Subgroups — C₂×#22581429 #31819593

Order 17792.

1488 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22581429	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×Dic₂₇₈	C₂	1	3	yes	maximal, complemented
5	C₂×#17984807	C₂	1	3	yes	maximal, complemented
6	C₂×#17984800	C₂	1	3	yes	maximal
7	C₂×#17984799	C₂	1	3	yes	maximal
8	C₂×C₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₂₇₈	C₂²	1	3	yes	—
10	C₂×Q₈×D₁₃₉	C₄	1	16	yes	complemented
11	C₂×Q₈×D₁₃₉	C₂²	1	8	yes	—
12	C₂²×C₄×D₁₃₉	C₂²	1	3	yes	—
13	Q₈×Dic₁₃₉	C₂²	1	8	yes	complemented
14	C₄×Dic₂₇₈	C₂²	1	24	yes	complemented
15	#17984807	C₂²	1	24	yes	complemented
16	#17984800	C₂²	1	24	yes	—
17	#17984799	C₂²	1	24	yes	—
18	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
19	C₂×Dic₁₃₉⋊C₄	C₂²	1	3	yes	—
20	C₂×Dic₁₃₉⋊C₄	C₂²	1	3	yes	—
21	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	—
23	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	—
24	C₂×C₄×Dic₁₃₉	C₂²	1	6	yes	—
25	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₃₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
28	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
29	C₂³×D₁₃₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₇₈	C₂×C₄	1	24	yes	—
31	C₂×Dic₂₇₈	C₂³	1	12	yes	—
32	Q₈×D₁₃₉	C₂×C₄	1	64	yes	complemented
33	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
35	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
36	C₂×C₄×D₁₃₉	C₂×C₄	1	24	yes	—
37	C₂×C₄×D₁₃₉	C₂³	1	12	yes	—
38	C₄⋊Dic₁₃₉	C₂³	1	12	yes	—
39	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
40	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
41	(C₂×D₁₃₉)⋊C₄	C₂³	1	4	yes	—
42	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
43	C₄×Dic₁₃₉	C₂³	1	12	yes	—
44	C₄×Dic₁₃₉	C₂³	1	24	yes	—
45	C₂×Q₈×C₁₃₉	C₂×C₄	1	8	yes	—
46	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
47	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
48	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
50	C₄⋊C₄×C₁₃₉	C₂³	1	12	yes	—
51	C₄²×C₁₃₉	C₂³	1	12	yes	—
52	C₂²×D₁₃₉	—	2	2	no	—
53	C₂²×D₁₃₉	—	2	2	no	—
54	C₂²×D₁₃₉	C₂²×C₄	1	4	yes	—
55	C₂²×D₁₃₉	C₂⁴	1	2	yes	—
56	Dic₂₇₈	C₂²×C₄	1	48	yes	—
57	C₂×Dic₁₃₉	—	2	2	no	—
58	C₂×Dic₁₃₉	—	2	6	no	—
59	C₂×Dic₁₃₉	C₂²×C₄	1	12	yes	—
60	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
61	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
62	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
63	C₄×D₁₃₉	C₂²×C₄	1	48	yes	—
64	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
65	Q₈×C₁₃₉	C₂²×C₄	1	16	yes	—
66	C₂×C₄×C₁₃₉	—	2	6	no	—
67	C₂×C₄×C₁₃₉	—	2	2	no	—
68	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
69	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
70	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
71	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
72	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
73	C₂²×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
74	C₂²×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
75	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
76	C₂×D₁₃₉	—	2	8	no	—
77	C₂×D₁₃₉	—	2	2	no	—
78	C₂×D₁₃₉	—	2	2	no	—
79	C₂×D₁₃₉	C₂³×C₄	1	4	yes	—
80	C₄×C₁₃₉	—	2	6	no	—
81	C₄×C₁₃₉	—	2	2	no	—
82	C₄×C₁₃₉	C₂³×C₄	1	12	yes	—
83	Dic₁₃₉	—	2	2	no	—
84	Dic₁₃₉	—	2	6	no	—
85	Dic₁₃₉	C₂³×C₄	1	12	yes	—
86	C₂×C₁₃₉	#519	1	4	yes	complemented
87	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
88	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
89	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
90	D₁₃₉	—	2	4	no	—
91	C₁₃₉	C₂×#519	1	1	yes	characteristic, complemented, Sylow
92	C₂×#519	—	139	1	no	maximal, Sylow
93	C₂³×Q₈	—	139	1	no	—
94	#519	—	139	16	no	—
95	C₂×C₄×Q₈	—	139	1	no	—
96	C₂×C₄×Q₈	—	139	3	no	—
97	C₂×C₄×Q₈	—	139	3	no	—
98	C₂×C₄×Q₈	D₁₃₉	1	1	yes	characteristic, complemented
99	C₂×#117	—	139	3	no	—
100	C₂×#117	—	139	3	no	—
101	C₂²×Q₈	—	139	16	no	—
102	C₂²×Q₈	—	139	8	no	—
103	C₂²×Q₈	—	139	3	no	—
104	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
105	C₂³×C₄	—	139	3	no	—
106	C₄×Q₈	—	139	8	no	—
107	C₄×Q₈	—	139	24	no	—
108	C₄×Q₈	—	139	24	no	—
109	C₄×Q₈	C₂×D₁₃₉	1	8	yes	complemented
110	#117	—	139	24	no	—
111	#117	—	139	24	no	—
112	C₂×C₄⋊C₄	—	139	3	no	—
113	C₂×C₄⋊C₄	—	139	3	no	—
114	C₂×C₄⋊C₄	—	139	3	no	—
115	C₂×C₄⋊C₄	C₂×D₁₃₉	1	3	yes	—
116	C₂×C₂²⋊C₄	—	139	3	no	—
117	C₂×C₂²⋊C₄	—	139	1	no	—
118	C₂×C₄²	—	139	3	no	—
119	C₂×C₄²	—	139	6	no	—
120	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
121	C₂⁴	—	139	1	no	—
122	C₂×Q₈	—	139	64	no	—
123	C₂×Q₈	—	139	24	no	—
124	C₂×Q₈	—	139	12	no	—
125	C₂×Q₈	C₄×D₁₃₉	1	8	yes	—
126	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
127	C₂²×C₄	—	139	1	no	—
128	C₂²×C₄	—	139	3	no	—
129	C₂²×C₄	—	139	3	no	—
130	C₂²×C₄	—	139	24	no	—
131	C₂²×C₄	—	139	12	no	—
132	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
133	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
135	C₄⋊C₄	—	139	12	no	—
136	C₄⋊C₄	—	139	12	no	—
137	C₄⋊C₄	—	139	12	no	—
138	C₄⋊C₄	C₂²×D₁₃₉	1	12	yes	—
139	C₂²⋊C₄	—	139	12	no	—
140	C₂²⋊C₄	—	139	4	no	—
141	C₄²	—	139	12	no	—
142	C₄²	—	139	24	no	—
143	C₄²	C₂²×D₁₃₉	1	12	yes	—
144	C₂³	—	278	2	no	—
145	C₂³	—	278	2	no	—
146	C₂³	—	139	4	no	—
147	C₂³	—	139	2	no	—
148	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
149	Q₈	—	139	48	no	—
150	Q₈	C₂×C₄×D₁₃₉	1	16	yes	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	278	2	no	—
154	C₂×C₄	—	278	6	no	—
155	C₂×C₄	—	139	2	no	—
156	C₂×C₄	—	139	6	no	—
157	C₂×C₄	—	139	12	no	—
158	C₂×C₄	—	139	6	no	—
159	C₂×C₄	—	139	48	no	—
160	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	—
161	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
162	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
163	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
164	C₂²	—	278	2	no	—
165	C₂²	—	278	2	no	—
166	C₂²	—	278	8	no	—
167	C₂²	—	139	4	no	—
168	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
169	C₂²	#17984889	1	2	yes	—
170	C₂²	#17984890	1	2	yes	—
171	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
172	C₄	—	2	6	no	—
173	C₄	—	2	2	no	—
174	C₄	—	278	2	no	—
175	C₄	—	278	6	no	—
176	C₄	—	139	12	no	—
177	C₄	C₂²×C₄×D₁₃₉	1	12	yes	—
178	C₂	—	278	4	no	—
179	C₂	#22581429	1	4	yes	direct product factor
180	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
181	C₂	C₂×#17984889	1	1	yes	characteristic
182	C₂	C₂×#17984890	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×C₄×Q₈×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂³×D₁₃₉	C₂³
cosocle	C₂²×C₁₃₉	C₂⁵
hypercenter	C₂×C₄×Q₈	D₁₃₉
hypocenter	C₁₃₉	C₂×#519
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22581429	C₂	direct (N × H)
3. C₂×Q₈×Dic₁₃₉	C₂	semidirect
4. C₂×C₄×Dic₂₇₈	C₂	semidirect
5. C₂×#17984807	C₂	semidirect
8. C₂×C₄×Q₈×C₁₃₉	C₂	semidirect
10. C₂×Q₈×D₁₃₉	C₄	semidirect
13. Q₈×Dic₁₃₉	C₂²	semidirect
14. C₄×Dic₂₇₈	C₂²	semidirect
15. #17984807	C₂²	semidirect
26. C₄×Q₈×C₁₃₉	C₂²	semidirect
32. Q₈×D₁₃₉	C₂×C₄	semidirect
86. C₂×C₁₃₉	#519	semidirect
91. C₁₃₉	C₂×#519	semidirect
98. C₂×C₄×Q₈	D₁₃₉	semidirect
109. C₄×Q₈	C₂×D₁₃₉	semidirect
179. C₂	#22581429	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×Q₈×D₁₃₉	2
#22581429	2
C₂×Q₈×Dic₁₃₉	2
C₂×C₄×Dic₂₇₈	2
C₂×#17984807	2
C₂×#17984800	2
C₂×#17984799	2
C₂×C₄×Q₈×C₁₃₉	2
C₂×#519	139

Subgroups — C2×#22581429 #31819593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581429 #31819593