Subgroups — C₂₉×#9721 #20304621

Order 7424.

498 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1980	C₂	1	2	yes	maximal, complemented
3	C₄×C₂₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1502	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1451	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#532	C₂²	1	2	yes	—
7	C₂₉×#500	C₄	1	1	yes	characteristic, complemented
8	C₂₉×#493	C₄	1	1	yes	characteristic, complemented
9	C₂₉×#486	C₄	1	2	yes	complemented
10	C₂₉×#466	—	2	1	no	—
11	C₂₉×#466	—	2	1	no	—
12	C₄×D₈×C₂₉	—	2	1	no	—
13	C₄×D₈×C₂₉	—	2	1	no	—
14	C₂₉×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₉×#106	—	2	1	no	—
16	C₂×C₂₉×#104	—	2	4	no	—
17	C₄×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
20	C₂₉×#340	—	2	1	no	—
21	C₂₉×#327	—	2	4	no	—
22	C₂₉×#322	—	2	1	no	—
23	C₂₉×#322	—	2	1	no	—
24	C₂×#41×C₂₉	—	2	2	no	—
25	C₂×D₈×C₂₉	—	2	1	no	—
26	C₂×D₈×C₂₉	—	2	1	no	—
27	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
28	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
30	C₂₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₂₉×#127	C₂×C₄	1	2	yes	—
32	C₂₉×#124	—	2	2	no	—
33	C₂₉×#121	—	2	2	no	—
34	C₂₉×#121	—	2	2	no	—
35	C₄×D₄×C₂₉	—	2	2	no	—
36	C₄×D₄×C₂₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
38	C₂×C₄²×C₂₉	—	2	2	no	—
39	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
40	C₂₉×#107	—	2	1	no	—
41	C₂₉×#107	—	2	1	no	—
42	C₂₉×#106	—	2	4	no	—
43	C₂₉×C₄⋊C₈	—	2	1	no	—
44	C₂₉×#104	—	4	8	no	—
45	C₂₉×#102	—	2	2	no	—
46	C₂₉×#102	—	2	2	no	—
47	C₂₉×C₂²⋊C₈	D₄	1	2	yes	—
48	C₂₉×#97	—	2	1	no	—
49	C₄×C₈×C₂₉	—	2	1	no	—
50	C₄×C₈×C₂₉	—	2	1	no	—
51	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
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	C₂₉×#95	—	2	2	no	—
56	#41×C₂₉	—	4	4	no	—
57	#41×C₂₉	—	4	4	no	—
58	C₂×Q₈×C₂₉	—	2	2	no	—
59	C₂×D₄×C₂₉	—	2	2	no	—
60	C₂×D₄×C₂₉	—	2	2	no	—
61	C₂×D₄×C₂₉	—	2	2	no	—
62	C₂×D₄×C₂₉	—	4	2	no	—
63	C₂²×C₄×C₂₉	—	4	2	no	—
64	C₂²×C₄×C₂₉	—	4	2	no	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
68	D₈×C₂₉	—	4	2	no	—
69	D₈×C₂₉	—	4	2	no	—
70	M₁₆×C₂₉	—	2	4	no	—
71	C₂×C₈×C₂₉	—	2	2	no	—
72	C₂×C₈×C₂₉	—	2	2	no	—
73	C₂×C₈×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	—	4	1	no	—
75	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₂₉	—	2	4	no	—
80	C₄⋊C₄×C₂₉	—	2	2	no	—
81	C₄⋊C₄×C₂₉	—	2	2	no	—
82	C₄⋊C₄×C₂₉	—	4	2	no	—
83	C₂²⋊C₄×C₂₉	—	4	2	no	—
84	C₂²⋊C₄×C₂₉	—	4	2	no	—
85	C₄²×C₂₉	—	4	4	no	—
86	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₉	#41	1	1	yes	characteristic
89	C₄²×C₂₉	#41	1	1	yes	characteristic
90	#9721	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₉	—	4	2	no	—
92	C₂³×C₂₉	#127	1	1	yes	characteristic
93	Q₈×C₂₉	—	4	4	no	—
94	D₄×C₂₉	—	4	2	no	—
95	D₄×C₂₉	—	4	2	no	—
96	D₄×C₂₉	—	4	4	no	—
97	D₄×C₂₉	—	8	2	no	—
98	D₄×C₂₉	—	8	2	no	—
99	D₄×C₂₉	—	8	2	no	—
100	C₂×C₄×C₂₉	—	2	1	no	—
101	C₂×C₄×C₂₉	—	2	1	no	—
102	C₂×C₄×C₂₉	—	4	1	no	—
103	C₂×C₄×C₂₉	—	4	4	no	—
104	C₂×C₄×C₂₉	—	4	2	no	—
105	C₂×C₄×C₂₉	—	8	2	no	—
106	C₂×C₄×C₂₉	—	8	2	no	—
107	C₂×C₄×C₂₉	—	8	4	no	—
108	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₂₉	#124	1	1	yes	characteristic
114	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	4	yes	—
115	C₈×C₂₉	—	2	1	no	—
116	C₈×C₂₉	—	2	1	no	—
117	C₈×C₂₉	—	2	1	no	—
118	C₈×C₂₉	—	4	2	no	—
119	C₈×C₂₉	—	4	2	no	—
120	C₈×C₂₉	#104	1	2	yes	complemented
121	#3169	C₂×C₂₉	1	1	yes	characteristic
122	#1980	C₂×C₂₉	1	2	yes	complemented
123	C₄×#106	C₂×C₂₉	1	1	yes	characteristic, complemented
124	#1502	C₂×C₂₉	1	2	yes	complemented
125	#1451	C₂×C₂₉	1	1	yes	characteristic, complemented
126	C₂²×C₂₉	—	2	1	no	—
127	C₂²×C₂₉	—	2	1	no	—
128	C₂²×C₂₉	—	8	2	no	—
129	C₂²×C₂₉	—	8	2	no	—
130	C₂²×C₂₉	—	8	2	no	—
131	C₂²×C₂₉	#390	1	1	yes	characteristic
132	C₂²×C₂₉	#496	1	2	yes	—
133	C₄×C₂₉	—	2	1	no	—
134	C₄×C₂₉	—	2	1	no	—
135	C₄×C₂₉	—	2	1	no	—
136	C₄×C₂₉	—	8	4	no	—
137	C₄×C₂₉	—	8	2	no	—
138	C₄×C₂₉	C₂×#104	1	1	yes	characteristic
139	C₄×C₂₉	#421	1	1	yes	characteristic
140	C₄×C₂₉	#442	1	2	yes	—
141	C₄×C₂₉	#469	1	2	yes	—
142	#532	C₂²×C₂₉	1	2	yes	—
143	#500	C₄×C₂₉	1	1	yes	characteristic, complemented
144	#493	C₄×C₂₉	1	1	yes	characteristic, complemented
145	#486	C₄×C₂₉	1	2	yes	complemented
146	#466	—	2	1	no	—
147	#466	—	2	1	no	—
148	C₄×D₈	—	2	1	no	—
149	C₄×D₈	—	2	1	no	—
150	#431	C₂²×C₂₉	1	1	yes	characteristic
151	C₂×#106	—	2	1	no	—
152	C₂×#104	—	2	4	no	—
153	C₄×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
155	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
156	#340	—	2	1	no	—
157	#327	—	2	4	no	—
158	#322	—	2	1	no	—
159	#322	—	2	1	no	—
160	C₂×C₂₉	—	2	2	no	—
161	C₂×C₂₉	—	16	2	no	—
162	C₂×C₂₉	#1972	1	1	yes	characteristic
163	C₂×C₂₉	#1976	1	1	yes	characteristic
164	C₂×C₂₉	#1980	1	1	yes	characteristic
165	C₂×#41	—	2	2	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
169	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
171	C₄⋊Q₈	C₂×C₄×C₂₉	1	2	yes	—
172	#127	C₂×C₄×C₂₉	1	2	yes	—
173	#124	—	2	2	no	—
174	#121	—	2	2	no	—
175	#121	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
181	#107	—	2	1	no	—
182	#107	—	2	1	no	—
183	#106	—	2	4	no	—
184	C₄⋊C₈	—	2	1	no	—
185	#104	—	4	8	no	—
186	#102	—	2	2	no	—
187	#102	—	2	2	no	—
188	C₂²⋊C₈	D₄×C₂₉	1	2	yes	—
189	#97	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
193	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
194	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
195	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
196	#95	—	2	2	no	—
197	C₂₉	#9721	1	1	yes	characteristic, direct product factor, Sylow
198	#41	—	4	4	no	—
199	#41	—	4	4	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
210	D₈	—	4	2	no	—
211	D₈	—	4	2	no	—
212	M₁₆	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₄²	—	4	4	no	—
228	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
229	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
230	C₄²	#41×C₂₉	1	1	yes	characteristic
231	C₄²	#41×C₂₉	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	C₂₉×#127	1	1	yes	characteristic
234	Q₈	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
251	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
252	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
253	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
254	C₂×C₄	C₂₉×#124	1	1	yes	characteristic
255	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	4	yes	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	C₂₉×#104	1	2	yes	complemented
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₂₉×#390	1	1	yes	characteristic
268	C₂²	C₂₉×#496	1	2	yes	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×C₂₉×#104	1	1	yes	characteristic
275	C₄	C₂₉×#421	1	1	yes	characteristic
276	C₄	C₂₉×#442	1	2	yes	—
277	C₄	C₂₉×#469	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	16	2	no	—
280	C₂	C₂₉×#1972	1	1	yes	characteristic
281	C₂	C₂₉×#1976	1	1	yes	characteristic
282	C₂	C₂₉×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	#121
commutator	C₄²	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	#390
prime radical	C₂₉×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₉×#1980	C₂	semidirect
3. C₄×C₂₉×#106	C₂	semidirect
4. C₂₉×#1502	C₂	semidirect
5. C₂₉×#1451	C₂	semidirect
7. C₂₉×#500	C₄	semidirect
8. C₂₉×#493	C₄	semidirect
9. C₂₉×#486	C₄	semidirect
90. #9721	C₂₉	direct (N × H)
120. C₈×C₂₉	#104	semidirect
122. #1980	C₂×C₂₉	semidirect
123. C₄×#106	C₂×C₂₉	semidirect
124. #1502	C₂×C₂₉	semidirect
125. #1451	C₂×C₂₉	semidirect
143. #500	C₄×C₂₉	semidirect
144. #493	C₄×C₂₉	semidirect
145. #486	C₄×C₂₉	semidirect
197. C₂₉	#9721	direct (N × H)
261. C₈	C₂₉×#104	semidirect

Maximal subgroups

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

subgroup	index
C₂₉×#3169	2
C₂₉×#1980	2
C₄×C₂₉×#106	2
C₂₉×#1502	2
C₂₉×#1451	2
#9721	29

Subgroups — C29×#9721 #20304621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#9721 #20304621