Subgroups — C₂²×C₂₉×#135 #16817291

Order 3712.

1504 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₉×#135	C₂	1	24	yes	maximal, direct product factor
3	C₂²×Q₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×SD₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
5	C₂²×D₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₉	—	2	12	no	—
8	C₂×#41×C₂₉	C₂²	1	24	yes	complemented
9	C₂²×Q₈×C₂₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₉	—	2	2	no	—
11	C₂²×D₄×C₂₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₉	—	2	2	no	—
13	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×#135	C₂²	1	64	yes	direct product factor
15	C₂×Q₁₆×C₂₉	C₂²	1	12	yes	complemented
16	C₂×SD₁₆×C₂₉	C₂²	1	24	yes	complemented
17	C₂×D₈×C₂₉	C₂²	1	12	yes	complemented
18	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₂₉	C₂²	1	12	yes	complemented
21	C₂⁴×C₂₉	—	2	2	no	—
22	C₂⁴×C₂₉	D₄	1	1	yes	characteristic
23	#41×C₂₉	—	2	48	no	—
24	#41×C₂₉	C₂³	1	32	yes	complemented
25	C₂×Q₈×C₂₉	—	2	6	no	—
26	C₂×Q₈×C₂₉	C₂³	1	12	yes	—
27	C₂×D₄×C₂₉	—	2	24	no	—
28	C₂×D₄×C₂₉	—	2	6	no	—
29	C₂×D₄×C₂₉	C₂³	1	12	yes	—
30	C₂²×C₄×C₂₉	—	2	2	no	—
31	C₂²×C₄×C₂₉	—	2	24	no	—
32	C₂²×C₄×C₂₉	D₄	1	6	yes	—
33	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₉	C₂³	1	6	yes	—
36	Q₁₆×C₂₉	C₂³	1	16	yes	complemented
37	SD₁₆×C₂₉	C₂³	1	32	yes	complemented
38	D₈×C₂₉	C₂³	1	16	yes	complemented
39	C₂×C₈×C₂₉	C₂³	1	6	yes	—
40	C₂×C₈×C₂₉	C₂³	1	6	yes	—
41	C₂×C₈×C₂₉	C₂³	1	16	yes	complemented
42	C₂³×C₂₉	—	2	4	no	—
43	C₂³×C₂₉	—	2	12	no	—
44	C₂³×C₂₉	—	4	8	no	—
45	C₂³×C₂₉	C₂×D₄	1	6	yes	—
46	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₂₉	—	2	12	no	—
48	Q₈×C₂₉	C₂⁴	1	8	yes	—
49	D₄×C₂₉	—	2	32	no	—
50	D₄×C₂₉	—	2	12	no	—
51	D₄×C₂₉	C₂⁴	1	8	yes	—
52	C₂×C₄×C₂₉	—	2	12	no	—
53	C₂×C₄×C₂₉	—	2	32	no	—
54	C₂×C₄×C₂₉	C₂×D₄	1	6	yes	—
55	C₂×C₄×C₂₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₂₉	C₂×D₄	1	12	yes	—
57	C₂×C₄×C₂₉	C₂⁴	1	3	yes	—
58	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
59	C₈×C₂₉	C₂⁴	1	4	yes	—
60	C₈×C₂₉	C₂⁴	1	4	yes	—
61	C₂²×#135	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₂₉	—	2	12	no	—
63	C₂²×C₂₉	—	2	8	no	—
64	C₂²×C₂₉	—	4	24	no	—
65	C₂²×C₂₉	#135	1	4	yes	direct product factor
66	C₂²×C₂₉	C₂²×D₄	1	4	yes	—
67	C₂²×C₂₉	C₂²×D₄	1	3	yes	—
68	C₄×C₂₉	—	2	8	no	—
69	C₄×C₂₉	C₂²×D₄	1	4	yes	—
70	C₄×C₂₉	C₂²×D₄	1	3	yes	—
71	C₄×C₂₉	C₂⁵	1	1	yes	characteristic
72	C₂²×#41	C₂×C₂₉	1	2	yes	complemented
73	C₂×#135	C₂×C₂₉	1	24	yes	direct product factor
74	C₂²×Q₁₆	C₂×C₂₉	1	1	yes	characteristic, complemented
75	C₂²×SD₁₆	C₂×C₂₉	1	2	yes	complemented
76	C₂²×D₈	C₂×C₂₉	1	1	yes	characteristic, complemented
77	C₂³×C₈	C₂×C₂₉	1	1	yes	characteristic, complemented
78	C₂×C₂₉	—	2	4	no	—
79	C₂×C₂₉	—	4	8	no	—
80	C₂×C₂₉	C₂×#135	1	6	yes	direct product factor
81	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
82	C₂×#41	—	2	12	no	—
83	C₂×#41	C₂²×C₂₉	1	24	yes	complemented
84	C₂²×Q₈	C₂²×C₂₉	1	2	yes	—
85	C₂²×D₄	—	2	2	no	—
86	C₂²×D₄	C₂²×C₂₉	1	2	yes	—
87	C₂³×C₄	—	2	2	no	—
88	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
89	#135	C₂²×C₂₉	1	64	yes	direct product factor
90	C₂×Q₁₆	C₂²×C₂₉	1	12	yes	complemented
91	C₂×SD₁₆	C₂²×C₂₉	1	24	yes	complemented
92	C₂×D₈	C₂²×C₂₉	1	12	yes	complemented
93	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₂₉	1	12	yes	complemented
96	C₂₉	C₂²×#135	1	1	yes	characteristic, direct product factor, Sylow
97	C₂⁴	—	2	2	no	—
98	C₂⁴	D₄×C₂₉	1	1	yes	characteristic
99	#41	—	2	48	no	—
100	#41	C₂³×C₂₉	1	32	yes	complemented
101	C₂×Q₈	—	2	6	no	—
102	C₂×Q₈	C₂³×C₂₉	1	12	yes	—
103	C₂×D₄	—	2	24	no	—
104	C₂×D₄	—	2	6	no	—
105	C₂×D₄	C₂³×C₂₉	1	12	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	24	no	—
108	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
109	C₂²×C₄	D₄×C₂₉	1	6	yes	—
110	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
111	C₂²×C₄	C₂³×C₂₉	1	6	yes	—
112	Q₁₆	C₂³×C₂₉	1	16	yes	complemented
113	SD₁₆	C₂³×C₂₉	1	32	yes	complemented
114	D₈	C₂³×C₂₉	1	16	yes	complemented
115	C₂×C₈	C₂³×C₂₉	1	6	yes	—
116	C₂×C₈	C₂³×C₂₉	1	6	yes	—
117	C₂×C₈	C₂³×C₂₉	1	16	yes	complemented
118	C₂³	—	2	4	no	—
119	C₂³	—	2	12	no	—
120	C₂³	—	4	8	no	—
121	C₂³	C₂×D₄×C₂₉	1	6	yes	—
122	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
123	Q₈	—	2	12	no	—
124	Q₈	C₂⁴×C₂₉	1	8	yes	—
125	D₄	—	2	32	no	—
126	D₄	—	2	12	no	—
127	D₄	C₂⁴×C₂₉	1	8	yes	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	—	2	32	no	—
130	C₂×C₄	C₂×D₄×C₂₉	1	6	yes	—
131	C₂×C₄	C₂×D₄×C₂₉	1	3	yes	—
132	C₂×C₄	C₂×D₄×C₂₉	1	12	yes	—
133	C₂×C₄	C₂⁴×C₂₉	1	3	yes	—
134	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
135	C₈	C₂⁴×C₂₉	1	4	yes	—
136	C₈	C₂⁴×C₂₉	1	4	yes	—
137	C₂²	—	2	12	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	4	24	no	—
140	C₂²	C₂₉×#135	1	4	yes	direct product factor
141	C₂²	C₂²×D₄×C₂₉	1	4	yes	—
142	C₂²	C₂²×D₄×C₂₉	1	3	yes	—
143	C₄	—	2	8	no	—
144	C₄	C₂²×D₄×C₂₉	1	4	yes	—
145	C₄	C₂²×D₄×C₂₉	1	3	yes	—
146	C₄	C₂⁵×C₂₉	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	—	4	8	no	—
149	C₂	C₂×C₂₉×#135	1	6	yes	direct product factor
150	C₂	C₂³×D₄×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₉	D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₉
socle	C₂³×C₂₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₉	C₂	semidirect
2. C₂×C₂₉×#135	C₂	direct (N × H)
3. C₂²×Q₁₆×C₂₉	C₂	semidirect
4. C₂²×SD₁₆×C₂₉	C₂	semidirect
5. C₂²×D₈×C₂₉	C₂	semidirect
6. C₂³×C₈×C₂₉	C₂	semidirect
8. C₂×#41×C₂₉	C₂²	semidirect
14. C₂₉×#135	C₂²	direct (N × H)
15. C₂×Q₁₆×C₂₉	C₂²	semidirect
16. C₂×SD₁₆×C₂₉	C₂²	semidirect
17. C₂×D₈×C₂₉	C₂²	semidirect
20. C₂²×C₈×C₂₉	C₂²	semidirect
24. #41×C₂₉	C₂³	semidirect
36. Q₁₆×C₂₉	C₂³	semidirect
37. SD₁₆×C₂₉	C₂³	semidirect
38. D₈×C₂₉	C₂³	semidirect
41. C₂×C₈×C₂₉	C₂³	semidirect
61. C₂²×#135	C₂₉	direct (N × H)
65. C₂²×C₂₉	#135	direct (N × H)
72. C₂²×#41	C₂×C₂₉	semidirect
73. C₂×#135	C₂×C₂₉	direct (N × H)
74. C₂²×Q₁₆	C₂×C₂₉	semidirect
75. C₂²×SD₁₆	C₂×C₂₉	semidirect
76. C₂²×D₈	C₂×C₂₉	semidirect
77. C₂³×C₈	C₂×C₂₉	semidirect
80. C₂×C₂₉	C₂×#135	direct (N × H)
83. C₂×#41	C₂²×C₂₉	semidirect
89. #135	C₂²×C₂₉	direct (N × H)
90. C₂×Q₁₆	C₂²×C₂₉	semidirect
91. C₂×SD₁₆	C₂²×C₂₉	semidirect
92. C₂×D₈	C₂²×C₂₉	semidirect
95. C₂²×C₈	C₂²×C₂₉	semidirect
96. C₂₉	C₂²×#135	direct (N × H)
100. #41	C₂³×C₂₉	semidirect
112. Q₁₆	C₂³×C₂₉	semidirect
113. SD₁₆	C₂³×C₂₉	semidirect
114. D₈	C₂³×C₂₉	semidirect
117. C₂×C₈	C₂³×C₂₉	semidirect
140. C₂²	C₂₉×#135	direct (N × H)
149. C₂	C₂×C₂₉×#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₉	2
C₂×C₂₉×#135	2
C₂²×Q₁₆×C₂₉	2
C₂²×SD₁₆×C₂₉	2
C₂²×D₈×C₂₉	2
C₂³×C₈×C₂₉	2
C₂²×#135	29

Subgroups — C22×C29×#135 #16817291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₉×#135 #16817291