Subgroups — #553×C₂₃₉ #29504457

Order 15296.

238 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#126×C₂₃₉	C₂	1	2	yes	maximal, complemented
2	#125×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#123×C₂₃₉	C₂	1	2	yes	maximal, complemented
4	#121×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	#121×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#121×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×C₂₃₉	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#117×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
13	C₂×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₂₃₉	C₂²	1	2	yes	—
19	C₂²×C₄×C₂₃₉	C₂²	1	2	yes	—
20	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
25	C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
26	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
27	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
28	C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
29	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
31	C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
33	C₄²×C₂₃₉	C₂²	1	2	yes	—
34	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
35	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
36	C₂³×C₂₃₉	C₂³	1	2	yes	—
37	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
38	D₄×C₂₃₉	—	2	4	no	—
39	D₄×C₂₃₉	—	2	2	no	—
40	D₄×C₂₃₉	—	2	2	no	—
41	D₄×C₂₃₉	—	4	2	no	—
42	C₂×C₄×C₂₃₉	—	2	2	no	—
43	C₂×C₄×C₂₃₉	—	2	2	no	—
44	C₂×C₄×C₂₃₉	—	4	2	no	—
45	C₂×C₄×C₂₃₉	—	4	2	no	—
46	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
48	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
50	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
53	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
54	C₂²×C₂₃₉	—	2	1	no	—
55	C₂²×C₂₃₉	—	2	1	no	—
56	C₂²×C₂₃₉	—	2	1	no	—
57	C₂²×C₂₃₉	—	2	1	no	—
58	C₂²×C₂₃₉	—	2	2	no	—
59	C₂²×C₂₃₉	—	2	2	no	—
60	C₂²×C₂₃₉	—	2	2	no	—
61	C₂²×C₂₃₉	—	2	1	no	—
62	C₂²×C₂₃₉	#41	1	2	yes	—
63	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
64	C₄×C₂₃₉	—	2	1	no	—
65	C₄×C₂₃₉	—	2	1	no	—
66	C₄×C₂₃₉	—	2	2	no	—
67	C₄×C₂₃₉	—	2	1	no	—
68	C₄×C₂₃₉	—	2	1	no	—
69	C₄×C₂₃₉	—	2	2	no	—
70	C₄×C₂₃₉	—	2	2	no	—
71	C₄×C₂₃₉	#41	1	2	yes	—
72	C₂×C₂₃₉	—	2	2	no	—
73	C₂×C₂₃₉	—	4	1	no	—
74	C₂×C₂₃₉	—	4	2	no	—
75	C₂×C₂₃₉	C₂×#41	1	1	yes	characteristic
76	C₂×C₂₃₉	C₂×#41	1	1	yes	characteristic
77	C₂×C₂₃₉	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂₃₉	#553	1	1	yes	characteristic, direct product factor, Sylow
79	#553	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	#126	C₂×C₂₃₉	1	2	yes	complemented
81	#125	C₂×C₂₃₉	1	1	yes	characteristic, complemented
82	#123	C₂×C₂₃₉	1	2	yes	complemented
83	#121	C₂×C₂₃₉	1	2	yes	complemented
84	#121	C₂×C₂₃₉	1	1	yes	characteristic, complemented
85	#121	C₂×C₂₃₉	1	1	yes	characteristic, complemented
86	C₄×D₄	C₂×C₂₃₉	1	2	yes	complemented
87	C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
88	C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
89	#117	C₂×C₂₃₉	1	1	yes	characteristic, complemented
90	C₂×C₄⋊C₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
91	C₂×D₄	C₂²×C₂₃₉	1	2	yes	—
92	C₂×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
93	C₂×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
94	C₂×D₄	C₂²×C₂₃₉	1	2	yes	—
95	C₂²×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×C₂₃₉	1	2	yes	—
98	C₂²×C₄	C₂²×C₂₃₉	1	2	yes	—
99	C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
100	C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
101	C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
102	C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
103	C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
104	C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
105	C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
106	C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
107	C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
108	C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
109	C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
110	C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
111	C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
112	C₄²	C₂²×C₂₃₉	1	2	yes	—
113	C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
114	C₂³	C₂³×C₂₃₉	1	1	yes	characteristic
115	C₂³	C₂³×C₂₃₉	1	2	yes	—
116	C₂³	C₂³×C₂₃₉	1	1	yes	characteristic
117	D₄	—	2	4	no	—
118	D₄	—	2	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	4	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
126	C₂×C₄	C₂³×C₂₃₉	1	2	yes	—
127	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
128	C₂×C₄	C₂³×C₂₃₉	1	2	yes	—
129	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
130	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
131	C₂×C₄	C₂³×C₂₃₉	1	2	yes	—
132	C₂×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#41×C₂₃₉	1	2	yes	—
142	C₂²	C₂⁴×C₂₃₉	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	#41×C₂₃₉	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	1	no	—
153	C₂	—	4	2	no	—
154	C₂	C₂×#41×C₂₃₉	1	1	yes	characteristic
155	C₂	C₂×#41×C₂₃₉	1	1	yes	characteristic
156	C₂	H(5,𝔽₂)×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #126×C₂₃₉	C₂	semidirect
2. #125×C₂₃₉	C₂	semidirect
3. #123×C₂₃₉	C₂	semidirect
4. #121×C₂₃₉	C₂	semidirect
5. #121×C₂₃₉	C₂	semidirect
6. #121×C₂₃₉	C₂	semidirect
7. C₄×D₄×C₂₃₉	C₂	semidirect
8. C₄×D₄×C₂₃₉	C₂	semidirect
9. C₄×D₄×C₂₃₉	C₂	semidirect
10. #117×C₂₃₉	C₂	semidirect
11. C₂×C₄⋊C₄×C₂₃₉	C₂	semidirect
78. C₂₃₉	#553	direct (N × H)
79. #553	C₂₃₉	direct (N × H)
80. #126	C₂×C₂₃₉	semidirect
81. #125	C₂×C₂₃₉	semidirect
82. #123	C₂×C₂₃₉	semidirect
83. #121	C₂×C₂₃₉	semidirect
84. #121	C₂×C₂₃₉	semidirect
85. #121	C₂×C₂₃₉	semidirect
86. C₄×D₄	C₂×C₂₃₉	semidirect
87. C₄×D₄	C₂×C₂₃₉	semidirect
88. C₄×D₄	C₂×C₂₃₉	semidirect
89. #117	C₂×C₂₃₉	semidirect
90. C₂×C₄⋊C₄	C₂×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#126×C₂₃₉	2
#125×C₂₃₉	2
#123×C₂₃₉	2
#121×C₂₃₉	2
C₄×D₄×C₂₃₉	2
#117×C₂₃₉	2
C₂×C₄⋊C₄×C₂₃₉	2
#553	239

Subgroups — #553×C239 #29504457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #553×C₂₃₉ #29504457