Subgroups — C₂₉×#28815 #20323715

Order 7424.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#501	C₂	1	2	yes	maximal
2	C₂₉×#2550	C₂	1	2	yes	maximal
3	C₂₉×#2059	C₂	1	2	yes	maximal
4	C₂₉×#2058	C₂	1	2	yes	maximal
5	C₂₉×#1949	C₂	1	2	yes	maximal
6	C₂₉×#1944	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×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₂₉×#125	C₂²	1	2	yes	—
12	C₂₉×#501	—	2	8	no	—
13	C₂×C₂₉×#107	C₂²	1	2	yes	—
14	C₂×C₂₉×#107	C₂²	1	2	yes	—
15	C₂×C₂₉×#106	C₂²	1	2	yes	—
16	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
18	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
19	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
20	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
23	C₂₉×#398	—	2	2	no	—
24	C₂₉×#395	—	2	2	no	—
25	C₂₉×#391	—	2	2	no	—
26	C₂₉×#389	—	2	2	no	—
27	C₂₉×#389	C₂²	1	2	yes	—
28	C₂₉×#389	C₂²	1	2	yes	—
29	C₂₉×#382	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
32	C₂₉×#340	C₂²	1	2	yes	—
33	C₂₉×#340	C₂²	1	2	yes	—
34	C₂₉×#340	C₂²	1	2	yes	—
35	C₂₉×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₂₉	—	2	2	no	—
37	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
40	C₂₉×C₄⋊Q₈	—	2	4	no	—
41	C₂₉×C₄⋊Q₈	—	4	2	no	—
42	C₂₉×#125	—	2	4	no	—
43	C₂₉×#125	—	4	2	no	—
44	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
54	C₂×C₄²×C₂₉	—	2	2	no	—
55	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
57	C₂₉×#107	—	2	4	no	—
58	C₂₉×#107	—	2	4	no	—
59	C₂₉×#106	—	2	4	no	—
60	C₂₉×#106	—	4	1	no	—
61	C₂₉×#106	—	4	1	no	—
62	C₂₉×#106	D₄	1	4	yes	—
63	C₂₉×#106	Q₈	1	4	yes	complemented
64	C₂₉×C₄⋊C₈	Q₈	1	4	yes	complemented
65	C₂₉×#97	—	2	4	no	—
66	C₄×C₈×C₂₉	Q₈	1	4	yes	complemented
67	C₂₉×#95	—	2	2	no	—
68	C₂₉×#95	—	2	2	no	—
69	C₂₉×#95	—	2	2	no	—
70	C₂₉×#95	—	2	2	no	—
71	C₂₉×#95	—	2	2	no	—
72	C₂₉×#95	D₄	1	1	yes	characteristic
73	C₂₉×#95	D₄	1	1	yes	characteristic
74	C₂×Q₈×C₂₉	—	4	4	no	—
75	C₂×Q₈×C₂₉	—	4	4	no	—
76	C₂×Q₈×C₂₉	—	8	2	no	—
77	C₂²×C₄×C₂₉	—	2	2	no	—
78	C₂²×C₄×C₂₉	—	2	2	no	—
79	C₂²×C₄×C₂₉	—	2	2	no	—
80	C₂²×C₄×C₂₉	—	2	2	no	—
81	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₂₉	—	2	2	no	—
87	C₂×C₈×C₂₉	—	2	1	no	—
88	C₂×C₈×C₂₉	—	2	1	no	—
89	C₂×C₈×C₂₉	—	2	2	no	—
90	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₂₉	—	2	2	no	—
96	C₄⋊C₄×C₂₉	—	2	2	no	—
97	C₄⋊C₄×C₂₉	—	2	2	no	—
98	C₄⋊C₄×C₂₉	—	2	2	no	—
99	C₄⋊C₄×C₂₉	—	2	2	no	—
100	C₄⋊C₄×C₂₉	—	2	2	no	—
101	C₄⋊C₄×C₂₉	—	4	2	no	—
102	C₄⋊C₄×C₂₉	—	4	4	no	—
103	C₄⋊C₄×C₂₉	—	4	4	no	—
104	C₄⋊C₄×C₂₉	—	4	4	no	—
105	C₄⋊C₄×C₂₉	—	4	4	no	—
106	C₄⋊C₄×C₂₉	—	8	2	no	—
107	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₂₉	C₂×Q₈	1	4	yes	—
109	C₄²×C₂₉	—	2	2	no	—
110	C₄²×C₂₉	—	2	2	no	—
111	C₄²×C₂₉	—	8	2	no	—
112	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
113	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
114	#28815	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₂₉	—	8	8	no	—
117	C₂×C₄×C₂₉	—	2	2	no	—
118	C₂×C₄×C₂₉	—	2	2	no	—
119	C₂×C₄×C₂₉	—	2	2	no	—
120	C₂×C₄×C₂₉	—	2	2	no	—
121	C₂×C₄×C₂₉	—	2	4	no	—
122	C₂×C₄×C₂₉	—	2	4	no	—
123	C₂×C₄×C₂₉	—	2	2	no	—
124	C₂×C₄×C₂₉	—	2	2	no	—
125	C₂×C₄×C₂₉	—	2	2	no	—
126	C₂×C₄×C₂₉	—	2	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	—	4	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	—	4	2	no	—
135	C₂×C₄×C₂₉	—	4	2	no	—
136	C₂×C₄×C₂₉	—	4	2	no	—
137	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₈×C₂₉	—	2	2	no	—
146	C₈×C₂₉	—	2	4	no	—
147	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
148	C₂×#501	C₂×C₂₉	1	2	yes	—
149	#2550	C₂×C₂₉	1	2	yes	—
150	#2059	C₂×C₂₉	1	2	yes	—
151	#2058	C₂×C₂₉	1	2	yes	—
152	#1949	C₂×C₂₉	1	2	yes	—
153	#1944	C₂×C₂₉	1	2	yes	—
154	#1846	C₂×C₂₉	1	1	yes	characteristic
155	C₄×#106	C₂×C₂₉	1	1	yes	characteristic
156	#1770	C₂×C₂₉	1	1	yes	characteristic
157	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂²×C₂₉	D₄²	1	1	yes	characteristic
159	C₂²×C₂₉	D₄×Q₈	1	1	yes	characteristic
160	C₂²×C₂₉	#575	1	2	yes	—
161	C₂²×C₂₉	#577	1	2	yes	—
162	C₄×C₂₉	—	2	2	no	—
163	C₄×C₂₉	—	2	2	no	—
164	C₄×C₂₉	—	4	4	no	—
165	C₄×C₂₉	—	4	2	no	—
166	C₄×C₂₉	—	4	2	no	—
167	C₄×C₂₉	—	4	4	no	—
168	C₄×C₂₉	—	8	2	no	—
169	C₄×C₂₉	—	8	2	no	—
170	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₄×C₂₉	#544	1	1	yes	characteristic
172	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
173	C₄×C₂₉	Q₈²	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₂₉	1	2	yes	—
175	C₂×#125	C₂²×C₂₉	1	2	yes	—
176	#501	—	2	8	no	—
177	C₂×#107	C₂²×C₂₉	1	2	yes	—
178	C₂×#107	C₂²×C₂₉	1	2	yes	—
179	C₂×#106	C₂²×C₂₉	1	2	yes	—
180	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
181	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
182	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
183	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
184	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
185	C₂×#97	C₂²×C₂₉	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
187	#398	—	2	2	no	—
188	#395	—	2	2	no	—
189	#391	—	2	2	no	—
190	#389	—	2	2	no	—
191	#389	C₂²×C₂₉	1	2	yes	—
192	#389	C₂²×C₂₉	1	2	yes	—
193	#382	C₂²×C₂₉	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
195	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
196	#340	C₂²×C₂₉	1	2	yes	—
197	#340	C₂²×C₂₉	1	2	yes	—
198	#340	C₂²×C₂₉	1	2	yes	—
199	#336	C₂²×C₂₉	1	2	yes	—
200	C₂×C₂₉	#2551	1	1	yes	characteristic
201	C₂×C₂₉	#3162	1	1	yes	characteristic
202	C₂×C₂₉	#3163	1	1	yes	characteristic
203	C₂×C₂₉	#3275	1	2	yes	—
204	C₂×C₂₉	#3388	1	2	yes	—
205	C₂²×Q₈	—	2	2	no	—
206	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
208	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	4	2	no	—
211	#125	—	2	4	no	—
212	#125	—	4	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
225	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
226	#107	—	2	4	no	—
227	#107	—	2	4	no	—
228	#106	—	2	4	no	—
229	#106	—	4	1	no	—
230	#106	—	4	1	no	—
231	#106	D₄×C₂₉	1	4	yes	—
232	#106	Q₈×C₂₉	1	4	yes	complemented
233	C₄⋊C₈	Q₈×C₂₉	1	4	yes	complemented
234	#97	—	2	4	no	—
235	C₄×C₈	Q₈×C₂₉	1	4	yes	complemented
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	D₄×C₂₉	1	1	yes	characteristic
242	#95	D₄×C₂₉	1	1	yes	characteristic
243	C₂₉	#28815	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	4	no	—
246	C₂×Q₈	—	8	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
261	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
262	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
263	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	8	2	no	—
277	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
278	C₄⋊C₄	C₂×Q₈×C₂₉	1	4	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	8	2	no	—
282	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
283	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
284	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
285	Q₈	—	8	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	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	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
307	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
308	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
309	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
310	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
311	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
312	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
313	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	4	no	—
316	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
317	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
318	C₂²	D₄²×C₂₉	1	1	yes	characteristic
319	C₂²	D₄×Q₈×C₂₉	1	1	yes	characteristic
320	C₂²	C₂₉×#575	1	2	yes	—
321	C₂²	C₂₉×#577	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
331	C₄	C₂₉×#544	1	1	yes	characteristic
332	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
333	C₄	Q₈²×C₂₉	1	1	yes	characteristic
334	C₂	C₂₉×#2551	1	1	yes	characteristic
335	C₂	C₂₉×#3162	1	1	yes	characteristic
336	C₂	C₂₉×#3163	1	1	yes	characteristic
337	C₂	C₂₉×#3275	1	2	yes	—
338	C₂	C₂₉×#3388	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
63. C₂₉×#106	Q₈	semidirect
64. C₂₉×C₄⋊C₈	Q₈	semidirect
66. C₄×C₈×C₂₉	Q₈	semidirect
114. #28815	C₂₉	direct (N × H)
232. #106	Q₈×C₂₉	semidirect
233. C₄⋊C₈	Q₈×C₂₉	semidirect
235. C₄×C₈	Q₈×C₂₉	semidirect
243. C₂₉	#28815	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#501	2
C₂₉×#2550	2
C₂₉×#2059	2
C₂₉×#2058	2
C₂₉×#1949	2
C₂₉×#1944	2
C₂₉×#1846	2
C₄×C₂₉×#106	2
C₂₉×#1770	2
#28815	29

Subgroups — C29×#28815 #20323715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28815 #20323715