Subgroups — C₂₉×#19688 #20314588

Order 7424.

724 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3210	C₂	1	2	yes	maximal, complemented
2	C₂₉×#3182	C₂	1	2	yes	maximal, complemented
3	C₂₉×#3169	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1723	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1722	C₂	1	2	yes	maximal, complemented
7	C₂₉×#1719	C₂	1	2	yes	maximal, complemented
8	C₂₉×#1715	C₂	1	2	yes	maximal, complemented
9	C₂₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×#533	C₂²	1	2	yes	—
11	C₂₉×#532	C₂²	1	2	yes	—
12	C₂₉×#500	C₂²	1	2	yes	—
13	C₂₉×#500	C₂²	1	1	yes	characteristic
14	C₂₉×#500	C₂²	1	1	yes	characteristic
15	C₂₉×#500	C₂²	1	1	yes	characteristic
16	C₂₉×#499	—	2	2	no	—
17	C₂₉×#498	C₂²	1	2	yes	—
18	C₂₉×#494	C₂²	1	2	yes	—
19	C₂₉×#493	C₂²	1	1	yes	characteristic
20	C₂₉×#492	C₂²	1	2	yes	—
21	C₂₉×#492	C₂²	1	2	yes	—
22	C₂₉×#487	C₂²	1	2	yes	—
23	C₂₉×#487	C₂²	1	2	yes	—
24	C₂₉×#486	C₂²	1	2	yes	—
25	C₂₉×#486	C₂²	1	2	yes	—
26	C₂₉×#486	C₂²	1	2	yes	—
27	C₂₉×#468	—	2	4	no	—
28	C₂₉×#466	—	2	2	no	—
29	C₂₉×#464	—	2	4	no	—
30	C₂₉×#452	—	2	4	no	—
31	C₂₉×#440	—	2	4	no	—
32	C₄×D₈×C₂₉	—	2	2	no	—
33	C₂₉×#431	C₂²	1	1	yes	characteristic
34	C₂₉×#427	—	2	2	no	—
35	C₄×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
37	C₂₉×#322	C₂²	1	2	yes	—
38	C₂₉×#322	C₂²	1	2	yes	—
39	C₂×#41×C₂₉	—	2	2	no	—
40	C₂×Q₁₆×C₂₉	—	2	4	no	—
41	C₂×SD₁₆×C₂₉	—	2	4	no	—
42	C₂×SD₁₆×C₂₉	—	2	4	no	—
43	C₂×D₈×C₂₉	—	2	2	no	—
44	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
45	C₂²×C₈×C₂₉	D₄	1	2	yes	—
46	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₉×#127	C₂³	1	2	yes	—
50	C₂₉×#125	—	2	2	no	—
51	C₂₉×#124	—	2	2	no	—
52	C₂₉×C₂²⋊Q₈	—	2	2	no	—
53	C₂₉×C₂²⋊Q₈	—	2	2	no	—
54	C₂₉×#121	—	2	2	no	—
55	C₂₉×#121	—	2	2	no	—
56	C₄×Q₈×C₂₉	—	2	2	no	—
57	C₄×Q₈×C₂₉	—	2	2	no	—
58	C₄×D₄×C₂₉	—	2	2	no	—
59	C₄×D₄×C₂₉	—	2	2	no	—
60	C₂₉×#117	—	2	2	no	—
61	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
62	C₂₉×#107	—	2	2	no	—
63	C₂₉×#107	—	2	2	no	—
64	C₂₉×#107	—	2	2	no	—
65	C₂₉×#107	—	2	4	no	—
66	C₂₉×#106	—	2	4	no	—
67	C₂₉×C₄⋊C₈	D₄	1	2	yes	—
68	C₂₉×#103	—	2	4	no	—
69	C₂₉×#103	—	2	4	no	—
70	C₂₉×#102	—	2	4	no	—
71	C₂₉×#102	—	2	4	no	—
72	C₂₉×#102	—	2	4	no	—
73	C₂₉×C₂²⋊C₈	D₄	1	2	yes	—
74	C₂₉×#97	D₄	1	2	yes	—
75	C₄×C₈×C₂₉	D₄	1	2	yes	—
76	C₄×C₈×C₂₉	C₂³	1	2	yes	—
77	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₂₉	C₂³	1	2	yes	—
79	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
80	#41×C₂₉	—	8	4	no	—
81	C₂×Q₈×C₂₉	—	2	2	no	—
82	C₂×Q₈×C₂₉	—	2	2	no	—
83	C₂×Q₈×C₂₉	—	2	2	no	—
84	C₂×D₄×C₂₉	—	2	2	no	—
85	C₂×D₄×C₂₉	—	2	2	no	—
86	C₂×D₄×C₂₉	—	2	2	no	—
87	C₂×D₄×C₂₉	—	4	2	no	—
88	C₂²×C₄×C₂₉	—	4	2	no	—
89	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₂₉	—	8	4	no	—
93	SD₁₆×C₂₉	—	4	8	no	—
94	SD₁₆×C₂₉	—	8	4	no	—
95	D₈×C₂₉	—	4	4	no	—
96	M₁₆×C₂₉	—	4	2	no	—
97	C₂×C₈×C₂₉	—	2	4	no	—
98	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₂₉	—	2	2	no	—
103	C₄⋊C₄×C₂₉	—	2	2	no	—
104	C₄⋊C₄×C₂₉	—	2	2	no	—
105	C₄⋊C₄×C₂₉	—	2	2	no	—
106	C₄⋊C₄×C₂₉	—	2	2	no	—
107	C₄⋊C₄×C₂₉	—	2	2	no	—
108	C₄⋊C₄×C₂₉	—	4	2	no	—
109	C₄⋊C₄×C₂₉	—	4	2	no	—
110	C₂²⋊C₄×C₂₉	—	4	2	no	—
111	C₂²⋊C₄×C₂₉	—	4	2	no	—
112	C₂²⋊C₄×C₂₉	—	4	2	no	—
113	C₄²×C₂₉	—	4	2	no	—
114	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₉	C₂⁴	1	1	yes	characteristic
118	#19688	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₂₉	—	4	2	no	—
120	C₂³×C₂₉	#127	1	1	yes	characteristic
121	Q₈×C₂₉	—	4	4	no	—
122	Q₈×C₂₉	—	4	4	no	—
123	Q₈×C₂₉	—	4	4	no	—
124	D₄×C₂₉	—	4	4	no	—
125	D₄×C₂₉	—	4	4	no	—
126	D₄×C₂₉	—	4	4	no	—
127	D₄×C₂₉	—	8	4	no	—
128	C₂×C₄×C₂₉	—	2	2	no	—
129	C₂×C₄×C₂₉	—	2	2	no	—
130	C₂×C₄×C₂₉	—	2	2	no	—
131	C₂×C₄×C₂₉	—	4	2	no	—
132	C₂×C₄×C₂₉	—	4	2	no	—
133	C₂×C₄×C₂₉	—	4	2	no	—
134	C₂×C₄×C₂₉	—	8	4	no	—
135	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
137	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
138	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₂₉	—	2	2	no	—
142	C₈×C₂₉	—	2	4	no	—
143	C₈×C₂₉	—	2	4	no	—
144	C₈×C₂₉	#137	1	4	yes	—
145	#3210	C₂×C₂₉	1	2	yes	complemented
146	#3182	C₂×C₂₉	1	2	yes	complemented
147	#3169	C₂×C₂₉	1	1	yes	characteristic
148	#3160	C₂×C₂₉	1	1	yes	characteristic, complemented
149	#1723	C₂×C₂₉	1	2	yes	complemented
150	#1722	C₂×C₂₉	1	2	yes	complemented
151	#1719	C₂×C₂₉	1	2	yes	complemented
152	#1715	C₂×C₂₉	1	2	yes	complemented
153	#1451	C₂×C₂₉	1	1	yes	characteristic, complemented
154	C₂²×C₂₉	—	2	1	no	—
155	C₂²×C₂₉	—	2	1	no	—
156	C₂²×C₂₉	—	2	1	no	—
157	C₂²×C₂₉	—	8	2	no	—
158	C₂²×C₂₉	—	8	2	no	—
159	C₂²×C₂₉	—	8	2	no	—
160	C₂²×C₂₉	C₂×#127	1	1	yes	characteristic
161	C₄×C₂₉	—	2	1	no	—
162	C₄×C₂₉	—	2	1	no	—
163	C₄×C₂₉	—	2	1	no	—
164	C₄×C₂₉	—	8	2	no	—
165	C₄×C₂₉	—	8	2	no	—
166	C₄×C₂₉	—	8	2	no	—
167	C₄×C₂₉	C₂×#137	1	2	yes	—
168	C₄×C₂₉	#575	1	2	yes	—
169	C₄×C₂₉	#576	1	2	yes	—
170	#533	C₂²×C₂₉	1	2	yes	—
171	#532	C₂²×C₂₉	1	2	yes	—
172	#500	C₂²×C₂₉	1	1	yes	characteristic
173	#500	C₂²×C₂₉	1	1	yes	characteristic
174	#500	C₂²×C₂₉	1	1	yes	characteristic
175	#500	C₂²×C₂₉	1	2	yes	—
176	#499	—	2	2	no	—
177	#498	C₂²×C₂₉	1	2	yes	—
178	#494	C₂²×C₂₉	1	2	yes	—
179	#493	C₂²×C₂₉	1	1	yes	characteristic
180	#492	C₂²×C₂₉	1	2	yes	—
181	#492	C₂²×C₂₉	1	2	yes	—
182	#487	C₂²×C₂₉	1	2	yes	—
183	#487	C₂²×C₂₉	1	2	yes	—
184	#486	C₂²×C₂₉	1	2	yes	—
185	#486	C₂²×C₂₉	1	2	yes	—
186	#486	C₂²×C₂₉	1	2	yes	—
187	#468	—	2	4	no	—
188	#466	—	2	2	no	—
189	#464	—	2	4	no	—
190	#452	—	2	4	no	—
191	#440	—	2	4	no	—
192	C₄×D₈	—	2	2	no	—
193	#431	C₂²×C₂₉	1	1	yes	characteristic
194	#427	—	2	2	no	—
195	C₄×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
197	#322	C₂²×C₂₉	1	2	yes	—
198	#322	C₂²×C₂₉	1	2	yes	—
199	C₂×C₂₉	—	4	1	no	—
200	C₂×C₂₉	—	16	2	no	—
201	C₂×C₂₉	#3150	1	1	yes	characteristic
202	C₂×C₂₉	#3152	1	1	yes	characteristic
203	C₂×C₂₉	#3155	1	1	yes	characteristic
204	C₂×#41	—	2	2	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×SD₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
210	C₂²×C₈	D₄×C₂₉	1	2	yes	—
211	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
214	#127	C₂³×C₂₉	1	2	yes	—
215	#125	—	2	2	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	2	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	#117	—	2	2	no	—
226	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	4	no	—
231	#106	—	2	4	no	—
232	C₄⋊C₈	D₄×C₂₉	1	2	yes	—
233	#103	—	2	4	no	—
234	#103	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	D₄×C₂₉	1	2	yes	—
239	#97	D₄×C₂₉	1	2	yes	—
240	C₄×C₈	D₄×C₂₉	1	2	yes	—
241	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₂₉	1	2	yes	—
244	C₄×C₈	C₂³×C₂₉	1	2	yes	—
245	C₂₉	#19688	1	1	yes	characteristic, direct product factor, Sylow
246	#41	—	8	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	4	8	no	—
260	SD₁₆	—	8	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
283	C₄²	C₂⁴×C₂₉	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	C₂₉×#127	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
301	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
302	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	4	no	—
308	C₈	—	2	4	no	—
309	C₈	C₂₉×#137	1	4	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×C₂₉×#127	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂×C₂₉×#137	1	2	yes	—
324	C₄	C₂₉×#575	1	2	yes	—
325	C₄	C₂₉×#576	1	2	yes	—
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₂₉×#3150	1	1	yes	characteristic
329	C₂	C₂₉×#3152	1	1	yes	characteristic
330	C₂	C₂₉×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₉
prime radical	C₂₉×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#3210	C₂	semidirect
2. C₂₉×#3182	C₂	semidirect
4. C₂₉×#3160	C₂	semidirect
5. C₂₉×#1723	C₂	semidirect
6. C₂₉×#1722	C₂	semidirect
7. C₂₉×#1719	C₂	semidirect
8. C₂₉×#1715	C₂	semidirect
9. C₂₉×#1451	C₂	semidirect
118. #19688	C₂₉	direct (N × H)
145. #3210	C₂×C₂₉	semidirect
146. #3182	C₂×C₂₉	semidirect
148. #3160	C₂×C₂₉	semidirect
149. #1723	C₂×C₂₉	semidirect
150. #1722	C₂×C₂₉	semidirect
151. #1719	C₂×C₂₉	semidirect
152. #1715	C₂×C₂₉	semidirect
153. #1451	C₂×C₂₉	semidirect
245. C₂₉	#19688	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×#3210	2
C₂₉×#3182	2
C₂₉×#3169	2
C₂₉×#3160	2
C₂₉×#1723	2
C₂₉×#1722	2
C₂₉×#1719	2
C₂₉×#1715	2
C₂₉×#1451	2
#19688	29

Subgroups — C29×#19688 #20314588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#19688 #20314588