Subgroups — C₂₉×#33214 #20328114

Order 7424.

794 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₉×#405	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2629	C₂	1	2	yes	maximal
4	C₂₉×#1992	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1944	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1941	C₂	1	2	yes	maximal, complemented
7	C₂₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#1809	C₂	1	2	yes	maximal
9	C₂₉×#1807	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₂×C₂₉×#125	C₂²	1	2	yes	—
13	C₂×C₂₉×#117	C₂²	1	1	yes	characteristic
14	C₂₉×#499	—	2	4	no	—
15	C₂×C₂₉×#107	C₂²	1	2	yes	—
16	C₂×C₂₉×#106	C₂²	1	2	yes	—
17	C₂×C₂₉×#106	C₂²	1	2	yes	—
18	C₂×C₂₉×#106	C₂²	1	2	yes	—
19	C₂₉×#405	—	2	4	no	—
20	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×#398	—	2	4	no	—
25	C₂₉×#396	—	2	4	no	—
26	C₂₉×#389	C₂²	1	4	yes	—
27	C₂₉×#384	C₂²	1	2	yes	—
28	C₂₉×#380	C₂²	1	4	yes	—
29	C₂₉×#340	C₂²	1	4	yes	—
30	C₂₉×#340	C₂²	1	4	yes	—
31	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₂₉	—	2	2	no	—
33	C₂×M₁₆×C₂₉	—	2	2	no	—
34	C₂×M₁₆×C₂₉	Q₈	1	4	yes	—
35	C₂²×C₈×C₂₉	—	2	2	no	—
36	C₂²×C₈×C₂₉	—	2	2	no	—
37	C₂²×C₈×C₂₉	Q₈	1	4	yes	—
38	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
42	C₂₉×#125	—	2	4	no	—
43	C₂₉×#125	—	2	4	no	—
44	C₂₉×#117	—	2	2	no	—
45	C₂₉×#117	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
47	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
48	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
49	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
52	C₂×C₂²⋊C₄×C₂₉	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₂₉	D₄	1	2	yes	—
54	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	C₂₉×#107	—	2	4	no	—
57	C₂₉×#106	—	2	4	no	—
58	C₂₉×#106	—	2	4	no	—
59	C₂₉×#106	—	2	4	no	—
60	C₂₉×#97	—	2	2	no	—
61	C₂₉×#97	—	2	2	no	—
62	C₄×C₈×C₂₉	—	2	2	no	—
63	C₄×C₈×C₂₉	D₄	1	4	yes	—
64	C₂₉×#95	—	2	4	no	—
65	C₂₉×#95	—	2	4	no	—
66	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	—	2	1	no	—
68	C₂²×C₄×C₂₉	—	2	1	no	—
69	C₂²×C₄×C₂₉	—	2	1	no	—
70	C₂²×C₄×C₂₉	—	2	1	no	—
71	C₂²×C₄×C₂₉	—	2	4	no	—
72	C₂²×C₄×C₂₉	—	2	4	no	—
73	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₂₉	—	2	8	no	—
80	C₂×C₈×C₂₉	—	2	1	no	—
81	C₂×C₈×C₂₉	—	2	1	no	—
82	C₂×C₈×C₂₉	—	2	1	no	—
83	C₂×C₈×C₂₉	—	2	1	no	—
84	C₂×C₈×C₂₉	—	2	8	no	—
85	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₂₉	#41	1	2	yes	—
92	C₂×C₈×C₂₉	#41	1	2	yes	—
93	C₄⋊C₄×C₂₉	—	2	4	no	—
94	C₄⋊C₄×C₂₉	—	2	4	no	—
95	C₄⋊C₄×C₂₉	—	2	4	no	—
96	C₄⋊C₄×C₂₉	—	2	4	no	—
97	C₄⋊C₄×C₂₉	—	2	4	no	—
98	C₄⋊C₄×C₂₉	—	4	8	no	—
99	C₄⋊C₄×C₂₉	—	8	4	no	—
100	C₄⋊C₄×C₂₉	—	8	4	no	—
101	C₂²⋊C₄×C₂₉	—	2	4	no	—
102	C₂²⋊C₄×C₂₉	—	8	4	no	—
103	C₄²×C₂₉	—	2	1	no	—
104	C₄²×C₂₉	—	2	1	no	—
105	C₄²×C₂₉	C₂×D₄	1	2	yes	—
106	C₄²×C₂₉	#41	1	2	yes	—
107	#33214	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₉	—	2	1	no	—
109	C₂³×C₂₉	—	2	1	no	—
110	C₂³×C₂₉	—	2	4	no	—
111	C₂³×C₂₉	C₄⋊Q₈	1	2	yes	—
112	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	—	2	1	no	—
114	C₂×C₄×C₂₉	—	2	1	no	—
115	C₂×C₄×C₂₉	—	2	4	no	—
116	C₂×C₄×C₂₉	—	2	2	no	—
117	C₂×C₄×C₂₉	—	2	2	no	—
118	C₂×C₄×C₂₉	—	2	2	no	—
119	C₂×C₄×C₂₉	—	2	2	no	—
120	C₂×C₄×C₂₉	—	2	2	no	—
121	C₂×C₄×C₂₉	—	2	4	no	—
122	C₂×C₄×C₂₉	—	4	4	no	—
123	C₂×C₄×C₂₉	—	4	4	no	—
124	C₂×C₄×C₂₉	—	4	4	no	—
125	C₂×C₄×C₂₉	—	4	4	no	—
126	C₂×C₄×C₂₉	—	4	4	no	—
127	C₂×C₄×C₂₉	—	4	4	no	—
128	C₂×C₄×C₂₉	#121	1	4	yes	—
129	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
136	C₈×C₂₉	—	2	2	no	—
137	C₈×C₂₉	—	2	2	no	—
138	C₈×C₂₉	C₂²⋊Q₈	1	4	yes	—
139	C₈×C₂₉	C₂²⋊Q₈	1	4	yes	—
140	C₂×#499	C₂×C₂₉	1	1	yes	characteristic, complemented
141	C₂×#405	C₂×C₂₉	1	1	yes	characteristic
142	#2629	C₂×C₂₉	1	2	yes	—
143	#1992	C₂×C₂₉	1	2	yes	complemented
144	#1944	C₂×C₂₉	1	2	yes	complemented
145	#1941	C₂×C₂₉	1	2	yes	complemented
146	#1844	C₂×C₂₉	1	1	yes	characteristic, complemented
147	#1809	C₂×C₂₉	1	2	yes	—
148	#1807	C₂×C₂₉	1	2	yes	—
149	C₂²×C₂₉	—	2	2	no	—
150	C₂²×C₂₉	—	2	4	no	—
151	C₂²×C₂₉	—	4	2	no	—
152	C₂²×C₂₉	—	4	2	no	—
153	C₂²×C₂₉	C₂×#121	1	1	yes	characteristic
154	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
155	C₂²×C₂₉	#532	1	1	yes	characteristic
156	C₂²×C₂₉	#577	1	4	yes	—
157	C₄×C₂₉	—	2	2	no	—
158	C₄×C₂₉	—	2	4	no	—
159	C₄×C₂₉	—	4	2	no	—
160	C₄×C₂₉	—	8	4	no	—
161	C₄×C₂₉	—	8	4	no	—
162	C₄×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₄×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
164	C₄×C₂₉	#533	1	1	yes	characteristic
165	C₄×C₂₉	#533	1	1	yes	characteristic
166	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
167	C₂³×C₈	C₂²×C₂₉	1	1	yes	characteristic
168	C₂×#125	C₂²×C₂₉	1	2	yes	—
169	C₂×#117	C₂²×C₂₉	1	1	yes	characteristic
170	#499	—	2	4	no	—
171	C₂×#107	C₂²×C₂₉	1	2	yes	—
172	C₂×#106	C₂²×C₂₉	1	2	yes	—
173	C₂×#106	C₂²×C₂₉	1	2	yes	—
174	C₂×#106	C₂²×C₂₉	1	2	yes	—
175	#405	—	2	4	no	—
176	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
177	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
180	#398	—	2	4	no	—
181	#396	—	2	4	no	—
182	#389	C₂²×C₂₉	1	4	yes	—
183	#384	C₂²×C₂₉	1	2	yes	—
184	#380	C₂²×C₂₉	1	4	yes	—
185	#340	C₂²×C₂₉	1	4	yes	—
186	#340	C₂²×C₂₉	1	4	yes	—
187	C₂×C₂₉	—	4	2	no	—
188	C₂×C₂₉	#2857	1	1	yes	characteristic
189	C₂×C₂₉	#3060	1	2	yes	—
190	C₂×C₂₉	#3163	1	2	yes	—
191	C₂×C₂₉	#3174	1	2	yes	—
192	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	Q₈×C₂₉	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	Q₈×C₂₉	1	4	yes	—
199	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
200	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
203	#125	—	2	4	no	—
204	#125	—	2	4	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	D₄×C₂₉	1	2	yes	—
215	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
217	#107	—	2	4	no	—
218	#106	—	2	4	no	—
219	#106	—	2	4	no	—
220	#106	—	2	4	no	—
221	#97	—	2	2	no	—
222	#97	—	2	2	no	—
223	C₄×C₈	—	2	2	no	—
224	C₄×C₈	D₄×C₂₉	1	4	yes	—
225	#95	—	2	4	no	—
226	#95	—	2	4	no	—
227	C₂₉	#33214	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
239	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
240	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
241	M₁₆	—	2	8	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	8	no	—
247	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
249	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
250	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
251	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
252	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
253	C₂×C₈	#41×C₂₉	1	2	yes	—
254	C₂×C₈	#41×C₂₉	1	2	yes	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	8	no	—
261	C₄⋊C₄	—	8	4	no	—
262	C₄⋊C₄	—	8	4	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	8	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	C₂×D₄×C₂₉	1	2	yes	—
268	C₄²	#41×C₂₉	1	2	yes	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	4	no	—
272	C₂³	C₂₉×C₄⋊Q₈	1	2	yes	—
273	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₂₉×#121	1	4	yes	—
290	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	C₂₉×C₂²⋊Q₈	1	4	yes	—
300	C₈	C₂₉×C₂²⋊Q₈	1	4	yes	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	C₂×C₂₉×#121	1	1	yes	characteristic
306	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
307	C₂²	C₂₉×#532	1	1	yes	characteristic
308	C₂²	C₂₉×#577	1	4	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	—	8	4	no	—
314	C₄	C₂×C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
315	C₄	C₂×C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
316	C₄	C₂₉×#533	1	1	yes	characteristic
317	C₄	C₂₉×#533	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	C₂₉×#2857	1	1	yes	characteristic
320	C₂	C₂₉×#3060	1	2	yes	—
321	C₂	C₂₉×#3163	1	2	yes	—
322	C₂	C₂₉×#3174	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
prime radical	C₂⁴×C₂₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₂₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×#499	C₂	semidirect
4. C₂₉×#1992	C₂	semidirect
5. C₂₉×#1944	C₂	semidirect
6. C₂₉×#1941	C₂	semidirect
7. C₂₉×#1844	C₂	semidirect
107. #33214	C₂₉	direct (N × H)
140. C₂×#499	C₂×C₂₉	semidirect
143. #1992	C₂×C₂₉	semidirect
144. #1944	C₂×C₂₉	semidirect
145. #1941	C₂×C₂₉	semidirect
146. #1844	C₂×C₂₉	semidirect
227. C₂₉	#33214	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×C₂₉×#499	2
C₂×C₂₉×#405	2
C₂₉×#2629	2
C₂₉×#1992	2
C₂₉×#1944	2
C₂₉×#1941	2
C₂₉×#1844	2
C₂₉×#1809	2
C₂₉×#1807	2
#33214	29

Subgroups — C29×#33214 #20328114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33214 #20328114