Subgroups — C₂₉×#19338 #20314238

Order 7424.

636 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3254	C₂	1	2	yes	maximal
2	C₂₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#2931	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1693	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1689	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1679	C₂	1	4	yes	maximal, complemented
7	C₂₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#1561	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₉×#131	—	2	1	no	—
10	C₂₉×#533	C₂²	1	2	yes	—
11	C₂₉×#532	C₂²	1	1	yes	characteristic
12	C₂₉×#500	C₂²	1	1	yes	characteristic
13	C₂₉×#494	C₂²	1	1	yes	characteristic
14	C₂₉×#487	C₂²	1	2	yes	—
15	C₂₉×#477	—	2	2	no	—
16	C₂₉×#474	C₂²	1	4	yes	—
17	C₂₉×#468	—	2	4	no	—
18	C₂₉×#467	—	2	1	no	—
19	C₂₉×#467	—	2	1	no	—
20	C₂₉×#461	C₂²	1	4	yes	—
21	C₂₉×#460	—	2	2	no	—
22	C₄×Q₁₆×C₂₉	—	2	1	no	—
23	C₄×Q₁₆×C₂₉	—	2	1	no	—
24	C₂₉×#436	—	2	1	no	—
25	C₂₉×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₂₉	—	2	1	no	—
27	C₈×D₄×C₂₉	C₂²	1	2	yes	—
28	C₂₉×#423	C₂²	1	1	yes	characteristic
29	C₂₉×#423	C₂²	1	2	yes	—
30	C₂₉×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
32	C₂₉×#334	C₂²	1	2	yes	—
33	C₂₉×#334	C₂²	1	2	yes	—
34	C₂₉×#332	C₂²	1	4	yes	—
35	C₂₉×#332	C₂²	1	4	yes	—
36	C₂×#41×C₂₉	—	2	1	no	—
37	C₂×Q₁₆×C₂₉	—	2	1	no	—
38	C₂×Q₁₆×C₂₉	—	2	1	no	—
39	C₂×SD₁₆×C₂₉	—	2	4	no	—
40	C₂₉×#131	—	4	4	no	—
41	C₂×M₁₆×C₂₉	—	2	1	no	—
42	C₂×M₁₆×C₂₉	—	2	2	no	—
43	C₂×M₁₆×C₂₉	—	2	2	no	—
44	C₂²×C₈×C₂₉	—	2	2	no	—
45	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
47	C₂₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
48	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₉×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₂₉×#127	D₄	1	1	yes	characteristic
51	C₂₉×#125	—	2	2	no	—
52	C₂₉×#124	—	2	1	no	—
53	C₂₉×C₂²⋊Q₈	—	2	2	no	—
54	C₂₉×C₂²⋊Q₈	—	2	2	no	—
55	C₂₉×#121	—	2	1	no	—
56	C₂₉×#121	D₄	1	2	yes	—
57	C₄×Q₈×C₂₉	—	2	2	no	—
58	C₄×Q₈×C₂₉	—	2	2	no	—
59	C₄×D₄×C₂₉	—	2	1	no	—
60	C₄×D₄×C₂₉	C₂³	1	2	yes	—
61	C₂₉×#117	—	2	2	no	—
62	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
63	C₂₉×#107	—	2	1	no	—
64	C₂₉×#107	—	2	1	no	—
65	C₂₉×#107	—	2	4	no	—
66	C₂₉×C₄⋊C₈	—	2	1	no	—
67	C₂₉×C₄⋊C₈	—	2	2	no	—
68	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
69	C₂₉×C₄⋊C₈	C₂³	1	4	yes	—
70	C₂₉×#103	—	2	2	no	—
71	C₂₉×#103	—	2	2	no	—
72	C₂₉×#103	—	2	4	no	—
73	C₂₉×#102	—	2	4	no	—
74	C₂₉×C₂²⋊C₈	—	2	2	no	—
75	C₂₉×C₂²⋊C₈	—	2	2	no	—
76	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
80	#41×C₂₉	—	4	2	no	—
81	#41×C₂₉	—	4	2	no	—
82	C₂×Q₈×C₂₉	—	2	1	no	—
83	C₂×Q₈×C₂₉	—	2	2	no	—
84	C₂×Q₈×C₂₉	—	2	2	no	—
85	C₂×D₄×C₂₉	—	2	2	no	—
86	C₂×D₄×C₂₉	—	2	1	no	—
87	C₂×D₄×C₂₉	—	2	1	no	—
88	C₂×D₄×C₂₉	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₂₉	—	2	2	no	—
90	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
93	Q₁₆×C₂₉	—	4	2	no	—
94	Q₁₆×C₂₉	—	4	2	no	—
95	SD₁₆×C₂₉	—	8	4	no	—
96	M₁₆×C₂₉	—	4	2	no	—
97	M₁₆×C₂₉	—	4	4	no	—
98	M₁₆×C₂₉	—	8	2	no	—
99	C₂×C₈×C₂₉	—	2	2	no	—
100	C₂×C₈×C₂₉	—	2	2	no	—
101	C₂×C₈×C₂₉	—	2	4	no	—
102	C₂×C₈×C₂₉	—	4	1	no	—
103	C₂×C₈×C₂₉	—	4	4	no	—
104	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₂₉	—	2	1	no	—
109	C₄⋊C₄×C₂₉	—	2	2	no	—
110	C₄⋊C₄×C₂₉	—	2	2	no	—
111	C₄⋊C₄×C₂₉	—	2	2	no	—
112	C₄⋊C₄×C₂₉	—	2	2	no	—
113	C₄⋊C₄×C₂₉	—	4	2	no	—
114	C₄⋊C₄×C₂₉	—	4	2	no	—
115	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₂₉	—	2	2	no	—
117	C₂²⋊C₄×C₂₉	—	2	2	no	—
118	C₂²⋊C₄×C₂₉	—	4	2	no	—
119	C₄²×C₂₉	—	4	2	no	—
120	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₂₉	#41	1	1	yes	characteristic
123	C₄²×C₂₉	C₂⁴	1	1	yes	characteristic
124	#19338	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₂₉	—	2	2	no	—
126	C₂³×C₂₉	#121	1	1	yes	characteristic
127	Q₈×C₂₉	—	4	2	no	—
128	Q₈×C₂₉	—	4	2	no	—
129	Q₈×C₂₉	—	4	2	no	—
130	Q₈×C₂₉	—	4	2	no	—
131	Q₈×C₂₉	—	4	2	no	—
132	D₄×C₂₉	—	2	4	no	—
133	D₄×C₂₉	—	4	2	no	—
134	D₄×C₂₉	—	4	4	no	—
135	D₄×C₂₉	—	4	2	no	—
136	C₂×C₄×C₂₉	—	2	1	no	—
137	C₂×C₄×C₂₉	—	2	1	no	—
138	C₂×C₄×C₂₉	—	2	2	no	—
139	C₂×C₄×C₂₉	—	2	2	no	—
140	C₂×C₄×C₂₉	—	2	2	no	—
141	C₂×C₄×C₂₉	—	4	2	no	—
142	C₂×C₄×C₂₉	—	4	2	no	—
143	C₂×C₄×C₂₉	—	4	2	no	—
144	C₂×C₄×C₂₉	—	4	2	no	—
145	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
151	C₈×C₂₉	—	2	1	no	—
152	C₈×C₂₉	—	2	1	no	—
153	C₈×C₂₉	—	2	1	no	—
154	C₈×C₂₉	—	4	2	no	—
155	C₈×C₂₉	—	4	4	no	—
156	C₈×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
157	#3254	C₂×C₂₉	1	2	yes	—
158	#3170	C₂×C₂₉	1	1	yes	characteristic, complemented
159	#2931	C₂×C₂₉	1	1	yes	characteristic
160	#1693	C₂×C₂₉	1	2	yes	complemented
161	#1689	C₂×C₂₉	1	2	yes	complemented
162	#1679	C₂×C₂₉	1	4	yes	complemented
163	#1566	C₂×C₂₉	1	1	yes	characteristic, complemented
164	#1561	C₂×C₂₉	1	2	yes	complemented
165	C₂²×C₂₉	—	2	1	no	—
166	C₂²×C₂₉	—	2	1	no	—
167	C₂²×C₂₉	—	2	1	no	—
168	C₂²×C₂₉	—	4	2	no	—
169	C₂²×C₂₉	—	4	2	no	—
170	C₂²×C₂₉	—	4	2	no	—
171	C₂²×C₂₉	C₂×#121	1	1	yes	characteristic
172	C₄×C₂₉	—	2	1	no	—
173	C₄×C₂₉	—	2	1	no	—
174	C₄×C₂₉	—	2	1	no	—
175	C₄×C₂₉	—	4	2	no	—
176	C₄×C₂₉	—	8	2	no	—
177	C₄×C₂₉	—	8	2	no	—
178	C₄×C₂₉	#473	1	2	yes	—
179	C₄×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₂₉	#575	1	1	yes	characteristic
181	C₄×C₂₉	#577	1	2	yes	—
182	C₂×#131	—	2	1	no	—
183	#533	C₂²×C₂₉	1	2	yes	—
184	#532	C₂²×C₂₉	1	1	yes	characteristic
185	#500	C₂²×C₂₉	1	1	yes	characteristic
186	#494	C₂²×C₂₉	1	1	yes	characteristic
187	#487	C₂²×C₂₉	1	2	yes	—
188	#477	—	2	2	no	—
189	#474	C₂²×C₂₉	1	4	yes	—
190	#468	—	2	4	no	—
191	#467	—	2	1	no	—
192	#467	—	2	1	no	—
193	#461	C₂²×C₂₉	1	4	yes	—
194	#460	—	2	2	no	—
195	C₄×Q₁₆	—	2	1	no	—
196	C₄×Q₁₆	—	2	1	no	—
197	#436	—	2	1	no	—
198	#436	C₂²×C₂₉	1	2	yes	—
199	C₈×D₄	—	2	1	no	—
200	C₈×D₄	C₂²×C₂₉	1	2	yes	—
201	#423	C₂²×C₂₉	1	1	yes	characteristic
202	#423	C₂²×C₂₉	1	2	yes	—
203	#408	—	2	1	no	—
204	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
205	#334	C₂²×C₂₉	1	2	yes	—
206	#334	C₂²×C₂₉	1	2	yes	—
207	#332	C₂²×C₂₉	1	4	yes	—
208	#332	C₂²×C₂₉	1	4	yes	—
209	C₂×C₂₉	—	4	1	no	—
210	C₂×C₂₉	—	8	2	no	—
211	C₂×C₂₉	#3036	1	1	yes	characteristic
212	C₂×C₂₉	#3060	1	1	yes	characteristic
213	C₂×C₂₉	C₂×#473	1	1	yes	characteristic
214	C₂×#41	—	2	1	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×SD₁₆	—	2	4	no	—
218	#131	—	4	4	no	—
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂²×C₈	—	2	2	no	—
223	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
224	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
225	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
226	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
227	C₄⋊Q₈	C₂³×C₂₉	1	2	yes	—
228	#127	D₄×C₂₉	1	1	yes	characteristic
229	#125	—	2	2	no	—
230	#124	—	2	1	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	#121	—	2	1	no	—
234	#121	D₄×C₂₉	1	2	yes	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	C₂³×C₂₉	1	2	yes	—
239	#117	—	2	2	no	—
240	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
241	#107	—	2	1	no	—
242	#107	—	2	1	no	—
243	#107	—	2	4	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
247	C₄⋊C₈	C₂³×C₂₉	1	4	yes	—
248	#103	—	2	2	no	—
249	#103	—	2	2	no	—
250	#103	—	2	4	no	—
251	#102	—	2	4	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
255	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
258	C₂₉	#19338	1	1	yes	characteristic, direct product factor, Sylow
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂×D₄×C₂₉	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
271	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
272	Q₁₆	—	4	2	no	—
273	Q₁₆	—	4	2	no	—
274	SD₁₆	—	8	4	no	—
275	M₁₆	—	4	2	no	—
276	M₁₆	—	4	4	no	—
277	M₁₆	—	8	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	4	no	—
283	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
301	C₄²	#41×C₂₉	1	1	yes	characteristic
302	C₄²	C₂⁴×C₂₉	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	C₂₉×#121	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
324	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
325	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
328	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	4	no	—
334	C₈	C₂₉×AGL(1,ℤ/8)	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	C₂×C₂₉×#121	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₂₉×#473	1	2	yes	—
349	C₄	C₂×C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₄	C₂₉×#575	1	1	yes	characteristic
351	C₄	C₂₉×#577	1	2	yes	—
352	C₂	—	4	1	no	—
353	C₂	—	8	2	no	—
354	C₂	C₂₉×#3036	1	1	yes	characteristic
355	C₂	C₂₉×#3060	1	1	yes	characteristic
356	C₂	C₂×C₂₉×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₉
prime radical	C₂₉×#532	C₂²
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×#3170	C₂	semidirect
4. C₂₉×#1693	C₂	semidirect
5. C₂₉×#1689	C₂	semidirect
6. C₂₉×#1679	C₂	semidirect
7. C₂₉×#1566	C₂	semidirect
8. C₂₉×#1561	C₂	semidirect
124. #19338	C₂₉	direct (N × H)
158. #3170	C₂×C₂₉	semidirect
160. #1693	C₂×C₂₉	semidirect
161. #1689	C₂×C₂₉	semidirect
162. #1679	C₂×C₂₉	semidirect
163. #1566	C₂×C₂₉	semidirect
164. #1561	C₂×C₂₉	semidirect
258. C₂₉	#19338	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×#3254	2
C₂₉×#3170	2
C₂₉×#2931	2
C₂₉×#1693	2
C₂₉×#1689	2
C₂₉×#1679	2
C₂₉×#1566	2
C₂₉×#1561	2
#19338	29

Subgroups — C29×#19338 #20314238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#19338 #20314238