Subgroups — C₂₉×#12831 #20307731

Order 7424.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#365	C₂	1	2	yes	maximal, complemented
3	C₂₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1807	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₂₉×M₃₂	—	2	4	no	—
8	C₂×C₂₉×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₂₉	C₄	1	2	yes	complemented
12	C₂×C₂₉×#108	C₂²	1	2	yes	—
13	C₂×C₂₉×#106	C₂²	1	2	yes	—
14	C₂₉×#407	—	2	1	no	—
15	C₂₉×#380	—	2	1	no	—
16	C₂₉×#365	—	2	4	no	—
17	C₂₉×#341	—	2	1	no	—
18	C₂₉×#340	—	2	1	no	—
19	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₂₉	—	2	2	no	—
21	C₂²×C₈×C₂₉	—	2	2	no	—
22	C₂²×C₈×C₂₉	—	2	2	no	—
23	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₉	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₂₉	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₂₉	—	4	1	no	—
31	C₂₉×M₃₂	—	2	2	no	—
32	C₂₉×M₃₂	—	2	4	no	—
33	C₂₉×M₃₂	D₄	1	4	yes	—
34	C₂×C₁₆×C₂₉	—	2	4	no	—
35	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
36	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
37	C₂₉×#108	—	2	2	no	—
38	C₂₉×#108	—	2	2	no	—
39	C₂₉×#106	—	2	2	no	—
40	C₂₉×#106	—	2	2	no	—
41	C₂₉×C₂²⋊C₈	—	4	1	no	—
42	C₂₉×C₂²⋊C₈	—	4	1	no	—
43	C₂₉×#95	—	4	1	no	—
44	C₂⁴×C₂₉	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	—	2	2	no	—
46	C₂²×C₄×C₂₉	—	2	2	no	—
47	C₂²×C₄×C₂₉	—	4	2	no	—
48	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₂₉	—	4	2	no	—
55	M₁₆×C₂₉	—	4	2	no	—
56	C₂×C₈×C₂₉	—	2	2	no	—
57	C₂×C₈×C₂₉	—	2	1	no	—
58	C₂×C₈×C₂₉	—	2	1	no	—
59	C₂×C₈×C₂₉	—	2	1	no	—
60	C₂×C₈×C₂₉	—	2	1	no	—
61	C₂×C₈×C₂₉	—	2	2	no	—
62	C₂×C₈×C₂₉	—	2	2	no	—
63	C₂×C₈×C₂₉	—	4	2	no	—
64	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₂₉	#41	1	2	yes	—
70	C₄⋊C₄×C₂₉	—	4	2	no	—
71	C₄⋊C₄×C₂₉	—	4	2	no	—
72	C₂²⋊C₄×C₂₉	—	8	2	no	—
73	C₁₆×C₂₉	—	2	2	no	—
74	C₁₆×C₂₉	C₂²⋊C₄	1	4	yes	complemented
75	#12831	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₉	—	2	2	no	—
77	C₂³×C₂₉	—	2	2	no	—
78	C₂³×C₂₉	—	4	1	no	—
79	C₂³×C₂₉	#106	1	1	yes	characteristic
80	C₂³×C₂₉	#106	1	1	yes	characteristic
81	C₂³×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₂₉	—	2	2	no	—
83	C₂×C₄×C₂₉	—	2	1	no	—
84	C₂×C₄×C₂₉	—	2	1	no	—
85	C₂×C₄×C₂₉	—	2	1	no	—
86	C₂×C₄×C₂₉	—	2	1	no	—
87	C₂×C₄×C₂₉	—	2	2	no	—
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	8	2	no	—
90	C₂×C₄×C₂₉	—	8	2	no	—
91	C₂×C₄×C₂₉	—	8	2	no	—
92	C₂×C₄×C₂₉	#106	1	1	yes	characteristic
93	C₂×C₄×C₂₉	#106	1	1	yes	characteristic
94	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	#121	1	2	yes	—
96	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₂₉	#135	1	2	yes	—
98	C₈×C₂₉	—	2	1	no	—
99	C₈×C₂₉	—	2	1	no	—
100	C₈×C₂₉	—	8	2	no	—
101	C₈×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₈×C₂₉	#117	1	1	yes	characteristic
103	C₈×C₂₉	C₂²⋊Q₈	1	2	yes	—
104	C₂²×M₃₂	C₂×C₂₉	1	1	yes	characteristic
105	C₂×#365	C₂×C₂₉	1	2	yes	complemented
106	#1810	C₂×C₂₉	1	1	yes	characteristic
107	#1807	C₂×C₂₉	1	1	yes	characteristic
108	#1383	C₂×C₂₉	1	2	yes	complemented
109	C₂²×C₂₉	—	2	1	no	—
110	C₂²×C₂₉	—	2	1	no	—
111	C₂²×C₂₉	—	2	1	no	—
112	C₂²×C₂₉	—	2	1	no	—
113	C₂²×C₂₉	—	2	1	no	—
114	C₂²×C₂₉	—	2	1	no	—
115	C₂²×C₂₉	—	2	1	no	—
116	C₂²×C₂₉	—	4	4	no	—
117	C₂²×C₂₉	#365	1	2	yes	complemented
118	C₂²×C₂₉	C₂×#106	1	1	yes	characteristic
119	C₂²×C₂₉	#465	1	2	yes	—
120	C₄×C₂₉	—	2	1	no	—
121	C₄×C₂₉	—	2	1	no	—
122	C₄×C₂₉	—	16	2	no	—
123	C₄×C₂₉	#380	1	1	yes	characteristic
124	C₄×C₂₉	#427	1	1	yes	characteristic
125	C₄×C₂₉	#465	1	2	yes	—
126	C₂³×C₈	C₂²×C₂₉	1	1	yes	characteristic
127	C₂×M₃₂	—	2	4	no	—
128	C₂×M₃₂	C₄×C₂₉	1	2	yes	complemented
129	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
131	C₂²×C₁₆	C₄×C₂₉	1	2	yes	complemented
132	C₂×#108	C₂²×C₂₉	1	2	yes	—
133	C₂×#106	C₂²×C₂₉	1	2	yes	—
134	#407	—	2	1	no	—
135	#380	—	2	1	no	—
136	#365	—	2	4	no	—
137	#341	—	2	1	no	—
138	#340	—	2	1	no	—
139	C₂×C₂₉	—	2	2	no	—
140	C₂×C₂₉	—	2	2	no	—
141	C₂×C₂₉	—	4	1	no	—
142	C₂×C₂₉	#1807	1	1	yes	characteristic
143	C₂×C₂₉	C₂×#365	1	1	yes	characteristic
144	C₂×C₂₉	#2155	1	1	yes	characteristic
145	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
146	C₂×M₁₆	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
152	C₂²×C₈	Q₈×C₂₉	1	1	yes	characteristic
153	C₂²×C₈	Q₈×C₂₉	1	1	yes	characteristic
154	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₂²⋊C₄	—	4	1	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	4	no	—
159	M₃₂	D₄×C₂₉	1	4	yes	—
160	C₂×C₁₆	—	2	4	no	—
161	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
162	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
163	#108	—	2	2	no	—
164	#108	—	2	2	no	—
165	#106	—	2	2	no	—
166	#106	—	2	2	no	—
167	C₂²⋊C₈	—	4	1	no	—
168	C₂²⋊C₈	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂₉	#12831	1	1	yes	characteristic, direct product factor, Sylow
171	C₂⁴	SD₁₆×C₂₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
177	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
178	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
179	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
192	C₂×C₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
195	C₂×C₈	C₂×Q₈×C₂₉	1	1	yes	characteristic
196	C₂×C₈	#41×C₂₉	1	2	yes	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	8	2	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂²⋊C₄×C₂₉	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂₉×#106	1	1	yes	characteristic
206	C₂³	C₂₉×#106	1	1	yes	characteristic
207	C₂³	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂₉×#106	1	1	yes	characteristic
219	C₂×C₄	C₂₉×#106	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
221	C₂×C₄	C₂₉×#121	1	2	yes	—
222	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
223	C₂×C₄	C₂₉×#135	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
228	C₈	C₂₉×#117	1	1	yes	characteristic
229	C₈	C₂₉×C₂²⋊Q₈	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	C₂₉×#365	1	2	yes	complemented
239	C₂²	C₂×C₂₉×#106	1	1	yes	characteristic
240	C₂²	C₂₉×#465	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	16	2	no	—
244	C₄	C₂₉×#380	1	1	yes	characteristic
245	C₄	C₂₉×#427	1	1	yes	characteristic
246	C₄	C₂₉×#465	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	C₂₉×#1807	1	1	yes	characteristic
251	C₂	C₂×C₂₉×#365	1	1	yes	characteristic
252	C₂	C₂₉×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
prime radical	C₂⁴×C₂₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ C₂³×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#365	C₂	semidirect
5. C₂₉×#1383	C₂	semidirect
8. C₂×C₂₉×M₃₂	C₄	semidirect
11. C₂²×C₁₆×C₂₉	C₄	semidirect
74. C₁₆×C₂₉	C₂²⋊C₄	semidirect
75. #12831	C₂₉	direct (N × H)
105. C₂×#365	C₂×C₂₉	semidirect
108. #1383	C₂×C₂₉	semidirect
117. C₂²×C₂₉	#365	semidirect
128. C₂×M₃₂	C₄×C₂₉	semidirect
131. C₂²×C₁₆	C₄×C₂₉	semidirect
170. C₂₉	#12831	direct (N × H)
201. C₁₆	C₂²⋊C₄×C₂₉	semidirect
238. C₂²	C₂₉×#365	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₉×M₃₂	2
C₂×C₂₉×#365	2
C₂₉×#1810	2
C₂₉×#1807	2
C₂₉×#1383	2
#12831	29

Subgroups — C29×#12831 #20307731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#12831 #20307731