Subgroups — C₂₉×#7644 #20302544

Order 7424.

402 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2295	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1839	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1272	C₂	1	2	yes	maximal
5	C₂₉×#1271	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
7	C₂×C₂₉×C₄⋊C₈	—	2	1	no	—
8	C₂₉×#391	C₄	1	1	yes	characteristic
9	C₂₉×#391	C₂²	1	1	yes	characteristic
10	C₂₉×#389	C₄	1	1	yes	characteristic
11	C₂₉×#389	C₂²	1	1	yes	characteristic
12	C₂₉×#387	—	2	1	no	—
13	C₂₉×#387	—	2	1	no	—
14	C₄×C₄⋊C₄×C₂₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₂₉	C₄	1	1	yes	characteristic
16	C₂₉×#376	—	2	1	no	—
17	C₄³×C₂₉	C₂²	1	1	yes	characteristic
18	C₂₉×#336	—	2	1	no	—
19	C₂₉×#336	—	2	1	no	—
20	C₂₉×#336	—	2	1	no	—
21	C₂₉×#336	C₂²	1	2	yes	—
22	C₂₉×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₉	—	2	2	no	—
25	C₂²×C₈×C₂₉	—	2	2	no	—
26	C₄×Q₈×C₂₉	—	2	2	no	—
27	C₄×Q₈×C₂₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₉	—	2	1	no	—
35	C₂×C₄²×C₂₉	—	2	1	no	—
36	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂₉×C₄⋊C₈	—	4	2	no	—
44	C₂₉×#95	—	2	1	no	—
45	C₂₉×#95	—	2	1	no	—
46	C₂₉×#95	—	2	1	no	—
47	C₂₉×#95	—	2	1	no	—
48	C₂₉×#95	—	2	1	no	—
49	C₂₉×#95	—	2	1	no	—
50	C₂₉×#95	—	2	1	no	—
51	C₂×Q₈×C₂₉	—	2	1	no	—
52	C₂×Q₈×C₂₉	—	4	1	no	—
53	C₂×Q₈×C₂₉	—	4	1	no	—
54	C₂×Q₈×C₂₉	M₁₆	1	2	yes	—
55	C₂²×C₄×C₂₉	—	2	1	no	—
56	C₂²×C₄×C₂₉	—	2	1	no	—
57	C₂²×C₄×C₂₉	—	2	1	no	—
58	C₂²×C₄×C₂₉	—	2	1	no	—
59	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₂₉	—	4	2	no	—
67	C₂×C₈×C₂₉	—	4	2	no	—
68	C₂×C₈×C₂₉	—	4	2	no	—
69	C₂×C₈×C₂₉	—	4	2	no	—
70	C₂×C₈×C₂₉	—	4	2	no	—
71	C₂×C₈×C₂₉	—	4	2	no	—
72	C₄⋊C₄×C₂₉	—	2	4	no	—
73	C₄⋊C₄×C₂₉	—	2	1	no	—
74	C₄⋊C₄×C₂₉	—	4	1	no	—
75	C₄⋊C₄×C₂₉	—	4	1	no	—
76	C₄⋊C₄×C₂₉	—	4	2	no	—
77	C₄⋊C₄×C₂₉	—	4	2	no	—
78	C₄⋊C₄×C₂₉	M₁₆	1	2	yes	—
79	C₄²×C₂₉	—	2	4	no	—
80	C₄²×C₂₉	—	2	1	no	—
81	C₄²×C₂₉	—	2	1	no	—
82	C₄²×C₂₉	—	2	2	no	—
83	C₄²×C₂₉	—	4	1	no	—
84	C₄²×C₂₉	—	4	1	no	—
85	C₄²×C₂₉	—	4	1	no	—
86	C₄²×C₂₉	—	4	1	no	—
87	C₄²×C₂₉	—	4	2	no	—
88	C₄²×C₂₉	SD₁₆	1	2	yes	—
89	C₄²×C₂₉	Q₁₆	1	2	yes	—
90	#7644	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	Q₈×C₂₉	—	4	2	no	—
93	Q₈×C₂₉	—	4	2	no	—
94	C₂×C₄×C₂₉	—	2	1	no	—
95	C₂×C₄×C₂₉	—	2	1	no	—
96	C₂×C₄×C₂₉	—	2	1	no	—
97	C₂×C₄×C₂₉	—	2	1	no	—
98	C₂×C₄×C₂₉	—	2	1	no	—
99	C₂×C₄×C₂₉	—	2	1	no	—
100	C₂×C₄×C₂₉	—	2	1	no	—
101	C₂×C₄×C₂₉	—	2	1	no	—
102	C₂×C₄×C₂₉	—	2	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	—	2	2	no	—
107	C₂×C₄×C₂₉	—	2	1	no	—
108	C₂×C₄×C₂₉	—	2	1	no	—
109	C₂×C₄×C₂₉	—	2	1	no	—
110	C₂×C₄×C₂₉	—	2	1	no	—
111	C₂×C₄×C₂₉	—	2	1	no	—
112	C₂×C₄×C₂₉	—	4	1	no	—
113	C₂×C₄×C₂₉	—	4	1	no	—
114	C₂×C₄×C₂₉	—	4	1	no	—
115	C₂×C₄×C₂₉	—	4	1	no	—
116	C₂×C₄×C₂₉	—	4	1	no	—
117	C₂×C₄×C₂₉	—	4	1	no	—
118	C₂×C₄×C₂₉	—	4	1	no	—
119	C₂×C₄×C₂₉	—	4	1	no	—
120	C₂×C₄×C₂₉	—	4	1	no	—
121	C₂×C₄×C₂₉	—	4	1	no	—
122	C₂×C₄×C₂₉	#103	1	2	yes	—
123	C₂×C₄×C₂₉	#103	1	2	yes	—
124	C₂×C₄×C₂₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄×C₂₉	#135	1	2	yes	—
129	C₈×C₂₉	—	8	2	no	—
130	C₈×C₂₉	—	8	2	no	—
131	#2295	C₂×C₂₉	1	1	yes	characteristic
132	#1839	C₂×C₂₉	1	1	yes	characteristic
133	#1831	C₂×C₂₉	1	1	yes	characteristic
134	#1272	C₂×C₂₉	1	2	yes	—
135	#1271	C₂×C₂₉	1	2	yes	—
136	C₂²×C₂₉	#407	1	1	yes	characteristic
137	C₂²×C₂₉	#410	1	1	yes	characteristic
138	C₂²×C₂₉	#413	1	1	yes	characteristic
139	C₂²×C₂₉	C₂×#103	1	1	yes	characteristic
140	C₂²×C₂₉	#416	1	1	yes	characteristic
141	C₂²×C₂₉	#421	1	1	yes	characteristic
142	C₂²×C₂₉	#421	1	1	yes	characteristic
143	C₄×C₂₉	—	2	2	no	—
144	C₄×C₂₉	—	2	1	no	—
145	C₄×C₂₉	—	2	1	no	—
146	C₄×C₂₉	—	2	2	no	—
147	C₄×C₂₉	—	2	1	no	—
148	C₄×C₂₉	—	2	1	no	—
149	C₄×C₂₉	—	2	2	no	—
150	C₄×C₂₉	—	4	2	no	—
151	C₄×C₂₉	—	4	1	no	—
152	C₄×C₂₉	—	4	2	no	—
153	C₄×C₂₉	—	4	1	no	—
154	C₄×C₂₉	—	8	1	no	—
155	C₄×C₂₉	—	8	1	no	—
156	C₂×C₄×Q₈	C₄×C₂₉	1	1	yes	characteristic
157	C₂×C₄⋊C₈	—	2	1	no	—
158	#391	C₄×C₂₉	1	1	yes	characteristic
159	#391	C₂²×C₂₉	1	1	yes	characteristic
160	#389	C₄×C₂₉	1	1	yes	characteristic
161	#389	C₂²×C₂₉	1	1	yes	characteristic
162	#387	—	2	1	no	—
163	#387	—	2	1	no	—
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	C₄×C₂₉	1	1	yes	characteristic
166	#376	—	2	1	no	—
167	C₄³	C₂²×C₂₉	1	1	yes	characteristic
168	#336	—	2	1	no	—
169	#336	—	2	1	no	—
170	#336	—	2	1	no	—
171	#336	C₂²×C₂₉	1	2	yes	—
172	#336	C₂²×C₂₉	1	2	yes	—
173	C₂×C₂₉	#1464	1	1	yes	characteristic
174	C₂×C₂₉	#1487	1	1	yes	characteristic
175	C₂×C₂₉	#1488	1	1	yes	characteristic
176	C₂×C₂₉	#1518	1	1	yes	characteristic
177	C₂×C₂₉	#1521	1	1	yes	characteristic
178	C₂×C₂₉	#1535	1	1	yes	characteristic
179	C₂×C₂₉	#1542	1	1	yes	characteristic
180	C₂²×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
200	C₄⋊C₈	—	4	2	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	C₂₉	#7644	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	4	1	no	—
212	C₂×Q₈	M₁₆×C₂₉	1	2	yes	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	M₁₆×C₂₉	1	2	yes	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	SD₁₆×C₂₉	1	2	yes	—
247	C₄²	Q₁₆×C₂₉	1	2	yes	—
248	C₂³	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂₉×#103	1	2	yes	—
280	C₂×C₄	C₂₉×#103	1	2	yes	—
281	C₂×C₄	C₂×M₁₆×C₂₉	1	1	yes	characteristic
282	C₂×C₄	C₂×M₁₆×C₂₉	1	1	yes	characteristic
283	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
284	C₂×C₄	C₂×Q₁₆×C₂₉	1	1	yes	characteristic
285	C₂×C₄	C₂₉×#135	1	2	yes	—
286	C₈	—	8	2	no	—
287	C₈	—	8	2	no	—
288	C₂²	C₂₉×#407	1	1	yes	characteristic
289	C₂²	C₂₉×#410	1	1	yes	characteristic
290	C₂²	C₂₉×#413	1	1	yes	characteristic
291	C₂²	C₂×C₂₉×#103	1	1	yes	characteristic
292	C₂²	C₂₉×#416	1	1	yes	characteristic
293	C₂²	C₂₉×#421	1	1	yes	characteristic
294	C₂²	C₂₉×#421	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₂	C₂₉×#1464	1	1	yes	characteristic
309	C₂	C₂₉×#1487	1	1	yes	characteristic
310	C₂	C₂₉×#1488	1	1	yes	characteristic
311	C₂	C₂₉×#1518	1	1	yes	characteristic
312	C₂	C₂₉×#1521	1	1	yes	characteristic
313	C₂	C₂₉×#1535	1	1	yes	characteristic
314	C₂	C₂₉×#1542	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₄³×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
90. #7644	C₂₉	direct (N × H)
208. C₂₉	#7644	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×#2295	2
C₂₉×#1839	2
C₂₉×#1831	2
C₂₉×#1272	2
C₂₉×#1271	2
#7644	29

Subgroups — C29×#7644 #20302544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#7644 #20302544