Subgroups — C₂₉×#28811 #20323711

Order 7424.

718 conjugacy classes of subgroups.

The subgroup lattice has 274 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	2	yes	maximal
2	C₂₉×#2552	C₂	1	2	yes	maximal
3	C₂₉×#2059	C₂	1	4	yes	maximal
4	C₂₉×#1947	C₂	1	2	yes	maximal
5	C₂₉×#1941	C₂	1	2	yes	maximal
6	C₂₉×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₉×#106	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₉×#125	C₂²	1	4	yes	—
10	C₂₉×#499	—	2	8	no	—
11	C₂×C₂₉×#107	C₂²	1	4	yes	—
12	C₂×C₂₉×#106	C₂²	1	2	yes	—
13	C₂×C₂₉×#106	C₂²	1	2	yes	—
14	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
15	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₂₉×#398	—	2	4	no	—
20	C₂₉×#389	—	2	2	no	—
21	C₂₉×#389	C₂²	1	4	yes	—
22	C₂₉×#384	—	2	2	no	—
23	C₂₉×#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#340	C₂²	1	4	yes	—
27	C₂₉×#340	C₂²	1	2	yes	—
28	C₂₉×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
32	C₂₉×#125	—	2	8	no	—
33	C₂₉×#125	—	4	4	no	—
34	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
35	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₂₉	—	2	2	no	—
44	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
46	C₂₉×#107	—	2	8	no	—
47	C₂₉×#106	—	2	4	no	—
48	C₂₉×#106	—	4	2	no	—
49	C₂₉×#106	D₄	1	4	yes	—
50	C₂₉×#106	Q₈	1	4	yes	—
51	C₂₉×C₄⋊C₈	Q₈	1	4	yes	—
52	C₄×C₈×C₂₉	—	2	4	no	—
53	C₄×C₈×C₂₉	Q₈	1	4	yes	—
54	C₂₉×#95	—	2	2	no	—
55	C₂₉×#95	—	2	4	no	—
56	C₂₉×#95	D₄	1	2	yes	—
57	C₂²×C₄×C₂₉	—	2	2	no	—
58	C₂²×C₄×C₂₉	—	2	2	no	—
59	C₂²×C₄×C₂₉	—	2	4	no	—
60	C₂²×C₄×C₂₉	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₂₉	—	2	2	no	—
65	C₂×C₈×C₂₉	—	2	1	no	—
66	C₂×C₈×C₂₉	—	2	1	no	—
67	C₂×C₈×C₂₉	—	2	2	no	—
68	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
71	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
73	C₄⋊C₄×C₂₉	—	2	4	no	—
74	C₄⋊C₄×C₂₉	—	2	2	no	—
75	C₄⋊C₄×C₂₉	—	2	2	no	—
76	C₄⋊C₄×C₂₉	—	2	4	no	—
77	C₄⋊C₄×C₂₉	—	2	4	no	—
78	C₄⋊C₄×C₂₉	—	4	8	no	—
79	C₄⋊C₄×C₂₉	—	4	8	no	—
80	C₄⋊C₄×C₂₉	—	4	4	no	—
81	C₄⋊C₄×C₂₉	—	8	2	no	—
82	C₄⋊C₄×C₂₉	—	8	4	no	—
83	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
84	C₄⋊C₄×C₂₉	C₂×Q₈	1	4	yes	—
85	C₄²×C₂₉	—	2	2	no	—
86	C₄²×C₂₉	—	2	2	no	—
87	C₄²×C₂₉	—	4	4	no	—
88	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
89	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
90	#28811	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₉	—	2	2	no	—
93	C₂×C₄×C₂₉	—	2	4	no	—
94	C₂×C₄×C₂₉	—	2	4	no	—
95	C₂×C₄×C₂₉	—	2	4	no	—
96	C₂×C₄×C₂₉	—	2	4	no	—
97	C₂×C₄×C₂₉	—	2	4	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	2	no	—
102	C₂×C₄×C₂₉	—	4	4	no	—
103	C₂×C₄×C₂₉	—	4	4	no	—
104	C₂×C₄×C₂₉	—	4	4	no	—
105	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
106	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
107	C₂×C₄×C₂₉	#135	1	4	yes	—
108	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
114	C₈×C₂₉	—	2	2	no	—
115	C₈×C₂₉	—	2	4	no	—
116	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
117	C₂×#499	C₂×C₂₉	1	2	yes	—
118	#2552	C₂×C₂₉	1	2	yes	—
119	#2059	C₂×C₂₉	1	4	yes	—
120	#1947	C₂×C₂₉	1	2	yes	—
121	#1941	C₂×C₂₉	1	2	yes	—
122	#1844	C₂×C₂₉	1	1	yes	characteristic
123	C₄×#106	C₂×C₂₉	1	1	yes	characteristic
124	C₈×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
125	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₂²×C₂₉	D₄²	1	1	yes	characteristic
127	C₂²×C₂₉	D₄×Q₈	1	1	yes	characteristic
128	C₂²×C₂₉	C₂×#135	1	2	yes	—
129	C₂²×C₂₉	#577	1	2	yes	—
130	C₄×C₂₉	—	2	2	no	—
131	C₄×C₂₉	—	4	4	no	—
132	C₄×C₂₉	—	4	4	no	—
133	C₄×C₂₉	—	4	4	no	—
134	C₄×C₂₉	—	8	4	no	—
135	C₄×C₂₉	#499	1	4	yes	complemented
136	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	C₄×C₂₉	#544	1	1	yes	characteristic
138	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
139	C₄×C₂₉	Q₈²	1	1	yes	characteristic
140	C₂×#125	C₂²×C₂₉	1	4	yes	—
141	#499	—	2	8	no	—
142	C₂×#107	C₂²×C₂₉	1	4	yes	—
143	C₂×#106	C₂²×C₂₉	1	2	yes	—
144	C₂×#106	C₂²×C₂₉	1	2	yes	—
145	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
146	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
150	#398	—	2	4	no	—
151	#389	—	2	2	no	—
152	#389	C₂²×C₂₉	1	4	yes	—
153	#384	—	2	2	no	—
154	#384	C₂²×C₂₉	1	2	yes	—
155	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
157	#340	C₂²×C₂₉	1	2	yes	—
158	#340	C₂²×C₂₉	1	4	yes	—
159	#336	C₂²×C₂₉	1	2	yes	—
160	C₂×C₂₉	#2551	1	1	yes	characteristic
161	C₂×C₂₉	C₂×#499	1	1	yes	characteristic
162	C₂×C₂₉	#3163	1	1	yes	characteristic
163	C₂×C₂₉	#3284	1	2	yes	—
164	C₂×C₂₉	#3381	1	2	yes	—
165	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
167	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
168	#125	—	2	8	no	—
169	#125	—	4	4	no	—
170	C₂×C₄⋊C₄	—	2	4	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
177	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
181	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
182	#107	—	2	8	no	—
183	#106	—	2	4	no	—
184	#106	—	4	2	no	—
185	#106	D₄×C₂₉	1	4	yes	—
186	#106	Q₈×C₂₉	1	4	yes	—
187	C₄⋊C₈	Q₈×C₂₉	1	4	yes	—
188	C₄×C₈	—	2	4	no	—
189	C₄×C₈	Q₈×C₂₉	1	4	yes	—
190	#95	—	2	2	no	—
191	#95	—	2	4	no	—
192	#95	D₄×C₂₉	1	2	yes	—
193	C₂₉	#28811	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂×D₄×C₂₉	1	4	yes	—
198	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
200	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
206	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
207	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
208	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
209	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	4	8	no	—
216	C₄⋊C₄	—	4	8	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	8	2	no	—
219	C₄⋊C₄	—	8	4	no	—
220	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
221	C₄⋊C₄	C₂×Q₈×C₂₉	1	4	yes	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	4	4	no	—
225	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
226	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
227	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
242	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
243	C₂×C₄	C₂₉×#135	1	4	yes	—
244	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
248	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
253	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
254	C₂²	D₄²×C₂₉	1	1	yes	characteristic
255	C₂²	D₄×Q₈×C₂₉	1	1	yes	characteristic
256	C₂²	C₂×C₂₉×#135	1	2	yes	—
257	C₂²	C₂₉×#577	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	C₂₉×#499	1	4	yes	complemented
264	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
265	C₄	C₂₉×#544	1	1	yes	characteristic
266	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
267	C₄	Q₈²×C₂₉	1	1	yes	characteristic
268	C₂	C₂₉×#2551	1	1	yes	characteristic
269	C₂	C₂×C₂₉×#499	1	1	yes	characteristic
270	C₂	C₂₉×#3163	1	1	yes	characteristic
271	C₂	C₂₉×#3284	1	2	yes	—
272	C₂	C₂₉×#3381	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
90. #28811	C₂₉	direct (N × H)
135. C₄×C₂₉	#499	semidirect
193. C₂₉	#28811	direct (N × H)
263. C₄	C₂₉×#499	semidirect

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×#2552	2
C₂₉×#2059	2
C₂₉×#1947	2
C₂₉×#1941	2
C₂₉×#1844	2
C₄×C₂₉×#106	2
C₈×C₄⋊C₄×C₂₉	2
#28811	29

Subgroups — C29×#28811 #20323711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28811 #20323711