Subgroups — C₂₉×#3005 #16816719

Order 3712.

758 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₉×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₉×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#525	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#452	C₂	1	4	yes	maximal, complemented
6	C₂₉×#449	C₂	1	4	yes	maximal, complemented
7	C₂₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#418	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₉	—	2	4	no	—
11	C₂×#41×C₂₉	—	2	4	no	—
12	C₂×#41×C₂₉	—	2	2	no	—
13	C₂×#41×C₂₉	C₂²	1	4	yes	—
14	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₂₉	—	2	1	no	—
17	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₂₉	—	2	1	no	—
19	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×#137	—	2	4	no	—
21	C₂₉×AGL(1,ℤ/8)	—	2	4	no	—
22	C₂×Q₁₆×C₂₉	C₂²	1	2	yes	complemented
23	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	complemented
24	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	complemented
25	C₂×D₈×C₂₉	C₂²	1	2	yes	complemented
26	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
27	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
28	C₂₉×#123	—	2	1	no	—
29	C₂₉×C₂²⋊Q₈	C₂²	1	2	yes	complemented
30	C₂₉×#121	C₂²	1	2	yes	complemented
31	C₂₉×#120	—	2	1	no	—
32	C₄×D₄×C₂₉	—	2	2	no	—
33	C₂₉×#117	C₂²	1	1	yes	characteristic
34	C₂₉×#103	C₂²	1	4	yes	complemented
35	C₂₉×#102	C₂²	1	4	yes	complemented
36	C₂₉×C₂²⋊C₈	C₂²	1	4	yes	complemented
37	C₂⁴×C₂₉	—	2	1	no	—
38	C₂⁴×C₂₉	D₄	1	1	yes	characteristic
39	#41×C₂₉	—	2	16	no	—
40	#41×C₂₉	—	2	8	no	—
41	#41×C₂₉	—	2	8	no	—
42	#41×C₂₉	—	2	2	no	—
43	#41×C₂₉	—	2	2	no	—
44	C₂×Q₈×C₂₉	—	2	2	no	—
45	C₂×Q₈×C₂₉	—	2	1	no	—
46	C₂×Q₈×C₂₉	D₄	1	4	yes	complemented
47	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
49	C₂×D₄×C₂₉	—	2	8	no	—
50	C₂×D₄×C₂₉	—	2	4	no	—
51	C₂×D₄×C₂₉	—	2	2	no	—
52	C₂×D₄×C₂₉	—	2	1	no	—
53	C₂×D₄×C₂₉	—	2	1	no	—
54	C₂×D₄×C₂₉	—	2	2	no	—
55	C₂×D₄×C₂₉	D₄	1	4	yes	complemented
56	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₂₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₉	—	2	8	no	—
59	C₂²×C₄×C₂₉	—	2	1	no	—
60	C₂²×C₄×C₂₉	—	2	4	no	—
61	C₂²×C₄×C₂₉	—	2	2	no	—
62	C₂²×C₄×C₂₉	—	2	2	no	—
63	C₂²×C₄×C₂₉	—	2	1	no	—
64	C₂²×C₄×C₂₉	D₄	1	2	yes	—
65	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
68	Q₁₆×C₂₉	—	2	4	no	—
69	SD₁₆×C₂₉	—	2	4	no	—
70	SD₁₆×C₂₉	—	2	4	no	—
71	D₈×C₂₉	—	2	4	no	—
72	M₁₆×C₂₉	—	2	2	no	—
73	M₁₆×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	C₂³	1	2	yes	—
75	C₂×C₈×C₂₉	C₂³	1	2	yes	—
76	C₄⋊C₄×C₂₉	—	2	2	no	—
77	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₂₉	—	2	1	no	—
79	C₂²⋊C₄×C₂₉	—	2	2	no	—
80	C₂²⋊C₄×C₂₉	—	2	2	no	—
81	C₄²×C₂₉	—	2	1	no	—
82	C₂³×C₂₉	—	2	2	no	—
83	C₂³×C₂₉	—	2	4	no	—
84	C₂³×C₂₉	—	2	2	no	—
85	C₂³×C₂₉	—	2	1	no	—
86	C₂³×C₂₉	—	4	2	no	—
87	C₂³×C₂₉	—	4	4	no	—
88	C₂³×C₂₉	C₂×D₄	1	2	yes	—
89	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₂₉	—	2	4	no	—
91	Q₈×C₂₉	—	2	2	no	—
92	Q₈×C₂₉	—	2	1	no	—
93	Q₈×C₂₉	—	2	1	no	—
94	Q₈×C₂₉	C₂×D₄	1	4	yes	—
95	D₄×C₂₉	—	2	16	no	—
96	D₄×C₂₉	—	2	4	no	—
97	D₄×C₂₉	—	2	2	no	—
98	D₄×C₂₉	—	2	1	no	—
99	D₄×C₂₉	—	2	1	no	—
100	D₄×C₂₉	—	4	4	no	—
101	D₄×C₂₉	—	4	2	no	—
102	D₄×C₂₉	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₂₉	—	2	16	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	—	2	4	no	—
106	C₂×C₄×C₂₉	—	2	2	no	—
107	C₂×C₄×C₂₉	—	2	4	no	—
108	C₂×C₄×C₂₉	—	2	2	no	—
109	C₂×C₄×C₂₉	—	2	2	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	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₂₉	—	2	2	no	—
119	C₈×C₂₉	—	2	2	no	—
120	#3005	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₂₉	—	2	2	no	—
122	C₂²×C₂₉	—	2	2	no	—
123	C₂²×C₂₉	—	2	4	no	—
124	C₂²×C₂₉	—	2	2	no	—
125	C₂²×C₂₉	—	2	2	no	—
126	C₂²×C₂₉	—	4	4	no	—
127	C₂²×C₂₉	—	4	8	no	—
128	C₂²×C₂₉	—	4	4	no	—
129	C₂²×C₂₉	—	4	1	no	—
130	C₂²×C₂₉	—	4	2	no	—
131	C₂²×C₂₉	#120	1	2	yes	—
132	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₂₉	—	2	2	no	—
134	C₄×C₂₉	—	2	4	no	—
135	C₄×C₂₉	—	4	1	no	—
136	C₄×C₂₉	—	4	2	no	—
137	C₄×C₂₉	#120	1	2	yes	—
138	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
140	C₂²×#41	C₂×C₂₉	1	1	yes	characteristic, complemented
141	C₂×#137	C₂×C₂₉	1	1	yes	characteristic, complemented
142	C₂×AGL(1,ℤ/8)	C₂×C₂₉	1	1	yes	characteristic, complemented
143	#525	C₂×C₂₉	1	1	yes	characteristic, complemented
144	#452	C₂×C₂₉	1	4	yes	complemented
145	#449	C₂×C₂₉	1	4	yes	complemented
146	#419	C₂×C₂₉	1	1	yes	characteristic, complemented
147	#418	C₂×C₂₉	1	1	yes	characteristic, complemented
148	#407	C₂×C₂₉	1	1	yes	characteristic, complemented
149	C₂×C₂₉	—	2	2	no	—
150	C₂×C₂₉	—	4	2	no	—
151	C₂×C₂₉	—	4	4	no	—
152	C₂×C₂₉	—	8	1	no	—
153	C₂×C₂₉	C₂×#120	1	1	yes	characteristic
154	C₂×C₂₉	#575	1	2	yes	—
155	C₂×#41	—	2	4	no	—
156	C₂×#41	—	2	4	no	—
157	C₂×#41	—	2	2	no	—
158	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
159	C₂×#41	C₂²×C₂₉	1	4	yes	—
160	C₂²×Q₈	C₂²×C₂₉	1	1	yes	characteristic
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
163	C₂³×C₄	—	2	1	no	—
164	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
165	#137	—	2	4	no	—
166	AGL(1,ℤ/8)	—	2	4	no	—
167	C₂×Q₁₆	C₂²×C₂₉	1	2	yes	complemented
168	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	complemented
169	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	complemented
170	C₂×D₈	C₂²×C₂₉	1	2	yes	complemented
171	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
172	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
173	#123	—	2	1	no	—
174	C₂²⋊Q₈	C₂²×C₂₉	1	2	yes	complemented
175	#121	C₂²×C₂₉	1	2	yes	complemented
176	#120	—	2	1	no	—
177	C₄×D₄	—	2	2	no	—
178	#117	C₂²×C₂₉	1	1	yes	characteristic
179	#103	C₂²×C₂₉	1	4	yes	complemented
180	#102	C₂²×C₂₉	1	4	yes	complemented
181	C₂²⋊C₈	C₂²×C₂₉	1	4	yes	complemented
182	C₂₉	#3005	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	2	1	no	—
184	C₂⁴	D₄×C₂₉	1	1	yes	characteristic
185	#41	—	2	16	no	—
186	#41	—	2	8	no	—
187	#41	—	2	8	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	D₄×C₂₉	1	4	yes	complemented
193	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
194	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	D₄×C₂₉	1	4	yes	complemented
202	C₂×D₄	C₂³×C₂₉	1	1	yes	characteristic
203	C₂×D₄	C₂³×C₂₉	1	2	yes	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	D₄×C₂₉	1	2	yes	—
211	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
213	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
214	Q₁₆	—	2	4	no	—
215	SD₁₆	—	2	4	no	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	4	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	C₂×C₈	C₂³×C₂₉	1	2	yes	—
221	C₂×C₈	C₂³×C₂₉	1	2	yes	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	4	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	C₂×D₄×C₂₉	1	2	yes	—
235	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	2	2	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	C₂×D₄×C₂₉	1	4	yes	—
241	D₄	—	2	16	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	1	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	2	no	—
248	D₄	C₂×D₄×C₂₉	1	4	yes	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
263	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	8	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂₉×#120	1	2	yes	—
277	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	2	no	—
282	C₄	C₂₉×#120	1	2	yes	—
283	C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
284	C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	4	no	—
288	C₂	—	8	1	no	—
289	C₂	C₂×C₂₉×#120	1	1	yes	characteristic
290	C₂	C₂₉×#575	1	2	yes	—

	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 ◁ C₂³×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#41×C₂₉	C₂	semidirect
2. C₂×C₂₉×#137	C₂	semidirect
3. C₂×C₂₉×AGL(1,ℤ/8)	C₂	semidirect
4. C₂₉×#525	C₂	semidirect
5. C₂₉×#452	C₂	semidirect
6. C₂₉×#449	C₂	semidirect
7. C₂₉×#419	C₂	semidirect
8. C₂₉×#418	C₂	semidirect
9. C₂₉×#407	C₂	semidirect
22. C₂×Q₁₆×C₂₉	C₂²	semidirect
23. C₂×SD₁₆×C₂₉	C₂²	semidirect
24. C₂×SD₁₆×C₂₉	C₂²	semidirect
25. C₂×D₈×C₂₉	C₂²	semidirect
29. C₂₉×C₂²⋊Q₈	C₂²	semidirect
30. C₂₉×#121	C₂²	semidirect
34. C₂₉×#103	C₂²	semidirect
35. C₂₉×#102	C₂²	semidirect
36. C₂₉×C₂²⋊C₈	C₂²	semidirect
46. C₂×Q₈×C₂₉	D₄	semidirect
55. C₂×D₄×C₂₉	D₄	semidirect
120. #3005	C₂₉	direct (N × H)
140. C₂²×#41	C₂×C₂₉	semidirect
141. C₂×#137	C₂×C₂₉	semidirect
142. C₂×AGL(1,ℤ/8)	C₂×C₂₉	semidirect
143. #525	C₂×C₂₉	semidirect
144. #452	C₂×C₂₉	semidirect
145. #449	C₂×C₂₉	semidirect
146. #419	C₂×C₂₉	semidirect
147. #418	C₂×C₂₉	semidirect
148. #407	C₂×C₂₉	semidirect
167. C₂×Q₁₆	C₂²×C₂₉	semidirect
168. C₂×SD₁₆	C₂²×C₂₉	semidirect
169. C₂×SD₁₆	C₂²×C₂₉	semidirect
170. C₂×D₈	C₂²×C₂₉	semidirect
174. C₂²⋊Q₈	C₂²×C₂₉	semidirect
175. #121	C₂²×C₂₉	semidirect
179. #103	C₂²×C₂₉	semidirect
180. #102	C₂²×C₂₉	semidirect
181. C₂²⋊C₈	C₂²×C₂₉	semidirect
182. C₂₉	#3005	direct (N × H)
192. C₂×Q₈	D₄×C₂₉	semidirect
201. C₂×D₄	D₄×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₉	2
C₂×C₂₉×#137	2
C₂×C₂₉×AGL(1,ℤ/8)	2
C₂₉×#525	2
C₂₉×#452	2
C₂₉×#449	2
C₂₉×#419	2
C₂₉×#418	2
C₂₉×#407	2
#3005	29

Subgroups — C29×#3005 #16816719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3005 #16816719