈×C₂²⋊C₄×C₁₃₉	2
C₄×#121×C₁₃₉	2
Q₈×#121	139

Subgroups — Q8×#121×C139 #48031137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#121×C₁₃₉ #48031137