Subgroups — C₂₉×#542 #14270837

Order 1856.

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	C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₉×#126	C₂	1	2	yes	maximal, complemented
3	C₂₉×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#124	C₂	1	2	yes	maximal, complemented
5	C₂₉×#123	C₂	1	2	yes	maximal, complemented
6	C₂₉×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
7	C₂₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#121	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₂₉	C₂²	1	2	yes	—
14	C₂×D₄×C₂₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₂₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₂₉	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
24	C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
25	C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
28	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
31	C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
32	C₄²×C₂₉	C₂²	1	1	yes	characteristic
33	C₄²×C₂₉	C₂²	1	1	yes	characteristic
34	C₄²×C₂₉	C₂²	1	2	yes	—
35	C₂³×C₂₉	C₂³	1	1	yes	characteristic
36	C₂³×C₂₉	C₂³	1	2	yes	—
37	Q₈×C₂₉	—	2	2	no	—
38	Q₈×C₂₉	—	4	2	no	—
39	D₄×C₂₉	—	2	2	no	—
40	D₄×C₂₉	—	4	2	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	—	4	2	no	—
43	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂³	1	2	yes	—
46	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂³	1	2	yes	—
49	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂³	1	2	yes	—
51	C₂×C₄×C₂₉	C₂³	1	1	yes	characteristic
52	C₂²×C₂₉	—	2	1	no	—
53	C₂²×C₂₉	—	2	2	no	—
54	C₂²×C₂₉	—	2	2	no	—
55	C₂²×C₂₉	—	2	1	no	—
56	C₂²×C₂₉	—	2	1	no	—
57	C₂²×C₂₉	—	2	2	no	—
58	C₂²×C₂₉	C₂⁴	1	1	yes	characteristic
59	C₄×C₂₉	—	2	1	no	—
60	C₄×C₂₉	—	2	2	no	—
61	C₄×C₂₉	—	2	1	no	—
62	C₄×C₂₉	—	2	2	no	—
63	C₄×C₂₉	—	2	2	no	—
64	C₄×C₂₉	—	2	1	no	—
65	C₄×C₂₉	—	2	1	no	—
66	C₄×C₂₉	—	2	1	no	—
67	C₄×C₂₉	#41	1	2	yes	—
68	#542	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×C₂₉	—	4	1	no	—
70	C₂×C₂₉	—	4	2	no	—
71	C₂×C₂₉	C₂×#41	1	1	yes	characteristic
72	C₂×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
73	C₂×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
74	C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
75	#126	C₂×C₂₉	1	2	yes	complemented
76	#124	C₂×C₂₉	1	1	yes	characteristic, complemented
77	#124	C₂×C₂₉	1	2	yes	complemented
78	#123	C₂×C₂₉	1	2	yes	complemented
79	C₂²⋊Q₈	C₂×C₂₉	1	2	yes	complemented
80	C₂²⋊Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
81	#121	C₂×C₂₉	1	1	yes	characteristic
82	C₄×Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
83	C₄×D₄	C₂×C₂₉	1	1	yes	characteristic, complemented
84	#117	C₂×C₂₉	1	1	yes	characteristic, complemented
85	C₂₉	#542	1	1	yes	characteristic, direct product factor, Sylow
86	C₂×Q₈	C₂²×C₂₉	1	1	yes	characteristic
87	C₂×Q₈	C₂²×C₂₉	1	2	yes	—
88	C₂×D₄	C₂²×C₂₉	1	1	yes	characteristic
89	C₂×D₄	C₂²×C₂₉	1	2	yes	—
90	C₂²×C₄	C₂²×C₂₉	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₂₉	1	2	yes	—
92	C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
93	C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
94	C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
95	C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
96	C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
97	C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
98	C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
99	C₂²⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
100	C₂²⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
101	C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
102	C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
103	C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
104	C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
105	C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
106	C₄²	C₂²×C₂₉	1	1	yes	characteristic
107	C₄²	C₂²×C₂₉	1	1	yes	characteristic
108	C₄²	C₂²×C₂₉	1	2	yes	—
109	C₂³	C₂³×C₂₉	1	1	yes	characteristic
110	C₂³	C₂³×C₂₉	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×C₂₉	1	1	yes	characteristic
118	C₂×C₄	C₂³×C₂₉	1	1	yes	characteristic
119	C₂×C₄	C₂³×C₂₉	1	2	yes	—
120	C₂×C₄	C₂³×C₂₉	1	1	yes	characteristic
121	C₂×C₄	C₂³×C₂₉	1	1	yes	characteristic
122	C₂×C₄	C₂³×C₂₉	1	2	yes	—
123	C₂×C₄	C₂³×C₂₉	1	1	yes	characteristic
124	C₂×C₄	C₂³×C₂₉	1	2	yes	—
125	C₂×C₄	C₂³×C₂₉	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×C₂₉	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×C₂₉	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×#41×C₂₉	1	1	yes	characteristic
145	C₂	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₉
prime radical	C₂₉×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×C₄⋊Q₈	C₂	semidirect
2. C₂₉×#126	C₂	semidirect
3. C₂₉×#124	C₂	semidirect
4. C₂₉×#124	C₂	semidirect
5. C₂₉×#123	C₂	semidirect
6. C₂₉×C₂²⋊Q₈	C₂	semidirect
7. C₂₉×C₂²⋊Q₈	C₂	semidirect
9. C₄×Q₈×C₂₉	C₂	semidirect
10. C₄×D₄×C₂₉	C₂	semidirect
11. C₂₉×#117	C₂	semidirect
68. #542	C₂₉	direct (N × H)
74. C₄⋊Q₈	C₂×C₂₉	semidirect
75. #126	C₂×C₂₉	semidirect
76. #124	C₂×C₂₉	semidirect
77. #124	C₂×C₂₉	semidirect
78. #123	C₂×C₂₉	semidirect
79. C₂²⋊Q₈	C₂×C₂₉	semidirect
80. C₂²⋊Q₈	C₂×C₂₉	semidirect
82. C₄×Q₈	C₂×C₂₉	semidirect
83. C₄×D₄	C₂×C₂₉	semidirect
84. #117	C₂×C₂₉	semidirect
85. C₂₉	#542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₉×C₄⋊Q₈	2
C₂₉×#126	2
C₂₉×#124	2
C₂₉×#123	2
C₂₉×C₂²⋊Q₈	2
C₂₉×#121	2
C₄×Q₈×C₂₉	2
C₄×D₄×C₂₉	2
C₂₉×#117	2
#542	29

Subgroups — C29×#542 #14270837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#542 #14270837