Subgroups — C₂×#22580851 #31808855

Order 17792.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₁₃₉⋊C₄	C₂	1	2	yes	maximal
4	#22580851	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃₉⋊#95	C₂	1	2	yes	maximal
6	C₂²×C₄²×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₉	C₂²	1	2	yes	—
8	C₂³×Dic₁₃₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₃₉	—	2	4	no	—
10	C₂×C₄⋊Dic₁₃₉	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
13	C₂×Dic₁₃₉⋊C₄	—	2	8	no	—
14	C₂×Dic₁₃₉⋊C₄	C₂²	1	8	yes	—
15	C₁₃₉⋊#95	C₂²	1	8	yes	—
16	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₃₉	—	2	4	no	—
20	C₂×C₄²×C₁₃₉	C₂²	1	4	yes	—
21	C₂²×Dic₁₃₉	—	2	4	no	—
22	C₂²×Dic₁₃₉	—	2	4	no	—
23	C₂²×Dic₁₃₉	—	2	4	no	—
24	C₂²×Dic₁₃₉	—	2	4	no	—
25	C₂²×Dic₁₃₉	—	2	4	no	—
26	C₂²×Dic₁₃₉	C₂×C₄	1	8	yes	—
27	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
28	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
29	C₄⋊Dic₁₃₉	—	2	8	no	—
30	C₄⋊Dic₁₃₉	C₂×C₄	1	16	yes	complemented
31	Dic₁₃₉⋊C₄	—	2	16	no	—
32	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₃₉	—	2	2	no	—
34	C₂²×C₄×C₁₃₉	—	2	2	no	—
35	C₂²×C₄×C₁₃₉	—	2	2	no	—
36	C₂²×C₄×C₁₃₉	—	2	2	no	—
37	C₂²×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₁₃₉	D₄	1	4	yes	—
39	C₂²×C₄×C₁₃₉	D₄	1	4	yes	—
40	C₂²×C₄×C₁₃₉	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₃₉	Q₈	1	4	yes	—
42	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
45	C₄²×C₁₃₉	—	2	8	no	—
46	C₂×Dic₁₃₉	—	2	16	no	—
47	C₂×Dic₁₃₉	—	2	4	no	—
48	C₂×Dic₁₃₉	—	2	4	no	—
49	C₂×Dic₁₃₉	—	2	4	no	—
50	C₂×Dic₁₃₉	—	2	4	no	—
51	C₂×Dic₁₃₉	—	2	4	no	—
52	C₂×Dic₁₃₉	—	4	8	no	—
53	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
54	C₂³×C₁₃₉	C₂²×C₄	1	2	yes	—
55	C₂³×C₁₃₉	C₂×D₄	1	2	yes	—
56	C₂³×C₁₃₉	C₂×D₄	1	2	yes	—
57	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
58	C₂³×C₁₃₉	C₂×Q₈	1	2	yes	—
59	C₂³×C₁₃₉	#41	1	2	yes	—
60	C₂³×C₁₃₉	#41	1	2	yes	—
61	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	—	2	8	no	—
63	C₂×C₄×C₁₃₉	—	2	8	no	—
64	C₂×C₄×C₁₃₉	—	2	2	no	—
65	C₂×C₄×C₁₃₉	—	2	2	no	—
66	C₂×C₄×C₁₃₉	—	2	2	no	—
67	C₂×C₄×C₁₃₉	—	2	2	no	—
68	C₂×C₄×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₁₃₉	C₄×D₄	1	4	yes	—
76	C₂²×C₁₃₉	C₄×Q₈	1	4	yes	—
77	C₂²×C₁₃₉	#121	1	4	yes	—
78	C₂²×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₁₃₉	#125	1	4	yes	—
80	C₂²×C₁₃₉	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₃₉	—	2	4	no	—
89	C₄×C₁₃₉	—	2	4	no	—
90	C₄×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	complemented
91	Dic₁₃₉	—	2	8	no	—
92	Dic₁₃₉	—	4	4	no	—
93	C₂×C₁₃₉	#389	1	8	yes	complemented
94	C₂×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₃₉	C₂×#121	1	1	yes	characteristic
98	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₃₉	C₂×#125	1	1	yes	characteristic
100	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₁₃₉	C₂×#389	1	1	yes	characteristic, complemented, Sylow
102	C₂×#389	—	139	1	no	maximal, Sylow
103	C₂²×C₄⋊C₄	—	139	2	no	—
104	C₂²×C₄⋊C₄	—	139	1	no	—
105	C₂²×C₄⋊C₄	—	139	1	no	—
106	C₂²×C₄²	D₁₃₉	1	1	yes	characteristic
107	#389	—	139	8	no	—
108	C₂×#95	—	139	2	no	—
109	C₂³×C₄	—	139	2	no	—
110	C₂³×C₄	—	139	2	no	—
111	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
112	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
113	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	278	4	no	—
115	C₂×C₄⋊C₄	—	278	8	no	—
116	C₂×C₄⋊C₄	—	139	8	no	—
117	C₂×C₄⋊C₄	—	139	4	no	—
118	C₂×C₄⋊C₄	—	139	8	no	—
119	C₂×C₄⋊C₄	—	139	4	no	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂×D₁₃₉	1	4	yes	—
122	#95	—	139	8	no	—
123	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	278	4	no	—
129	C₂²×C₄	—	278	4	no	—
130	C₂²×C₄	—	278	4	no	—
131	C₂²×C₄	—	278	4	no	—
132	C₂²×C₄	—	278	4	no	—
133	C₂²×C₄	—	139	4	no	—
134	C₂²×C₄	—	139	8	no	—
135	C₂²×C₄	—	139	4	no	—
136	C₂²×C₄	C₄×D₁₃₉	1	4	yes	—
137	C₂²×C₄	C₂²⋊D₁₃₉	1	4	yes	—
138	C₂²×C₄	D₅₅₆	1	4	yes	—
139	C₂²×C₄	Dic₂₇₈	1	4	yes	—
140	C₂²×C₄	Dic₂₇₈	1	4	yes	—
141	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
142	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
143	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
144	C₄⋊C₄	—	278	8	no	—
145	C₄⋊C₄	—	278	16	no	—
146	C₄⋊C₄	—	139	16	no	—
147	C₄²	—	2	8	no	—
148	C₂³	C₂×C₄×D₁₃₉	1	2	yes	—
149	C₂³	C₂×C₂²⋊D₁₃₉	1	2	yes	—
150	C₂³	C₂×D₅₅₆	1	2	yes	—
151	C₂³	C₂×Dic₂₇₈	1	2	yes	—
152	C₂³	C₂×Dic₂₇₈	1	2	yes	—
153	C₂³	#14555785	1	2	yes	—
154	C₂³	#14555785	1	2	yes	—
155	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	278	16	no	—
163	C₂×C₄	—	278	4	no	—
164	C₂×C₄	—	278	4	no	—
165	C₂×C₄	—	278	4	no	—
166	C₂×C₄	—	278	4	no	—
167	C₂×C₄	—	278	4	no	—
168	C₂×C₄	—	556	8	no	—
169	C₂×C₄	—	139	8	no	—
170	C₂×C₄	Dic₁₃₉⋊C₄	1	16	yes	complemented
171	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
172	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
173	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
174	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
175	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
176	C₂²	C₂×Dic₁₃₉⋊C₄	1	4	yes	—
177	C₂²	C₄×D₅₅₆	1	4	yes	—
178	C₂²	C₄×Dic₂₇₈	1	4	yes	—
179	C₂²	#17984816	1	4	yes	—
180	C₂²	#17984821	1	4	yes	—
181	C₂²	#17984834	1	4	yes	—
182	C₂²	#17984842	1	4	yes	—
183	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
184	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
185	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
186	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
187	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
188	C₂²	C₂×#14555785	1	1	yes	characteristic
189	C₂²	C₂×#14555785	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	278	8	no	—
193	C₄	—	556	4	no	—
194	C₄	C₂×Dic₁₃₉⋊C₄	1	8	yes	complemented
195	C₂	#22580851	1	8	yes	direct product factor
196	C₂	C₂²×Dic₁₃₉⋊C₄	1	1	yes	characteristic
197	C₂	C₂×C₄×D₅₅₆	1	1	yes	characteristic
198	C₂	C₂×C₄×Dic₂₇₈	1	1	yes	characteristic
199	C₂	C₂×#17984816	1	1	yes	characteristic
200	C₂	C₂×#17984821	1	1	yes	characteristic
201	C₂	C₂×#17984834	1	1	yes	characteristic
202	C₂	C₂×#17984842	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Frattini	C₂³	C₂³×D₁₃₉
Fitting	C₂²×C₄²×C₁₃₉	C₂
socle, prime radical	C₂⁴×C₁₃₉	C₂³
cosocle	C₂³×C₁₃₉	C₂⁴
hypercenter	C₂²×C₄²	D₁₃₉
hypocenter	C₁₃₉	C₂×#389
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #22580851	C₂	direct (N × H)
10. C₂×C₄⋊Dic₁₃₉	C₄	semidirect
30. C₄⋊Dic₁₃₉	C₂×C₄	semidirect
68. C₂×C₄×C₁₃₉	C₄⋊C₄	semidirect
90. C₄×C₁₃₉	C₂×C₄⋊C₄	semidirect
93. C₂×C₁₃₉	#389	semidirect
101. C₁₃₉	C₂×#389	semidirect
170. C₂×C₄	Dic₁₃₉⋊C₄	semidirect
194. C₄	C₂×Dic₁₃₉⋊C₄	semidirect
195. C₂	#22580851	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₃₉	2
C₂²×Dic₁₃₉⋊C₄	2
#22580851	2
C₂×C₁₃₉⋊#95	2
C₂²×C₄²×C₁₃₉	2
C₂×#389	139

Subgroups — C2×#22580851 #31808855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22580851 #31808855