Subgroups — C₂₉×#10805 #20305705

Order 7424.

674 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#344	C₂	1	2	yes	maximal
3	C₂×C₂₉×#339	C₂	1	2	yes	maximal
4	C₂×C₂₉×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₉	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₂₉×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₂₉	—	2	2	no	—
12	C₄×M₁₆×C₂₉	—	2	2	no	—
13	C₂×C₂₉×#97	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₂₉	C₄	1	2	yes	complemented
15	C₂₉×#344	C₄	1	8	yes	complemented
16	C₂₉×#339	—	2	4	no	—
17	C₂₉×#338	—	2	2	no	—
18	C₂₉×H(3,ℤ/4)	—	2	2	no	—
19	C₂³×C₄×C₂₉	—	2	1	no	—
20	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₂₉	—	2	1	no	—
22	C₂×M₁₆×C₂₉	—	2	1	no	—
23	C₂×M₁₆×C₂₉	—	2	2	no	—
24	C₂×M₁₆×C₂₉	—	2	2	no	—
25	C₂×M₁₆×C₂₉	—	2	2	no	—
26	C₂×M₁₆×C₂₉	—	2	2	no	—
27	C₂×M₁₆×C₂₉	—	2	2	no	—
28	C₂×M₁₆×C₂₉	—	2	2	no	—
29	C₂×M₁₆×C₂₉	—	2	2	no	—
30	C₂×M₁₆×C₂₉	—	2	2	no	—
31	C₂×M₁₆×C₂₉	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₂₉	C₂×C₄	1	4	yes	—
33	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
34	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
35	C₂²×C₈×C₂₉	—	2	2	no	—
36	C₂²×C₈×C₂₉	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₂₉	C₂×C₄	1	2	yes	—
38	C₂₉×#117	—	2	2	no	—
39	C₂₉×#117	—	2	2	no	—
40	C₂₉×#117	—	2	2	no	—
41	C₂₉×#117	C₂×C₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₉	—	2	2	no	—
44	C₂×C₄²×C₂₉	—	2	1	no	—
45	C₂×C₄²×C₂₉	—	2	1	no	—
46	C₂×C₄²×C₂₉	—	2	1	no	—
47	C₂×C₄²×C₂₉	—	2	2	no	—
48	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₂₉	D₄	1	2	yes	—
51	C₂×C₄²×C₂₉	Q₈	1	2	yes	—
52	C₂₉×#97	—	2	4	no	—
53	C₄×C₈×C₂₉	—	2	4	no	—
54	C₂⁴×C₂₉	#41	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	—	2	1	no	—
56	C₂²×C₄×C₂₉	—	2	1	no	—
57	C₂²×C₄×C₂₉	—	2	2	no	—
58	C₂²×C₄×C₂₉	—	2	2	no	—
59	C₂²×C₄×C₂₉	—	4	4	no	—
60	C₂²×C₄×C₂₉	—	4	2	no	—
61	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂×D₄	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	C₂²×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
68	C₂²×C₄×C₂₉	#41	1	2	yes	—
69	C₂²×C₄×C₂₉	#41	1	2	yes	—
70	C₂²×C₄×C₂₉	#41	1	2	yes	—
71	M₁₆×C₂₉	—	2	4	no	—
72	M₁₆×C₂₉	—	2	4	no	—
73	M₁₆×C₂₉	—	2	4	no	—
74	M₁₆×C₂₉	—	2	4	no	—
75	M₁₆×C₂₉	—	2	4	no	—
76	M₁₆×C₂₉	—	2	4	no	—
77	M₁₆×C₂₉	—	4	2	no	—
78	M₁₆×C₂₉	—	4	2	no	—
79	C₂×C₈×C₂₉	—	2	2	no	—
80	C₂×C₈×C₂₉	—	2	2	no	—
81	C₂×C₈×C₂₉	—	2	2	no	—
82	C₂×C₈×C₂₉	—	2	2	no	—
83	C₂×C₈×C₂₉	—	2	4	no	—
84	C₂×C₈×C₂₉	—	4	2	no	—
85	C₂×C₈×C₂₉	—	4	2	no	—
86	C₂×C₈×C₂₉	C₄²	1	4	yes	—
87	C₂×C₈×C₂₉	C₂²⋊C₄	1	4	yes	—
88	C₄⋊C₄×C₂₉	—	2	4	no	—
89	C₄⋊C₄×C₂₉	—	4	2	no	—
90	C₂²⋊C₄×C₂₉	—	2	4	no	—
91	C₂²⋊C₄×C₂₉	—	4	2	no	—
92	C₄²×C₂₉	—	2	1	no	—
93	C₄²×C₂₉	—	2	1	no	—
94	C₄²×C₂₉	—	2	2	no	—
95	C₄²×C₂₉	—	2	2	no	—
96	C₄²×C₂₉	—	2	4	no	—
97	C₄²×C₂₉	—	4	2	no	—
98	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
99	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
100	#10805	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂³×C₂₉	—	4	2	no	—
102	C₂³×C₂₉	#117	1	1	yes	characteristic
103	C₂³×C₂₉	#117	1	2	yes	—
104	C₂³×C₂₉	#121	1	1	yes	characteristic
105	C₂³×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₂₉	#126	1	2	yes	—
107	C₂×C₄×C₂₉	—	2	4	no	—
108	C₂×C₄×C₂₉	—	2	2	no	—
109	C₂×C₄×C₂₉	—	2	2	no	—
110	C₂×C₄×C₂₉	—	4	4	no	—
111	C₂×C₄×C₂₉	—	4	4	no	—
112	C₂×C₄×C₂₉	—	4	2	no	—
113	C₂×C₄×C₂₉	—	4	4	no	—
114	C₂×C₄×C₂₉	—	8	4	no	—
115	C₂×C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₉	#117	1	1	yes	characteristic
119	C₂×C₄×C₂₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₂₉	#117	1	2	yes	—
121	C₂×C₄×C₂₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₂₉	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₂₉	#123	1	2	yes	—
129	C₂×C₄×C₂₉	#123	1	2	yes	—
130	C₂×C₄×C₂₉	#124	1	2	yes	—
131	C₂×C₄×C₂₉	#125	1	2	yes	—
132	C₂×C₄×C₂₉	#126	1	2	yes	—
133	C₈×C₂₉	—	2	4	no	—
134	C₈×C₂₉	—	2	4	no	—
135	C₈×C₂₉	—	4	4	no	—
136	C₂×C₄×M₁₆	C₂×C₂₉	1	1	yes	characteristic
137	C₂×#344	C₂×C₂₉	1	2	yes	—
138	C₂×#339	C₂×C₂₉	1	2	yes	—
139	C₂×#338	C₂×C₂₉	1	1	yes	characteristic
140	C₂×H(3,ℤ/4)	C₂×C₂₉	1	1	yes	characteristic
141	C₂²×C₂₉	—	2	2	no	—
142	C₂²×C₂₉	—	2	4	no	—
143	C₂²×C₂₉	—	4	1	no	—
144	C₂²×C₂₉	—	4	1	no	—
145	C₂²×C₂₉	—	4	2	no	—
146	C₂²×C₂₉	#376	1	1	yes	characteristic
147	C₂²×C₂₉	#380	1	1	yes	characteristic
148	C₂²×C₂₉	#383	1	1	yes	characteristic
149	C₂²×C₂₉	#387	1	2	yes	—
150	C₂²×C₂₉	#388	1	2	yes	—
151	C₄×C₂₉	—	2	4	no	—
152	C₄×C₂₉	—	4	2	no	—
153	C₄×C₂₉	—	8	4	no	—
154	C₄×C₂₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₂₉	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₂₉	#381	1	1	yes	characteristic
157	C₄×C₂₉	#382	1	1	yes	characteristic
158	C₄×C₂₉	#387	1	2	yes	—
159	C₄×C₂₉	#388	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₂₉	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
163	C₂×#117	C₂²×C₂₉	1	2	yes	—
164	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
165	C₄×M₁₆	—	2	2	no	—
166	C₄×M₁₆	—	2	2	no	—
167	C₂×#97	C₄×C₂₉	1	2	yes	complemented
168	C₂×C₄×C₈	C₄×C₂₉	1	2	yes	complemented
169	#344	C₄×C₂₉	1	8	yes	complemented
170	#339	—	2	4	no	—
171	#338	—	2	2	no	—
172	H(3,ℤ/4)	—	2	2	no	—
173	C₂×C₂₉	—	2	1	no	—
174	C₂×C₂₉	—	2	2	no	—
175	C₂×C₂₉	—	2	1	no	—
176	C₂×C₂₉	—	2	2	no	—
177	C₂×C₂₉	#1434	1	1	yes	characteristic
178	C₂×C₂₉	#1841	1	2	yes	—
179	C₂³×C₄	—	2	1	no	—
180	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₂₉	1	4	yes	—
192	C₂×M₁₆	C₂×C₄×C₂₉	1	4	yes	—
193	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
194	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	C₂×C₄×C₂₉	1	2	yes	—
197	C₂²×C₈	C₂×C₄×C₂₉	1	2	yes	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	C₂×C₄×C₂₉	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
209	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₂₉	1	2	yes	—
211	C₂×C₄²	Q₈×C₂₉	1	2	yes	—
212	#97	—	2	4	no	—
213	C₄×C₈	—	2	4	no	—
214	C₂₉	#10805	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	#41×C₂₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×Q₈×C₂₉	1	1	yes	characteristic
228	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
229	C₂²×C₄	#41×C₂₉	1	2	yes	—
230	C₂²×C₄	#41×C₂₉	1	2	yes	—
231	C₂²×C₄	#41×C₂₉	1	2	yes	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	2	no	—
239	M₁₆	—	4	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	C₄²×C₂₉	1	4	yes	—
248	C₂×C₈	C₂²⋊C₄×C₂₉	1	4	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
260	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
261	C₂³	—	4	2	no	—
262	C₂³	C₂₉×#117	1	1	yes	characteristic
263	C₂³	C₂₉×#117	1	2	yes	—
264	C₂³	C₂₉×#121	1	1	yes	characteristic
265	C₂³	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
266	C₂³	C₂₉×#126	1	2	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	8	4	no	—
275	C₂×C₄	C₂×C₄²×C₂₉	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
278	C₂×C₄	C₂₉×#117	1	1	yes	characteristic
279	C₂×C₄	C₂₉×#117	1	1	yes	characteristic
280	C₂×C₄	C₂₉×#117	1	1	yes	characteristic
281	C₂×C₄	C₂₉×#117	1	2	yes	—
282	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
284	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
285	C₂×C₄	C₄×Q₈×C₂₉	1	2	yes	—
286	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
287	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
288	C₂×C₄	C₂₉×#123	1	2	yes	—
289	C₂×C₄	C₂₉×#123	1	2	yes	—
290	C₂×C₄	C₂₉×#124	1	2	yes	—
291	C₂×C₄	C₂₉×#125	1	2	yes	—
292	C₂×C₄	C₂₉×#126	1	2	yes	—
293	C₈	—	2	4	no	—
294	C₈	—	2	4	no	—
295	C₈	—	4	4	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂₉×#376	1	1	yes	characteristic
302	C₂²	C₂₉×#380	1	1	yes	characteristic
303	C₂²	C₂₉×#383	1	1	yes	characteristic
304	C₂²	C₂₉×#387	1	2	yes	—
305	C₂²	C₂₉×#388	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
310	C₄	C₄×C₄⋊C₄×C₂₉	1	1	yes	characteristic
311	C₄	C₂₉×#381	1	1	yes	characteristic
312	C₄	C₂₉×#382	1	1	yes	characteristic
313	C₄	C₂₉×#387	1	2	yes	—
314	C₄	C₂₉×#388	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	C₂₉×#1434	1	1	yes	characteristic
320	C₂	C₂₉×#1841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₉	#117
commutator	C₂×C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂³×C₄	C₂³×C₂₉
socle	C₂²×C₂₉	#383
prime radical	C₂⁴×C₂₉	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂²×M₁₆×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₂₉×#97	C₄	semidirect
14. C₂×C₄×C₈×C₂₉	C₄	semidirect
15. C₂₉×#344	C₄	semidirect
100. #10805	C₂₉	direct (N × H)
167. C₂×#97	C₄×C₂₉	semidirect
168. C₂×C₄×C₈	C₄×C₂₉	semidirect
169. #344	C₄×C₂₉	semidirect
214. C₂₉	#10805	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₂₉	2
C₂×C₂₉×#344	2
C₂×C₂₉×#339	2
C₂×C₂₉×#338	2
C₂×C₂₉×H(3,ℤ/4)	2
#10805	29

Subgroups — C29×#10805 #20305705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#10805 #20305705