Subgroups — C₂₉×#8208 #20303108

Order 7424.

538 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#436	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₉×#102	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₉×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₉×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₂₉	—	2	1	no	—
9	C₂×C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
10	C₂₉×#436	—	2	2	no	—
11	C₂₉×#436	C₄	1	4	yes	complemented
12	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₉×#102	C₄	1	2	yes	—
15	C₂×C₂₉×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₂₉×#97	—	2	1	no	—
17	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂₉×#340	C₄	1	2	yes	—
22	C₂₉×#336	—	2	1	no	—
23	C₂₉×D₄⋊C₈	C₄	1	4	yes	complemented
24	C₂₉×#322	—	2	2	no	—
25	C₂²×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₉	—	2	1	no	—
27	C₂×M₁₆×C₂₉	—	2	4	no	—
28	C₂×M₁₆×C₂₉	—	4	2	no	—
29	C₂×M₁₆×C₂₉	—	4	2	no	—
30	C₂²×C₈×C₂₉	—	2	1	no	—
31	C₂²×C₈×C₂₉	—	2	1	no	—
32	C₂²×C₈×C₂₉	—	2	1	no	—
33	C₂²×C₈×C₂₉	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
36	C₄×D₄×C₂₉	—	2	1	no	—
37	C₄×D₄×C₂₉	—	2	1	no	—
38	C₄×D₄×C₂₉	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₂₉	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₂₉	—	2	1	no	—
43	C₂×C₄²×C₂₉	—	2	1	no	—
44	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₂₉×C₄⋊C₈	—	2	1	no	—
46	C₂₉×C₄⋊C₈	—	2	1	no	—
47	C₂₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₂₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₂₉×#102	—	2	4	no	—
50	C₂₉×C₂²⋊C₈	—	2	4	no	—
51	C₂₉×#97	—	4	2	no	—
52	C₄×C₈×C₂₉	—	2	1	no	—
53	C₄×C₈×C₂₉	—	2	1	no	—
54	C₄×C₈×C₂₉	—	2	1	no	—
55	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
56	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₂₉	D₄	1	2	yes	—
58	C₂₉×#95	—	2	1	no	—
59	C₂⁴×C₂₉	—	2	1	no	—
60	C₂×D₄×C₂₉	—	2	1	no	—
61	C₂×D₄×C₂₉	—	2	2	no	—
62	C₂×D₄×C₂₉	—	2	2	no	—
63	C₂×D₄×C₂₉	C₄²	1	2	yes	—
64	C₂²×C₄×C₂₉	—	2	1	no	—
65	C₂²×C₄×C₂₉	—	2	2	no	—
66	C₂²×C₄×C₂₉	—	2	2	no	—
67	C₂²×C₄×C₂₉	—	2	2	no	—
68	C₂²×C₄×C₂₉	—	4	1	no	—
69	C₂²×C₄×C₂₉	—	4	1	no	—
70	C₂²×C₄×C₂₉	—	4	1	no	—
71	C₂²×C₄×C₂₉	—	4	1	no	—
72	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
75	M₁₆×C₂₉	—	4	8	no	—
76	C₂×C₈×C₂₉	—	2	2	no	—
77	C₂×C₈×C₂₉	—	2	2	no	—
78	C₂×C₈×C₂₉	—	2	1	no	—
79	C₂×C₈×C₂₉	—	2	1	no	—
80	C₂×C₈×C₂₉	—	2	1	no	—
81	C₂×C₈×C₂₉	—	2	1	no	—
82	C₂×C₈×C₂₉	—	2	2	no	—
83	C₂×C₈×C₂₉	—	2	2	no	—
84	C₂×C₈×C₂₉	—	2	2	no	—
85	C₂×C₈×C₂₉	—	4	2	no	—
86	C₂×C₈×C₂₉	—	4	1	no	—
87	C₂×C₈×C₂₉	—	4	1	no	—
88	C₂×C₈×C₂₉	—	4	1	no	—
89	C₂×C₈×C₂₉	—	4	1	no	—
90	C₂×C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
91	C₂×C₈×C₂₉	C₂²⋊C₄	1	2	yes	—
92	C₂×C₈×C₂₉	M₁₆	1	4	yes	complemented
93	C₄⋊C₄×C₂₉	—	2	1	no	—
94	C₄⋊C₄×C₂₉	—	2	2	no	—
95	C₄⋊C₄×C₂₉	—	2	2	no	—
96	C₄⋊C₄×C₂₉	C₄²	1	2	yes	—
97	C₂²⋊C₄×C₂₉	—	2	2	no	—
98	C₂²⋊C₄×C₂₉	—	2	2	no	—
99	C₄²×C₂₉	—	4	1	no	—
100	C₄²×C₂₉	—	4	1	no	—
101	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₂₉	#41	1	1	yes	characteristic
105	#8208	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₉	—	2	2	no	—
107	C₂³×C₂₉	—	4	2	no	—
108	C₂³×C₂₉	—	4	1	no	—
109	C₂³×C₂₉	—	4	1	no	—
110	C₂³×C₂₉	—	4	2	no	—
111	C₂³×C₂₉	C₄×D₄	1	1	yes	characteristic
112	D₄×C₂₉	—	2	2	no	—
113	D₄×C₂₉	—	2	2	no	—
114	D₄×C₂₉	—	4	2	no	—
115	C₂×C₄×C₂₉	—	2	2	no	—
116	C₂×C₄×C₂₉	—	2	2	no	—
117	C₂×C₄×C₂₉	—	4	1	no	—
118	C₂×C₄×C₂₉	—	4	1	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	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₉	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₂₉	#131	1	2	yes	—
134	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₂₉	#137	1	1	yes	characteristic
138	C₈×C₂₉	—	2	2	no	—
139	C₈×C₂₉	—	2	2	no	—
140	C₈×C₂₉	—	2	4	no	—
141	C₈×C₂₉	—	4	2	no	—
142	C₈×C₂₉	—	4	2	no	—
143	C₈×C₂₉	—	4	2	no	—
144	C₂×#436	C₂×C₂₉	1	1	yes	characteristic
145	#1770	C₂×C₂₉	1	1	yes	characteristic, complemented
146	C₄×#102	C₂×C₂₉	1	1	yes	characteristic
147	C₂×D₄⋊C₈	C₂×C₂₉	1	1	yes	characteristic
148	C₂×#322	C₂×C₂₉	1	1	yes	characteristic, complemented
149	#1279	C₂×C₂₉	1	1	yes	characteristic, complemented
150	#1277	C₂×C₂₉	1	1	yes	characteristic, complemented
151	C₂²×C₂₉	—	4	2	no	—
152	C₂²×C₂₉	—	4	2	no	—
153	C₂²×C₂₉	—	4	2	no	—
154	C₂²×C₂₉	—	8	2	no	—
155	C₂²×C₂₉	—	8	2	no	—
156	C₂²×C₂₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₂₉	#436	1	2	yes	—
158	C₂²×C₂₉	#440	1	1	yes	characteristic
159	C₂²×C₂₉	#442	1	1	yes	characteristic
160	C₂²×C₂₉	#444	1	2	yes	—
161	C₄×C₂₉	—	2	1	no	—
162	C₄×C₂₉	—	2	1	no	—
163	C₄×C₂₉	—	4	2	no	—
164	C₄×C₂₉	—	8	2	no	—
165	C₄×C₂₉	C₄×M₁₆	1	1	yes	characteristic
166	C₄×C₂₉	#405	1	1	yes	characteristic
167	C₄×C₂₉	#417	1	1	yes	characteristic
168	C₄×C₂₉	#418	1	1	yes	characteristic
169	C₄×C₂₉	#435	1	2	yes	—
170	C₄×C₂₉	#440	1	1	yes	characteristic
171	C₄×C₂₉	#442	1	1	yes	characteristic
172	C₂²×M₁₆	—	2	1	no	—
173	C₂×C₄×D₄	C₂²×C₂₉	1	1	yes	characteristic
174	#436	—	2	2	no	—
175	#436	C₄×C₂₉	1	4	yes	complemented
176	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
178	C₂×#102	C₄×C₂₉	1	2	yes	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₂×#97	—	2	1	no	—
181	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
185	#340	C₄×C₂₉	1	2	yes	—
186	#336	—	2	1	no	—
187	D₄⋊C₈	C₄×C₂₉	1	4	yes	complemented
188	#322	—	2	2	no	—
189	C₂×C₂₉	—	8	2	no	—
190	C₂×C₂₉	#1592	1	2	yes	—
191	C₂×C₂₉	#1594	1	2	yes	—
192	C₂×C₂₉	#1753	1	1	yes	characteristic
193	C₂×C₂₉	#1762	1	1	yes	characteristic
194	C₂×C₂₉	#1765	1	1	yes	characteristic
195	C₂²×D₄	C₂×C₄×C₂₉	1	1	yes	characteristic
196	C₂³×C₄	—	2	1	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	4	2	no	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₂×C₄×C₂₉	1	2	yes	—
204	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	C₂×C₄×C₂₉	1	2	yes	—
209	C₄×D₄	C₂×C₄×C₂₉	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
218	C₄⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
219	#102	—	2	4	no	—
220	C₂²⋊C₈	—	2	4	no	—
221	#97	—	4	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
226	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
227	C₄×C₈	D₄×C₂₉	1	2	yes	—
228	#95	—	2	1	no	—
229	C₂₉	#8208	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₄²×C₂₉	1	2	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
245	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
246	M₁₆	—	4	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	C₂²⋊C₄×C₂₉	1	2	yes	—
262	C₂×C₈	C₂²⋊C₄×C₂₉	1	2	yes	—
263	C₂×C₈	M₁₆×C₂₉	1	4	yes	complemented
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	C₄²×C₂₉	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
273	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
275	C₄²	#41×C₂₉	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₄×D₄×C₂₉	1	1	yes	characteristic
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	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	C₂×C₄²×C₂₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
298	C₂×C₄	C₂₉×#117	1	1	yes	characteristic
299	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
301	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
302	C₂×C₄	C₂×M₁₆×C₂₉	1	2	yes	—
303	C₂×C₄	C₂₉×#131	1	2	yes	—
304	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₄	C₂₉×#137	1	1	yes	characteristic
308	C₈	—	2	4	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₄×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
320	C₂²	C₂₉×#436	1	2	yes	—
321	C₂²	C₂₉×#440	1	1	yes	characteristic
322	C₂²	C₂₉×#442	1	1	yes	characteristic
323	C₂²	C₂₉×#444	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₄×M₁₆×C₂₉	1	1	yes	characteristic
329	C₄	C₂₉×#405	1	1	yes	characteristic
330	C₄	C₂₉×#417	1	1	yes	characteristic
331	C₄	C₂₉×#418	1	1	yes	characteristic
332	C₄	C₂₉×#435	1	2	yes	—
333	C₄	C₂₉×#440	1	1	yes	characteristic
334	C₄	C₂₉×#442	1	1	yes	characteristic
335	C₂	—	8	2	no	—
336	C₂	C₂₉×#1592	1	2	yes	—
337	C₂	C₂₉×#1594	1	2	yes	—
338	C₂	C₂₉×#1753	1	1	yes	characteristic
339	C₂	C₂₉×#1762	1	1	yes	characteristic
340	C₂	C₂₉×#1765	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
socle	C₂³×C₂₉	C₄×D₄
prime radical	C₂²×D₄×C₂₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₉×#1770	C₂	semidirect
5. C₂×C₂₉×#322	C₂	semidirect
6. C₂₉×#1279	C₂	semidirect
7. C₂₉×#1277	C₂	semidirect
11. C₂₉×#436	C₄	semidirect
23. C₂₉×D₄⋊C₈	C₄	semidirect
92. C₂×C₈×C₂₉	M₁₆	semidirect
105. #8208	C₂₉	direct (N × H)
145. #1770	C₂×C₂₉	semidirect
148. C₂×#322	C₂×C₂₉	semidirect
149. #1279	C₂×C₂₉	semidirect
150. #1277	C₂×C₂₉	semidirect
175. #436	C₄×C₂₉	semidirect
187. D₄⋊C₈	C₄×C₂₉	semidirect
229. C₂₉	#8208	direct (N × H)
263. C₂×C₈	M₁₆×C₂₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₉×#436	2
C₂₉×#1770	2
C₄×C₂₉×#102	2
C₂×C₂₉×D₄⋊C₈	2
C₂×C₂₉×#322	2
C₂₉×#1279	2
C₂₉×#1277	2
#8208	29

Subgroups — C29×#8208 #20303108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#8208 #20303108