Subgroups — C₂₉×#3062 #16816776

Order 3712.

504 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#537	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#469	C₂	1	2	yes	maximal, complemented
6	C₂₉×#468	C₂	1	2	yes	maximal, complemented
7	C₂₉×#466	C₂	1	2	yes	maximal, complemented
8	C₂₉×#465	C₂	1	2	yes	maximal, complemented
9	C₂₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₉×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
15	C₂₉×AGL(1,ℤ/8)	—	2	4	no	—
16	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	—
17	C₂×D₈×C₂₉	C₂²	1	2	yes	—
18	C₂₉×#131	—	2	4	no	—
19	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₂₉	C₂²	1	2	yes	complemented
21	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₂₉	C₂²	1	2	yes	complemented
23	C₂₉×#127	—	2	1	no	—
24	C₂₉×#124	—	2	1	no	—
25	C₂₉×C₂²⋊Q₈	—	2	1	no	—
26	C₂₉×C₂²⋊Q₈	C₂²	1	2	yes	complemented
27	C₂₉×#121	—	2	1	no	—
28	C₂₉×#121	—	2	2	no	—
29	C₂₉×#121	C₂²	1	2	yes	—
30	C₂₉×#121	C₂²	1	2	yes	—
31	C₂₉×#121	C₂²	1	2	yes	complemented
32	C₂₉×#120	—	2	2	no	—
33	C₂₉×#117	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
35	C₂₉×#107	C₂²	1	2	yes	complemented
36	C₂₉×#106	C₂²	1	2	yes	complemented
37	C₂₉×#103	C₂²	1	1	yes	characteristic, complemented
38	C₂₉×#103	C₂²	1	1	yes	characteristic, complemented
39	C₂₉×#102	C₂²	1	2	yes	—
40	C₂₉×#102	C₂²	1	2	yes	—
41	C₂₉×#102	C₂²	1	1	yes	characteristic, complemented
42	C₂₉×#102	C₂²	1	1	yes	characteristic, complemented
43	C₂⁴×C₂₉	—	2	1	no	—
44	#41×C₂₉	—	2	2	no	—
45	#41×C₂₉	—	2	2	no	—
46	#41×C₂₉	—	2	2	no	—
47	#41×C₂₉	—	2	2	no	—
48	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
49	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₉	—	2	1	no	—
51	C₂×D₄×C₂₉	—	2	2	no	—
52	C₂×D₄×C₂₉	—	2	2	no	—
53	C₂×D₄×C₂₉	—	2	2	no	—
54	C₂×D₄×C₂₉	—	2	2	no	—
55	C₂×D₄×C₂₉	—	2	1	no	—
56	C₂×D₄×C₂₉	—	2	2	no	—
57	C₂×D₄×C₂₉	D₄	1	2	yes	—
58	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
59	C₂×D₄×C₂₉	C₂³	1	2	yes	—
60	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	—	2	1	no	—
62	C₂²×C₄×C₂₉	—	2	1	no	—
63	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
64	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
65	SD₁₆×C₂₉	—	2	4	no	—
66	D₈×C₂₉	—	2	4	no	—
67	M₁₆×C₂₉	—	2	4	no	—
68	M₁₆×C₂₉	D₄	1	4	yes	complemented
69	C₂×C₈×C₂₉	—	2	2	no	—
70	C₂×C₈×C₂₉	—	2	4	no	—
71	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₂₉	C₂³	1	2	yes	—
74	C₄⋊C₄×C₂₉	—	2	1	no	—
75	C₄⋊C₄×C₂₉	—	2	2	no	—
76	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
77	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₂₉	—	2	1	no	—
80	C₂²⋊C₄×C₂₉	—	2	2	no	—
81	C₂²⋊C₄×C₂₉	—	2	2	no	—
82	C₂²⋊C₄×C₂₉	—	2	2	no	—
83	C₂²⋊C₄×C₂₉	—	2	2	no	—
84	C₄²×C₂₉	—	2	1	no	—
85	C₂³×C₂₉	—	2	1	no	—
86	C₂³×C₂₉	—	2	2	no	—
87	C₂³×C₂₉	—	4	2	no	—
88	C₂³×C₂₉	—	8	1	no	—
89	C₂³×C₂₉	—	8	1	no	—
90	C₂³×C₂₉	C₂×D₄	1	2	yes	—
91	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
92	Q₈×C₂₉	—	2	2	no	—
93	Q₈×C₂₉	—	2	1	no	—
94	Q₈×C₂₉	—	2	1	no	—
95	D₄×C₂₉	—	2	4	no	—
96	D₄×C₂₉	—	2	2	no	—
97	D₄×C₂₉	—	2	1	no	—
98	D₄×C₂₉	—	2	2	no	—
99	D₄×C₂₉	—	2	2	no	—
100	D₄×C₂₉	—	2	1	no	—
101	D₄×C₂₉	—	4	2	no	—
102	D₄×C₂₉	—	4	2	no	—
103	D₄×C₂₉	—	8	2	no	—
104	D₄×C₂₉	—	8	2	no	—
105	D₄×C₂₉	—	8	2	no	—
106	C₂×C₄×C₂₉	—	2	2	no	—
107	C₂×C₄×C₂₉	—	2	2	no	—
108	C₂×C₄×C₂₉	—	2	1	no	—
109	C₂×C₄×C₂₉	—	2	1	no	—
110	C₂×C₄×C₂₉	—	2	2	no	—
111	C₂×C₄×C₂₉	—	2	1	no	—
112	C₂×C₄×C₂₉	—	4	2	no	—
113	C₂×C₄×C₂₉	—	8	1	no	—
114	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₂₉	#41	1	2	yes	—
118	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
119	C₈×C₂₉	—	2	1	no	—
120	C₈×C₂₉	—	2	1	no	—
121	C₈×C₂₉	C₂×D₄	1	4	yes	—
122	#3062	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₂₉	—	2	1	no	—
124	C₂²×C₂₉	—	2	1	no	—
125	C₂²×C₂₉	—	2	2	no	—
126	C₂²×C₂₉	—	2	2	no	—
127	C₂²×C₂₉	—	2	2	no	—
128	C₂²×C₂₉	—	4	1	no	—
129	C₂²×C₂₉	—	4	2	no	—
130	C₂²×C₂₉	—	4	1	no	—
131	C₂²×C₂₉	—	4	2	no	—
132	C₂²×C₂₉	—	4	2	no	—
133	C₂²×C₂₉	—	4	1	no	—
134	C₂²×C₂₉	—	8	4	no	—
135	C₂²×C₂₉	#121	1	2	yes	—
136	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₂₉	—	2	2	no	—
138	C₄×C₂₉	—	4	1	no	—
139	C₄×C₂₉	—	4	2	no	—
140	C₄×C₂₉	—	4	1	no	—
141	C₄×C₂₉	—	4	1	no	—
142	C₄×C₂₉	#121	1	2	yes	—
143	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
145	C₂×AGL(1,ℤ/8)	C₂×C₂₉	1	1	yes	characteristic, complemented
146	C₂×#131	C₂×C₂₉	1	1	yes	characteristic, complemented
147	#537	C₂×C₂₉	1	1	yes	characteristic, complemented
148	#535	C₂×C₂₉	1	1	yes	characteristic, complemented
149	#469	C₂×C₂₉	1	2	yes	complemented
150	#468	C₂×C₂₉	1	2	yes	complemented
151	#466	C₂×C₂₉	1	2	yes	complemented
152	#465	C₂×C₂₉	1	2	yes	complemented
153	#428	C₂×C₂₉	1	1	yes	characteristic, complemented
154	#416	C₂×C₂₉	1	1	yes	characteristic, complemented
155	C₂×#102	C₂×C₂₉	1	1	yes	characteristic, complemented
156	C₂×C₂₉	—	2	2	no	—
157	C₂×C₂₉	—	4	2	no	—
158	C₂×C₂₉	—	8	1	no	—
159	C₂×C₂₉	—	8	2	no	—
160	C₂×C₂₉	C₂×#121	1	1	yes	characteristic
161	C₂×C₂₉	#576	1	1	yes	characteristic
162	C₂×C₂₉	#577	1	1	yes	characteristic
163	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
164	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
165	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
166	AGL(1,ℤ/8)	—	2	4	no	—
167	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	—
168	C₂×D₈	C₂²×C₂₉	1	2	yes	—
169	#131	—	2	4	no	—
170	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
171	C₂×M₁₆	C₂²×C₂₉	1	2	yes	complemented
172	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
173	C₂²×C₈	C₂²×C₂₉	1	2	yes	complemented
174	#127	—	2	1	no	—
175	#124	—	2	1	no	—
176	C₂²⋊Q₈	—	2	1	no	—
177	C₂²⋊Q₈	C₂²×C₂₉	1	2	yes	complemented
178	#121	—	2	1	no	—
179	#121	—	2	2	no	—
180	#121	C₂²×C₂₉	1	2	yes	—
181	#121	C₂²×C₂₉	1	2	yes	—
182	#121	C₂²×C₂₉	1	2	yes	complemented
183	#120	—	2	2	no	—
184	#117	C₂²×C₂₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
186	#107	C₂²×C₂₉	1	2	yes	complemented
187	#106	C₂²×C₂₉	1	2	yes	complemented
188	#103	C₂²×C₂₉	1	1	yes	characteristic, complemented
189	#103	C₂²×C₂₉	1	1	yes	characteristic, complemented
190	#102	C₂²×C₂₉	1	2	yes	—
191	#102	C₂²×C₂₉	1	2	yes	—
192	#102	C₂²×C₂₉	1	1	yes	characteristic, complemented
193	#102	C₂²×C₂₉	1	1	yes	characteristic, complemented
194	C₂₉	#3062	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	—	2	1	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	C₂×Q₈	D₄×C₂₉	1	1	yes	characteristic
201	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	D₄×C₂₉	1	2	yes	—
210	C₂×D₄	D₄×C₂₉	1	1	yes	characteristic
211	C₂×D₄	C₂³×C₂₉	1	2	yes	—
212	C₂×D₄	C₂³×C₂₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
216	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	D₄×C₂₉	1	4	yes	complemented
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
224	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
225	C₂×C₈	C₂³×C₂₉	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
229	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
230	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
231	C₂²⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₄²	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	8	1	no	—
241	C₂³	—	8	1	no	—
242	C₂³	C₂×D₄×C₂₉	1	2	yes	—
243	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	2	1	no	—
246	Q₈	—	2	1	no	—
247	D₄	—	2	4	no	—
248	D₄	—	2	2	no	—
249	D₄	—	2	1	no	—
250	D₄	—	2	2	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	8	2	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
269	C₂×C₄	#41×C₂₉	1	2	yes	—
270	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	C₂×D₄×C₂₉	1	4	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	8	4	no	—
286	C₂²	C₂₉×#121	1	2	yes	—
287	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	C₂₉×#121	1	2	yes	—
294	C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
295	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	8	2	no	—
299	C₂	—	8	1	no	—
300	C₂	C₂×C₂₉×#121	1	1	yes	characteristic
301	C₂	C₂₉×#576	1	1	yes	characteristic
302	C₂	C₂₉×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₂₉×#131	C₂	semidirect
3. C₂₉×#537	C₂	semidirect
4. C₂₉×#535	C₂	semidirect
5. C₂₉×#469	C₂	semidirect
6. C₂₉×#468	C₂	semidirect
7. C₂₉×#466	C₂	semidirect
8. C₂₉×#465	C₂	semidirect
9. C₂₉×#428	C₂	semidirect
10. C₂₉×#416	C₂	semidirect
11. C₂×C₂₉×#102	C₂	semidirect
20. C₂×M₁₆×C₂₉	C₂²	semidirect
22. C₂²×C₈×C₂₉	C₂²	semidirect
26. C₂₉×C₂²⋊Q₈	C₂²	semidirect
31. C₂₉×#121	C₂²	semidirect
35. C₂₉×#107	C₂²	semidirect
36. C₂₉×#106	C₂²	semidirect
37. C₂₉×#103	C₂²	semidirect
38. C₂₉×#103	C₂²	semidirect
41. C₂₉×#102	C₂²	semidirect
42. C₂₉×#102	C₂²	semidirect
68. M₁₆×C₂₉	D₄	semidirect
122. #3062	C₂₉	direct (N × H)
145. C₂×AGL(1,ℤ/8)	C₂×C₂₉	semidirect
146. C₂×#131	C₂×C₂₉	semidirect
147. #537	C₂×C₂₉	semidirect
148. #535	C₂×C₂₉	semidirect
149. #469	C₂×C₂₉	semidirect
150. #468	C₂×C₂₉	semidirect
151. #466	C₂×C₂₉	semidirect
152. #465	C₂×C₂₉	semidirect
153. #428	C₂×C₂₉	semidirect
154. #416	C₂×C₂₉	semidirect
155. C₂×#102	C₂×C₂₉	semidirect
171. C₂×M₁₆	C₂²×C₂₉	semidirect
173. C₂²×C₈	C₂²×C₂₉	semidirect
177. C₂²⋊Q₈	C₂²×C₂₉	semidirect
182. #121	C₂²×C₂₉	semidirect
186. #107	C₂²×C₂₉	semidirect
187. #106	C₂²×C₂₉	semidirect
188. #103	C₂²×C₂₉	semidirect
189. #103	C₂²×C₂₉	semidirect
192. #102	C₂²×C₂₉	semidirect
193. #102	C₂²×C₂₉	semidirect
194. C₂₉	#3062	direct (N × H)
220. M₁₆	D₄×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×AGL(1,ℤ/8)	2
C₂×C₂₉×#131	2
C₂₉×#537	2
C₂₉×#535	2
C₂₉×#469	2
C₂₉×#468	2
C₂₉×#466	2
C₂₉×#465	2
C₂₉×#428	2
C₂₉×#416	2
C₂×C₂₉×#102	2
#3062	29

Subgroups — C29×#3062 #16816776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3062 #16816776