Subgroups — C₁₃₉×#29286 #48023882

Order 35584.

756 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₉	C₂	1	2	yes	maximal
2	#2086×C₁₃₉	C₂	1	4	yes	maximal
3	#1988×C₁₃₉	C₂	1	4	yes	maximal
4	#1947×C₁₃₉	C₂	1	4	yes	maximal
5	#1842×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₁₃₉	—	2	2	no	—
8	C₂×#107×C₁₃₉	C₂²	1	4	yes	—
9	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
11	#391×C₁₃₉	—	2	2	no	—
12	#389×C₁₃₉	—	2	2	no	—
13	#389×C₁₃₉	C₂²	1	4	yes	—
14	#384×C₁₃₉	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
16	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
17	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
18	#340×C₁₃₉	C₂²	1	8	yes	—
19	#340×C₁₃₉	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₃₉	—	2	2	no	—
21	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
22	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
23	C₄⋊Q₈×C₁₃₉	Q₈	1	8	yes	complemented
24	C₄×Q₈×C₁₃₉	—	4	8	no	—
25	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
27	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
28	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
30	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
31	C₂×C₄²×C₁₃₉	—	2	4	no	—
32	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
37	#107×C₁₃₉	—	4	4	no	—
38	C₄⋊C₈×C₁₃₉	—	2	8	no	—
39	C₄⋊C₈×C₁₃₉	—	4	2	no	—
40	#95×C₁₃₉	—	2	4	no	—
41	#95×C₁₃₉	—	2	4	no	—
42	C₂×Q₈×C₁₃₉	—	2	8	no	—
43	C₂×Q₈×C₁₃₉	—	4	4	no	—
44	C₂×Q₈×C₁₃₉	—	4	4	no	—
45	C₂²×C₄×C₁₃₉	—	2	4	no	—
46	C₂²×C₄×C₁₃₉	—	2	4	no	—
47	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₁₃₉	—	2	8	no	—
53	C₂×C₈×C₁₃₉	—	2	4	no	—
54	C₄⋊C₄×C₁₃₉	—	2	8	no	—
55	C₄⋊C₄×C₁₃₉	—	2	4	no	—
56	C₄⋊C₄×C₁₃₉	—	2	4	no	—
57	C₄⋊C₄×C₁₃₉	—	2	4	no	—
58	C₄⋊C₄×C₁₃₉	—	4	4	no	—
59	C₄⋊C₄×C₁₃₉	—	4	8	no	—
60	C₄⋊C₄×C₁₃₉	—	4	4	no	—
61	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	8	yes	—
62	C₄²×C₁₃₉	—	2	4	no	—
63	C₄²×C₁₃₉	—	2	1	no	—
64	C₄²×C₁₃₉	—	2	1	no	—
65	C₄²×C₁₃₉	—	4	1	no	—
66	C₄²×C₁₃₉	—	4	8	no	—
67	C₄²×C₁₃₉	D₈	1	4	yes	—
68	C₄²×C₁₃₉	C₂×Q₈	1	4	yes	—
69	C₄²×C₁₃₉	#41	1	4	yes	—
70	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₁₃₉	—	4	16	no	—
72	C₂×C₄×C₁₃₉	—	2	1	no	—
73	C₂×C₄×C₁₃₉	—	2	2	no	—
74	C₂×C₄×C₁₃₉	—	2	2	no	—
75	C₂×C₄×C₁₃₉	—	2	1	no	—
76	C₂×C₄×C₁₃₉	—	2	8	no	—
77	C₂×C₄×C₁₃₉	—	2	1	no	—
78	C₂×C₄×C₁₃₉	—	2	2	no	—
79	C₂×C₄×C₁₃₉	—	4	4	no	—
80	C₂×C₄×C₁₃₉	—	4	4	no	—
81	C₂×C₄×C₁₃₉	—	4	4	no	—
82	C₂×C₄×C₁₃₉	—	4	4	no	—
83	C₂×C₄×C₁₃₉	—	4	4	no	—
84	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
85	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
86	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₁₃₉	C₂×D₈	1	4	yes	—
88	C₂×C₄×C₁₃₉	#137	1	4	yes	—
89	C₂×C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
91	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
93	C₈×C₁₃₉	—	4	4	no	—
94	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	2	yes	—
95	C₂²×C₁₃₉	#537	1	1	yes	characteristic
96	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
97	C₂²×C₁₃₉	C₂×#137	1	2	yes	—
98	C₂²×C₁₃₉	#575	1	1	yes	characteristic
99	C₄×C₁₃₉	—	2	4	no	—
100	C₄×C₁₃₉	—	2	2	no	—
101	C₄×C₁₃₉	—	4	1	no	—
102	C₄×C₁₃₉	—	4	8	no	—
103	C₄×C₁₃₉	—	8	4	no	—
104	C₄×C₁₃₉	#474	1	8	yes	—
105	C₄×C₁₃₉	#557	1	2	yes	—
106	C₄×C₁₃₉	Q₈²	1	1	yes	characteristic
107	C₄×C₁₃₉	#559	1	1	yes	characteristic
108	C₂×C₁₃₉	#2582	1	1	yes	characteristic
109	C₂×C₁₃₉	C₂×#474	1	2	yes	—
110	C₂×C₁₃₉	#3078	1	2	yes	—
111	C₂×C₁₃₉	#3245	1	1	yes	characteristic
112	C₂×C₁₃₉	#3255	1	1	yes	characteristic
113	#29286	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₁₃₉	#29286	1	1	yes	characteristic, direct product factor, Sylow
115	C₄×C₄⋊Q₈	C₂×C₁₃₉	1	2	yes	—
116	#2086	C₂×C₁₃₉	1	4	yes	—
117	#1988	C₂×C₁₃₉	1	4	yes	—
118	#1947	C₂×C₁₃₉	1	4	yes	—
119	#1842	C₂×C₁₃₉	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	2	yes	—
121	C₂×C₄×Q₈	—	2	2	no	—
122	C₂×#107	C₂²×C₁₃₉	1	4	yes	—
123	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
124	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
125	#391	—	2	2	no	—
126	#389	—	2	2	no	—
127	#389	C₂²×C₁₃₉	1	4	yes	—
128	#384	C₂²×C₁₃₉	1	2	yes	—
129	C₄×C₄⋊C₄	—	2	2	no	—
130	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
131	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
132	#340	C₂²×C₁₃₉	1	8	yes	—
133	#340	C₂²×C₁₃₉	1	4	yes	—
134	C₂²×Q₈	—	2	2	no	—
135	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	Q₈×C₁₃₉	1	8	yes	complemented
138	C₄×Q₈	—	4	8	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	4	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
144	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
145	C₂×C₄²	—	2	4	no	—
146	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
147	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
148	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
150	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
151	#107	—	4	4	no	—
152	C₄⋊C₈	—	2	8	no	—
153	C₄⋊C₈	—	4	2	no	—
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	C₂×Q₈	—	2	8	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
163	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
164	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
165	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	2	4	no	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	8	yes	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	4	1	no	—
180	C₄²	—	4	8	no	—
181	C₄²	D₈×C₁₃₉	1	4	yes	—
182	C₄²	C₂×Q₈×C₁₃₉	1	4	yes	—
183	C₄²	#41×C₁₃₉	1	4	yes	—
184	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
185	Q₈	—	4	16	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
199	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
200	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
201	C₂×C₄	C₂×D₈×C₁₃₉	1	4	yes	—
202	C₂×C₄	#137×C₁₃₉	1	4	yes	—
203	C₂×C₄	C₂²×Q₈×C₁₃₉	1	2	yes	—
204	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
205	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
206	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
207	C₈	—	4	4	no	—
208	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	2	yes	—
209	C₂²	#537×C₁₃₉	1	1	yes	characteristic
210	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
211	C₂²	C₂×#137×C₁₃₉	1	2	yes	—
212	C₂²	#575×C₁₃₉	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	—	4	1	no	—
217	C₄	—	8	4	no	—
218	C₄	#474×C₁₃₉	1	8	yes	—
219	C₄	#557×C₁₃₉	1	2	yes	—
220	C₄	Q₈²×C₁₃₉	1	1	yes	characteristic
221	C₄	#559×C₁₃₉	1	1	yes	characteristic
222	C₂	#2582×C₁₃₉	1	1	yes	characteristic
223	C₂	C₂×#474×C₁₃₉	1	2	yes	—
224	C₂	#3078×C₁₃₉	1	2	yes	—
225	C₂	#3245×C₁₃₉	1	1	yes	characteristic
226	C₂	#3255×C₁₃₉	1	1	yes	characteristic

	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
23. C₄⋊Q₈×C₁₃₉	Q₈	semidirect
113. #29286	C₁₃₉	direct (N × H)
114. C₁₃₉	#29286	direct (N × H)
137. C₄⋊Q₈	Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₃₉	2
#2086×C₁₃₉	2
#1988×C₁₃₉	2
#1947×C₁₃₉	2
#1842×C₁₃₉	2
#29286	139

Subgroups — C139×#29286 #48023882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29286 #48023882