Subgroups — C₂₉×#8668 #20303568

Order 7424.

432 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2965	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1806	C₂	1	2	yes	maximal
3	C₄×C₂₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1303	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₂₉	C₂²	1	1	yes	characteristic
7	C₂₉×#446	C₄	1	1	yes	characteristic, complemented
8	C₂₉×#446	C₄	1	1	yes	characteristic, complemented
9	C₂₉×#436	C₄	1	1	yes	characteristic, complemented
10	C₂₉×#436	C₄	1	1	yes	characteristic, complemented
11	C₂₉×#435	—	2	1	no	—
12	C₂₉×#435	—	2	1	no	—
13	C₂₉×#431	—	2	1	no	—
14	C₂₉×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₉×#107	—	2	1	no	—
16	C₂₉×#423	C₂²	1	2	yes	—
17	C₂×C₂₉×#104	—	2	2	no	—
18	C₄×M₁₆×C₂₉	—	2	1	no	—
19	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
21	C₂₉×#340	—	2	1	no	—
22	C₂₉×#339	—	2	2	no	—
23	C₂₉×#334	—	2	1	no	—
24	C₂₉×#334	—	2	1	no	—
25	C₂×#41×C₂₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₉	—	2	1	no	—
27	C₂×M₁₆×C₂₉	—	2	1	no	—
28	C₂×M₁₆×C₂₉	—	2	2	no	—
29	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
31	C₄×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₂₉	Q₈	1	1	yes	characteristic
33	C₄×D₄×C₂₉	—	2	1	no	—
34	C₄×D₄×C₂₉	—	2	1	no	—
35	C₄×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₂₉	Q₈	1	1	yes	characteristic
37	C₂₉×#117	—	2	1	no	—
38	C₂₉×#117	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
40	C₂×C₄²×C₂₉	—	2	1	no	—
41	C₂×C₄²×C₂₉	—	2	2	no	—
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂₉×#107	—	2	4	no	—
44	C₂₉×C₄⋊C₈	—	2	1	no	—
45	C₂₉×C₄⋊C₈	—	2	1	no	—
46	C₂₉×C₄⋊C₈	—	2	2	no	—
47	C₂₉×C₄⋊C₈	—	2	2	no	—
48	C₂₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂₉×#104	—	4	4	no	—
51	C₂₉×C₂²⋊C₈	—	2	1	no	—
52	C₂₉×C₂²⋊C₈	—	2	1	no	—
53	C₂₉×C₂²⋊C₈	—	2	1	no	—
54	C₂₉×#97	—	2	1	no	—
55	C₂₉×#97	—	2	1	no	—
56	C₄×C₈×C₂₉	—	2	1	no	—
57	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
61	C₂₉×#95	—	2	2	no	—
62	#41×C₂₉	—	2	2	no	—
63	#41×C₂₉	—	2	2	no	—
64	C₂×Q₈×C₂₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₂₉	—	2	1	no	—
66	C₂×D₄×C₂₉	C₄⋊C₄	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	—	4	2	no	—
71	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
74	M₁₆×C₂₉	—	4	2	no	—
75	M₁₆×C₂₉	—	4	2	no	—
76	M₁₆×C₂₉	—	4	2	no	—
77	M₁₆×C₂₉	—	4	2	no	—
78	C₂×C₈×C₂₉	—	2	2	no	—
79	C₂×C₈×C₂₉	—	2	1	no	—
80	C₂×C₈×C₂₉	—	2	1	no	—
81	C₂×C₈×C₂₉	—	4	1	no	—
82	C₂×C₈×C₂₉	—	4	2	no	—
83	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₂₉	—	2	1	no	—
88	C₄⋊C₄×C₂₉	—	2	1	no	—
89	C₄⋊C₄×C₂₉	—	2	4	no	—
90	C₄⋊C₄×C₂₉	—	4	2	no	—
91	C₄⋊C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
93	C₂²⋊C₄×C₂₉	—	2	1	no	—
94	C₂²⋊C₄×C₂₉	—	2	1	no	—
95	C₂²⋊C₄×C₂₉	—	2	1	no	—
96	C₄²×C₂₉	—	2	1	no	—
97	C₄²×C₂₉	—	2	1	no	—
98	C₄²×C₂₉	—	2	1	no	—
99	C₄²×C₂₉	—	4	2	no	—
100	C₄²×C₂₉	—	4	2	no	—
101	C₄²×C₂₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₂₉	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₂₉	#41	1	1	yes	characteristic
105	#8668	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₉	—	2	1	no	—
107	C₂³×C₂₉	#121	1	1	yes	characteristic
108	Q₈×C₂₉	—	2	2	no	—
109	D₄×C₂₉	—	2	2	no	—
110	D₄×C₂₉	—	4	1	no	—
111	D₄×C₂₉	—	4	1	no	—
112	C₂×C₄×C₂₉	—	2	1	no	—
113	C₂×C₄×C₂₉	—	2	1	no	—
114	C₂×C₄×C₂₉	—	2	1	no	—
115	C₂×C₄×C₂₉	—	2	1	no	—
116	C₂×C₄×C₂₉	—	2	1	no	—
117	C₂×C₄×C₂₉	—	4	1	no	—
118	C₂×C₄×C₂₉	—	4	2	no	—
119	C₂×C₄×C₂₉	—	4	1	no	—
120	C₂×C₄×C₂₉	—	4	1	no	—
121	C₂×C₄×C₂₉	—	4	2	no	—
122	C₂×C₄×C₂₉	—	4	4	no	—
123	C₂×C₄×C₂₉	—	8	2	no	—
124	C₂×C₄×C₂₉	—	8	2	no	—
125	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₂₉	#137	1	2	yes	—
133	C₈×C₂₉	—	2	1	no	—
134	C₈×C₂₉	—	2	1	no	—
135	C₈×C₂₉	—	2	1	no	—
136	C₈×C₂₉	—	4	1	no	—
137	C₈×C₂₉	—	4	1	no	—
138	C₈×C₂₉	—	8	2	no	—
139	C₈×C₂₉	#104	1	2	yes	complemented
140	#2965	C₂×C₂₉	1	1	yes	characteristic
141	#1806	C₂×C₂₉	1	2	yes	—
142	C₄×#107	C₂×C₂₉	1	1	yes	characteristic, complemented
143	#1566	C₂×C₂₉	1	1	yes	characteristic, complemented
144	#1303	C₂×C₂₉	1	2	yes	complemented
145	C₂²×C₂₉	—	2	1	no	—
146	C₂²×C₂₉	—	2	1	no	—
147	C₂²×C₂₉	—	4	1	no	—
148	C₂²×C₂₉	—	4	1	no	—
149	C₂²×C₂₉	—	4	1	no	—
150	C₂²×C₂₉	#380	1	1	yes	characteristic
151	C₂²×C₂₉	#469	1	1	yes	characteristic
152	C₂²×C₂₉	#470	1	1	yes	characteristic
153	C₄×C₂₉	—	2	1	no	—
154	C₄×C₂₉	—	2	1	no	—
155	C₄×C₂₉	—	2	1	no	—
156	C₄×C₂₉	—	4	1	no	—
157	C₄×C₂₉	—	4	1	no	—
158	C₄×C₂₉	—	4	1	no	—
159	C₄×C₂₉	—	8	4	no	—
160	C₄×C₂₉	C₂×#104	1	1	yes	characteristic
161	C₄×C₂₉	#421	1	1	yes	characteristic
162	C₄×C₂₉	#428	1	2	yes	—
163	C₄×C₂₉	#468	1	2	yes	—
164	C₄×#41	C₂²×C₂₉	1	1	yes	characteristic
165	#446	C₄×C₂₉	1	1	yes	characteristic, complemented
166	#446	C₄×C₂₉	1	1	yes	characteristic, complemented
167	#436	C₄×C₂₉	1	1	yes	characteristic, complemented
168	#436	C₄×C₂₉	1	1	yes	characteristic, complemented
169	#435	—	2	1	no	—
170	#435	—	2	1	no	—
171	#431	—	2	1	no	—
172	#431	C₂²×C₂₉	1	1	yes	characteristic
173	C₂×#107	—	2	1	no	—
174	#423	C₂²×C₂₉	1	2	yes	—
175	C₂×#104	—	2	2	no	—
176	C₄×M₁₆	—	2	1	no	—
177	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
178	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
179	#340	—	2	1	no	—
180	#339	—	2	2	no	—
181	#334	—	2	1	no	—
182	#334	—	2	1	no	—
183	C₂×C₂₉	—	2	1	no	—
184	C₂×C₂₉	—	2	1	no	—
185	C₂×C₂₉	—	8	1	no	—
186	C₂×C₂₉	#1806	1	1	yes	characteristic
187	C₂×C₂₉	#1809	1	1	yes	characteristic
188	C₂×C₂₉	#1814	1	1	yes	characteristic
189	C₂×#41	D₄×C₂₉	1	1	yes	characteristic
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
195	C₄×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
196	C₄×Q₈	Q₈×C₂₉	1	1	yes	characteristic
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	C₂×C₄×C₂₉	1	1	yes	characteristic
200	C₄×D₄	Q₈×C₂₉	1	1	yes	characteristic
201	#117	—	2	1	no	—
202	#117	D₄×C₂₉	1	1	yes	characteristic
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
213	C₄⋊C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
214	#104	—	4	4	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	#97	—	2	1	no	—
219	#97	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
222	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
223	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
224	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
225	#95	—	2	2	no	—
226	C₂₉	#8668	1	1	yes	characteristic, direct product factor, Sylow
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	C₂×Q₈	C₄⋊C₄×C₂₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
238	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	2	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	C₂²×C₄×C₂₉	1	1	yes	characteristic
267	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
268	C₄²	C₂×Q₈×C₂₉	1	1	yes	characteristic
269	C₄²	#41×C₂₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	C₂₉×#121	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
290	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
291	C₂×C₄	C₂₉×#117	1	1	yes	characteristic
292	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
293	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
294	C₂×C₄	C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
295	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	2	yes	—
296	C₂×C₄	C₂₉×#137	1	2	yes	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	8	2	no	—
303	C₈	C₂₉×#104	1	2	yes	complemented
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂₉×#380	1	1	yes	characteristic
310	C₂²	C₂₉×#469	1	1	yes	characteristic
311	C₂²	C₂₉×#470	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	8	4	no	—
319	C₄	C₂×C₂₉×#104	1	1	yes	characteristic
320	C₄	C₂₉×#421	1	1	yes	characteristic
321	C₄	C₂₉×#428	1	2	yes	—
322	C₄	C₂₉×#468	1	2	yes	—
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	1	no	—
326	C₂	C₂₉×#1806	1	1	yes	characteristic
327	C₂	C₂₉×#1809	1	1	yes	characteristic
328	C₂	C₂₉×#1814	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	#380
prime radical	C₂×#41×C₂₉	D₄
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
3. C₄×C₂₉×#107	C₂	semidirect
4. C₂₉×#1566	C₂	semidirect
5. C₂₉×#1303	C₂	semidirect
7. C₂₉×#446	C₄	semidirect
8. C₂₉×#446	C₄	semidirect
9. C₂₉×#436	C₄	semidirect
10. C₂₉×#436	C₄	semidirect
105. #8668	C₂₉	direct (N × H)
139. C₈×C₂₉	#104	semidirect
142. C₄×#107	C₂×C₂₉	semidirect
143. #1566	C₂×C₂₉	semidirect
144. #1303	C₂×C₂₉	semidirect
165. #446	C₄×C₂₉	semidirect
166. #446	C₄×C₂₉	semidirect
167. #436	C₄×C₂₉	semidirect
168. #436	C₄×C₂₉	semidirect
226. C₂₉	#8668	direct (N × H)
303. 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×#2965	2
C₂₉×#1806	2
C₄×C₂₉×#107	2
C₂₉×#1566	2
C₂₉×#1303	2
#8668	29

Subgroups — C29×#8668 #20303568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#8668 #20303568