Subgroups — C₂×#22580910 #31814818

Order 17792.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555768	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#17984799	C₂	1	1	yes	maximal, characteristic
4	#22580910	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17984752	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#14555768	C₄	1	8	yes	complemented
11	C₂×#14555768	C₂²	1	4	yes	—
12	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₁₃₉	C₄	1	8	yes	complemented
15	C₂×C₈×D₁₃₉	C₂²	1	4	yes	—
16	#17984799	C₄	1	4	yes	—
17	#17984799	C₂²	1	4	yes	—
18	C₂×Dic₁₃₉⋊C₄	C₄	1	2	yes	—
19	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	2	yes	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
21	#17984752	C₂²	1	4	yes	complemented
22	C₄×C₁₃₉⋊C₈	C₂²	1	4	yes	complemented
23	C₈×Dic₁₃₉	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
27	#97×C₁₃₉	C₂²	1	4	yes	complemented
28	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
30	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
35	#14555768	C₂×C₄	1	16	yes	complemented
36	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
39	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
40	C₈×D₁₃₉	C₂×C₄	1	16	yes	complemented
41	Dic₁₃₉⋊C₄	C₂×C₄	1	8	yes	—
42	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₃₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
50	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
51	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
52	C₄²×C₁₃₉	C₂³	1	2	yes	—
53	C₂²×D₁₃₉	—	2	4	no	—
54	C₂²×D₁₃₉	C₄²	1	4	yes	—
55	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
56	C₂×Dic₁₃₉	—	2	4	no	—
57	C₂×Dic₁₃₉	C₄²	1	4	yes	—
58	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
60	C₄×D₁₃₉	C₄²	1	8	yes	—
61	C₄×D₁₃₉	C₂²×C₄	1	4	yes	—
62	C₄×D₁₃₉	C₂²×C₄	1	4	yes	—
63	C₁₃₉⋊C₈	C₂²×C₄	1	4	yes	—
64	C₁₃₉⋊C₈	C₂²×C₄	1	4	yes	—
65	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	—	2	4	no	—
67	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
72	C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
73	C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
74	C₂²×C₁₃₉	C₂×C₄²	1	2	yes	—
75	C₂²×C₁₃₉	#131	1	4	yes	—
76	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₁₃₉	—	2	4	no	—
78	C₂×D₁₃₉	—	2	8	no	—
79	C₂×D₁₃₉	C₂×C₄²	1	4	yes	—
80	C₄×C₁₃₉	—	2	4	no	—
81	C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
82	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
84	Dic₁₃₉	—	2	4	no	—
85	Dic₁₃₉	C₂×C₄²	1	4	yes	—
86	C₂×C₁₃₉	#405	1	4	yes	complemented
87	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₁₃₉	C₂×#131	1	2	yes	—
89	D₁₃₉	—	2	4	no	—
90	C₁₃₉	C₂×#405	1	1	yes	characteristic, complemented, Sylow
91	C₂×#405	—	139	1	no	maximal, Sylow
92	C₂²×M₁₆	—	139	1	no	—
93	C₂³×C₈	—	139	1	no	—
94	C₂×#117	—	139	1	no	—
95	#405	—	139	8	no	—
96	C₂×#97	—	139	1	no	—
97	C₂×#97	D₁₃₉	1	1	yes	characteristic, complemented
98	C₂×C₄×C₈	—	139	1	no	—
99	C₂×C₄×C₈	—	139	1	no	—
100	C₂³×C₄	—	139	1	no	—
101	C₂×M₁₆	—	139	8	no	—
102	C₂×M₁₆	—	139	4	no	—
103	C₂²×C₈	—	139	1	no	—
104	C₂²×C₈	—	139	1	no	—
105	C₂²×C₈	—	139	8	no	—
106	C₂²×C₈	—	139	4	no	—
107	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
109	#117	—	139	4	no	—
110	#117	—	139	4	no	—
111	C₂×C₄⋊C₄	—	139	2	no	—
112	C₂×C₂²⋊C₄	—	139	2	no	—
113	C₂×C₄²	—	139	1	no	—
114	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
115	#97	—	139	4	no	—
116	#97	C₂×D₁₃₉	1	4	yes	complemented
117	C₄×C₈	—	139	4	no	—
118	C₄×C₈	—	139	4	no	—
119	C₂⁴	—	139	1	no	—
120	C₂²×C₄	—	139	2	no	—
121	C₂²×C₄	—	139	1	no	—
122	C₂²×C₄	—	139	2	no	—
123	C₂²×C₄	—	139	4	no	—
124	C₂²×C₄	—	139	4	no	—
125	C₂²×C₄	—	139	2	no	—
126	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
127	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
128	M₁₆	—	139	16	no	—
129	C₂×C₈	—	139	4	no	—
130	C₂×C₈	—	139	4	no	—
131	C₂×C₈	—	139	2	no	—
132	C₂×C₈	—	139	2	no	—
133	C₂×C₈	—	139	16	no	—
134	C₂×C₈	C₄×D₁₃₉	1	4	yes	—
135	C₂×C₈	C₄×D₁₃₉	1	4	yes	—
136	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
137	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
138	C₄⋊C₄	—	139	8	no	—
139	C₂²⋊C₄	—	139	8	no	—
140	C₄²	—	139	2	no	—
141	C₄²	—	139	2	no	—
142	C₄²	C₄×D₁₃₉	1	2	yes	—
143	C₄²	C₂²×D₁₃₉	1	2	yes	—
144	C₂³	—	278	4	no	—
145	C₂³	—	139	4	no	—
146	C₂³	—	139	2	no	—
147	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	278	4	no	—
150	C₂×C₄	—	139	4	no	—
151	C₂×C₄	—	139	4	no	—
152	C₂×C₄	—	139	2	no	—
153	C₂×C₄	—	139	8	no	—
154	C₂×C₄	—	139	4	no	—
155	C₂×C₄	—	139	4	no	—
156	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
160	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
161	C₈	—	139	4	no	—
162	C₈	—	139	4	no	—
163	C₈	C₂×C₄×D₁₃₉	1	4	yes	—
164	C₈	C₂×C₄×D₁₃₉	1	4	yes	—
165	C₂²	—	278	8	no	—
166	C₂²	—	278	4	no	—
167	C₂²	—	139	4	no	—
168	C₂²	C₄²×D₁₃₉	1	2	yes	—
169	C₂²	#17984850	1	4	yes	—
170	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	278	4	no	—
173	C₄	—	139	4	no	—
174	C₄	C₄²×D₁₃₉	1	2	yes	—
175	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
177	C₂	—	278	4	no	—
178	C₂	#22580910	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₁₃₉	1	1	yes	characteristic
180	C₂	C₂×#17984850	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂×C₁₃₉	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#97×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂²×C₄
prime radical	C₂³×D₁₃₉	C₂×C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	C₂×#405
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. #22580910	C₂	direct (N × H)
5. C₂×#17984752	C₂	semidirect
6. C₂×C₄×C₁₃₉⋊C₈	C₂	semidirect
7. C₂×C₈×Dic₁₃₉	C₂	semidirect
8. C₂×#97×C₁₃₉	C₂	semidirect
10. C₂×#14555768	C₄	semidirect
14. C₂×C₈×D₁₃₉	C₄	semidirect
21. #17984752	C₂²	semidirect
22. C₄×C₁₃₉⋊C₈	C₂²	semidirect
23. C₈×Dic₁₃₉	C₂²	semidirect
27. #97×C₁₃₉	C₂²	semidirect
35. #14555768	C₂×C₄	semidirect
40. C₈×D₁₃₉	C₂×C₄	semidirect
86. C₂×C₁₃₉	#405	semidirect
90. C₁₃₉	C₂×#405	semidirect
97. C₂×#97	D₁₃₉	semidirect
116. #97	C₂×D₁₃₉	semidirect
178. C₂	#22580910	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×#14555768	2
C₂²×C₈×D₁₃₉	2
C₂×#17984799	2
#22580910	2
C₂×#17984752	2
C₂×C₄×C₁₃₉⋊C₈	2
C₂×C₈×Dic₁₃₉	2
C₂×#97×C₁₃₉	2
C₂×#405	139

Subgroups — C2×#22580910 #31814818

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22580910 #31814818