Subgroups — C₂×C₂₉×#407 #16816591

Order 3712.

1008 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
3	C₂₉×#407	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₉×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂⁵×C₂₉	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₂₉	—	2	8	no	—
7	C₂³×C₄×C₂₉	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₂₉	C₄	1	4	yes	—
9	C₂³×C₄×C₂₉	C₄	1	2	yes	—
10	C₂³×C₄×C₂₉	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₉	—	2	8	no	—
14	C₂×M₁₆×C₂₉	C₂²	1	8	yes	complemented
15	C₂²×C₈×C₂₉	C₂²	1	4	yes	complemented
16	C₂₉×C₂²⋊C₈	C₂²	1	16	yes	complemented
17	C₂⁴×C₂₉	—	2	4	no	—
18	C₂⁴×C₂₉	—	2	8	no	—
19	C₂⁴×C₂₉	C₂×C₄	1	4	yes	—
20	C₂⁴×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×C₂₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₉	—	2	4	no	—
23	C₂²×C₄×C₂₉	—	2	16	no	—
24	C₂²×C₄×C₂₉	—	2	8	no	—
25	C₂²×C₄×C₂₉	—	2	8	no	—
26	C₂²×C₄×C₂₉	—	2	16	no	—
27	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₉	C₂×C₄	1	8	yes	—
30	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₉	D₄	1	8	yes	—
34	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
37	M₁₆×C₂₉	—	2	16	no	—
38	C₂×C₈×C₂₉	—	2	8	no	—
39	C₂×C₈×C₂₉	C₂³	1	8	yes	complemented
40	C₂³×C₂₉	—	2	32	no	—
41	C₂³×C₂₉	—	2	8	no	—
42	C₂³×C₂₉	—	2	4	no	—
43	C₂³×C₂₉	—	2	4	no	—
44	C₂³×C₂₉	—	2	4	no	—
45	C₂³×C₂₉	—	2	16	no	—
46	C₂³×C₂₉	C₂²⋊C₄	1	4	yes	—
47	C₂³×C₂₉	M₁₆	1	8	yes	complemented
48	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₉	C₂²×C₄	1	4	yes	—
50	C₂³×C₂₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	—	2	8	no	—
52	C₂×C₄×C₂₉	—	2	4	no	—
53	C₂×C₄×C₂₉	—	2	16	no	—
54	C₂×C₄×C₂₉	—	2	8	no	—
55	C₂×C₄×C₂₉	—	2	8	no	—
56	C₂×C₄×C₂₉	C₂²⋊C₄	1	8	yes	—
57	C₂×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₂₉	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₂₉	C₂×D₄	1	8	yes	—
63	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
64	C₈×C₂₉	—	2	8	no	—
65	C₂×#407	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂²×C₂₉	—	2	32	no	—
67	C₂²×C₂₉	—	2	4	no	—
68	C₂²×C₂₉	—	2	4	no	—
69	C₂²×C₂₉	—	2	4	no	—
70	C₂²×C₂₉	—	2	8	no	—
71	C₂²×C₂₉	—	2	16	no	—
72	C₂²×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
73	C₂²×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
74	C₂²×C₂₉	C₂×M₁₆	1	4	yes	—
75	C₂²×C₂₉	C₂×M₁₆	1	8	yes	complemented
76	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₂₉	—	2	4	no	—
78	C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
79	C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
80	C₄×C₂₉	C₂²×D₄	1	2	yes	—
81	C₂⁴×C₄	C₂×C₂₉	1	1	yes	characteristic
82	C₂²×M₁₆	C₂×C₂₉	1	2	yes	complemented
83	#407	C₂×C₂₉	1	8	yes	direct product factor
84	C₂×C₂²⋊C₈	C₂×C₂₉	1	4	yes	complemented
85	C₂×C₂₉	—	2	4	no	—
86	C₂×C₂₉	—	2	8	no	—
87	C₂×C₂₉	#407	1	4	yes	direct product factor
88	C₂×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₂₉	C₂²×M₁₆	1	2	yes	—
90	C₂⁵	C₄×C₂₉	1	1	yes	characteristic
91	C₂³×C₄	—	2	8	no	—
92	C₂³×C₄	C₄×C₂₉	1	1	yes	characteristic
93	C₂³×C₄	C₄×C₂₉	1	4	yes	—
94	C₂³×C₄	C₄×C₂₉	1	2	yes	—
95	C₂³×C₄	C₂²×C₂₉	1	2	yes	—
96	C₂³×C₄	C₂²×C₂₉	1	4	yes	—
97	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
98	C₂×M₁₆	—	2	8	no	—
99	C₂×M₁₆	C₂²×C₂₉	1	8	yes	complemented
100	C₂²×C₈	C₂²×C₂₉	1	4	yes	complemented
101	C₂²⋊C₈	C₂²×C₂₉	1	16	yes	complemented
102	C₂₉	C₂×#407	1	1	yes	characteristic, direct product factor, Sylow
103	C₂⁴	—	2	4	no	—
104	C₂⁴	—	2	8	no	—
105	C₂⁴	C₂×C₄×C₂₉	1	4	yes	—
106	C₂⁴	C₂×C₄×C₂₉	1	1	yes	characteristic
107	C₂⁴	C₂×C₄×C₂₉	1	2	yes	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	16	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	16	no	—
113	C₂²×C₄	C₂×C₄×C₂₉	1	4	yes	—
114	C₂²×C₄	C₂×C₄×C₂₉	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₂₉	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₂₉	1	8	yes	—
117	C₂²×C₄	C₂×C₄×C₂₉	1	2	yes	—
118	C₂²×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
119	C₂²×C₄	D₄×C₂₉	1	8	yes	—
120	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
121	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
123	M₁₆	—	2	16	no	—
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	C₂³×C₂₉	1	8	yes	complemented
126	C₂³	—	2	32	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	16	no	—
132	C₂³	C₂²⋊C₄×C₂₉	1	4	yes	—
133	C₂³	M₁₆×C₂₉	1	8	yes	complemented
134	C₂³	C₂²×C₄×C₂₉	1	2	yes	—
135	C₂³	C₂²×C₄×C₂₉	1	4	yes	—
136	C₂³	C₂²×C₄×C₂₉	1	1	yes	characteristic
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	16	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	C₂²⋊C₄×C₂₉	1	8	yes	—
143	C₂×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
144	C₂×C₄	C₂²×C₄×C₂₉	1	4	yes	—
145	C₂×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
146	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
147	C₂×C₄	C₂×D₄×C₂₉	1	4	yes	—
148	C₂×C₄	C₂×D₄×C₂₉	1	8	yes	—
149	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
150	C₈	—	2	8	no	—
151	C₂²	—	2	32	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	16	no	—
157	C₂²	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
158	C₂²	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
159	C₂²	C₂×M₁₆×C₂₉	1	4	yes	—
160	C₂²	C₂×M₁₆×C₂₉	1	8	yes	complemented
161	C₂²	C₂³×C₄×C₂₉	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
164	C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
165	C₄	C₂²×D₄×C₂₉	1	2	yes	—
166	C₂	—	2	4	no	—
167	C₂	—	2	8	no	—
168	C₂	C₂₉×#407	1	4	yes	direct product factor
169	C₂	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
170	C₂	C₂²×M₁₆×C₂₉	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂³
commutator	C₂²	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂³×C₂₉	C₂²×C₄
prime radical	C₂⁵×C₂₉	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₂₉	C₂	semidirect
3. C₂₉×#407	C₂	direct (N × H)
4. C₂×C₂₉×C₂²⋊C₈	C₂	semidirect
14. C₂×M₁₆×C₂₉	C₂²	semidirect
15. C₂²×C₈×C₂₉	C₂²	semidirect
16. C₂₉×C₂²⋊C₈	C₂²	semidirect
39. C₂×C₈×C₂₉	C₂³	semidirect
47. C₂³×C₂₉	M₁₆	semidirect
65. C₂×#407	C₂₉	direct (N × H)
75. C₂²×C₂₉	C₂×M₁₆	semidirect
82. C₂²×M₁₆	C₂×C₂₉	semidirect
83. #407	C₂×C₂₉	direct (N × H)
84. C₂×C₂²⋊C₈	C₂×C₂₉	semidirect
87. C₂×C₂₉	#407	direct (N × H)
99. C₂×M₁₆	C₂²×C₂₉	semidirect
100. C₂²×C₈	C₂²×C₂₉	semidirect
101. C₂²⋊C₈	C₂²×C₂₉	semidirect
102. C₂₉	C₂×#407	direct (N × H)
125. C₂×C₈	C₂³×C₂₉	semidirect
133. C₂³	M₁₆×C₂₉	semidirect
160. C₂²	C₂×M₁₆×C₂₉	semidirect
168. C₂	C₂₉×#407	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₂₉	2
C₂²×M₁₆×C₂₉	2
C₂₉×#407	2
C₂×C₂₉×C₂²⋊C₈	2
C₂×#407	29

Subgroups — C2×C29×#407 #16816591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#407 #16816591