Subgroups — C₂₉×#28834 #20323734

Order 7424.

594 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2570	C₂	1	2	yes	maximal
2	C₂₉×#2085	C₂	1	2	yes	maximal
3	C₂₉×#2084	C₂	1	2	yes	maximal
4	C₂₉×#1949	C₂	1	2	yes	maximal
5	C₂₉×#1943	C₂	1	2	yes	maximal
6	C₂₉×#1847	C₂	1	2	yes	maximal
7	C₂₉×#1846	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂₉×#106	C₂	1	1	yes	maximal, characteristic
9	C₂₉×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₉×#107	C₂²	1	2	yes	—
11	C₂×C₂₉×#107	C₂²	1	2	yes	—
12	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
13	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
14	C₂×C₂₉×#106	C₂²	1	2	yes	—
15	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₉×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×#400	—	2	2	no	—
21	C₂₉×#398	—	2	2	no	—
22	C₂₉×#389	C₂²	1	2	yes	—
23	C₂₉×#389	C₂²	1	2	yes	—
24	C₂₉×#387	—	2	2	no	—
25	C₂₉×#387	—	2	2	no	—
26	C₂₉×#384	C₂²	1	2	yes	—
27	C₂₉×#382	C₂²	1	2	yes	—
28	C₂₉×#382	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
31	C₂₉×#340	C₂²	1	2	yes	—
32	C₂₉×#340	C₂²	1	2	yes	—
33	C₂₉×#340	C₂²	1	2	yes	—
34	C₂₉×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₉	—	2	2	no	—
48	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
49	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
50	C₂₉×#107	—	2	4	no	—
51	C₂₉×#107	—	4	2	no	—
52	C₂₉×#106	—	2	2	no	—
53	C₂₉×#106	—	2	2	no	—
54	C₂₉×#106	—	4	1	no	—
55	C₂₉×#106	—	4	2	no	—
56	C₂₉×#106	—	4	1	no	—
57	C₂₉×C₄⋊C₈	Q₈	1	4	yes	—
58	C₂₉×#97	—	2	4	no	—
59	C₄×C₈×C₂₉	Q₈	1	4	yes	—
60	C₂₉×#95	—	2	2	no	—
61	C₂₉×#95	—	2	2	no	—
62	C₂₉×#95	—	2	2	no	—
63	C₂₉×#95	—	2	2	no	—
64	C₂₉×#95	—	2	2	no	—
65	C₂₉×#95	—	2	2	no	—
66	C₂₉×#95	—	2	2	no	—
67	C₂₉×#95	D₄	1	1	yes	characteristic
68	C₂₉×#95	D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₉	—	2	2	no	—
70	C₂²×C₄×C₂₉	—	2	2	no	—
71	C₂²×C₄×C₂₉	—	2	2	no	—
72	C₂²×C₄×C₂₉	—	2	2	no	—
73	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₂₉	—	2	2	no	—
79	C₂×C₈×C₂₉	—	2	1	no	—
80	C₂×C₈×C₂₉	—	2	2	no	—
81	C₂×C₈×C₂₉	—	2	1	no	—
82	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
84	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₂₉	#41	1	2	yes	—
86	C₂×C₈×C₂₉	#41	1	2	yes	—
87	C₄⋊C₄×C₂₉	—	2	2	no	—
88	C₄⋊C₄×C₂₉	—	2	2	no	—
89	C₄⋊C₄×C₂₉	—	2	2	no	—
90	C₄⋊C₄×C₂₉	—	2	2	no	—
91	C₄⋊C₄×C₂₉	—	2	2	no	—
92	C₄⋊C₄×C₂₉	—	2	2	no	—
93	C₄⋊C₄×C₂₉	—	2	2	no	—
94	C₄⋊C₄×C₂₉	—	2	2	no	—
95	C₄⋊C₄×C₂₉	—	4	2	no	—
96	C₄⋊C₄×C₂₉	—	4	4	no	—
97	C₄⋊C₄×C₂₉	—	4	4	no	—
98	C₄⋊C₄×C₂₉	—	8	2	no	—
99	C₄⋊C₄×C₂₉	—	8	2	no	—
100	C₄²×C₂₉	—	2	2	no	—
101	C₄²×C₂₉	—	2	2	no	—
102	C₄²×C₂₉	—	8	2	no	—
103	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
104	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
105	#28834	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	—	2	2	no	—
108	C₂×C₄×C₂₉	—	2	2	no	—
109	C₂×C₄×C₂₉	—	2	2	no	—
110	C₂×C₄×C₂₉	—	2	2	no	—
111	C₂×C₄×C₂₉	—	2	2	no	—
112	C₂×C₄×C₂₉	—	2	2	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	2	2	no	—
115	C₂×C₄×C₂₉	—	4	2	no	—
116	C₂×C₄×C₂₉	—	4	2	no	—
117	C₂×C₄×C₂₉	—	4	2	no	—
118	C₂×C₄×C₂₉	—	4	2	no	—
119	C₂×C₄×C₂₉	—	4	2	no	—
120	C₂×C₄×C₂₉	—	4	2	no	—
121	C₂×C₄×C₂₉	—	4	2	no	—
122	C₂×C₄×C₂₉	—	4	2	no	—
123	C₂×C₄×C₂₉	—	4	2	no	—
124	C₂×C₄×C₂₉	—	4	2	no	—
125	C₂×C₄×C₂₉	—	4	2	no	—
126	C₂×C₄×C₂₉	—	4	2	no	—
127	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₈×C₂₉	—	2	2	no	—
136	C₈×C₂₉	—	2	4	no	—
137	C₈×C₂₉	#125	1	4	yes	—
138	#2570	C₂×C₂₉	1	2	yes	—
139	#2085	C₂×C₂₉	1	2	yes	—
140	#2084	C₂×C₂₉	1	2	yes	—
141	#1949	C₂×C₂₉	1	2	yes	—
142	#1943	C₂×C₂₉	1	2	yes	—
143	#1847	C₂×C₂₉	1	2	yes	—
144	#1846	C₂×C₂₉	1	1	yes	characteristic
145	C₄×#106	C₂×C₂₉	1	1	yes	characteristic
146	#1770	C₂×C₂₉	1	1	yes	characteristic
147	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₂₉	#547	1	1	yes	characteristic
149	C₂²×C₂₉	#550	1	1	yes	characteristic
150	C₂²×C₂₉	#575	1	2	yes	—
151	C₂²×C₂₉	#577	1	2	yes	—
152	C₄×C₂₉	—	2	2	no	—
153	C₄×C₂₉	—	2	2	no	—
154	C₄×C₂₉	—	4	2	no	—
155	C₄×C₂₉	—	4	2	no	—
156	C₄×C₂₉	—	8	2	no	—
157	C₄×C₂₉	—	8	2	no	—
158	C₄×C₂₉	—	8	2	no	—
159	C₄×C₂₉	—	8	2	no	—
160	C₄×C₂₉	C₂×#125	1	1	yes	characteristic
161	C₄×C₂₉	#544	1	1	yes	characteristic
162	C₄×C₂₉	#554	1	1	yes	characteristic
163	C₄×C₂₉	#557	1	1	yes	characteristic
164	C₂×#107	C₂²×C₂₉	1	2	yes	—
165	C₂×#107	C₂²×C₂₉	1	2	yes	—
166	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
167	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
168	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
169	C₂×#106	C₂²×C₂₉	1	2	yes	—
170	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
172	C₂×#97	C₂²×C₂₉	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
174	#400	—	2	2	no	—
175	#398	—	2	2	no	—
176	#389	C₂²×C₂₉	1	2	yes	—
177	#389	C₂²×C₂₉	1	2	yes	—
178	#387	—	2	2	no	—
179	#387	—	2	2	no	—
180	#384	C₂²×C₂₉	1	2	yes	—
181	#382	C₂²×C₂₉	1	2	yes	—
182	#382	C₂²×C₂₉	1	2	yes	—
183	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
184	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
185	#340	C₂²×C₂₉	1	2	yes	—
186	#340	C₂²×C₂₉	1	2	yes	—
187	#340	C₂²×C₂₉	1	2	yes	—
188	#336	C₂²×C₂₉	1	2	yes	—
189	C₂×C₂₉	#2552	1	1	yes	characteristic
190	C₂×C₂₉	#3162	1	1	yes	characteristic
191	C₂×C₂₉	#3163	1	1	yes	characteristic
192	C₂×C₂₉	#3351	1	2	yes	—
193	C₂×C₂₉	#3401	1	2	yes	—
194	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
196	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
208	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
209	#107	—	2	4	no	—
210	#107	—	4	2	no	—
211	#106	—	2	2	no	—
212	#106	—	2	2	no	—
213	#106	—	4	1	no	—
214	#106	—	4	2	no	—
215	#106	—	4	1	no	—
216	C₄⋊C₈	Q₈×C₂₉	1	4	yes	—
217	#97	—	2	4	no	—
218	C₄×C₈	Q₈×C₂₉	1	4	yes	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	D₄×C₂₉	1	1	yes	characteristic
227	#95	D₄×C₂₉	1	1	yes	characteristic
228	C₂₉	#28834	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
237	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
243	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
244	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
245	C₂×C₈	#41×C₂₉	1	2	yes	—
246	C₂×C₈	#41×C₂₉	1	2	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	8	2	no	—
263	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
264	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
265	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
287	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
288	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
289	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
290	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
291	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
292	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	2	4	no	—
296	C₈	C₂₉×#125	1	4	yes	—
297	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
298	C₂²	C₂₉×#547	1	1	yes	characteristic
299	C₂²	C₂₉×#550	1	1	yes	characteristic
300	C₂²	C₂₉×#575	1	2	yes	—
301	C₂²	C₂₉×#577	1	2	yes	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×C₂₉×#125	1	1	yes	characteristic
311	C₄	C₂₉×#544	1	1	yes	characteristic
312	C₄	C₂₉×#554	1	1	yes	characteristic
313	C₄	C₂₉×#557	1	1	yes	characteristic
314	C₂	C₂₉×#2552	1	1	yes	characteristic
315	C₂	C₂₉×#3162	1	1	yes	characteristic
316	C₂	C₂₉×#3163	1	1	yes	characteristic
317	C₂	C₂₉×#3351	1	2	yes	—
318	C₂	C₂₉×#3401	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
105. #28834	C₂₉	direct (N × H)
228. C₂₉	#28834	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×#2570	2
C₂₉×#2085	2
C₂₉×#2084	2
C₂₉×#1949	2
C₂₉×#1943	2
C₂₉×#1847	2
C₂₉×#1846	2
C₄×C₂₉×#106	2
C₂₉×#1770	2
#28834	29

Subgroups — C29×#28834 #20323734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28834 #20323734