Subgroups — C₂₉×#11659 #20306559

Order 7424.

364 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1332	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1331	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#135	C₄	1	1	yes	characteristic
7	C₄×#41×C₂₉	—	2	1	no	—
8	C₄×Q₁₆×C₂₉	C₄	1	1	yes	characteristic
9	C₄×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₄×SD₁₆×C₂₉	—	2	1	no	—
11	C₄×SD₁₆×C₂₉	—	2	1	no	—
12	C₄×D₈×C₂₉	C₄	1	1	yes	characteristic
13	C₄×D₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×#432	—	2	1	no	—
15	C₂₉×#427	C₄	1	1	yes	characteristic
16	C₂₉×#416	—	2	1	no	—
17	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
18	C₂₉×C₄⋊C₁₆	—	2	1	no	—
19	C₂₉×C₂²⋊C₁₆	—	2	1	no	—
20	C₄×C₁₆×C₂₉	C₂²	1	2	yes	—
21	C₂₉×#335	C₂²	1	2	yes	—
22	C₂₉×#334	—	2	1	no	—
23	C₂×#41×C₂₉	—	2	1	no	—
24	C₂₉×#135	—	2	2	no	—
25	C₂₉×#135	—	2	2	no	—
26	C₂×Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂×SD₁₆×C₂₉	—	2	1	no	—
28	C₂×D₈×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₂₉	—	2	1	no	—
32	C₄×Q₈×C₂₉	—	2	1	no	—
33	C₄×D₄×C₂₉	—	2	1	no	—
34	C₄×D₄×C₂₉	—	2	1	no	—
35	C₄×D₄×C₂₉	—	4	1	no	—
36	C₄×D₄×C₂₉	—	4	1	no	—
37	C₂₉×#117	—	2	1	no	—
38	C₂₉×#117	—	4	1	no	—
39	C₂×C₄²×C₂₉	—	4	1	no	—
40	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₂₉	—	2	2	no	—
42	C₂₉×#107	C₂×C₄	1	1	yes	characteristic
43	C₂₉×#107	C₂×C₄	1	1	yes	characteristic
44	C₂₉×#106	—	2	1	no	—
45	C₂₉×C₄⋊C₈	—	2	2	no	—
46	C₂₉×#103	—	2	1	no	—
47	C₂₉×#103	—	2	1	no	—
48	C₂₉×#102	—	2	1	no	—
49	C₂₉×#102	—	2	1	no	—
50	C₂₉×C₂²⋊C₈	—	4	1	no	—
51	C₂₉×#97	—	4	1	no	—
52	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
56	#41×C₂₉	—	2	4	no	—
57	#41×C₂₉	—	4	2	no	—
58	C₂×Q₈×C₂₉	—	2	1	no	—
59	C₂×D₄×C₂₉	—	2	1	no	—
60	C₂×D₄×C₂₉	—	4	1	no	—
61	C₂²×C₄×C₂₉	—	4	1	no	—
62	C₂²×C₄×C₂₉	—	4	1	no	—
63	C₂²×C₄×C₂₉	—	4	1	no	—
64	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
67	Q₁₆×C₂₉	—	2	1	no	—
68	Q₁₆×C₂₉	M₁₆	1	2	yes	—
69	SD₁₆×C₂₉	—	2	4	no	—
70	D₈×C₂₉	—	2	1	no	—
71	D₈×C₂₉	M₁₆	1	2	yes	—
72	C₂×C₈×C₂₉	—	2	1	no	—
73	C₂×C₈×C₂₉	—	2	1	no	—
74	C₂×C₈×C₂₉	—	4	1	no	—
75	C₂×C₈×C₂₉	—	4	2	no	—
76	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₂₉	—	2	1	no	—
81	C₄⋊C₄×C₂₉	—	2	1	no	—
82	C₄⋊C₄×C₂₉	—	4	1	no	—
83	C₄⋊C₄×C₂₉	—	4	1	no	—
84	C₂²⋊C₄×C₂₉	—	4	1	no	—
85	C₂²⋊C₄×C₂₉	—	4	1	no	—
86	C₂²⋊C₄×C₂₉	—	4	1	no	—
87	C₄²×C₂₉	—	4	1	no	—
88	C₄²×C₂₉	—	4	1	no	—
89	C₄²×C₂₉	—	4	1	no	—
90	C₄²×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₂₉	D₈	1	1	yes	characteristic
92	C₄²×C₂₉	SD₁₆	1	1	yes	characteristic
93	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
94	C₁₆×C₂₉	—	4	2	no	—
95	#11659	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₉	—	4	1	no	—
97	C₂³×C₂₉	#102	1	1	yes	characteristic
98	Q₈×C₂₉	—	2	2	no	—
99	Q₈×C₂₉	—	4	1	no	—
100	D₄×C₂₉	—	2	2	no	—
101	D₄×C₂₉	—	4	4	no	—
102	D₄×C₂₉	—	4	1	no	—
103	C₂×C₄×C₂₉	—	2	1	no	—
104	C₂×C₄×C₂₉	—	2	1	no	—
105	C₂×C₄×C₂₉	—	2	2	no	—
106	C₂×C₄×C₂₉	—	4	1	no	—
107	C₂×C₄×C₂₉	—	4	4	no	—
108	C₂×C₄×C₂₉	—	4	1	no	—
109	C₂×C₄×C₂₉	—	4	1	no	—
110	C₂×C₄×C₂₉	—	4	1	no	—
111	C₂×C₄×C₂₉	—	8	2	no	—
112	C₂×C₄×C₂₉	—	8	2	no	—
113	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
114	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
115	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
116	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₈×C₂₉	—	2	1	no	—
120	C₈×C₂₉	—	2	1	no	—
121	C₈×C₂₉	—	2	1	no	—
122	C₈×C₂₉	—	8	2	no	—
123	C₈×C₂₉	C₂×M₁₆	1	1	yes	characteristic
124	C₈×C₂₉	C₂×M₁₆	1	1	yes	characteristic
125	C₄×#135	C₂×C₂₉	1	1	yes	characteristic
126	#2162	C₂×C₂₉	1	1	yes	characteristic, complemented
127	#1568	C₂×C₂₉	1	1	yes	characteristic, complemented
128	#1332	C₂×C₂₉	1	2	yes	complemented
129	#1331	C₂×C₂₉	1	2	yes	complemented
130	C₂²×C₂₉	—	2	1	no	—
131	C₂²×C₂₉	—	2	1	no	—
132	C₂²×C₂₉	—	2	1	no	—
133	C₂²×C₂₉	—	4	2	no	—
134	C₂²×C₂₉	—	8	1	no	—
135	C₂²×C₂₉	—	8	1	no	—
136	C₂²×C₂₉	C₂×#102	1	1	yes	characteristic
137	C₄×C₂₉	—	2	1	no	—
138	C₄×C₂₉	—	2	1	no	—
139	C₄×C₂₉	—	2	1	no	—
140	C₄×C₂₉	—	2	1	no	—
141	C₄×C₂₉	—	4	2	no	—
142	C₄×C₂₉	—	8	1	no	—
143	C₄×C₂₉	—	8	1	no	—
144	C₄×C₂₉	#407	1	1	yes	characteristic
145	C₄×C₂₉	#421	1	1	yes	characteristic
146	C₄×C₂₉	#508	1	2	yes	—
147	C₂×#135	C₄×C₂₉	1	1	yes	characteristic
148	C₄×#41	—	2	1	no	—
149	C₄×Q₁₆	C₄×C₂₉	1	1	yes	characteristic
150	C₄×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
151	C₄×SD₁₆	—	2	1	no	—
152	C₄×SD₁₆	—	2	1	no	—
153	C₄×D₈	C₄×C₂₉	1	1	yes	characteristic
154	C₄×D₈	C₂²×C₂₉	1	1	yes	characteristic
155	#432	—	2	1	no	—
156	#427	C₄×C₂₉	1	1	yes	characteristic
157	#416	—	2	1	no	—
158	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
159	C₄⋊C₁₆	—	2	1	no	—
160	C₂²⋊C₁₆	—	2	1	no	—
161	C₄×C₁₆	C₂²×C₂₉	1	2	yes	—
162	#335	C₂²×C₂₉	1	2	yes	—
163	#334	—	2	1	no	—
164	C₂×C₂₉	—	4	1	no	—
165	C₂×C₂₉	—	8	2	no	—
166	C₂×C₂₉	#1486	1	1	yes	characteristic
167	C₂×C₂₉	#2138	1	1	yes	characteristic
168	C₂×C₂₉	#2143	1	1	yes	characteristic
169	C₂×#41	—	2	1	no	—
170	#135	—	2	2	no	—
171	#135	—	2	2	no	—
172	C₂×Q₁₆	C₂×C₄×C₂₉	1	1	yes	characteristic
173	C₂×SD₁₆	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₂₉	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
176	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
177	C₄×Q₈	—	2	1	no	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×D₄	—	2	1	no	—
180	C₄×D₄	—	2	1	no	—
181	C₄×D₄	—	4	1	no	—
182	C₄×D₄	—	4	1	no	—
183	#117	—	2	1	no	—
184	#117	—	4	1	no	—
185	C₂×C₄²	—	4	1	no	—
186	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
187	C₂×C₁₆	—	2	2	no	—
188	#107	C₂×C₄×C₂₉	1	1	yes	characteristic
189	#107	C₂×C₄×C₂₉	1	1	yes	characteristic
190	#106	—	2	1	no	—
191	C₄⋊C₈	—	2	2	no	—
192	#103	—	2	1	no	—
193	#103	—	2	1	no	—
194	#102	—	2	1	no	—
195	#102	—	2	1	no	—
196	C₂²⋊C₈	—	4	1	no	—
197	#97	—	4	1	no	—
198	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
199	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
200	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
202	C₂₉	#11659	1	1	yes	characteristic, direct product factor, Sylow
203	#41	—	2	4	no	—
204	#41	—	4	2	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
212	C₂²×C₄	D₈×C₂₉	1	1	yes	characteristic
213	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
214	Q₁₆	—	2	1	no	—
215	Q₁₆	M₁₆×C₂₉	1	2	yes	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	1	no	—
218	D₈	M₁₆×C₂₉	1	2	yes	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
225	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₂²⋊C₄	—	4	1	no	—
232	C₂²⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
238	C₄²	D₈×C₂₉	1	1	yes	characteristic
239	C₄²	SD₁₆×C₂₉	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
241	C₁₆	—	4	2	no	—
242	C₂³	—	4	1	no	—
243	C₂³	C₂₉×#102	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
260	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
261	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
262	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
263	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
264	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	8	2	no	—
269	C₈	C₂×M₁₆×C₂₉	1	1	yes	characteristic
270	C₈	C₂×M₁₆×C₂₉	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	8	1	no	—
276	C₂²	—	8	1	no	—
277	C₂²	C₂×C₂₉×#102	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	2	no	—
283	C₄	—	8	1	no	—
284	C₄	—	8	1	no	—
285	C₄	C₂₉×#407	1	1	yes	characteristic
286	C₄	C₂₉×#421	1	1	yes	characteristic
287	C₄	C₂₉×#508	1	2	yes	—
288	C₂	—	4	1	no	—
289	C₂	—	8	2	no	—
290	C₂	C₂₉×#1486	1	1	yes	characteristic
291	C₂	C₂₉×#2138	1	1	yes	characteristic
292	C₂	C₂₉×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₉
Frattini, cosocle	C₄×C₈	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×#102
prime radical	C₂×C₂₉×#135	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
2. C₂₉×#2162	C₂	semidirect
3. C₂₉×#1568	C₂	semidirect
4. C₂₉×#1332	C₂	semidirect
5. C₂₉×#1331	C₂	semidirect
95. #11659	C₂₉	direct (N × H)
126. #2162	C₂×C₂₉	semidirect
127. #1568	C₂×C₂₉	semidirect
128. #1332	C₂×C₂₉	semidirect
129. #1331	C₂×C₂₉	semidirect
202. C₂₉	#11659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₉×#135	2
C₂₉×#2162	2
C₂₉×#1568	2
C₂₉×#1332	2
C₂₉×#1331	2
#11659	29

Subgroups — C29×#11659 #20306559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#11659 #20306559