Subgroups — C₂₉×#28825 #20323725

Order 7424.

634 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3164	C₂	1	4	yes	maximal
2	C₂₉×#2060	C₂	1	4	yes	maximal
3	C₂₉×#1952	C₂	1	4	yes	maximal
4	C₂₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1772	C₂	1	2	yes	maximal
6	C₂₉×#544	C₂²	1	4	yes	—
7	C₂₉×#501	—	2	8	no	—
8	C₂₉×#500	—	2	4	no	—
9	C₂₉×#499	—	2	4	no	—
10	C₂×C₂₉×#108	C₂²	1	2	yes	—
11	C₂₉×#428	C₂²	1	8	yes	—
12	C₂₉×#427	C₂²	1	2	yes	—
13	C₂×C₂₉×#107	C₂²	1	4	yes	—
14	C₂₉×#405	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₂₉×#384	C₂²	1	2	yes	—
17	C₂₉×#343	C₂²	1	4	yes	—
18	C₂₉×#339	C₂²	1	4	yes	—
19	C₂×M₁₆×C₂₉	C₂³	1	4	yes	—
20	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
21	C₂₉×C₄⋊Q₈	—	2	8	no	—
22	C₂₉×#125	—	2	8	no	—
23	C₂₉×C₂²⋊Q₈	—	4	4	no	—
24	C₂₉×#117	C₂³	1	4	yes	—
25	C₂×C₄⋊C₄×C₂₉	—	4	2	no	—
26	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
27	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
28	C₂₉×#108	Q₈	1	8	yes	complemented
29	C₂₉×#107	—	2	4	no	—
30	C₂₉×#107	—	2	4	no	—
31	C₂₉×#107	—	2	8	no	—
32	C₂₉×#107	D₄	1	2	yes	—
33	C₂₉×#107	D₄	1	2	yes	—
34	C₂₉×#106	—	2	8	no	—
35	C₂₉×#106	D₄	1	4	yes	—
36	C₂₉×#97	—	2	4	no	—
37	C₄×C₈×C₂₉	—	2	4	no	—
38	C₄×C₈×C₂₉	Q₈	1	4	yes	complemented
39	C₂×Q₈×C₂₉	—	4	4	no	—
40	C₂²×C₄×C₂₉	—	2	1	no	—
41	C₂²×C₄×C₂₉	—	2	4	no	—
42	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₂₉	—	2	4	no	—
44	M₁₆×C₂₉	C₂×Q₈	1	8	yes	—
45	C₂×C₈×C₂₉	—	2	4	no	—
46	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
49	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
50	C₄⋊C₄×C₂₉	—	2	8	no	—
51	C₄⋊C₄×C₂₉	—	4	4	no	—
52	C₄⋊C₄×C₂₉	—	4	8	no	—
53	C₄⋊C₄×C₂₉	—	8	4	no	—
54	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
55	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
56	C₂²⋊C₄×C₂₉	—	2	4	no	—
57	C₄²×C₂₉	—	2	4	no	—
58	C₄²×C₂₉	—	2	1	no	—
59	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
60	#28825	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
62	Q₈×C₂₉	—	16	4	no	—
63	C₂×C₄×C₂₉	—	2	2	no	—
64	C₂×C₄×C₂₉	—	2	2	no	—
65	C₂×C₄×C₂₉	—	2	2	no	—
66	C₂×C₄×C₂₉	—	2	4	no	—
67	C₂×C₄×C₂₉	—	2	4	no	—
68	C₂×C₄×C₂₉	—	4	4	no	—
69	C₂×C₄×C₂₉	—	8	4	no	—
70	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
71	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²×Q₈	1	2	yes	—
73	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
74	C₈×C₂₉	—	2	8	no	—
75	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
76	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
77	#3164	C₂×C₂₉	1	4	yes	—
78	#2060	C₂×C₂₉	1	4	yes	—
79	#1952	C₂×C₂₉	1	4	yes	—
80	#1845	C₂×C₂₉	1	1	yes	characteristic
81	#1772	C₂×C₂₉	1	2	yes	—
82	C₂²×C₂₉	—	2	1	no	—
83	C₂²×C₂₉	—	2	1	no	—
84	C₂²×C₂₉	#544	1	1	yes	characteristic
85	C₂²×C₂₉	D₄²	1	1	yes	characteristic
86	C₂²×C₂₉	Q₈²	1	1	yes	characteristic
87	C₄×C₂₉	—	2	4	no	—
88	C₄×C₂₉	—	4	4	no	—
89	C₄×C₂₉	—	4	4	no	—
90	C₄×C₂₉	—	8	4	no	—
91	C₄×C₂₉	C₂×C₄⋊Q₈	1	2	yes	—
92	C₄×C₂₉	D₄×Q₈	1	2	yes	—
93	#544	C₂²×C₂₉	1	4	yes	—
94	#501	—	2	8	no	—
95	#500	—	2	4	no	—
96	#499	—	2	4	no	—
97	C₂×#108	C₂²×C₂₉	1	2	yes	—
98	#428	C₂²×C₂₉	1	8	yes	—
99	#427	C₂²×C₂₉	1	2	yes	—
100	C₂×#107	C₂²×C₂₉	1	4	yes	—
101	#405	C₂²×C₂₉	1	4	yes	—
102	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
103	#384	C₂²×C₂₉	1	2	yes	—
104	#343	C₂²×C₂₉	1	4	yes	—
105	#339	C₂²×C₂₉	1	4	yes	—
106	C₂×C₂₉	—	2	1	no	—
107	C₂×C₂₉	—	2	1	no	—
108	C₂×C₂₉	#2551	1	1	yes	characteristic
109	C₂×C₂₉	#3292	1	1	yes	characteristic
110	C₂×C₂₉	#3293	1	1	yes	characteristic
111	C₂×M₁₆	C₂³×C₂₉	1	4	yes	—
112	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
113	C₄⋊Q₈	—	2	8	no	—
114	#125	—	2	8	no	—
115	C₂²⋊Q₈	—	4	4	no	—
116	#117	C₂³×C₂₉	1	4	yes	—
117	C₂×C₄⋊C₄	—	4	2	no	—
118	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
119	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
120	#108	Q₈×C₂₉	1	8	yes	complemented
121	#107	—	2	4	no	—
122	#107	—	2	4	no	—
123	#107	—	2	8	no	—
124	#107	D₄×C₂₉	1	2	yes	—
125	#107	D₄×C₂₉	1	2	yes	—
126	#106	—	2	8	no	—
127	#106	D₄×C₂₉	1	4	yes	—
128	#97	—	2	4	no	—
129	C₄×C₈	—	2	4	no	—
130	C₄×C₈	Q₈×C₂₉	1	4	yes	complemented
131	C₂₉	#28825	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×Q₈	—	4	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
136	M₁₆	—	2	4	no	—
137	M₁₆	C₂×Q₈×C₂₉	1	8	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
141	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
142	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
148	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
153	C₂³	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
154	Q₈	—	16	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
163	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×Q₈×C₂₉	1	2	yes	—
165	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
168	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₂₉×#544	1	1	yes	characteristic
172	C₂²	D₄²×C₂₉	1	1	yes	characteristic
173	C₂²	Q₈²×C₂₉	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	8	4	no	—
178	C₄	C₂×C₂₉×C₄⋊Q₈	1	2	yes	—
179	C₄	D₄×Q₈×C₂₉	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	C₂₉×#2551	1	1	yes	characteristic
183	C₂	C₂₉×#3292	1	1	yes	characteristic
184	C₂	C₂₉×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
prime radical	C₂³×C₂₉	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
28. C₂₉×#108	Q₈	semidirect
38. C₄×C₈×C₂₉	Q₈	semidirect
60. #28825	C₂₉	direct (N × H)
120. #108	Q₈×C₂₉	semidirect
130. C₄×C₈	Q₈×C₂₉	semidirect
131. C₂₉	#28825	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×#3164	2
C₂₉×#2060	2
C₂₉×#1952	2
C₂₉×#1845	2
C₂₉×#1772	2
#28825	29

Subgroups — C29×#28825 #20323725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28825 #20323725