Subgroups — #542×C₄₁₉ #41000035

Order 26816.

216 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#126×C₄₁₉	C₂	1	2	yes	maximal, complemented
3	#124×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#124×C₄₁₉	C₂	1	2	yes	maximal, complemented
5	#123×C₄₁₉	C₂	1	2	yes	maximal, complemented
6	C₂²⋊Q₈×C₄₁₉	C₂	1	2	yes	maximal, complemented
7	C₂²⋊Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#121×C₄₁₉	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#117×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₄₁₉	C₂²	1	2	yes	—
14	C₂×D₄×C₄₁₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₄₁₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₄₁₉	C₂²	1	2	yes	—
18	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₄₁₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₄₁₉	C₂²	1	2	yes	—
24	C₄⋊C₄×C₄₁₉	C₂²	1	2	yes	—
25	C₂²⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₁₉	C₂²	1	2	yes	—
28	C₂²⋊C₄×C₄₁₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₄₁₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₄₁₉	C₂²	1	2	yes	—
31	C₂²⋊C₄×C₄₁₉	C₂²	1	2	yes	—
32	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
33	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
34	C₄²×C₄₁₉	C₂²	1	2	yes	—
35	C₂³×C₄₁₉	C₂³	1	1	yes	characteristic
36	C₂³×C₄₁₉	C₂³	1	2	yes	—
37	Q₈×C₄₁₉	—	2	2	no	—
38	Q₈×C₄₁₉	—	4	2	no	—
39	D₄×C₄₁₉	—	2	2	no	—
40	D₄×C₄₁₉	—	4	2	no	—
41	C₂×C₄×C₄₁₉	—	2	2	no	—
42	C₂×C₄×C₄₁₉	—	4	2	no	—
43	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
46	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
49	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
51	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄₁₉	—	2	1	no	—
53	C₂²×C₄₁₉	—	2	2	no	—
54	C₂²×C₄₁₉	—	2	2	no	—
55	C₂²×C₄₁₉	—	2	1	no	—
56	C₂²×C₄₁₉	—	2	1	no	—
57	C₂²×C₄₁₉	—	2	2	no	—
58	C₂²×C₄₁₉	C₂⁴	1	1	yes	characteristic
59	C₄×C₄₁₉	—	2	1	no	—
60	C₄×C₄₁₉	—	2	2	no	—
61	C₄×C₄₁₉	—	2	1	no	—
62	C₄×C₄₁₉	—	2	2	no	—
63	C₄×C₄₁₉	—	2	2	no	—
64	C₄×C₄₁₉	—	2	1	no	—
65	C₄×C₄₁₉	—	2	1	no	—
66	C₄×C₄₁₉	—	2	1	no	—
67	C₄×C₄₁₉	#41	1	2	yes	—
68	C₂×C₄₁₉	—	4	1	no	—
69	C₂×C₄₁₉	—	4	2	no	—
70	C₂×C₄₁₉	C₂×#41	1	1	yes	characteristic
71	C₂×C₄₁₉	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₄₁₉	2₋¹⁺⁴	1	1	yes	characteristic
73	C₄₁₉	#542	1	1	yes	characteristic, direct product factor, Sylow
74	#542	C₄₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₄⋊Q₈	C₂×C₄₁₉	1	1	yes	characteristic, complemented
76	#126	C₂×C₄₁₉	1	2	yes	complemented
77	#124	C₂×C₄₁₉	1	1	yes	characteristic, complemented
78	#124	C₂×C₄₁₉	1	2	yes	complemented
79	#123	C₂×C₄₁₉	1	2	yes	complemented
80	C₂²⋊Q₈	C₂×C₄₁₉	1	2	yes	complemented
81	C₂²⋊Q₈	C₂×C₄₁₉	1	1	yes	characteristic, complemented
82	#121	C₂×C₄₁₉	1	1	yes	characteristic
83	C₄×Q₈	C₂×C₄₁₉	1	1	yes	characteristic, complemented
84	C₄×D₄	C₂×C₄₁₉	1	1	yes	characteristic, complemented
85	#117	C₂×C₄₁₉	1	1	yes	characteristic, complemented
86	C₂×Q₈	C₂²×C₄₁₉	1	1	yes	characteristic
87	C₂×Q₈	C₂²×C₄₁₉	1	2	yes	—
88	C₂×D₄	C₂²×C₄₁₉	1	1	yes	characteristic
89	C₂×D₄	C₂²×C₄₁₉	1	2	yes	—
90	C₂²×C₄	C₂²×C₄₁₉	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄₁₉	1	2	yes	—
92	C₄⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
93	C₄⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
94	C₄⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
95	C₄⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
96	C₄⋊C₄	C₂²×C₄₁₉	1	2	yes	—
97	C₄⋊C₄	C₂²×C₄₁₉	1	2	yes	—
98	C₄⋊C₄	C₂²×C₄₁₉	1	2	yes	—
99	C₂²⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
100	C₂²⋊C₄	C₂²×C₄₁₉	1	1	yes	characteristic
101	C₂²⋊C₄	C₂²×C₄₁₉	1	2	yes	—
102	C₂²⋊C₄	C₂²×C₄₁₉	1	2	yes	—
103	C₂²⋊C₄	C₂²×C₄₁₉	1	2	yes	—
104	C₂²⋊C₄	C₂²×C₄₁₉	1	2	yes	—
105	C₂²⋊C₄	C₂²×C₄₁₉	1	2	yes	—
106	C₄²	C₂²×C₄₁₉	1	1	yes	characteristic
107	C₄²	C₂²×C₄₁₉	1	1	yes	characteristic
108	C₄²	C₂²×C₄₁₉	1	2	yes	—
109	C₂³	C₂³×C₄₁₉	1	1	yes	characteristic
110	C₂³	C₂³×C₄₁₉	1	2	yes	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	2	no	—
113	D₄	—	2	2	no	—
114	D₄	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
118	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
119	C₂×C₄	C₂³×C₄₁₉	1	2	yes	—
120	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
122	C₂×C₄	C₂³×C₄₁₉	1	2	yes	—
123	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
124	C₂×C₄	C₂³×C₄₁₉	1	2	yes	—
125	C₂×C₄	C₂³×C₄₁₉	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂⁴×C₄₁₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	#41×C₄₁₉	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×#41×C₄₁₉	1	1	yes	characteristic
145	C₂	H(5,𝔽₂)×C₄₁₉	1	1	yes	characteristic
146	C₂	2₋¹⁺⁴×C₄₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₄₁₉
prime radical	#121×C₄₁₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₄₁₉	C₂	semidirect
2. #126×C₄₁₉	C₂	semidirect
3. #124×C₄₁₉	C₂	semidirect
4. #124×C₄₁₉	C₂	semidirect
5. #123×C₄₁₉	C₂	semidirect
6. C₂²⋊Q₈×C₄₁₉	C₂	semidirect
7. C₂²⋊Q₈×C₄₁₉	C₂	semidirect
9. C₄×Q₈×C₄₁₉	C₂	semidirect
10. C₄×D₄×C₄₁₉	C₂	semidirect
11. #117×C₄₁₉	C₂	semidirect
73. C₄₁₉	#542	direct (N × H)
74. #542	C₄₁₉	direct (N × H)
75. C₄⋊Q₈	C₂×C₄₁₉	semidirect
76. #126	C₂×C₄₁₉	semidirect
77. #124	C₂×C₄₁₉	semidirect
78. #124	C₂×C₄₁₉	semidirect
79. #123	C₂×C₄₁₉	semidirect
80. C₂²⋊Q₈	C₂×C₄₁₉	semidirect
81. C₂²⋊Q₈	C₂×C₄₁₉	semidirect
83. C₄×Q₈	C₂×C₄₁₉	semidirect
84. C₄×D₄	C₂×C₄₁₉	semidirect
85. #117	C₂×C₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₄₁₉	2
#126×C₄₁₉	2
#124×C₄₁₉	2
#123×C₄₁₉	2
C₂²⋊Q₈×C₄₁₉	2
#121×C₄₁₉	2
C₄×Q₈×C₄₁₉	2
C₄×D₄×C₄₁₉	2
#117×C₄₁₉	2
#542	419

Subgroups — #542×C419 #41000035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #542×C₄₁₉ #41000035