Subgroups — C₂₉×#2972 #16816686

Order 3712.

500 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₈×D₄×C₂₉	C₂	1	8	yes	maximal, complemented
3	C₂₉×#431	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₉×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×C₂₉	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₉	C₂²	1	8	yes	complemented
8	C₄×D₄×C₂₉	C₄	1	4	yes	—
9	C₄×D₄×C₂₉	C₂²	1	4	yes	—
10	C₂×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊C₄×C₂₉	C₄	1	2	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
13	C₂₉×C₄⋊C₈	C₂²	1	4	yes	complemented
14	C₂₉×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂₉×C₂²⋊C₈	C₂²	1	8	yes	complemented
16	C₄×C₈×C₂₉	C₂²	1	4	yes	complemented
17	C₂⁴×C₂₉	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₂₉	C₈	1	8	yes	complemented
19	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₉	—	4	4	no	—
21	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
27	C₂×C₈×C₂₉	—	4	8	no	—
28	C₂×C₈×C₂₉	C₂³	1	8	yes	—
29	C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
30	C₂²⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
31	C₄²×C₂₉	C₂×C₄	1	2	yes	—
32	C₄²×C₂₉	C₂³	1	2	yes	—
33	C₂³×C₂₉	—	4	2	no	—
34	C₂³×C₂₉	—	4	2	no	—
35	C₂³×C₂₉	C₂×C₈	1	8	yes	complemented
36	C₂³×C₂₉	C₂²×C₄	1	4	yes	—
37	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₂₉	C₂×C₈	1	16	yes	complemented
39	C₂×C₄×C₂₉	—	2	8	no	—
40	C₂×C₄×C₂₉	—	2	2	no	—
41	C₂×C₄×C₂₉	—	4	4	no	—
42	C₂×C₄×C₂₉	—	4	2	no	—
43	C₂×C₄×C₂₉	—	4	1	no	—
44	C₂×C₄×C₂₉	C₂×C₈	1	4	yes	complemented
45	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
50	C₈×C₂₉	—	2	8	no	—
51	#2972	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₂₉	—	2	1	no	—
53	C₂²×C₂₉	—	2	4	no	—
54	C₂²×C₂₉	—	2	1	no	—
55	C₂²×C₂₉	—	2	4	no	—
56	C₂²×C₂₉	—	4	8	no	—
57	C₂²×C₂₉	C₂²×C₈	1	2	yes	—
58	C₂²×C₂₉	C₂²×C₈	1	8	yes	—
59	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₂₉	—	2	1	no	—
61	C₄×C₂₉	—	2	4	no	—
62	C₄×C₂₉	—	2	2	no	—
63	C₄×C₂₉	C₂²×C₈	1	4	yes	—
64	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
65	C₂×C₄×D₄	C₂×C₂₉	1	1	yes	characteristic
66	C₈×D₄	C₂×C₂₉	1	8	yes	complemented
67	#431	C₂×C₂₉	1	2	yes	complemented
68	C₂×C₂²⋊C₈	C₂×C₂₉	1	4	yes	complemented
69	C₂×C₂₉	—	2	2	no	—
70	C₂×C₂₉	—	2	8	no	—
71	C₂×C₂₉	#518	1	1	yes	characteristic
72	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
73	C₂×C₂₉	#568	1	1	yes	characteristic
74	C₂²×D₄	C₄×C₂₉	1	1	yes	characteristic
75	C₂³×C₄	C₂²×C₂₉	1	2	yes	—
76	C₂²×C₈	C₂²×C₂₉	1	8	yes	complemented
77	C₄×D₄	C₄×C₂₉	1	4	yes	—
78	C₄×D₄	C₂²×C₂₉	1	4	yes	—
79	C₂×C₄⋊C₄	C₄×C₂₉	1	1	yes	characteristic
80	C₂×C₂²⋊C₄	C₄×C₂₉	1	2	yes	—
81	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
82	C₄⋊C₈	C₂²×C₂₉	1	4	yes	complemented
83	C₂²⋊C₈	C₂²×C₂₉	1	4	yes	—
84	C₂²⋊C₈	C₂²×C₂₉	1	8	yes	complemented
85	C₄×C₈	C₂²×C₂₉	1	4	yes	complemented
86	C₂₉	#2972	1	1	yes	characteristic, direct product factor, Sylow
87	C₂⁴	C₂×C₄×C₂₉	1	2	yes	—
88	C₂×D₄	C₈×C₂₉	1	8	yes	complemented
89	C₂×D₄	C₂×C₄×C₂₉	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
92	C₂²×C₄	C₂×C₄×C₂₉	1	2	yes	—
93	C₂²×C₄	C₂×C₄×C₂₉	1	4	yes	—
94	C₂²×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
97	C₂×C₈	—	4	8	no	—
98	C₂×C₈	C₂³×C₂₉	1	8	yes	—
99	C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
100	C₂²⋊C₄	C₂×C₄×C₂₉	1	8	yes	—
101	C₄²	C₂×C₄×C₂₉	1	2	yes	—
102	C₄²	C₂³×C₂₉	1	2	yes	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	C₂×C₈×C₂₉	1	8	yes	complemented
106	C₂³	C₂²×C₄×C₂₉	1	4	yes	—
107	C₂³	C₂²×C₄×C₂₉	1	1	yes	characteristic
108	D₄	C₂×C₈×C₂₉	1	16	yes	complemented
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	C₂×C₈×C₂₉	1	4	yes	complemented
115	C₂×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
117	C₂×C₄	C₂²×C₄×C₂₉	1	4	yes	—
118	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
119	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	8	no	—
126	C₂²	C₂²×C₈×C₂₉	1	2	yes	—
127	C₂²	C₂²×C₈×C₂₉	1	8	yes	—
128	C₂²	C₂³×C₄×C₂₉	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂²×C₈×C₂₉	1	4	yes	—
133	C₄	C₂₉×H(5,𝔽₂)	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	2	8	no	—
136	C₂	C₂₉×#518	1	1	yes	characteristic
137	C₂	C₂³×C₈×C₂₉	1	1	yes	characteristic
138	C₂	C₂₉×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂⁴
commutator	C₂	C₂³×C₈×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂²×D₄×C₂₉	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×D₄×C₂₉	C₂	semidirect
3. C₂₉×#431	C₂	semidirect
4. C₂×C₂₉×C₂²⋊C₈	C₂	semidirect
7. C₂²×C₈×C₂₉	C₂²	semidirect
13. C₂₉×C₄⋊C₈	C₂²	semidirect
15. C₂₉×C₂²⋊C₈	C₂²	semidirect
16. C₄×C₈×C₂₉	C₂²	semidirect
18. C₂×D₄×C₂₉	C₈	semidirect
35. C₂³×C₂₉	C₂×C₈	semidirect
38. D₄×C₂₉	C₂×C₈	semidirect
44. C₂×C₄×C₂₉	C₂×C₈	semidirect
51. #2972	C₂₉	direct (N × H)
66. C₈×D₄	C₂×C₂₉	semidirect
67. #431	C₂×C₂₉	semidirect
68. C₂×C₂²⋊C₈	C₂×C₂₉	semidirect
76. C₂²×C₈	C₂²×C₂₉	semidirect
82. C₄⋊C₈	C₂²×C₂₉	semidirect
84. C₂²⋊C₈	C₂²×C₂₉	semidirect
85. C₄×C₈	C₂²×C₂₉	semidirect
86. C₂₉	#2972	direct (N × H)
88. C₂×D₄	C₈×C₂₉	semidirect
105. C₂³	C₂×C₈×C₂₉	semidirect
108. D₄	C₂×C₈×C₂₉	semidirect
114. C₂×C₄	C₂×C₈×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₉	2
C₈×D₄×C₂₉	2
C₂₉×#431	2
C₂×C₂₉×C₂²⋊C₈	2
#2972	29

Subgroups — C29×#2972 #16816686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#2972 #16816686