Subgroups — C₂₉×#19289 #20314189

Order 7424.

630 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1698	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1696	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1688	C₂	1	2	yes	maximal, complemented
7	C₂₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#1551	C₂	1	2	yes	maximal, complemented
9	C₂₉×#532	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
11	C₂₉×#500	C₂²	1	1	yes	characteristic
12	C₂₉×#493	C₂²	1	1	yes	characteristic
13	C₂₉×#486	C₂²	1	2	yes	—
14	C₂₉×#480	—	2	2	no	—
15	C₂₉×#477	C₂²	1	4	yes	—
16	C₂₉×#466	—	2	1	no	—
17	C₂₉×#466	—	2	1	no	—
18	C₂₉×#460	C₂²	1	4	yes	—
19	C₂₉×#448	—	2	2	no	—
20	C₂₉×#446	—	2	1	no	—
21	C₂₉×#446	—	2	1	no	—
22	C₂₉×#446	C₂²	1	2	yes	—
23	C₂₉×#446	C₂²	1	2	yes	—
24	C₂₉×#440	—	2	4	no	—
25	C₄×D₈×C₂₉	—	2	1	no	—
26	C₄×D₈×C₂₉	—	2	1	no	—
27	C₂₉×#432	C₂²	1	2	yes	—
28	C₂×C₂₉×C₄⋊C₈	—	2	1	no	—
29	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₂₉×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
32	C₂₉×#335	C₂²	1	2	yes	—
33	C₂₉×#334	C₂²	1	2	yes	—
34	C₂₉×#333	C₂²	1	4	yes	—
35	C₂₉×#331	C₂²	1	4	yes	—
36	C₂×#41×C₂₉	—	2	2	no	—
37	C₂²×Q₈×C₂₉	D₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₂₉	—	2	4	no	—
39	C₂×D₈×C₂₉	—	2	1	no	—
40	C₂×D₈×C₂₉	—	2	1	no	—
41	C₂²×C₈×C₂₉	—	2	1	no	—
42	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
44	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
45	C₂₉×#127	C₂³	1	2	yes	—
46	C₂₉×#124	—	2	2	no	—
47	C₂₉×#121	—	2	2	no	—
48	C₂₉×#121	—	2	2	no	—
49	C₄×Q₈×C₂₉	—	2	1	no	—
50	C₄×Q₈×C₂₉	—	2	1	no	—
51	C₄×Q₈×C₂₉	D₄	1	2	yes	—
52	C₄×Q₈×C₂₉	C₂³	1	2	yes	—
53	C₄×D₄×C₂₉	—	2	2	no	—
54	C₄×D₄×C₂₉	—	2	2	no	—
55	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₂₉	—	2	1	no	—
58	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
59	C₂₉×#107	—	2	1	no	—
60	C₂₉×#107	—	2	1	no	—
61	C₂₉×#107	—	2	4	no	—
62	C₂₉×C₄⋊C₈	—	2	2	no	—
63	C₂₉×C₄⋊C₈	—	2	2	no	—
64	C₂₉×C₄⋊C₈	—	2	1	no	—
65	C₂₉×C₄⋊C₈	C₂³	1	4	yes	—
66	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
67	C₂₉×#103	—	2	4	no	—
68	C₂₉×#102	—	2	2	no	—
69	C₂₉×#102	—	2	2	no	—
70	C₂₉×#102	—	2	4	no	—
71	C₂₉×C₂²⋊C₈	—	2	2	no	—
72	C₂₉×#97	—	2	2	no	—
73	C₂₉×#97	—	2	2	no	—
74	C₂₉×#97	—	2	2	no	—
75	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
79	#41×C₂₉	—	8	4	no	—
80	C₂×Q₈×C₂₉	—	2	1	no	—
81	C₂×Q₈×C₂₉	—	2	2	no	—
82	C₂×Q₈×C₂₉	—	4	2	no	—
83	C₂×Q₈×C₂₉	C₂×D₄	1	2	yes	—
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	—	2	1	no	—
89	C₂²×C₄×C₂₉	—	2	1	no	—
90	C₂²×C₄×C₂₉	—	4	2	no	—
91	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
94	SD₁₆×C₂₉	—	4	8	no	—
95	D₈×C₂₉	—	4	2	no	—
96	D₈×C₂₉	—	4	2	no	—
97	C₂×C₈×C₂₉	—	2	2	no	—
98	C₂×C₈×C₂₉	—	2	2	no	—
99	C₂×C₈×C₂₉	—	2	4	no	—
100	C₂×C₈×C₂₉	—	4	1	no	—
101	C₂×C₈×C₂₉	—	4	2	no	—
102	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₂₉	—	2	2	no	—
107	C₄⋊C₄×C₂₉	—	2	1	no	—
108	C₄⋊C₄×C₂₉	—	2	2	no	—
109	C₄⋊C₄×C₂₉	—	2	2	no	—
110	C₄⋊C₄×C₂₉	—	2	2	no	—
111	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₂₉	—	4	2	no	—
113	C₂²⋊C₄×C₂₉	—	4	2	no	—
114	C₄²×C₂₉	—	2	2	no	—
115	C₄²×C₂₉	—	2	2	no	—
116	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₉	#41	1	1	yes	characteristic
119	C₄²×C₂₉	C₂⁴	1	1	yes	characteristic
120	#19289	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₉	—	4	2	no	—
122	C₂³×C₂₉	#121	1	1	yes	characteristic
123	Q₈×C₂₉	—	2	4	no	—
124	Q₈×C₂₉	—	4	2	no	—
125	Q₈×C₂₉	—	8	2	no	—
126	D₄×C₂₉	—	4	2	no	—
127	D₄×C₂₉	—	4	2	no	—
128	D₄×C₂₉	—	4	4	no	—
129	D₄×C₂₉	—	8	2	no	—
130	D₄×C₂₉	—	8	4	no	—
131	C₂×C₄×C₂₉	—	2	2	no	—
132	C₂×C₄×C₂₉	—	2	1	no	—
133	C₂×C₄×C₂₉	—	2	1	no	—
134	C₂×C₄×C₂₉	—	2	2	no	—
135	C₂×C₄×C₂₉	—	2	2	no	—
136	C₂×C₄×C₂₉	—	2	2	no	—
137	C₂×C₄×C₂₉	—	4	2	no	—
138	C₂×C₄×C₂₉	—	4	2	no	—
139	C₂×C₄×C₂₉	—	4	2	no	—
140	C₂×C₄×C₂₉	—	8	4	no	—
141	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₂₉	—	2	1	no	—
148	C₈×C₂₉	—	2	1	no	—
149	C₈×C₂₉	—	2	1	no	—
150	C₈×C₂₉	—	4	2	no	—
151	C₈×C₂₉	—	4	4	no	—
152	C₈×C₂₉	#137	1	2	yes	—
153	#3169	C₂×C₂₉	1	1	yes	characteristic
154	#3112	C₂×C₂₉	1	2	yes	complemented
155	C₂×#446	C₂×C₂₉	1	1	yes	characteristic, complemented
156	#1698	C₂×C₂₉	1	4	yes	complemented
157	#1696	C₂×C₂₉	1	2	yes	complemented
158	#1688	C₂×C₂₉	1	2	yes	complemented
159	#1568	C₂×C₂₉	1	1	yes	characteristic, complemented
160	#1551	C₂×C₂₉	1	2	yes	complemented
161	C₂²×C₂₉	—	2	1	no	—
162	C₂²×C₂₉	—	2	1	no	—
163	C₂²×C₂₉	—	8	2	no	—
164	C₂²×C₂₉	—	8	2	no	—
165	C₂²×C₂₉	—	8	2	no	—
166	C₂²×C₂₉	#472	1	2	yes	—
167	C₂²×C₂₉	C₂×#121	1	1	yes	characteristic
168	C₄×C₂₉	—	2	1	no	—
169	C₄×C₂₉	—	2	1	no	—
170	C₄×C₂₉	—	2	1	no	—
171	C₄×C₂₉	—	2	1	no	—
172	C₄×C₂₉	—	4	2	no	—
173	C₄×C₂₉	—	4	2	no	—
174	C₄×C₂₉	—	8	2	no	—
175	C₄×C₂₉	C₂×#137	1	1	yes	characteristic
176	C₄×C₂₉	#575	1	1	yes	characteristic
177	C₄×C₂₉	#575	1	2	yes	—
178	#532	C₂²×C₂₉	1	2	yes	—
179	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
180	#500	C₂²×C₂₉	1	1	yes	characteristic
181	#493	C₂²×C₂₉	1	1	yes	characteristic
182	#486	C₂²×C₂₉	1	2	yes	—
183	#480	—	2	2	no	—
184	#477	C₂²×C₂₉	1	4	yes	—
185	#466	—	2	1	no	—
186	#466	—	2	1	no	—
187	#460	C₂²×C₂₉	1	4	yes	—
188	#448	—	2	2	no	—
189	#446	—	2	1	no	—
190	#446	—	2	1	no	—
191	#446	C₂²×C₂₉	1	2	yes	—
192	#446	C₂²×C₂₉	1	2	yes	—
193	#440	—	2	4	no	—
194	C₄×D₈	—	2	1	no	—
195	C₄×D₈	—	2	1	no	—
196	#432	C₂²×C₂₉	1	2	yes	—
197	C₂×C₄⋊C₈	—	2	1	no	—
198	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
199	C₂×#97	—	2	1	no	—
200	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
201	#335	C₂²×C₂₉	1	2	yes	—
202	#334	C₂²×C₂₉	1	2	yes	—
203	#333	C₂²×C₂₉	1	4	yes	—
204	#331	C₂²×C₂₉	1	4	yes	—
205	C₂×C₂₉	—	2	2	no	—
206	C₂×C₂₉	—	16	2	no	—
207	C₂×C₂₉	#3037	1	1	yes	characteristic
208	C₂×C₂₉	#3054	1	1	yes	characteristic
209	C₂×C₂₉	C₂×#472	1	1	yes	characteristic
210	C₂×#41	—	2	2	no	—
211	C₂²×Q₈	D₄×C₂₉	1	1	yes	characteristic
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	—	2	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
217	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
218	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
219	#127	C₂³×C₂₉	1	2	yes	—
220	#124	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	—	2	1	no	—
225	C₄×Q₈	D₄×C₂₉	1	2	yes	—
226	C₄×Q₈	C₂³×C₂₉	1	2	yes	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#107	—	2	4	no	—
236	C₄⋊C₈	—	2	2	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
240	C₄⋊C₈	C₂³×C₂₉	1	4	yes	—
241	#103	—	2	4	no	—
242	#102	—	2	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	4	no	—
245	C₂²⋊C₈	—	2	2	no	—
246	#97	—	2	2	no	—
247	#97	—	2	2	no	—
248	#97	—	2	2	no	—
249	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
250	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
253	C₂₉	#19289	1	1	yes	characteristic, direct product factor, Sylow
254	#41	—	8	4	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	C₂×D₄×C₂₉	1	2	yes	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
268	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
269	SD₁₆	—	4	8	no	—
270	D₈	—	4	2	no	—
271	D₈	—	4	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
293	C₄²	#41×C₂₉	1	1	yes	characteristic
294	C₄²	C₂⁴×C₂₉	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	C₂₉×#121	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
316	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
317	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
320	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	4	2	no	—
325	C₈	—	4	4	no	—
326	C₈	C₂₉×#137	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	C₂₉×#472	1	2	yes	—
333	C₂²	C₂×C₂₉×#121	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	8	2	no	—
341	C₄	C₂×C₂₉×#137	1	1	yes	characteristic
342	C₄	C₂₉×#575	1	1	yes	characteristic
343	C₄	C₂₉×#575	1	2	yes	—
344	C₂	—	2	2	no	—
345	C₂	—	16	2	no	—
346	C₂	C₂₉×#3037	1	1	yes	characteristic
347	C₂	C₂₉×#3054	1	1	yes	characteristic
348	C₂	C₂×C₂₉×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×#121
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
2. C₂₉×#3112	C₂	semidirect
3. C₂×C₂₉×#446	C₂	semidirect
4. C₂₉×#1698	C₂	semidirect
5. C₂₉×#1696	C₂	semidirect
6. C₂₉×#1688	C₂	semidirect
7. C₂₉×#1568	C₂	semidirect
8. C₂₉×#1551	C₂	semidirect
120. #19289	C₂₉	direct (N × H)
154. #3112	C₂×C₂₉	semidirect
155. C₂×#446	C₂×C₂₉	semidirect
156. #1698	C₂×C₂₉	semidirect
157. #1696	C₂×C₂₉	semidirect
158. #1688	C₂×C₂₉	semidirect
159. #1568	C₂×C₂₉	semidirect
160. #1551	C₂×C₂₉	semidirect
253. C₂₉	#19289	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×#3169	2
C₂₉×#3112	2
C₂×C₂₉×#446	2
C₂₉×#1698	2
C₂₉×#1696	2
C₂₉×#1688	2
C₂₉×#1568	2
C₂₉×#1551	2
#19289	29

Subgroups — C29×#19289 #20314189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#19289 #20314189