Subgroups — C₂₉×#28833 #20323733

Order 7424.

594 conjugacy classes of subgroups.

The subgroup lattice has 334 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×#2083	C₂	1	2	yes	maximal
4	C₂₉×#1949	C₂	1	2	yes	maximal
5	C₂₉×#1944	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×#107	C₂	1	1	yes	maximal, characteristic
9	C₂₉×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
11	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
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	2	yes	—
16	C₂×C₂₉×#106	C₂²	1	2	yes	—
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	2	yes	—
36	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
43	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
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×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₂₉	—	2	2	no	—
49	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
51	C₂₉×#107	—	2	2	no	—
52	C₂₉×#107	—	2	2	no	—
53	C₂₉×#107	—	4	1	no	—
54	C₂₉×#107	—	4	2	no	—
55	C₂₉×#107	—	4	1	no	—
56	C₂₉×#106	—	2	4	no	—
57	C₂₉×#106	—	4	2	no	—
58	C₂₉×C₄⋊C₈	Q₈	1	4	yes	—
59	C₂₉×#97	—	2	4	no	—
60	C₄×C₈×C₂₉	Q₈	1	4	yes	—
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	—	2	2	no	—
68	C₂₉×#95	D₄	1	2	yes	—
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	2	no	—
80	C₂×C₈×C₂₉	—	2	1	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	1	no	—
96	C₄⋊C₄×C₂₉	—	4	1	no	—
97	C₄⋊C₄×C₂₉	—	4	4	no	—
98	C₄⋊C₄×C₂₉	—	4	4	no	—
99	C₄⋊C₄×C₂₉	—	8	2	no	—
100	C₄⋊C₄×C₂₉	—	8	2	no	—
101	C₄²×C₂₉	—	2	2	no	—
102	C₄²×C₂₉	—	2	2	no	—
103	C₄²×C₂₉	—	8	2	no	—
104	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
105	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
106	#28833	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	—	2	1	no	—
109	C₂×C₄×C₂₉	—	2	1	no	—
110	C₂×C₄×C₂₉	—	2	1	no	—
111	C₂×C₄×C₂₉	—	2	1	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	—	2	2	no	—
116	C₂×C₄×C₂₉	—	2	2	no	—
117	C₂×C₄×C₂₉	—	2	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	—	4	2	no	—
128	C₂×C₄×C₂₉	—	4	2	no	—
129	C₂×C₄×C₂₉	—	4	2	no	—
130	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
138	C₈×C₂₉	—	2	2	no	—
139	C₈×C₂₉	—	2	4	no	—
140	C₈×C₂₉	#125	1	4	yes	—
141	#2570	C₂×C₂₉	1	2	yes	—
142	#2085	C₂×C₂₉	1	2	yes	—
143	#2083	C₂×C₂₉	1	2	yes	—
144	#1949	C₂×C₂₉	1	2	yes	—
145	#1944	C₂×C₂₉	1	2	yes	—
146	#1847	C₂×C₂₉	1	2	yes	—
147	#1846	C₂×C₂₉	1	1	yes	characteristic
148	C₄×#107	C₂×C₂₉	1	1	yes	characteristic
149	#1770	C₂×C₂₉	1	1	yes	characteristic
150	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂²×C₂₉	#547	1	1	yes	characteristic
152	C₂²×C₂₉	#550	1	1	yes	characteristic
153	C₂²×C₂₉	#575	1	1	yes	characteristic
154	C₂²×C₂₉	#575	1	1	yes	characteristic
155	C₂²×C₂₉	#576	1	1	yes	characteristic
156	C₂²×C₂₉	#578	1	1	yes	characteristic
157	C₄×C₂₉	—	2	2	no	—
158	C₄×C₂₉	—	2	2	no	—
159	C₄×C₂₉	—	4	2	no	—
160	C₄×C₂₉	—	4	2	no	—
161	C₄×C₂₉	—	8	2	no	—
162	C₄×C₂₉	—	8	2	no	—
163	C₄×C₂₉	—	8	2	no	—
164	C₄×C₂₉	—	8	2	no	—
165	C₄×C₂₉	C₂×#125	1	1	yes	characteristic
166	C₄×C₂₉	#544	1	1	yes	characteristic
167	C₄×C₂₉	#554	1	1	yes	characteristic
168	C₄×C₂₉	#557	1	1	yes	characteristic
169	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
170	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
171	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
172	C₂×#107	C₂²×C₂₉	1	2	yes	—
173	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
174	C₂×#106	C₂²×C₂₉	1	2	yes	—
175	C₂×#106	C₂²×C₂₉	1	2	yes	—
176	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
177	C₂×#97	C₂²×C₂₉	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
179	#400	—	2	2	no	—
180	#398	—	2	2	no	—
181	#389	C₂²×C₂₉	1	2	yes	—
182	#389	C₂²×C₂₉	1	2	yes	—
183	#387	—	2	2	no	—
184	#387	—	2	2	no	—
185	#384	C₂²×C₂₉	1	2	yes	—
186	#382	C₂²×C₂₉	1	2	yes	—
187	#382	C₂²×C₂₉	1	2	yes	—
188	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
189	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
190	#340	C₂²×C₂₉	1	2	yes	—
191	#340	C₂²×C₂₉	1	2	yes	—
192	#340	C₂²×C₂₉	1	2	yes	—
193	#336	C₂²×C₂₉	1	2	yes	—
194	C₂×C₂₉	#2552	1	1	yes	characteristic
195	C₂×C₂₉	#3162	1	1	yes	characteristic
196	C₂×C₂₉	#3164	1	1	yes	characteristic
197	C₂×C₂₉	#3352	1	1	yes	characteristic
198	C₂×C₂₉	#3355	1	1	yes	characteristic
199	C₂×C₂₉	#3402	1	1	yes	characteristic
200	C₂×C₂₉	#3403	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
203	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
216	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	4	1	no	—
220	#107	—	4	2	no	—
221	#107	—	4	1	no	—
222	#106	—	2	4	no	—
223	#106	—	4	2	no	—
224	C₄⋊C₈	Q₈×C₂₉	1	4	yes	—
225	#97	—	2	4	no	—
226	C₄×C₈	Q₈×C₂₉	1	4	yes	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	D₄×C₂₉	1	2	yes	—
235	C₂₉	#28833	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
250	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
251	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
252	C₂×C₈	#41×C₂₉	1	2	yes	—
253	C₂×C₈	#41×C₂₉	1	2	yes	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	8	2	no	—
267	C₄⋊C₄	—	8	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	8	2	no	—
271	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
272	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
273	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
297	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
298	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
300	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
302	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
303	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	4	no	—
306	C₈	C₂₉×#125	1	4	yes	—
307	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
308	C₂²	C₂₉×#547	1	1	yes	characteristic
309	C₂²	C₂₉×#550	1	1	yes	characteristic
310	C₂²	C₂₉×#575	1	1	yes	characteristic
311	C₂²	C₂₉×#575	1	1	yes	characteristic
312	C₂²	C₂₉×#576	1	1	yes	characteristic
313	C₂²	C₂₉×#578	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₂×C₂₉×#125	1	1	yes	characteristic
323	C₄	C₂₉×#544	1	1	yes	characteristic
324	C₄	C₂₉×#554	1	1	yes	characteristic
325	C₄	C₂₉×#557	1	1	yes	characteristic
326	C₂	C₂₉×#2552	1	1	yes	characteristic
327	C₂	C₂₉×#3162	1	1	yes	characteristic
328	C₂	C₂₉×#3164	1	1	yes	characteristic
329	C₂	C₂₉×#3352	1	1	yes	characteristic
330	C₂	C₂₉×#3355	1	1	yes	characteristic
331	C₂	C₂₉×#3402	1	1	yes	characteristic
332	C₂	C₂₉×#3403	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
106. #28833	C₂₉	direct (N × H)
235. C₂₉	#28833	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×#2083	2
C₂₉×#1949	2
C₂₉×#1944	2
C₂₉×#1847	2
C₂₉×#1846	2
C₄×C₂₉×#107	2
C₂₉×#1770	2
#28833	29

Subgroups — C29×#28833 #20323733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28833 #20323733