Subgroups — #31814829

Order 17792.

556 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984848	C₂	1	2	yes	maximal
2	C₄×#14555785	C₂	1	1	yes	maximal, characteristic
3	#22580910	C₂	1	4	yes	maximal, complemented
4	#22580908	C₂	1	2	yes	maximal, complemented
5	C₄×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580905	C₂	1	2	yes	maximal, complemented
7	#97×D₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555785	C₂²	1	1	yes	characteristic
10	#17984848	C₄	1	16	yes	complemented
11	C₂×#14555769	C₂²	1	2	yes	—
12	C₂×#14555768	C₂²	1	4	yes	—
13	C₂×C₈×D₁₃₉	C₂²	1	4	yes	—
14	C₄×Dic₂₇₈	C₄	1	2	yes	—
15	C₄×D₅₅₆	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₃₉	C₄	1	4	yes	—
17	#17984801	C₂²	1	1	yes	characteristic
18	#17984799	C₂²	1	2	yes	—
19	C₄²×D₁₃₉	C₂²	1	2	yes	—
20	#17984753	C₂²	1	2	yes	—
21	#17984752	C₂²	1	4	yes	—
22	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
23	C₈×Dic₁₃₉	C₂²	1	4	yes	—
24	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
26	#97×C₁₃₉	C₂²	1	1	yes	characteristic
27	#97×C₁₃₉	C₂²	1	1	yes	characteristic
28	#97×C₁₃₉	C₂²	1	2	yes	—
29	#14555785	C₂×C₄	1	8	yes	—
30	C₂×Dic₂₇₈	C₂×C₄	1	1	yes	characteristic
31	C₂×D₅₅₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
35	#14555769	C₂×C₄	1	8	yes	complemented
36	#14555768	C₂×C₄	1	16	yes	complemented
37	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
38	C₈×D₁₃₉	C₂×C₄	1	16	yes	complemented
39	C₄⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
40	Dic₁₃₉⋊C₄	C₂×C₄	1	4	yes	—
41	C₂²⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
42	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
43	C₄×Dic₁₃₉	C₂×C₄	1	4	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	8	yes	complemented
48	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
49	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
50	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
51	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
54	Dic₂₇₈	C₄²	1	4	yes	—
55	D₅₅₆	C₄²	1	4	yes	—
56	C₂²⋊D₁₃₉	C₄²	1	8	yes	—
57	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
59	C₄×D₁₃₉	—	4	4	no	—
60	C₄×D₁₃₉	C₂²×C₄	1	8	yes	—
61	C₁₃₉⋊C₈	C₂²×C₄	1	8	yes	—
62	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	—	4	2	no	—
64	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
70	C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
71	C₂²×C₁₃₉	—	2	2	no	—
72	C₂²×C₁₃₉	C₂×C₄²	1	2	yes	—
73	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
74	C₂×D₁₃₉	—	2	4	no	—
75	C₂×D₁₃₉	C₂×C₄²	1	4	yes	—
76	C₄×C₁₃₉	—	2	2	no	—
77	C₄×C₁₃₉	—	2	2	no	—
78	C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
79	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
80	Dic₁₃₉	—	2	4	no	—
81	Dic₁₃₉	C₂×C₄²	1	4	yes	—
82	C₂×C₁₃₉	—	2	2	no	—
83	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
84	C₂×C₁₃₉	#568	1	2	yes	—
85	D₁₃₉	—	2	4	no	—
86	C₁₃₉	#2875	1	1	yes	characteristic, complemented, Sylow
87	#2875	—	139	1	no	maximal, Sylow
88	C₂×#131	—	139	2	no	—
89	C₄×#41	—	139	1	no	—
90	#405	—	139	2	no	—
91	#405	—	139	4	no	—
92	C₄×M₁₆	—	139	1	no	—
93	C₄×M₁₆	—	139	2	no	—
94	C₂×#97	—	139	2	no	—
95	C₂×#97	D₁₃₉	1	1	yes	characteristic, complemented
96	C₂×#41	—	139	1	no	—
97	#131	—	139	16	no	—
98	C₂×M₁₆	—	139	2	no	—
99	C₂×M₁₆	—	139	4	no	—
100	C₂²×C₈	—	139	4	no	—
101	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
102	C₄×Q₈	—	139	2	no	—
103	C₄×D₄	—	139	4	no	—
104	C₄×D₄	—	139	2	no	—
105	#117	—	139	1	no	—
106	#117	—	139	2	no	—
107	C₂×C₄²	—	139	2	no	—
108	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
109	#97	—	139	4	no	—
110	#97	—	139	2	no	—
111	#97	C₂×D₁₃₉	1	1	yes	characteristic
112	#97	C₂×D₁₃₉	1	1	yes	characteristic
113	#97	C₂×D₁₃₉	1	2	yes	—
114	C₄×C₈	—	139	4	no	—
115	C₄×C₈	—	139	2	no	—
116	#41	—	139	8	no	—
117	C₂×Q₈	—	139	1	no	—
118	C₂×D₄	—	139	2	no	—
119	C₂×D₄	—	139	1	no	—
120	C₂²×C₄	—	139	4	no	—
121	C₂²×C₄	—	139	2	no	—
122	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
123	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
124	M₁₆	—	139	8	no	—
125	M₁₆	—	139	16	no	—
126	C₂×C₈	—	139	4	no	—
127	C₂×C₈	—	139	16	no	—
128	C₂×C₈	C₄×D₁₃₉	1	8	yes	complemented
129	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
130	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
131	C₄⋊C₄	—	139	4	no	—
132	C₄⋊C₄	—	139	2	no	—
133	C₂²⋊C₄	—	139	2	no	—
134	C₂²⋊C₄	—	139	4	no	—
135	C₄²	—	139	4	no	—
136	C₄²	—	139	2	no	—
137	C₄²	C₄×D₁₃₉	1	2	yes	—
138	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
139	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
140	C₂³	—	139	2	no	—
141	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
142	Q₈	—	139	4	no	—
143	D₄	—	139	8	no	—
144	D₄	—	139	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	556	4	no	—
147	C₂×C₄	—	139	4	no	—
148	C₂×C₄	—	139	2	no	—
149	C₂×C₄	—	139	8	no	—
150	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
152	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
154	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
155	C₈	—	139	8	no	—
156	C₈	C₂×C₄×D₁₃₉	1	4	yes	—
157	C₈	C₂×C₄×D₁₃₉	1	4	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	—	278	4	no	—
160	C₂²	—	139	4	no	—
161	C₂²	C₄²×D₁₃₉	1	2	yes	—
162	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	278	4	no	—
166	C₄	—	139	4	no	—
167	C₄	C₄²×D₁₃₉	1	2	yes	—
168	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	—	278	4	no	—
171	C₂	C₂×C₄²×D₁₃₉	1	1	yes	characteristic
172	C₂	#22581719	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂²×C₄²
Fitting	C₂×#97×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂³×C₄
prime radical	C₂×#14555785	C₂²
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	#2875
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. #22580910	C₂	semidirect
4. #22580908	C₂	semidirect
5. C₄×#14555769	C₂	semidirect
6. #22580905	C₂	semidirect
7. #97×D₁₃₉	C₂	semidirect
8. C₂×#97×C₁₃₉	C₂	semidirect
10. #17984848	C₄	semidirect
35. #14555769	C₂×C₄	semidirect
36. #14555768	C₂×C₄	semidirect
38. C₈×D₁₃₉	C₂×C₄	semidirect
47. C₂×C₈×C₁₃₉	C₂×C₄	semidirect
86. C₁₃₉	#2875	semidirect
95. C₂×#97	D₁₃₉	semidirect
128. C₂×C₈	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984848	2
C₄×#14555785	2
#22580910	2
#22580908	2
C₄×#14555769	2
#22580905	2
#97×D₁₃₉	2
C₂×#97×C₁₃₉	2
#2875	139