Subgroups — C₂₉×#33484 #20328384

Order 7424.

552 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	C₈×#41×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2183	C₂	1	2	yes	maximal, complemented
3	C₂₉×#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
5	C₂₉×#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#2163	C₂	1	2	yes	maximal, complemented
7	C₄×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#346	C₂	1	2	yes	maximal, complemented
9	C₄×#41×C₂₉	C₄	1	1	yes	characteristic
10	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₂₉	C₂²	1	4	yes	—
12	C₈×Q₈×C₂₉	C₄	1	1	yes	characteristic
13	C₈×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₈×D₄×C₂₉	C₄	1	2	yes	—
15	C₈×D₄×C₂₉	C₄	1	1	yes	characteristic
16	C₈×D₄×C₂₉	C₂²	1	1	yes	characteristic
17	C₈×D₄×C₂₉	C₂²	1	2	yes	—
18	C₂₉×#431	C₄	1	2	yes	—
19	C₂₉×#431	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
22	C₂₉×C₄⋊C₁₆	C₂²	1	2	yes	—
23	C₂₉×C₄⋊C₁₆	C₂²	1	4	yes	—
24	C₂₉×C₂²⋊C₁₆	C₂²	1	4	yes	—
25	C₂₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
26	C₂₉×#346	C₂²	1	2	yes	—
27	C₂₉×#346	C₂²	1	1	yes	characteristic
28	C₂₉×#346	C₂²	1	4	yes	—
29	C₂₉×#346	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₂₉	C₂²	1	2	yes	—
31	C₂×#41×C₂₉	C₈	1	1	yes	characteristic
32	C₂²×C₈×C₂₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₂₉	C₈	1	1	yes	characteristic
37	C₄×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄×C₂₉	C₈	1	2	yes	—
39	C₄×D₄×C₂₉	C₈	1	1	yes	characteristic
40	C₄×D₄×C₂₉	C₂×C₄	1	2	yes	—
41	C₄×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂₉×#117	C₈	1	1	yes	characteristic
43	C₂₉×#117	C₈	1	2	yes	—
44	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
46	C₂₉×M₃₂	—	2	4	no	—
47	C₂×C₁₆×C₂₉	—	2	8	no	—
48	C₂×C₁₆×C₂₉	C₂³	1	4	yes	—
49	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
50	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
51	C₂₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₂₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
56	C₂₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₂₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₂₉	C₂³	1	2	yes	—
64	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
65	#41×C₂₉	—	2	4	no	—
66	C₂×Q₈×C₂₉	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₂₉	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₂₉	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₂₉	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₂₉	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₂₉	—	2	4	no	—
76	C₂×C₈×C₂₉	—	2	2	no	—
77	C₂×C₈×C₂₉	—	2	4	no	—
78	C₂×C₈×C₂₉	—	2	2	no	—
79	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₂₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₂₉	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
87	C₄⋊C₄×C₂₉	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₂₉	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₂₉	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₂₉	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₂₉	C₂×C₈	1	2	yes	—
92	C₄²×C₂₉	C₂×C₈	1	2	yes	—
93	C₄²×C₂₉	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₂₉	C₂×C₈	1	2	yes	—
95	C₄²×C₂₉	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
98	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₂₉	—	2	4	no	—
100	C₁₆×C₂₉	—	2	2	no	—
101	C₁₆×C₂₉	#41	1	4	yes	complemented
102	#33484	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₂₉	C₂²×C₈	1	1	yes	characteristic
104	C₂³×C₂₉	C₂²×C₈	1	2	yes	—
105	Q₈×C₂₉	—	2	2	no	—
106	D₄×C₂₉	—	2	4	no	—
107	D₄×C₂₉	M₃₂	1	4	yes	complemented
108	C₂×C₄×C₂₉	—	2	4	no	—
109	C₂×C₄×C₂₉	—	2	2	no	—
110	C₂×C₄×C₂₉	—	2	1	no	—
111	C₂×C₄×C₂₉	—	2	2	no	—
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	4	2	no	—
115	C₂×C₄×C₂₉	—	4	1	no	—
116	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₂₉	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₂₉	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₂₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₂₉	—	2	2	no	—
126	C₈×C₂₉	—	2	1	no	—
127	C₈×C₂₉	—	2	1	no	—
128	C₈×C₂₉	—	2	2	no	—
129	C₈×C₂₉	#131	1	2	yes	—
130	C₈×C₂₉	C₂×#41	1	2	yes	—
131	C₈×#41	C₂×C₂₉	1	1	yes	characteristic
132	#2183	C₂×C₂₉	1	2	yes	complemented
133	#2168	C₂×C₂₉	1	4	yes	complemented
134	D₄×C₁₆	C₂×C₂₉	1	2	yes	complemented
135	#2163	C₂×C₂₉	1	1	yes	characteristic, complemented
136	#2163	C₂×C₂₉	1	2	yes	complemented
137	C₄×M₃₂	C₂×C₂₉	1	1	yes	characteristic, complemented
138	C₂×#346	C₂×C₂₉	1	2	yes	complemented
139	C₂²×C₂₉	—	2	2	no	—
140	C₂²×C₂₉	—	2	2	no	—
141	C₂²×C₂₉	—	2	1	no	—
142	C₂²×C₂₉	—	2	1	no	—
143	C₂²×C₂₉	—	2	1	no	—
144	C₂²×C₂₉	C₂×M₃₂	1	4	yes	—
145	C₂²×C₂₉	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₂₉	—	2	1	no	—
147	C₄×C₂₉	—	2	2	no	—
148	C₄×C₂₉	—	2	1	no	—
149	C₄×C₂₉	—	2	2	no	—
150	C₄×C₂₉	—	2	1	no	—
151	C₄×C₂₉	—	2	2	no	—
152	C₄×C₂₉	—	2	1	no	—
153	C₄×C₂₉	C₂×M₃₂	1	2	yes	—
154	C₄×C₂₉	C₄×#41	1	1	yes	characteristic
155	C₄×C₂₉	C₂×#131	1	1	yes	characteristic
156	C₄×#41	C₄×C₂₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
158	C₂²×C₁₆	C₂²×C₂₉	1	4	yes	—
159	C₈×Q₈	C₄×C₂₉	1	1	yes	characteristic
160	C₈×Q₈	C₂²×C₂₉	1	1	yes	characteristic
161	C₈×D₄	C₄×C₂₉	1	2	yes	—
162	C₈×D₄	C₄×C₂₉	1	1	yes	characteristic
163	C₈×D₄	C₂²×C₂₉	1	1	yes	characteristic
164	C₈×D₄	C₂²×C₂₉	1	2	yes	—
165	#431	C₄×C₂₉	1	1	yes	characteristic
166	#431	C₄×C₂₉	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
169	C₄⋊C₁₆	C₂²×C₂₉	1	2	yes	—
170	C₄⋊C₁₆	C₂²×C₂₉	1	4	yes	—
171	C₂²⋊C₁₆	C₂²×C₂₉	1	4	yes	—
172	C₂²⋊C₁₆	C₂²×C₂₉	1	2	yes	—
173	#346	C₂²×C₂₉	1	1	yes	characteristic
174	#346	C₂²×C₂₉	1	2	yes	—
175	#346	C₂²×C₂₉	1	4	yes	—
176	#346	C₂²×C₂₉	1	1	yes	characteristic
177	C₄×C₁₆	C₂²×C₂₉	1	2	yes	—
178	C₂×C₂₉	—	2	4	no	—
179	C₂×C₂₉	—	4	1	no	—
180	C₂×C₂₉	C₈×#41	1	1	yes	characteristic
181	C₂×C₂₉	C₂²×M₃₂	1	1	yes	characteristic
182	C₂×C₂₉	#3407	1	1	yes	characteristic
183	C₂×#41	C₈×C₂₉	1	1	yes	characteristic
184	C₂²×C₈	C₂×C₄×C₂₉	1	2	yes	—
185	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
187	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
188	C₄×Q₈	C₈×C₂₉	1	1	yes	characteristic
189	C₄×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
190	C₄×D₄	C₈×C₂₉	1	2	yes	—
191	C₄×D₄	C₈×C₂₉	1	1	yes	characteristic
192	C₄×D₄	C₂×C₄×C₂₉	1	2	yes	—
193	C₄×D₄	C₂×C₄×C₂₉	1	1	yes	characteristic
194	#117	C₈×C₂₉	1	1	yes	characteristic
195	#117	C₈×C₂₉	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
198	M₃₂	—	2	4	no	—
199	C₂×C₁₆	—	2	8	no	—
200	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
201	C₂×C₁₆	C₂³×C₂₉	1	4	yes	—
202	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
203	C₄⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
204	C₄⋊C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
205	C₄⋊C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
206	C₄⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
207	C₂²⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
208	C₂²⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
209	C₂²⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
210	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
211	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
212	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
213	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
216	C₄×C₈	C₂³×C₂₉	1	2	yes	—
217	C₂₉	#33484	1	1	yes	characteristic, direct product factor, Sylow
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×C₈×C₂₉	1	1	yes	characteristic
220	C₂×D₄	C₂×C₈×C₂₉	1	2	yes	—
221	C₂×D₄	C₂×C₈×C₂₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₈×C₂₉	1	2	yes	—
223	C₂²×C₄	C₂×C₈×C₂₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₈×C₂₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₈×C₂₉	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂²×C₄×C₂₉	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₄×C₂₉	1	2	yes	—
236	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
237	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
238	C₄⋊C₄	C₂×C₈×C₂₉	1	2	yes	—
239	C₄⋊C₄	C₂×C₈×C₂₉	1	1	yes	characteristic
240	C₄⋊C₄	C₂×C₈×C₂₉	1	2	yes	—
241	C₄⋊C₄	C₂×C₈×C₂₉	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×C₈×C₂₉	1	2	yes	—
243	C₂²⋊C₄	C₂×C₈×C₂₉	1	2	yes	—
244	C₂²⋊C₄	C₂×C₈×C₂₉	1	2	yes	—
245	C₄²	C₂×C₈×C₂₉	1	2	yes	—
246	C₄²	C₂×C₈×C₂₉	1	1	yes	characteristic
247	C₄²	C₂×C₈×C₂₉	1	2	yes	—
248	C₄²	C₂×C₈×C₂₉	1	1	yes	characteristic
249	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
250	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
251	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
252	C₁₆	—	2	4	no	—
253	C₁₆	—	2	2	no	—
254	C₁₆	#41×C₂₉	1	4	yes	complemented
255	C₂³	C₂²×C₈×C₂₉	1	1	yes	characteristic
256	C₂³	C₂²×C₈×C₂₉	1	2	yes	—
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	C₂₉×M₃₂	1	4	yes	complemented
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₈×C₂₉	1	2	yes	—
271	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₈×C₂₉	1	2	yes	—
273	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₈×C₂₉	1	2	yes	—
275	C₂×C₄	C₂²×C₈×C₂₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	C₂₉×#131	1	2	yes	—
282	C₈	C₂×#41×C₂₉	1	2	yes	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	C₂×C₂₉×M₃₂	1	4	yes	—
289	C₂²	C₂³×C₈×C₂₉	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	C₂×C₂₉×M₃₂	1	2	yes	—
298	C₄	C₄×#41×C₂₉	1	1	yes	characteristic
299	C₄	C₂×C₂₉×#131	1	1	yes	characteristic
300	C₂	—	2	4	no	—
301	C₂	—	4	1	no	—
302	C₂	C₈×#41×C₂₉	1	1	yes	characteristic
303	C₂	C₂²×C₂₉×M₃₂	1	1	yes	characteristic
304	C₂	C₂₉×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₉	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂³×C₈
prime radical	C₂×#41×C₂₉	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₉×#2183	C₂	semidirect
3. C₂₉×#2168	C₂	semidirect
4. D₄×C₁₆×C₂₉	C₂	semidirect
5. C₂₉×#2163	C₂	semidirect
6. C₂₉×#2163	C₂	semidirect
7. C₄×C₂₉×M₃₂	C₂	semidirect
8. C₂×C₂₉×#346	C₂	semidirect
101. C₁₆×C₂₉	#41	semidirect
102. #33484	C₂₉	direct (N × H)
107. D₄×C₂₉	M₃₂	semidirect
132. #2183	C₂×C₂₉	semidirect
133. #2168	C₂×C₂₉	semidirect
134. D₄×C₁₆	C₂×C₂₉	semidirect
135. #2163	C₂×C₂₉	semidirect
136. #2163	C₂×C₂₉	semidirect
137. C₄×M₃₂	C₂×C₂₉	semidirect
138. C₂×#346	C₂×C₂₉	semidirect
217. C₂₉	#33484	direct (N × H)
254. C₁₆	#41×C₂₉	semidirect
259. D₄	C₂₉×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₂₉	2
C₂₉×#2183	2
C₂₉×#2168	2
D₄×C₁₆×C₂₉	2
C₂₉×#2163	2
C₄×C₂₉×M₃₂	2
C₂×C₂₉×#346	2
#33484	29

Subgroups — C29×#33484 #20328384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33484 #20328384