Subgroups — C₂₉×#20765 #20315665

Order 7424.

926 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×#382	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1836	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1835	C₂	1	4	yes	maximal, complemented
6	C₂₉×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₂₉	C₂²	1	2	yes	—
8	C₂×C₂₉×#117	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₂₉	C₄	1	8	yes	complemented
12	C₂×C₂₉×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
14	C₂₉×#384	C₂²	1	2	yes	—
15	C₂₉×#382	—	2	2	no	—
16	C₂₉×#382	C₂²	1	2	yes	—
17	C₂₉×#382	C₂²	1	2	yes	—
18	C₂₉×#382	C₂²	1	2	yes	—
19	C₂₉×#380	—	2	2	no	—
20	C₄×C₄⋊C₄×C₂₉	—	2	2	no	—
21	C₂×C₂₉×#95	—	2	2	no	—
22	C₂₉×#340	C₂²	1	16	yes	—
23	C₂³×C₄×C₂₉	—	2	2	no	—
24	C₂³×C₄×C₂₉	D₄	1	2	yes	—
25	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₂₉	—	2	4	no	—
27	C₂×M₁₆×C₂₉	—	2	4	no	—
28	C₂×M₁₆×C₂₉	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
30	C₂₉×#117	—	2	8	no	—
31	C₂×C₄⋊C₄×C₂₉	—	2	8	no	—
32	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
36	C₂×C₂²⋊C₄×C₂₉	—	2	2	no	—
37	C₂×C₄²×C₂₉	—	2	2	no	—
38	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₂₉	D₄	1	2	yes	—
40	C₂×C₄²×C₂₉	Q₈	1	4	yes	—
41	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂₉×#97	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₂₉	C₂×C₄	1	8	yes	complemented
45	C₂₉×#95	—	2	4	no	—
46	C₂₉×#95	—	2	4	no	—
47	C₂₉×#95	—	2	4	no	—
48	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	—	2	4	no	—
50	C₂²×C₄×C₂₉	—	2	4	no	—
51	C₂²×C₄×C₂₉	—	4	4	no	—
52	C₂²×C₄×C₂₉	—	4	8	no	—
53	C₂²×C₄×C₂₉	—	4	4	no	—
54	C₂²×C₄×C₂₉	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₂₉	#41	1	4	yes	—
61	C₂²×C₄×C₂₉	#41	1	4	yes	—
62	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₂₉	—	2	16	no	—
64	C₂×C₈×C₂₉	—	2	4	no	—
65	C₂×C₈×C₂₉	—	2	4	no	—
66	C₂×C₈×C₂₉	C₂²×C₄	1	8	yes	—
67	C₄⋊C₄×C₂₉	—	2	8	no	—
68	C₄⋊C₄×C₂₉	—	2	8	no	—
69	C₄⋊C₄×C₂₉	—	4	8	no	—
70	C₄⋊C₄×C₂₉	—	4	4	no	—
71	C₂²⋊C₄×C₂₉	—	4	4	no	—
72	C₄²×C₂₉	—	4	4	no	—
73	C₄²×C₂₉	C₄⋊C₄	1	8	yes	—
74	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
75	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
77	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
78	#20765	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₉	—	2	4	no	—
80	C₂³×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₂₉	#124	1	2	yes	—
82	C₂³×C₂₉	#124	1	2	yes	—
83	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₉	—	2	4	no	—
85	C₂×C₄×C₂₉	—	2	4	no	—
86	C₂×C₄×C₂₉	—	4	8	no	—
87	C₂×C₄×C₂₉	—	4	4	no	—
88	C₂×C₄×C₂₉	—	4	4	no	—
89	C₂×C₄×C₂₉	—	4	8	no	—
90	C₂×C₄×C₂₉	—	4	8	no	—
91	C₂×C₄×C₂₉	—	8	8	no	—
92	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₂₉	#117	1	8	yes	—
96	C₂×C₄×C₂₉	#124	1	2	yes	—
97	C₂×C₄×C₂₉	#124	1	2	yes	—
98	C₂×C₄×C₂₉	#125	1	8	yes	—
99	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
102	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
103	C₈×C₂₉	—	2	8	no	—
104	C₂×C₄×M₁₆	C₂×C₂₉	1	1	yes	characteristic
105	#2285	C₂×C₂₉	1	1	yes	characteristic
106	C₂×#382	C₂×C₂₉	1	1	yes	characteristic
107	#1836	C₂×C₂₉	1	4	yes	complemented
108	#1835	C₂×C₂₉	1	4	yes	complemented
109	#1736	C₂×C₂₉	1	4	yes	—
110	C₂²×C₂₉	—	2	4	no	—
111	C₂²×C₂₉	—	2	4	no	—
112	C₂²×C₂₉	—	2	4	no	—
113	C₂²×C₂₉	#382	1	4	yes	—
114	C₂²×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₂₉	C₂×#124	1	1	yes	characteristic
116	C₂²×C₂₉	C₂×#124	1	1	yes	characteristic
117	C₂²×C₂₉	#575	1	4	yes	—
118	C₄×C₂₉	—	2	4	no	—
119	C₄×C₂₉	—	2	4	no	—
120	C₄×C₂₉	—	8	4	no	—
121	C₄×C₂₉	—	8	4	no	—
122	C₄×C₂₉	#382	1	4	yes	—
123	C₄×C₂₉	C₂×#117	1	2	yes	—
124	C₄×C₂₉	C₂×#125	1	2	yes	—
125	C₂²×M₁₆	C₂²×C₂₉	1	2	yes	—
126	C₂×#117	C₂²×C₂₉	1	2	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
128	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
129	C₄×M₁₆	C₄×C₂₉	1	8	yes	complemented
130	C₂×#97	C₂²×C₂₉	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
132	#384	C₂²×C₂₉	1	2	yes	—
133	#382	—	2	2	no	—
134	#382	C₂²×C₂₉	1	2	yes	—
135	#382	C₂²×C₂₉	1	2	yes	—
136	#382	C₂²×C₂₉	1	2	yes	—
137	#380	—	2	2	no	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₂×#95	—	2	2	no	—
140	#340	C₂²×C₂₉	1	16	yes	—
141	C₂×C₂₉	—	2	4	no	—
142	C₂×C₂₉	C₂×#382	1	1	yes	characteristic
143	C₂×C₂₉	#2911	1	2	yes	—
144	C₂×C₂₉	#3140	1	2	yes	—
145	C₂×C₂₉	#3142	1	2	yes	—
146	C₂³×C₄	—	2	2	no	—
147	C₂³×C₄	D₄×C₂₉	1	2	yes	—
148	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	C₂×C₄×C₂₉	1	8	yes	—
152	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
153	#117	—	2	8	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
158	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
162	C₂×C₄²	D₄×C₂₉	1	2	yes	—
163	C₂×C₄²	Q₈×C₂₉	1	4	yes	—
164	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
166	#97	C₂×C₄×C₂₉	1	8	yes	complemented
167	C₄×C₈	C₂×C₄×C₂₉	1	8	yes	complemented
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	#95	—	2	4	no	—
171	C₂₉	#20765	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	C₄⋊C₄×C₂₉	1	8	yes	—
179	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
184	C₂²×C₄	#41×C₂₉	1	4	yes	—
185	C₂²×C₄	#41×C₂₉	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
187	M₁₆	—	2	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂²×C₄×C₂₉	1	8	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₂²⋊C₄	—	4	4	no	—
196	C₄²	—	4	4	no	—
197	C₄²	C₄⋊C₄×C₂₉	1	8	yes	—
198	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
199	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
200	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
201	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
204	C₂³	C₂₉×#124	1	2	yes	—
205	C₂³	C₂₉×#124	1	2	yes	—
206	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	—	8	8	no	—
215	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
216	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
218	C₂×C₄	C₂₉×#117	1	8	yes	—
219	C₂×C₄	C₂₉×#124	1	2	yes	—
220	C₂×C₄	C₂₉×#124	1	2	yes	—
221	C₂×C₄	C₂₉×#125	1	8	yes	—
222	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
223	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
224	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
225	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
226	C₈	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	C₂₉×#382	1	4	yes	—
231	C₂²	C₂²×C₄⋊C₄×C₂₉	1	1	yes	characteristic
232	C₂²	C₂×C₂₉×#124	1	1	yes	characteristic
233	C₂²	C₂×C₂₉×#124	1	1	yes	characteristic
234	C₂²	C₂₉×#575	1	4	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₂₉×#382	1	4	yes	—
240	C₄	C₂×C₂₉×#117	1	2	yes	—
241	C₄	C₂×C₂₉×#125	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×C₂₉×#382	1	1	yes	characteristic
244	C₂	C₂₉×#2911	1	2	yes	—
245	C₂	C₂₉×#3140	1	2	yes	—
246	C₂	C₂₉×#3142	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
prime radical	C₂⁴×C₂₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂²×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₉×#1836	C₂	semidirect
5. C₂₉×#1835	C₂	semidirect
11. C₄×M₁₆×C₂₉	C₄	semidirect
43. C₂₉×#97	C₂×C₄	semidirect
44. C₄×C₈×C₂₉	C₂×C₄	semidirect
78. #20765	C₂₉	direct (N × H)
107. #1836	C₂×C₂₉	semidirect
108. #1835	C₂×C₂₉	semidirect
129. C₄×M₁₆	C₄×C₂₉	semidirect
166. #97	C₂×C₄×C₂₉	semidirect
167. C₄×C₈	C₂×C₄×C₂₉	semidirect
171. C₂₉	#20765	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×C₄×M₁₆×C₂₉	2
C₂₉×#2285	2
C₂×C₂₉×#382	2
C₂₉×#1836	2
C₂₉×#1835	2
C₂₉×#1736	2
#20765	29

Subgroups — C29×#20765 #20315665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#20765 #20315665