Subgroups — C₂₉×#28733 #20323633

Order 7424.

666 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₉×#3142	C₂	1	2	yes	maximal
3	C₂₉×#2931	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#2078	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1998	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1815	C₂	1	4	yes	maximal, complemented
7	C₄×C₂₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#131	C₂²	1	2	yes	—
9	C₂₉×#532	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×#125	C₂²	1	2	yes	—
11	C₂₉×#499	—	2	2	no	—
12	C₂₉×#499	—	2	2	no	—
13	C₂₉×#491	—	2	2	no	—
14	C₂₉×#490	—	2	2	no	—
15	C₂₉×#488	—	2	4	no	—
16	C₂₉×#436	—	2	2	no	—
17	C₈×D₄×C₂₉	—	2	2	no	—
18	C₂×C₂₉×#108	C₂²	1	2	yes	—
19	C₂×C₂₉×#107	C₂²	1	2	yes	—
20	C₂×C₂₉×#106	C₂²	1	2	yes	—
21	C₂₉×#423	C₂²	1	1	yes	characteristic
22	C₂₉×#417	C₂²	1	8	yes	—
23	C₂₉×#408	C₂²	1	2	yes	—
24	C₄×M₁₆×C₂₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#343	C₂²	1	8	yes	—
27	C₂₉×#341	C₂²	1	2	yes	—
28	C₂×#41×C₂₉	C₂³	1	2	yes	—
29	C₂₉×#131	—	2	8	no	—
30	C₂×M₁₆×C₂₉	—	2	2	no	—
31	C₂×M₁₆×C₂₉	C₂³	1	4	yes	—
32	C₂×M₁₆×C₂₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₉	—	2	2	no	—
34	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
36	C₂₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
37	C₂₉×#127	D₄	1	1	yes	characteristic
38	C₂₉×#125	—	2	2	no	—
39	C₂₉×#125	—	2	2	no	—
40	C₂₉×#125	—	2	4	no	—
41	C₂₉×#124	D₄	1	2	yes	—
42	C₂₉×#121	—	2	2	no	—
43	C₄×D₄×C₂₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
45	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
46	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
47	C₂₉×#108	—	2	4	no	—
48	C₂₉×#107	—	2	4	no	—
49	C₂₉×#106	—	2	2	no	—
50	C₂₉×#106	—	2	2	no	—
51	C₂₉×C₄⋊C₈	—	2	2	no	—
52	C₂₉×#103	—	2	8	no	—
53	C₂₉×#102	—	2	8	no	—
54	C₂₉×C₂²⋊C₈	—	2	4	no	—
55	C₂₉×#97	—	2	2	no	—
56	C₄×C₈×C₂₉	—	2	2	no	—
57	C₄×C₈×C₂₉	D₄	1	2	yes	—
58	C₄×C₈×C₂₉	D₄	1	2	yes	—
59	#41×C₂₉	—	2	4	no	—
60	#41×C₂₉	—	2	4	no	—
61	C₂×Q₈×C₂₉	C₂×D₄	1	2	yes	—
62	C₂×D₄×C₂₉	—	2	2	no	—
63	C₂×D₄×C₂₉	—	2	2	no	—
64	C₂×D₄×C₂₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₉	—	2	2	no	—
66	C₂²×C₄×C₂₉	—	2	4	no	—
67	C₂²×C₄×C₂₉	#41	1	2	yes	—
68	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₂₉	—	2	4	no	—
70	M₁₆×C₂₉	—	2	4	no	—
71	M₁₆×C₂₉	—	2	4	no	—
72	M₁₆×C₂₉	—	4	4	no	—
73	C₂×C₈×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	—	2	2	no	—
75	C₂×C₈×C₂₉	—	2	4	no	—
76	C₂×C₈×C₂₉	—	4	4	no	—
77	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₂₉	#41	1	4	yes	—
81	C₄⋊C₄×C₂₉	—	2	2	no	—
82	C₄⋊C₄×C₂₉	—	2	4	no	—
83	C₄⋊C₄×C₂₉	—	2	4	no	—
84	C₄⋊C₄×C₂₉	—	4	4	no	—
85	C₄⋊C₄×C₂₉	—	4	4	no	—
86	C₂²⋊C₄×C₂₉	—	2	4	no	—
87	C₄²×C₂₉	C₂×D₄	1	2	yes	—
88	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
90	#28733	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₉	—	2	2	no	—
92	C₂³×C₂₉	C₂×#41	1	1	yes	characteristic
93	Q₈×C₂₉	—	2	4	no	—
94	D₄×C₂₉	—	2	4	no	—
95	D₄×C₂₉	—	4	4	no	—
96	D₄×C₂₉	—	4	4	no	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	4	2	no	—
99	C₂×C₄×C₂₉	—	4	2	no	—
100	C₂×C₄×C₂₉	—	4	2	no	—
101	C₂×C₄×C₂₉	—	4	4	no	—
102	C₂×C₄×C₂₉	—	8	4	no	—
103	C₂×C₄×C₂₉	#121	1	2	yes	—
104	C₂×C₄×C₂₉	#121	1	2	yes	—
105	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
108	C₂×C₄×C₂₉	2₋¹⁺⁴	1	2	yes	—
109	C₈×C₂₉	—	2	2	no	—
110	C₈×C₂₉	—	4	2	no	—
111	C₈×C₂₉	—	4	4	no	—
112	C₈×C₂₉	#124	1	2	yes	—
113	C₈×C₂₉	#124	1	2	yes	—
114	C₂×#499	C₂×C₂₉	1	1	yes	characteristic, complemented
115	#3142	C₂×C₂₉	1	2	yes	—
116	#2931	C₂×C₂₉	1	1	yes	characteristic
117	#2078	C₂×C₂₉	1	4	yes	complemented
118	#1998	C₂×C₂₉	1	2	yes	complemented
119	#1815	C₂×C₂₉	1	4	yes	complemented
120	C₄×#108	C₂×C₂₉	1	1	yes	characteristic, complemented
121	C₂²×C₂₉	—	2	2	no	—
122	C₂²×C₂₉	—	4	2	no	—
123	C₂²×C₂₉	—	4	4	no	—
124	C₂²×C₂₉	C₂×#121	1	1	yes	characteristic
125	C₂²×C₂₉	#556	1	2	yes	—
126	C₄×C₂₉	—	2	2	no	—
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	4	2	no	—
129	C₄×C₂₉	—	8	4	no	—
130	C₄×C₂₉	C₂×#124	1	1	yes	characteristic
131	C₄×C₂₉	#536	1	1	yes	characteristic
132	C₄×C₂₉	#547	1	2	yes	—
133	C₂×#131	C₂²×C₂₉	1	2	yes	—
134	#532	C₂²×C₂₉	1	1	yes	characteristic
135	C₂×#125	C₂²×C₂₉	1	2	yes	—
136	#499	—	2	2	no	—
137	#499	—	2	2	no	—
138	#491	—	2	2	no	—
139	#490	—	2	2	no	—
140	#488	—	2	4	no	—
141	#436	—	2	2	no	—
142	C₈×D₄	—	2	2	no	—
143	C₂×#108	C₂²×C₂₉	1	2	yes	—
144	C₂×#107	C₂²×C₂₉	1	2	yes	—
145	C₂×#106	C₂²×C₂₉	1	2	yes	—
146	#423	C₂²×C₂₉	1	1	yes	characteristic
147	#417	C₂²×C₂₉	1	8	yes	—
148	#408	C₂²×C₂₉	1	2	yes	—
149	C₄×M₁₆	C₂²×C₂₉	1	2	yes	—
150	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
151	#343	C₂²×C₂₉	1	8	yes	—
152	#341	C₂²×C₂₉	1	2	yes	—
153	C₂×C₂₉	—	2	2	no	—
154	C₂×C₂₉	—	8	2	no	—
155	C₂×C₂₉	#2547	1	1	yes	characteristic
156	C₂×C₂₉	#3067	1	2	yes	—
157	C₂×#41	C₂³×C₂₉	1	2	yes	—
158	#131	—	2	8	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	C₂³×C₂₉	1	2	yes	—
161	C₂×M₁₆	C₂³×C₂₉	1	4	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
165	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
166	#127	D₄×C₂₉	1	1	yes	characteristic
167	#125	—	2	2	no	—
168	#125	—	2	2	no	—
169	#125	—	2	4	no	—
170	#124	D₄×C₂₉	1	2	yes	—
171	#121	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
175	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
176	#108	—	2	4	no	—
177	#107	—	2	4	no	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#103	—	2	8	no	—
182	#102	—	2	8	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	#97	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	D₄×C₂₉	1	2	yes	—
187	C₄×C₈	D₄×C₂₉	1	2	yes	—
188	C₂₉	#28733	1	1	yes	characteristic, direct product factor, Sylow
189	#41	—	2	4	no	—
190	#41	—	2	4	no	—
191	C₂×Q₈	C₂×D₄×C₂₉	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	C₂×D₄×C₂₉	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	#41×C₂₉	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
210	C₂×C₈	#41×C₂₉	1	4	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₂₉	1	2	yes	—
219	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	C₂×#41×C₂₉	1	1	yes	characteristic
222	Q₈	—	2	4	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	C₂₉×#121	1	2	yes	—
233	C₂×C₄	C₂₉×#121	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
236	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
237	C₂×C₄	C₂₉×2₋¹⁺⁴	1	2	yes	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	4	no	—
241	C₈	C₂₉×#124	1	2	yes	—
242	C₈	C₂₉×#124	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×C₂₉×#121	1	1	yes	characteristic
247	C₂²	C₂₉×#556	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	C₂×C₂₉×#124	1	1	yes	characteristic
253	C₄	C₂₉×#536	1	1	yes	characteristic
254	C₄	C₂₉×#547	1	2	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	8	2	no	—
257	C₂	C₂₉×#2547	1	1	yes	characteristic
258	C₂	C₂₉×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
prime radical	C₂₉×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×#499	C₂	semidirect
4. C₂₉×#2078	C₂	semidirect
5. C₂₉×#1998	C₂	semidirect
6. C₂₉×#1815	C₂	semidirect
7. C₄×C₂₉×#108	C₂	semidirect
90. #28733	C₂₉	direct (N × H)
114. C₂×#499	C₂×C₂₉	semidirect
117. #2078	C₂×C₂₉	semidirect
118. #1998	C₂×C₂₉	semidirect
119. #1815	C₂×C₂₉	semidirect
120. C₄×#108	C₂×C₂₉	semidirect
188. C₂₉	#28733	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#499	2
C₂₉×#3142	2
C₂₉×#2931	2
C₂₉×#2078	2
C₂₉×#1998	2
C₂₉×#1815	2
C₄×C₂₉×#108	2
#28733	29

Subgroups — C29×#28733 #20323633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28733 #20323633