Subgroups — C₂₉×#9630 #20304530

Order 7424.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#1848	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1770	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1571	C₂	1	2	yes	maximal
4	C₂₉×#1564	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1303	C₂	1	2	yes	maximal
6	C₂₉×#431	C₂²	1	2	yes	—
7	C₂×C₂₉×#108	—	2	1	no	—
8	C₂×C₂₉×#108	—	2	1	no	—
9	C₂₉×#423	C₂²	1	2	yes	—
10	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₉×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×#336	—	2	1	no	—
15	C₂₉×#335	—	2	4	no	—
16	C₂₉×#335	C₄	1	2	yes	complemented
17	C₂₉×#334	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₂₉	—	2	2	no	—
19	C₂²×C₈×C₂₉	—	2	1	no	—
20	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
23	C₂×C₄²×C₂₉	—	2	1	no	—
24	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
25	C₂₉×#108	—	4	2	no	—
26	C₂₉×#108	—	4	2	no	—
27	C₂₉×C₄⋊C₈	—	2	2	no	—
28	C₂₉×C₄⋊C₈	—	2	1	no	—
29	C₂₉×C₄⋊C₈	—	2	2	no	—
30	C₂₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₂₉×C₄⋊C₈	D₄	1	2	yes	—
32	C₂₉×C₂²⋊C₈	—	2	2	no	—
33	C₂₉×#97	—	4	2	no	—
34	C₄×C₈×C₂₉	—	2	2	no	—
35	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₂₉	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₂₉	Q₈	1	1	yes	characteristic
39	C₂₉×#95	Q₈	1	2	yes	—
40	C₂²×C₄×C₂₉	—	2	2	no	—
41	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
44	M₁₆×C₂₉	—	4	2	no	—
45	M₁₆×C₂₉	—	4	2	no	—
46	C₂×C₈×C₂₉	—	2	1	no	—
47	C₂×C₈×C₂₉	—	2	1	no	—
48	C₂×C₈×C₂₉	—	2	4	no	—
49	C₂×C₈×C₂₉	—	2	2	no	—
50	C₂×C₈×C₂₉	—	4	1	no	—
51	C₂×C₈×C₂₉	—	4	1	no	—
52	C₂×C₈×C₂₉	—	4	2	no	—
53	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₂₉	D₈	1	2	yes	—
58	C₂×C₈×C₂₉	Q₁₆	1	2	yes	—
59	C₄⋊C₄×C₂₉	—	4	2	no	—
60	C₄²×C₂₉	—	4	1	no	—
61	C₄²×C₂₉	—	4	1	no	—
62	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₂₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₂₉	#41	1	1	yes	characteristic
66	#9630	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₉	—	2	2	no	—
69	C₂×C₄×C₂₉	—	4	1	no	—
70	C₂×C₄×C₂₉	—	4	2	no	—
71	C₂×C₄×C₂₉	—	4	2	no	—
72	C₂×C₄×C₂₉	—	4	2	no	—
73	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₂₉	#125	1	1	yes	characteristic
78	C₂×C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₂₉	#137	1	1	yes	characteristic
83	C₈×C₂₉	—	2	1	no	—
84	C₈×C₂₉	—	2	1	no	—
85	C₈×C₂₉	—	4	4	no	—
86	C₈×C₂₉	—	4	2	no	—
87	C₈×C₂₉	—	8	2	no	—
88	C₈×C₂₉	#107	1	2	yes	—
89	C₈×C₂₉	#107	1	2	yes	—
90	#1848	C₂×C₂₉	1	1	yes	characteristic
91	#1770	C₂×C₂₉	1	1	yes	characteristic
92	#1571	C₂×C₂₉	1	2	yes	—
93	#1564	C₂×C₂₉	1	1	yes	characteristic
94	#1303	C₂×C₂₉	1	2	yes	—
95	C₂²×C₂₉	—	2	1	no	—
96	C₂²×C₂₉	—	2	1	no	—
97	C₂²×C₂₉	#389	1	1	yes	characteristic
98	C₂²×C₂₉	#474	1	1	yes	characteristic
99	C₂²×C₂₉	#477	1	1	yes	characteristic
100	C₄×C₂₉	—	2	1	no	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	2	1	no	—
103	C₄×C₂₉	—	8	2	no	—
104	C₄×C₂₉	C₂×#107	1	1	yes	characteristic
105	C₄×C₂₉	#428	1	1	yes	characteristic
106	C₄×C₂₉	#444	1	2	yes	—
107	C₄×C₂₉	#459	1	1	yes	characteristic
108	C₄×C₂₉	#462	1	1	yes	characteristic
109	#431	C₂²×C₂₉	1	2	yes	—
110	C₂×#108	—	2	1	no	—
111	C₂×#108	—	2	1	no	—
112	#423	C₂²×C₂₉	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
114	C₂×#97	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
116	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
117	#336	—	2	1	no	—
118	#335	—	2	4	no	—
119	#335	C₄×C₂₉	1	2	yes	complemented
120	#334	C₄×C₂₉	1	2	yes	complemented
121	C₂×C₂₉	—	2	1	no	—
122	C₂×C₂₉	—	2	1	no	—
123	C₂×C₂₉	#1946	1	1	yes	characteristic
124	C₂×C₂₉	#1947	1	1	yes	characteristic
125	C₂×C₂₉	#1953	1	1	yes	characteristic
126	C₂×M₁₆	—	2	2	no	—
127	C₂²×C₈	—	2	1	no	—
128	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
133	#108	—	4	2	no	—
134	#108	—	4	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	C₂×C₄×C₂₉	1	4	yes	—
139	C₄⋊C₈	D₄×C₂₉	1	2	yes	—
140	C₂²⋊C₈	—	2	2	no	—
141	#97	—	4	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
145	C₄×C₈	Q₈×C₂₉	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₂₉	1	1	yes	characteristic
147	#95	Q₈×C₂₉	1	2	yes	—
148	C₂₉	#9630	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₂₉	1	1	yes	characteristic
152	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
153	M₁₆	—	4	2	no	—
154	M₁₆	—	4	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
163	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
164	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
165	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
166	C₂×C₈	D₈×C₂₉	1	2	yes	—
167	C₂×C₈	Q₁₆×C₂₉	1	2	yes	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
173	C₄²	C₂×Q₈×C₂₉	1	1	yes	characteristic
174	C₄²	#41×C₂₉	1	1	yes	characteristic
175	C₂³	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
182	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
183	C₂×C₄	C₄×Q₈×C₂₉	1	1	yes	characteristic
184	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
185	C₂×C₄	C₂₉×#125	1	1	yes	characteristic
186	C₂×C₄	C₂₉×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
188	C₂×C₄	C₂×Q₁₆×C₂₉	1	1	yes	characteristic
189	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₄	C₂₉×#137	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	4	2	no	—
195	C₈	—	8	2	no	—
196	C₈	C₂₉×#107	1	2	yes	—
197	C₈	C₂₉×#107	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₂₉×#389	1	1	yes	characteristic
201	C₂²	C₂₉×#474	1	1	yes	characteristic
202	C₂²	C₂₉×#477	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	8	2	no	—
207	C₄	C₂×C₂₉×#107	1	1	yes	characteristic
208	C₄	C₂₉×#428	1	1	yes	characteristic
209	C₄	C₂₉×#444	1	2	yes	—
210	C₄	C₂₉×#459	1	1	yes	characteristic
211	C₄	C₂₉×#462	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₂₉×#1946	1	1	yes	characteristic
215	C₂	C₂₉×#1947	1	1	yes	characteristic
216	C₂	C₂₉×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	#121
commutator	C₄²	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	#389
prime radical	C₂³×C₂₉	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₂₉×#335	C₄	semidirect
17. C₂₉×#334	C₄	semidirect
66. #9630	C₂₉	direct (N × H)
119. #335	C₄×C₂₉	semidirect
120. #334	C₄×C₂₉	semidirect
148. C₂₉	#9630	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×#1848	2
C₂₉×#1770	2
C₂₉×#1571	2
C₂₉×#1564	2
C₂₉×#1303	2
#9630	29

Subgroups — C29×#9630 #20304530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#9630 #20304530