Subgroups — C₂₉×#13029 #20307929

Order 7424.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#2139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1970	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×#131	—	2	1	no	—
11	C₂₉×#509	—	2	2	no	—
12	C₂₉×#509	C₄	1	4	yes	complemented
13	C₂×C₂₉×SD₃₂	—	2	1	no	—
14	C₂×C₂₉×D₁₆	—	2	1	no	—
15	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₂₉×#443	—	2	2	no	—
18	C₂₉×#443	—	2	2	no	—
19	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
20	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×#104	—	2	1	no	—
22	C₂×C₂₉×#102	—	2	1	no	—
23	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×#390	—	2	1	no	—
25	C₂₉×#361	—	2	2	no	—
26	C₂₉×#358	—	2	1	no	—
27	C₂₉×#357	—	2	1	no	—
28	C₂×#41×C₂₉	—	2	1	no	—
29	C₂²×D₄×C₂₉	—	2	1	no	—
30	C₂₉×#135	—	2	1	no	—
31	C₂₉×#135	—	2	1	no	—
32	C₂₉×#135	C₂×C₄	1	2	yes	—
33	C₂₉×#135	D₄	1	2	yes	—
34	C₂×Q₁₆×C₂₉	D₄	1	1	yes	characteristic
35	C₂×SD₁₆×C₂₉	—	2	1	no	—
36	C₂×D₈×C₂₉	—	2	1	no	—
37	C₂×D₈×C₂₉	—	2	1	no	—
38	C₂×D₈×C₂₉	—	2	1	no	—
39	C₂×D₈×C₂₉	—	2	1	no	—
40	C₂×D₈×C₂₉	—	2	1	no	—
41	C₂×D₈×C₂₉	C₂×C₄	1	2	yes	—
42	C₂×D₈×C₂₉	D₄	1	1	yes	characteristic
43	C₂₉×#131	—	2	4	no	—
44	C₂₉×#131	—	4	2	no	—
45	C₂×M₁₆×C₂₉	—	2	1	no	—
46	C₂×M₁₆×C₂₉	—	4	1	no	—
47	C₂²×C₈×C₂₉	—	4	1	no	—
48	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₂₉	—	4	2	no	—
52	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
53	C₂₉×SD₃₂	—	2	4	no	—
54	C₂₉×D₁₆	—	2	4	no	—
55	C₂₉×M₃₂	—	2	1	no	—
56	C₂₉×M₃₂	—	2	1	no	—
57	C₂₉×M₃₂	—	2	1	no	—
58	C₂₉×M₃₂	C₂×C₄	1	2	yes	—
59	C₂×C₁₆×C₂₉	—	2	1	no	—
60	C₂×C₁₆×C₂₉	—	2	1	no	—
61	C₂₉×#108	—	2	1	no	—
62	C₂₉×#108	D₄	1	2	yes	—
63	C₂₉×#107	—	2	1	no	—
64	C₂₉×#107	—	2	1	no	—
65	C₂₉×#104	—	2	4	no	—
66	C₂₉×#102	—	4	1	no	—
67	C₂₉×#102	—	4	1	no	—
68	C₄×C₈×C₂₉	—	2	1	no	—
69	C₄×C₈×C₂₉	—	2	1	no	—
70	C₂⁴×C₂₉	—	4	1	no	—
71	#41×C₂₉	—	2	2	no	—
72	#41×C₂₉	—	2	2	no	—
73	#41×C₂₉	—	4	1	no	—
74	#41×C₂₉	—	4	1	no	—
75	C₂×Q₈×C₂₉	—	2	1	no	—
76	C₂×D₄×C₂₉	—	2	2	no	—
77	C₂×D₄×C₂₉	—	2	1	no	—
78	C₂×D₄×C₂₉	—	4	1	no	—
79	C₂×D₄×C₂₉	—	4	1	no	—
80	C₂×D₄×C₂₉	—	4	1	no	—
81	C₂×D₄×C₂₉	—	4	1	no	—
82	C₂×D₄×C₂₉	—	4	1	no	—
83	C₂×D₄×C₂₉	—	4	1	no	—
84	C₂²×C₄×C₂₉	—	2	1	no	—
85	C₂²×C₄×C₂₉	—	4	1	no	—
86	C₂²×C₄×C₂₉	—	4	1	no	—
87	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₂₉	—	2	1	no	—
89	Q₁₆×C₂₉	C₂²⋊C₄	1	2	yes	complemented
90	SD₁₆×C₂₉	—	2	2	no	—
91	SD₁₆×C₂₉	—	2	2	no	—
92	D₈×C₂₉	—	2	2	no	—
93	D₈×C₂₉	—	2	2	no	—
94	D₈×C₂₉	—	2	1	no	—
95	D₈×C₂₉	—	4	1	no	—
96	D₈×C₂₉	—	4	1	no	—
97	D₈×C₂₉	C₂²⋊C₄	1	2	yes	complemented
98	M₁₆×C₂₉	—	2	2	no	—
99	M₁₆×C₂₉	—	4	4	no	—
100	M₁₆×C₂₉	—	4	1	no	—
101	C₂×C₈×C₂₉	—	2	1	no	—
102	C₂×C₈×C₂₉	—	2	1	no	—
103	C₂×C₈×C₂₉	—	4	4	no	—
104	C₂×C₈×C₂₉	—	4	1	no	—
105	C₂×C₈×C₂₉	D₈	1	1	yes	characteristic
106	C₂×C₈×C₂₉	D₈	1	1	yes	characteristic
107	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
113	C₄⋊C₄×C₂₉	—	4	1	no	—
114	C₄⋊C₄×C₂₉	—	4	1	no	—
115	C₂²⋊C₄×C₂₉	—	8	4	no	—
116	C₄²×C₂₉	—	2	1	no	—
117	C₄²×C₂₉	—	2	1	no	—
118	C₁₆×C₂₉	—	2	2	no	—
119	C₁₆×C₂₉	—	4	1	no	—
120	#13029	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂³×C₂₉	—	4	1	no	—
122	C₂³×C₂₉	—	4	2	no	—
123	C₂³×C₂₉	—	8	2	no	—
124	C₂³×C₂₉	—	8	2	no	—
125	C₂³×C₂₉	—	8	1	no	—
126	C₂³×C₂₉	—	8	1	no	—
127	C₂³×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₂₉	—	2	2	no	—
129	Q₈×C₂₉	—	4	1	no	—
130	D₄×C₂₉	—	2	2	no	—
131	D₄×C₂₉	—	4	2	no	—
132	D₄×C₂₉	—	4	2	no	—
133	D₄×C₂₉	—	4	2	no	—
134	D₄×C₂₉	—	4	2	no	—
135	D₄×C₂₉	—	4	1	no	—
136	D₄×C₂₉	—	8	1	no	—
137	D₄×C₂₉	—	8	1	no	—
138	C₂×C₄×C₂₉	—	2	2	no	—
139	C₂×C₄×C₂₉	—	4	1	no	—
140	C₂×C₄×C₂₉	—	4	1	no	—
141	C₂×C₄×C₂₉	—	4	1	no	—
142	C₂×C₄×C₂₉	—	4	2	no	—
143	C₂×C₄×C₂₉	—	4	2	no	—
144	C₂×C₄×C₂₉	—	8	2	no	—
145	C₂×C₄×C₂₉	—	8	1	no	—
146	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
149	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₂₉	#135	1	1	yes	characteristic
151	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₈×C₂₉	—	2	1	no	—
153	C₈×C₂₉	—	2	1	no	—
154	C₈×C₂₉	—	4	2	no	—
155	C₈×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₈×C₂₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₂₉	#121	1	1	yes	characteristic
158	C₈×C₂₉	#123	1	1	yes	characteristic
159	C₂×#509	C₂×C₂₉	1	1	yes	characteristic
160	C₂×#443	C₂×C₂₉	1	1	yes	characteristic, complemented
161	C₂×#361	C₂×C₂₉	1	1	yes	characteristic, complemented
162	#2139	C₂×C₂₉	1	1	yes	characteristic, complemented
163	#1970	C₂×C₂₉	1	1	yes	characteristic, complemented
164	#1387	C₂×C₂₉	1	1	yes	characteristic, complemented
165	#1376	C₂×C₂₉	1	1	yes	characteristic, complemented
166	C₂²×C₂₉	—	2	1	no	—
167	C₂²×C₂₉	—	2	1	no	—
168	C₂²×C₂₉	—	4	2	no	—
169	C₂²×C₂₉	—	8	4	no	—
170	C₂²×C₂₉	—	8	1	no	—
171	C₂²×C₂₉	—	8	1	no	—
172	C₂²×C₂₉	—	8	2	no	—
173	C₂²×C₂₉	—	16	2	no	—
174	C₂²×C₂₉	—	16	1	no	—
175	C₂²×C₂₉	—	16	1	no	—
176	C₂²×C₂₉	#442	1	1	yes	characteristic
177	C₂²×C₂₉	#449	1	1	yes	characteristic
178	C₂²×C₂₉	#459	1	1	yes	characteristic
179	C₄×C₂₉	—	4	2	no	—
180	C₄×C₂₉	—	4	2	no	—
181	C₄×C₂₉	—	16	1	no	—
182	C₄×C₂₉	#386	1	1	yes	characteristic
183	C₄×C₂₉	C₄×D₈	1	1	yes	characteristic
184	C₄×C₂₉	#447	1	1	yes	characteristic
185	C₄×C₂₉	#463	1	1	yes	characteristic
186	C₂×#135	C₂²×C₂₉	1	1	yes	characteristic
187	C₂²×D₈	C₂²×C₂₉	1	1	yes	characteristic
188	C₂×#131	—	2	1	no	—
189	#509	—	2	2	no	—
190	#509	C₄×C₂₉	1	4	yes	complemented
191	C₂×SD₃₂	—	2	1	no	—
192	C₂×D₁₆	—	2	1	no	—
193	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
194	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
195	#443	—	2	2	no	—
196	#443	—	2	2	no	—
197	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
198	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
199	C₂×#104	—	2	1	no	—
200	C₂×#102	—	2	1	no	—
201	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
202	#390	—	2	1	no	—
203	#361	—	2	2	no	—
204	#358	—	2	1	no	—
205	#357	—	2	1	no	—
206	C₂×C₂₉	—	2	1	no	—
207	C₂×C₂₉	—	2	1	no	—
208	C₂×C₂₉	—	8	2	no	—
209	C₂×C₂₉	—	16	2	no	—
210	C₂×C₂₉	#1879	1	1	yes	characteristic
211	C₂×C₂₉	#2212	1	1	yes	characteristic
212	C₂×C₂₉	#2213	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	C₂×C₄×C₂₉	1	2	yes	—
218	#135	D₄×C₂₉	1	2	yes	—
219	C₂×Q₁₆	D₄×C₂₉	1	1	yes	characteristic
220	C₂×SD₁₆	—	2	1	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	C₂×C₄×C₂₉	1	2	yes	—
227	C₂×D₈	D₄×C₂₉	1	1	yes	characteristic
228	#131	—	2	4	no	—
229	#131	—	4	2	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₂×M₁₆	—	4	1	no	—
232	C₂²×C₈	—	4	1	no	—
233	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	4	2	no	—
237	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
238	SD₃₂	—	2	4	no	—
239	D₁₆	—	2	4	no	—
240	M₃₂	—	2	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	C₂×C₄×C₂₉	1	2	yes	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	#108	—	2	1	no	—
247	#108	D₄×C₂₉	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	—	2	1	no	—
250	#104	—	2	4	no	—
251	#102	—	4	1	no	—
252	#102	—	4	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₂₉	#13029	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	4	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
274	Q₁₆	—	2	1	no	—
275	Q₁₆	C₂²⋊C₄×C₂₉	1	2	yes	complemented
276	SD₁₆	—	2	2	no	—
277	SD₁₆	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	1	no	—
281	D₈	—	4	1	no	—
282	D₈	—	4	1	no	—
283	D₈	C₂²⋊C₄×C₂₉	1	2	yes	complemented
284	M₁₆	—	2	2	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	4	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	D₈×C₂₉	1	1	yes	characteristic
292	C₂×C₈	D₈×C₂₉	1	1	yes	characteristic
293	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
297	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
298	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	8	4	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	4	1	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	—	8	2	no	—
310	C₂³	—	8	1	no	—
311	C₂³	—	8	1	no	—
312	C₂³	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
332	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
333	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
334	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
335	C₂×C₄	C₂₉×#135	1	1	yes	characteristic
336	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	4	2	no	—
340	C₈	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
341	C₈	C₄×D₄×C₂₉	1	1	yes	characteristic
342	C₈	C₂₉×#121	1	1	yes	characteristic
343	C₈	C₂₉×#123	1	1	yes	characteristic
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	8	4	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	2	no	—
352	C₂²	—	16	1	no	—
353	C₂²	—	16	1	no	—
354	C₂²	C₂₉×#442	1	1	yes	characteristic
355	C₂²	C₂₉×#449	1	1	yes	characteristic
356	C₂²	C₂₉×#459	1	1	yes	characteristic
357	C₄	—	4	2	no	—
358	C₄	—	4	2	no	—
359	C₄	—	16	1	no	—
360	C₄	C₂₉×#386	1	1	yes	characteristic
361	C₄	C₄×D₈×C₂₉	1	1	yes	characteristic
362	C₄	C₂₉×#447	1	1	yes	characteristic
363	C₄	C₂₉×#463	1	1	yes	characteristic
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	8	2	no	—
367	C₂	—	16	2	no	—
368	C₂	C₂₉×#1879	1	1	yes	characteristic
369	C₂	C₂₉×#2212	1	1	yes	characteristic
370	C₂	C₂₉×#2213	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	#459
commutator	C₂×C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
prime radical	C₂×C₂₉×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ 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. C₂×C₂₉×#443	C₂	semidirect
3. C₂×C₂₉×#361	C₂	semidirect
4. C₂₉×#2139	C₂	semidirect
5. C₂₉×#1970	C₂	semidirect
6. C₂₉×#1387	C₂	semidirect
7. C₂₉×#1376	C₂	semidirect
12. C₂₉×#509	C₄	semidirect
89. Q₁₆×C₂₉	C₂²⋊C₄	semidirect
97. D₈×C₂₉	C₂²⋊C₄	semidirect
120. #13029	C₂₉	direct (N × H)
160. C₂×#443	C₂×C₂₉	semidirect
161. C₂×#361	C₂×C₂₉	semidirect
162. #2139	C₂×C₂₉	semidirect
163. #1970	C₂×C₂₉	semidirect
164. #1387	C₂×C₂₉	semidirect
165. #1376	C₂×C₂₉	semidirect
190. #509	C₄×C₂₉	semidirect
255. C₂₉	#13029	direct (N × H)
275. Q₁₆	C₂²⋊C₄×C₂₉	semidirect
283. D₈	C₂²⋊C₄×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×#509	2
C₂×C₂₉×#443	2
C₂×C₂₉×#361	2
C₂₉×#2139	2
C₂₉×#1970	2
C₂₉×#1387	2
C₂₉×#1376	2
#13029	29

Subgroups — C29×#13029 #20307929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#13029 #20307929