Subgroups — C₂₉×#33797 #20328697

Order 7424.

460 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#3160	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#2237	C₂	1	4	yes	maximal
8	C₂₉×#2229	C₂	1	4	yes	maximal, complemented
9	C₂₉×#2162	C₂	1	1	yes	maximal, characteristic
10	C₂₉×#533	—	2	1	no	—
11	C₂₉×#533	—	2	1	no	—
12	C₂₉×#500	C₂²	1	1	yes	characteristic
13	C₂₉×#500	C₂²	1	1	yes	characteristic
14	C₂₉×#500	C₂²	1	1	yes	characteristic
15	C₂₉×#499	—	2	1	no	—
16	C₂₉×#499	C₂²	1	2	yes	—
17	C₂₉×#498	—	2	1	no	—
18	C₂₉×#494	C₂²	1	1	yes	characteristic
19	C₂₉×#487	—	2	1	no	—
20	C₂₉×#467	—	2	1	no	—
21	C₂₉×#467	C₂²	1	2	yes	—
22	C₄×Q₁₆×C₂₉	—	2	1	no	—
23	C₄×Q₁₆×C₂₉	C₂²	1	2	yes	—
24	C₂₉×#427	—	2	1	no	—
25	C₂₉×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×#367	C₂²	1	4	yes	—
28	C₂₉×#366	C₂²	1	4	yes	—
29	C₂₉×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₂₉×#358	C₂²	1	4	yes	—
31	C₂₉×#358	C₂²	1	4	yes	—
32	C₂₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₂₉	—	2	1	no	—
36	C₂×Q₁₆×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
39	C₂₉×C₄⋊Q₈	—	2	1	no	—
40	C₂₉×C₄⋊Q₈	—	2	1	no	—
41	C₂₉×C₄⋊Q₈	—	2	1	no	—
42	C₂₉×C₄⋊Q₈	—	2	1	no	—
43	C₂₉×#125	—	2	2	no	—
44	C₂₉×#125	—	4	1	no	—
45	C₂₉×C₂²⋊Q₈	—	2	2	no	—
46	C₂₉×C₂²⋊Q₈	—	4	1	no	—
47	C₂₉×C₂²⋊Q₈	—	4	1	no	—
48	C₂₉×C₂²⋊Q₈	—	4	1	no	—
49	C₄×Q₈×C₂₉	—	2	2	no	—
50	C₄×Q₈×C₂₉	—	4	1	no	—
51	C₄×Q₈×C₂₉	—	4	1	no	—
52	C₄×Q₈×C₂₉	—	4	1	no	—
53	C₂₉×#117	—	2	2	no	—
54	C₂₉×#117	—	4	1	no	—
55	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
56	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
57	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
58	C₂₉×#107	—	2	1	no	—
59	C₂₉×#107	—	2	1	no	—
60	C₂₉×#107	—	2	1	no	—
61	C₂₉×#107	C₂³	1	2	yes	—
62	C₂₉×#107	C₂³	1	2	yes	—
63	C₂₉×#107	C₂³	1	2	yes	—
64	C₂₉×#106	—	2	2	no	—
65	C₂₉×#106	—	2	2	no	—
66	C₂₉×#103	—	2	2	no	—
67	C₂₉×#103	—	2	2	no	—
68	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₂₉	—	2	2	no	—
73	C₂×Q₈×C₂₉	—	4	1	no	—
74	C₂×Q₈×C₂₉	—	4	1	no	—
75	C₂×Q₈×C₂₉	—	4	1	no	—
76	C₂²×C₄×C₂₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₂₉	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₂₉	—	2	4	no	—
80	Q₁₆×C₂₉	—	4	2	no	—
81	C₂×C₈×C₂₉	—	2	2	no	—
82	C₂×C₈×C₂₉	—	2	2	no	—
83	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₂₉	—	2	2	no	—
88	C₄⋊C₄×C₂₉	—	2	2	no	—
89	C₄⋊C₄×C₂₉	—	2	2	no	—
90	C₄⋊C₄×C₂₉	—	4	1	no	—
91	C₄⋊C₄×C₂₉	—	4	1	no	—
92	C₄⋊C₄×C₂₉	—	4	2	no	—
93	C₄⋊C₄×C₂₉	—	4	2	no	—
94	C₄⋊C₄×C₂₉	—	4	2	no	—
95	C₄⋊C₄×C₂₉	—	4	2	no	—
96	C₄⋊C₄×C₂₉	—	4	1	no	—
97	C₄⋊C₄×C₂₉	—	4	1	no	—
98	C₄⋊C₄×C₂₉	—	4	1	no	—
99	C₂²⋊C₄×C₂₉	—	4	2	no	—
100	C₂²⋊C₄×C₂₉	—	4	2	no	—
101	C₄²×C₂₉	—	4	2	no	—
102	C₄²×C₂₉	—	4	2	no	—
103	C₄²×C₂₉	D₈	1	1	yes	characteristic
104	C₄²×C₂₉	D₈	1	1	yes	characteristic
105	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
107	C₁₆×C₂₉	—	2	2	no	—
108	C₁₆×C₂₉	—	2	2	no	—
109	#33797	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₉	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₂₉	—	4	2	no	—
112	Q₈×C₂₉	—	4	2	no	—
113	Q₈×C₂₉	—	8	2	no	—
114	Q₈×C₂₉	—	8	2	no	—
115	Q₈×C₂₉	—	8	2	no	—
116	C₂×C₄×C₂₉	—	2	1	no	—
117	C₂×C₄×C₂₉	—	2	1	no	—
118	C₂×C₄×C₂₉	—	2	2	no	—
119	C₂×C₄×C₂₉	—	2	2	no	—
120	C₂×C₄×C₂₉	—	4	2	no	—
121	C₂×C₄×C₂₉	—	4	2	no	—
122	C₂×C₄×C₂₉	—	4	2	no	—
123	C₂×C₄×C₂₉	—	4	2	no	—
124	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₂₉	—	2	1	no	—
131	C₈×C₂₉	—	2	1	no	—
132	C₈×C₂₉	#137	1	2	yes	—
133	C₈×C₂₉	2₋¹⁺⁴	1	2	yes	—
134	#3170	C₂×C₂₉	1	1	yes	characteristic
135	#3160	C₂×C₂₉	1	1	yes	characteristic
136	#2254	C₂×C₂₉	1	1	yes	characteristic, complemented
137	#2252	C₂×C₂₉	1	1	yes	characteristic, complemented
138	#2241	C₂×C₂₉	1	1	yes	characteristic, complemented
139	#2241	C₂×C₂₉	1	1	yes	characteristic, complemented
140	#2237	C₂×C₂₉	1	4	yes	—
141	#2229	C₂×C₂₉	1	4	yes	complemented
142	#2162	C₂×C₂₉	1	1	yes	characteristic
143	C₂²×C₂₉	—	2	1	no	—
144	C₂²×C₂₉	—	2	1	no	—
145	C₂²×C₂₉	—	2	1	no	—
146	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₂₉	—	2	1	no	—
148	C₄×C₂₉	—	2	1	no	—
149	C₄×C₂₉	—	2	1	no	—
150	C₄×C₂₉	—	8	2	no	—
151	C₄×C₂₉	—	8	2	no	—
152	C₄×C₂₉	—	8	2	no	—
153	C₄×C₂₉	—	8	2	no	—
154	C₄×C₂₉	#508	1	2	yes	—
155	C₄×C₂₉	#510	1	2	yes	—
156	C₄×C₂₉	#536	1	1	yes	characteristic
157	C₄×C₂₉	C₂×#137	1	1	yes	characteristic
158	#533	—	2	1	no	—
159	#533	—	2	1	no	—
160	#500	C₂²×C₂₉	1	1	yes	characteristic
161	#500	C₂²×C₂₉	1	1	yes	characteristic
162	#500	C₂²×C₂₉	1	1	yes	characteristic
163	#499	—	2	1	no	—
164	#499	C₂²×C₂₉	1	2	yes	—
165	#498	—	2	1	no	—
166	#494	C₂²×C₂₉	1	1	yes	characteristic
167	#487	—	2	1	no	—
168	#467	—	2	1	no	—
169	#467	C₂²×C₂₉	1	2	yes	—
170	C₄×Q₁₆	—	2	1	no	—
171	C₄×Q₁₆	C₂²×C₂₉	1	2	yes	—
172	#427	—	2	1	no	—
173	#427	C₂²×C₂₉	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
175	#367	C₂²×C₂₉	1	4	yes	—
176	#366	C₂²×C₂₉	1	4	yes	—
177	C₄⋊C₁₆	C₂²×C₂₉	1	2	yes	—
178	#358	C₂²×C₂₉	1	4	yes	—
179	#358	C₂²×C₂₉	1	4	yes	—
180	C₂²⋊C₁₆	C₂²×C₂₉	1	2	yes	—
181	C₄×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
182	C₄×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
183	C₂×C₂₉	—	4	1	no	—
184	C₂×C₂₉	#3226	1	1	yes	characteristic
185	C₂×C₂₉	C₂×#508	1	1	yes	characteristic
186	C₂×C₂₉	C₂×#510	1	1	yes	characteristic
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	C₂³×C₂₉	1	2	yes	—
189	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	2	1	no	—
192	C₄⋊Q₈	—	2	1	no	—
193	C₄⋊Q₈	—	2	1	no	—
194	C₄⋊Q₈	—	2	1	no	—
195	#125	—	2	2	no	—
196	#125	—	4	1	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	4	1	no	—
199	C₂²⋊Q₈	—	4	1	no	—
200	C₂²⋊Q₈	—	4	1	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	#117	—	2	2	no	—
206	#117	—	4	1	no	—
207	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
209	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	C₂³×C₂₉	1	2	yes	—
214	#107	C₂³×C₂₉	1	2	yes	—
215	#107	C₂³×C₂₉	1	2	yes	—
216	#106	—	2	2	no	—
217	#106	—	2	2	no	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
221	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
224	C₂₉	#33797	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂²×C₄	D₈×C₂₉	1	1	yes	characteristic
230	C₂²×C₄	D₈×C₂₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
232	Q₁₆	—	2	4	no	—
233	Q₁₆	—	4	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
239	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₄⋊C₄	—	4	1	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	D₈×C₂₉	1	1	yes	characteristic
257	C₄²	D₈×C₂₉	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₂³	C₂×D₈×C₂₉	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	2	no	—
266	Q₈	—	8	2	no	—
267	Q₈	—	8	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	C₂₉×#137	1	2	yes	—
285	C₈	C₂₉×2₋¹⁺⁴	1	2	yes	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	C₂₉×#508	1	2	yes	—
298	C₄	C₂₉×#510	1	2	yes	—
299	C₄	C₂₉×#536	1	1	yes	characteristic
300	C₄	C₂×C₂₉×#137	1	1	yes	characteristic
301	C₂	—	4	1	no	—
302	C₂	C₂₉×#3226	1	1	yes	characteristic
303	C₂	C₂×C₂₉×#508	1	1	yes	characteristic
304	C₂	C₂×C₂₉×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
prime radical	C₂³×C₂₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₉×#2254	C₂	semidirect
4. C₂₉×#2252	C₂	semidirect
5. C₂₉×#2241	C₂	semidirect
6. C₂₉×#2241	C₂	semidirect
8. C₂₉×#2229	C₂	semidirect
109. #33797	C₂₉	direct (N × H)
136. #2254	C₂×C₂₉	semidirect
137. #2252	C₂×C₂₉	semidirect
138. #2241	C₂×C₂₉	semidirect
139. #2241	C₂×C₂₉	semidirect
141. #2229	C₂×C₂₉	semidirect
224. C₂₉	#33797	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×#3170	2
C₂₉×#3160	2
C₂₉×#2254	2
C₂₉×#2252	2
C₂₉×#2241	2
C₂₉×#2237	2
C₂₉×#2229	2
C₂₉×#2162	2
#33797	29

Subgroups — C29×#33797 #20328697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33797 #20328697