Subgroups — C₂₉×#21709 #20316609

Order 7424.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#444	C₂	1	4	yes	maximal, complemented
3	C₂₉×#2931	C₂	1	2	yes	maximal, complemented
4	C₂₉×#1974	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1963	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#135	C₂²	1	4	yes	—
9	C₂×C₂₉×#131	C₂²	1	4	yes	—
10	C₂₉×#532	C₂²	1	2	yes	—
11	C₂₉×#497	C₄	1	4	yes	complemented
12	C₂₉×#496	C₄	1	4	yes	complemented
13	C₂₉×#444	—	2	16	no	—
14	C₂₉×#436	—	2	4	no	—
15	C₈×D₄×C₂₉	—	2	4	no	—
16	C₂×C₂₉×#108	C₂²	1	4	yes	—
17	C₂₉×#423	C₂²	1	2	yes	—
18	C₂×C₂₉×#104	C₂²	1	8	yes	—
19	C₂₉×#408	C₂²	1	4	yes	—
20	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×#376	C₂²	1	1	yes	characteristic
25	C₂×#41×C₂₉	C₂³	1	4	yes	—
26	C₂₉×#135	—	2	8	no	—
27	C₂₉×#135	—	2	8	no	—
28	C₂×Q₁₆×C₂₉	C₂×C₄	1	4	yes	—
29	C₂×SD₁₆×C₂₉	C₂×C₄	1	8	yes	—
30	C₂×D₈×C₂₉	C₂×C₄	1	4	yes	—
31	C₂₉×#131	—	2	16	no	—
32	C₂×M₁₆×C₂₉	—	2	4	no	—
33	C₂×M₁₆×C₂₉	C₂³	1	4	yes	—
34	C₂²×C₈×C₂₉	—	2	4	no	—
35	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₂₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₂₉×#127	C₂×C₄	1	2	yes	—
40	C₂₉×#124	C₂×C₄	1	4	yes	—
41	C₂₉×#121	—	2	4	no	—
42	C₄×D₄×C₂₉	—	2	4	no	—
43	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₂₉×#108	—	2	8	no	—
46	C₂₉×C₄⋊C₈	—	2	4	no	—
47	C₂₉×#104	—	2	16	no	—
48	C₂₉×C₂²⋊C₈	—	2	8	no	—
49	C₂₉×#97	C₂×C₄	1	4	yes	—
50	C₂₉×#97	D₄	1	8	yes	complemented
51	C₄×C₈×C₂₉	—	2	4	no	—
52	C₄×C₈×C₂₉	D₄	1	4	yes	—
53	C₂₉×#95	—	2	2	no	—
54	#41×C₂₉	—	2	8	no	—
55	#41×C₂₉	—	2	8	no	—
56	C₂×Q₈×C₂₉	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₂₉	—	2	4	no	—
58	C₂×D₄×C₂₉	—	2	4	no	—
59	C₂×D₄×C₂₉	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₂₉	—	2	2	no	—
61	C₂²×C₄×C₂₉	—	2	4	no	—
62	C₂²×C₄×C₂₉	#41	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
64	Q₁₆×C₂₉	—	2	8	no	—
65	SD₁₆×C₂₉	—	2	16	no	—
66	D₈×C₂₉	—	2	8	no	—
67	M₁₆×C₂₉	—	2	8	no	—
68	M₁₆×C₂₉	—	4	8	no	—
69	C₂×C₈×C₂₉	—	2	2	no	—
70	C₂×C₈×C₂₉	—	2	4	no	—
71	C₂×C₈×C₂₉	—	4	8	no	—
72	C₂×C₈×C₂₉	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₂₉	#41	1	4	yes	—
78	C₄⋊C₄×C₂₉	—	2	4	no	—
79	C₂²⋊C₄×C₂₉	—	2	8	no	—
80	C₄²×C₂₉	—	2	2	no	—
81	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₂₉	C₂×D₄	1	4	yes	—
83	C₄²×C₂₉	C₂×D₄	1	2	yes	—
84	#21709	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₉	—	2	4	no	—
86	C₂³×C₂₉	C₂×#41	1	1	yes	characteristic
87	Q₈×C₂₉	—	2	8	no	—
88	D₄×C₂₉	—	2	8	no	—
89	D₄×C₂₉	—	4	8	no	—
90	D₄×C₂₉	—	4	8	no	—
91	C₂×C₄×C₂₉	—	2	4	no	—
92	C₂×C₄×C₂₉	—	2	4	no	—
93	C₂×C₄×C₂₉	—	4	2	no	—
94	C₂×C₄×C₂₉	—	4	4	no	—
95	C₂×C₄×C₂₉	—	4	4	no	—
96	C₂×C₄×C₂₉	—	4	4	no	—
97	C₂×C₄×C₂₉	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
100	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
102	C₈×C₂₉	—	2	2	no	—
103	C₈×C₂₉	—	4	4	no	—
104	C₈×C₂₉	C₄×D₄	1	8	yes	—
105	C₈×C₂₉	#127	1	4	yes	—
106	#3150	C₂×C₂₉	1	1	yes	characteristic
107	C₂×#444	C₂×C₂₉	1	4	yes	complemented
108	#2931	C₂×C₂₉	1	2	yes	complemented
109	#1974	C₂×C₂₉	1	4	yes	complemented
110	#1963	C₂×C₂₉	1	2	yes	complemented
111	#1849	C₂×C₂₉	1	1	yes	characteristic, complemented
112	#1726	C₂×C₂₉	1	1	yes	characteristic, complemented
113	C₂²×C₂₉	—	2	2	no	—
114	C₂²×C₂₉	—	4	4	no	—
115	C₂²×C₂₉	—	4	8	no	—
116	C₂²×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂²×C₂₉	#532	1	2	yes	—
118	C₄×C₂₉	—	2	4	no	—
119	C₄×C₂₉	—	4	4	no	—
120	C₄×C₂₉	—	4	4	no	—
121	C₄×C₂₉	C₂×C₄×D₄	1	2	yes	—
122	C₄×C₂₉	C₂×#127	1	1	yes	characteristic
123	C₄×C₂₉	#532	1	1	yes	characteristic
124	C₂×#135	C₂²×C₂₉	1	4	yes	—
125	C₂×#131	C₂²×C₂₉	1	4	yes	—
126	#532	C₂²×C₂₉	1	2	yes	—
127	#497	C₄×C₂₉	1	4	yes	complemented
128	#496	C₄×C₂₉	1	4	yes	complemented
129	#444	—	2	16	no	—
130	#436	—	2	4	no	—
131	C₈×D₄	—	2	4	no	—
132	C₂×#108	C₂²×C₂₉	1	4	yes	—
133	#423	C₂²×C₂₉	1	2	yes	—
134	C₂×#104	C₂²×C₂₉	1	8	yes	—
135	#408	C₂²×C₂₉	1	4	yes	—
136	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
137	C₂×#97	C₂²×C₂₉	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
140	#376	C₂²×C₂₉	1	1	yes	characteristic
141	C₂×C₂₉	—	2	2	no	—
142	C₂×C₂₉	—	8	4	no	—
143	C₂×C₂₉	C₄×#127	1	1	yes	characteristic
144	C₂×C₂₉	#2955	1	2	yes	—
145	C₂×#41	C₂³×C₂₉	1	4	yes	—
146	#135	—	2	8	no	—
147	#135	—	2	8	no	—
148	C₂×Q₁₆	C₂×C₄×C₂₉	1	4	yes	—
149	C₂×SD₁₆	C₂×C₄×C₂₉	1	8	yes	—
150	C₂×D₈	C₂×C₄×C₂₉	1	4	yes	—
151	#131	—	2	16	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	C₂³×C₂₉	1	4	yes	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
157	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
158	C₄⋊Q₈	C₂×C₄×C₂₉	1	2	yes	—
159	#127	C₂×C₄×C₂₉	1	2	yes	—
160	#124	C₂×C₄×C₂₉	1	4	yes	—
161	#121	—	2	4	no	—
162	C₄×D₄	—	2	4	no	—
163	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
164	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
165	#108	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	#104	—	2	16	no	—
168	C₂²⋊C₈	—	2	8	no	—
169	#97	C₂×C₄×C₂₉	1	4	yes	—
170	#97	D₄×C₂₉	1	8	yes	complemented
171	C₄×C₈	—	2	4	no	—
172	C₄×C₈	D₄×C₂₉	1	4	yes	—
173	#95	—	2	2	no	—
174	C₂₉	#21709	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	2	8	no	—
176	#41	—	2	8	no	—
177	C₂×Q₈	C₂²×C₄×C₂₉	1	4	yes	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	C₂²×C₄×C₂₉	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	#41×C₂₉	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
185	Q₁₆	—	2	8	no	—
186	SD₁₆	—	2	16	no	—
187	D₈	—	2	8	no	—
188	M₁₆	—	2	8	no	—
189	M₁₆	—	4	8	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	C₂²×C₄×C₂₉	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
196	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
197	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
198	C₂×C₈	#41×C₂₉	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
203	C₄²	C₂×D₄×C₂₉	1	4	yes	—
204	C₄²	C₂×D₄×C₂₉	1	2	yes	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×#41×C₂₉	1	1	yes	characteristic
207	Q₈	—	2	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	4	8	no	—
210	D₄	—	4	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₄×D₄×C₂₉	1	4	yes	—
218	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
220	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
221	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	4	4	no	—
224	C₈	C₄×D₄×C₂₉	1	8	yes	—
225	C₈	C₂₉×#127	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
230	C₂²	C₂₉×#532	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	4	no	—
233	C₄	—	4	4	no	—
234	C₄	C₂×C₄×D₄×C₂₉	1	2	yes	—
235	C₄	C₂×C₂₉×#127	1	1	yes	characteristic
236	C₄	C₂₉×#532	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	8	4	no	—
239	C₂	C₄×C₂₉×#127	1	1	yes	characteristic
240	C₂	C₂₉×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂×C₄×D₄
prime radical	C₂₉×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#444	C₂	semidirect
3. C₂₉×#2931	C₂	semidirect
4. C₂₉×#1974	C₂	semidirect
5. C₂₉×#1963	C₂	semidirect
6. C₂₉×#1849	C₂	semidirect
7. C₂₉×#1726	C₂	semidirect
11. C₂₉×#497	C₄	semidirect
12. C₂₉×#496	C₄	semidirect
50. C₂₉×#97	D₄	semidirect
84. #21709	C₂₉	direct (N × H)
107. C₂×#444	C₂×C₂₉	semidirect
108. #2931	C₂×C₂₉	semidirect
109. #1974	C₂×C₂₉	semidirect
110. #1963	C₂×C₂₉	semidirect
111. #1849	C₂×C₂₉	semidirect
112. #1726	C₂×C₂₉	semidirect
127. #497	C₄×C₂₉	semidirect
128. #496	C₄×C₂₉	semidirect
170. #97	D₄×C₂₉	semidirect
174. C₂₉	#21709	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×#3150	2
C₂×C₂₉×#444	2
C₂₉×#2931	2
C₂₉×#1974	2
C₂₉×#1963	2
C₂₉×#1849	2
C₂₉×#1726	2
#21709	29

Subgroups — C29×#21709 #20316609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#21709 #20316609