Subgroups — C₂₉×#28814 #20323714

Order 7424.

718 conjugacy classes of subgroups.

The subgroup lattice has 300 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×#1948	C₂	1	2	yes	maximal
5	C₂₉×#1942	C₂	1	2	yes	maximal
6	C₂₉×#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₉×#107	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	2	yes	—
12	C₂×C₂₉×#107	C₂²	1	2	yes	—
13	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₂₉×#106	C₂²	1	4	yes	—
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	2	yes	—
30	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
40	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₉	—	2	2	no	—
45	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
47	C₂₉×#107	—	2	4	no	—
48	C₂₉×#107	—	4	2	no	—
49	C₂₉×#107	D₄	1	4	yes	—
50	C₂₉×#107	Q₈	1	4	yes	—
51	C₂₉×#106	—	2	8	no	—
52	C₂₉×C₄⋊C₈	Q₈	1	4	yes	—
53	C₄×C₈×C₂₉	—	2	4	no	—
54	C₄×C₈×C₂₉	Q₈	1	4	yes	—
55	C₂₉×#95	—	2	2	no	—
56	C₂₉×#95	—	2	4	no	—
57	C₂₉×#95	D₄	1	2	yes	—
58	C₂²×C₄×C₂₉	—	2	2	no	—
59	C₂²×C₄×C₂₉	—	2	2	no	—
60	C₂²×C₄×C₂₉	—	2	4	no	—
61	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₂₉	—	2	2	no	—
67	C₂×C₈×C₂₉	—	2	1	no	—
68	C₂×C₈×C₂₉	—	2	1	no	—
69	C₂×C₈×C₂₉	—	2	2	no	—
70	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
75	C₄⋊C₄×C₂₉	—	2	2	no	—
76	C₄⋊C₄×C₂₉	—	2	2	no	—
77	C₄⋊C₄×C₂₉	—	2	2	no	—
78	C₄⋊C₄×C₂₉	—	2	2	no	—
79	C₄⋊C₄×C₂₉	—	2	4	no	—
80	C₄⋊C₄×C₂₉	—	2	4	no	—
81	C₄⋊C₄×C₂₉	—	4	8	no	—
82	C₄⋊C₄×C₂₉	—	4	8	no	—
83	C₄⋊C₄×C₂₉	—	4	4	no	—
84	C₄⋊C₄×C₂₉	—	8	2	no	—
85	C₄⋊C₄×C₂₉	—	8	4	no	—
86	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₂₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₂₉	—	2	2	no	—
89	C₄²×C₂₉	—	2	2	no	—
90	C₄²×C₂₉	—	4	4	no	—
91	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
92	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
93	#28814	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	—	2	1	no	—
96	C₂×C₄×C₂₉	—	2	1	no	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	2	2	no	—
99	C₂×C₄×C₂₉	—	2	4	no	—
100	C₂×C₄×C₂₉	—	2	4	no	—
101	C₂×C₄×C₂₉	—	2	2	no	—
102	C₂×C₄×C₂₉	—	2	2	no	—
103	C₂×C₄×C₂₉	—	2	2	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	—	4	2	no	—
106	C₂×C₄×C₂₉	—	4	2	no	—
107	C₂×C₄×C₂₉	—	4	2	no	—
108	C₂×C₄×C₂₉	—	4	2	no	—
109	C₂×C₄×C₂₉	—	4	4	no	—
110	C₂×C₄×C₂₉	—	4	4	no	—
111	C₂×C₄×C₂₉	—	4	4	no	—
112	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
113	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₂₉	#135	1	2	yes	—
115	C₂×C₄×C₂₉	#135	1	2	yes	—
116	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
122	C₈×C₂₉	—	2	2	no	—
123	C₈×C₂₉	—	2	4	no	—
124	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
125	C₂×#499	C₂×C₂₉	1	2	yes	—
126	#2552	C₂×C₂₉	1	2	yes	—
127	#2059	C₂×C₂₉	1	4	yes	—
128	#1948	C₂×C₂₉	1	2	yes	—
129	#1942	C₂×C₂₉	1	2	yes	—
130	#1845	C₂×C₂₉	1	1	yes	characteristic
131	C₄×#107	C₂×C₂₉	1	1	yes	characteristic
132	C₈×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
133	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₂²×C₂₉	D₄²	1	1	yes	characteristic
135	C₂²×C₂₉	D₄×Q₈	1	1	yes	characteristic
136	C₂²×C₂₉	C₂×#135	1	1	yes	characteristic
137	C₂²×C₂₉	C₂×#135	1	1	yes	characteristic
138	C₂²×C₂₉	#576	1	1	yes	characteristic
139	C₂²×C₂₉	#578	1	1	yes	characteristic
140	C₄×C₂₉	—	2	2	no	—
141	C₄×C₂₉	—	4	4	no	—
142	C₄×C₂₉	—	4	2	no	—
143	C₄×C₂₉	—	4	4	no	—
144	C₄×C₂₉	—	4	2	no	—
145	C₄×C₂₉	—	8	4	no	—
146	C₄×C₂₉	#499	1	4	yes	complemented
147	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₄×C₂₉	#544	1	1	yes	characteristic
149	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
150	C₄×C₂₉	Q₈²	1	1	yes	characteristic
151	C₂×#125	C₂²×C₂₉	1	4	yes	—
152	#499	—	2	8	no	—
153	C₂×#107	C₂²×C₂₉	1	2	yes	—
154	C₂×#107	C₂²×C₂₉	1	2	yes	—
155	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
156	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
157	C₂×#106	C₂²×C₂₉	1	4	yes	—
158	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
161	#398	—	2	4	no	—
162	#389	—	2	2	no	—
163	#389	C₂²×C₂₉	1	4	yes	—
164	#384	—	2	2	no	—
165	#384	C₂²×C₂₉	1	2	yes	—
166	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
167	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
168	#340	C₂²×C₂₉	1	2	yes	—
169	#340	C₂²×C₂₉	1	4	yes	—
170	#336	C₂²×C₂₉	1	2	yes	—
171	C₂×C₂₉	#2551	1	1	yes	characteristic
172	C₂×C₂₉	C₂×#499	1	1	yes	characteristic
173	C₂×C₂₉	#3164	1	1	yes	characteristic
174	C₂×C₂₉	#3283	1	1	yes	characteristic
175	C₂×C₂₉	#3287	1	1	yes	characteristic
176	C₂×C₂₉	#3380	1	1	yes	characteristic
177	C₂×C₂₉	#3383	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
180	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
181	#125	—	2	8	no	—
182	#125	—	4	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
191	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
195	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
196	#107	—	2	4	no	—
197	#107	—	4	2	no	—
198	#107	D₄×C₂₉	1	4	yes	—
199	#107	Q₈×C₂₉	1	4	yes	—
200	#106	—	2	8	no	—
201	C₄⋊C₈	Q₈×C₂₉	1	4	yes	—
202	C₄×C₈	—	2	4	no	—
203	C₄×C₈	Q₈×C₂₉	1	4	yes	—
204	#95	—	2	2	no	—
205	#95	—	2	4	no	—
206	#95	D₄×C₂₉	1	2	yes	—
207	C₂₉	#28814	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
221	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
222	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
223	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
224	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	4	8	no	—
232	C₄⋊C₄	—	4	8	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	8	2	no	—
235	C₄⋊C₄	—	8	4	no	—
236	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
237	C₄⋊C₄	C₂×Q₈×C₂₉	1	4	yes	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	4	no	—
241	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
242	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
243	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
262	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
263	C₂×C₄	C₂₉×#135	1	2	yes	—
264	C₂×C₄	C₂₉×#135	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
267	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
269	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	2	4	no	—
273	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
274	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
275	C₂²	D₄²×C₂₉	1	1	yes	characteristic
276	C₂²	D₄×Q₈×C₂₉	1	1	yes	characteristic
277	C₂²	C₂×C₂₉×#135	1	1	yes	characteristic
278	C₂²	C₂×C₂₉×#135	1	1	yes	characteristic
279	C₂²	C₂₉×#576	1	1	yes	characteristic
280	C₂²	C₂₉×#578	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	4	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	4	no	—
286	C₄	—	8	4	no	—
287	C₄	C₂₉×#499	1	4	yes	complemented
288	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
289	C₄	C₂₉×#544	1	1	yes	characteristic
290	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
291	C₄	Q₈²×C₂₉	1	1	yes	characteristic
292	C₂	C₂₉×#2551	1	1	yes	characteristic
293	C₂	C₂×C₂₉×#499	1	1	yes	characteristic
294	C₂	C₂₉×#3164	1	1	yes	characteristic
295	C₂	C₂₉×#3283	1	1	yes	characteristic
296	C₂	C₂₉×#3287	1	1	yes	characteristic
297	C₂	C₂₉×#3380	1	1	yes	characteristic
298	C₂	C₂₉×#3383	1	1	yes	characteristic

	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
93. #28814	C₂₉	direct (N × H)
146. C₄×C₂₉	#499	semidirect
207. C₂₉	#28814	direct (N × H)
287. 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×#1948	2
C₂₉×#1942	2
C₂₉×#1845	2
C₄×C₂₉×#107	2
C₈×C₄⋊C₄×C₂₉	2
#28814	29

Subgroups — C29×#28814 #20323714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28814 #20323714