Subgroups — C₂₉×#10162 #20305062

Order 7424.

510 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3206	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#3141	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1548	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#1492	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#532	C₂²	1	1	yes	characteristic
9	C₂×C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂₉×#492	—	2	1	no	—
12	C₂₉×#489	—	2	2	no	—
13	C₂₉×#487	—	2	2	no	—
14	C₂₉×#487	—	2	1	no	—
15	C₂₉×#487	—	2	2	no	—
16	C₂₉×#487	—	2	1	no	—
17	C₂₉×#486	—	2	1	no	—
18	C₂₉×#461	—	2	2	no	—
19	C₂₉×#448	—	2	2	no	—
20	C₂₉×#431	C₂²	1	1	yes	characteristic
21	C₂₉×#423	C₂²	1	1	yes	characteristic
22	C₂₉×#417	—	2	2	no	—
23	C₂×C₂₉×#103	—	2	2	no	—
24	C₄×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#343	—	2	2	no	—
27	C₂₉×#333	—	2	2	no	—
28	C₂₉×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₂₉	—	2	1	no	—
30	C₂²×Q₈×C₂₉	—	2	1	no	—
31	C₂²×Q₈×C₂₉	D₄	1	2	yes	—
32	C₂×SD₁₆×C₂₉	—	2	4	no	—
33	C₂×M₁₆×C₂₉	—	2	1	no	—
34	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
35	C₂²×C₈×C₂₉	—	2	1	no	—
36	C₂₉×C₄⋊Q₈	—	2	1	no	—
37	C₂₉×C₄⋊Q₈	—	2	1	no	—
38	C₂₉×C₄⋊Q₈	—	2	1	no	—
39	C₂₉×C₄⋊Q₈	—	2	1	no	—
40	C₂₉×C₄⋊Q₈	—	2	1	no	—
41	C₂₉×C₄⋊Q₈	—	2	1	no	—
42	C₂₉×C₄⋊Q₈	—	2	1	no	—
43	C₂₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₂₉×C₄⋊Q₈	D₄	1	2	yes	—
45	C₂₉×#127	D₄	1	1	yes	characteristic
46	C₂₉×#124	—	2	1	no	—
47	C₂₉×#121	—	2	1	no	—
48	C₂₉×#121	—	2	1	no	—
49	C₄×D₄×C₂₉	—	2	1	no	—
50	C₄×D₄×C₂₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
55	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	C₂₉×C₄⋊C₈	—	2	1	no	—
57	C₂₉×C₄⋊C₈	—	2	1	no	—
58	C₂₉×C₄⋊C₈	D₄	1	2	yes	—
59	C₂₉×#103	—	2	4	no	—
60	C₂₉×#103	—	2	8	no	—
61	C₂₉×#103	—	2	4	no	—
62	C₂₉×#102	—	2	4	no	—
63	C₂₉×C₂²⋊C₈	—	2	1	no	—
64	C₂₉×#97	—	2	1	no	—
65	C₄×C₈×C₂₉	—	2	1	no	—
66	C₄×C₈×C₂₉	—	2	1	no	—
67	C₄×C₈×C₂₉	—	2	1	no	—
68	C₄×C₈×C₂₉	D₄	1	2	yes	—
69	#41×C₂₉	—	4	2	no	—
70	#41×C₂₉	—	4	2	no	—
71	C₂×Q₈×C₂₉	—	2	1	no	—
72	C₂×Q₈×C₂₉	—	2	2	no	—
73	C₂×Q₈×C₂₉	—	2	2	no	—
74	C₂×Q₈×C₂₉	—	2	2	no	—
75	C₂×Q₈×C₂₉	—	4	4	no	—
76	C₂×Q₈×C₂₉	—	4	1	no	—
77	C₂×Q₈×C₂₉	—	8	2	no	—
78	C₂×D₄×C₂₉	—	2	1	no	—
79	C₂×D₄×C₂₉	—	2	1	no	—
80	C₂×D₄×C₂₉	—	2	1	no	—
81	C₂×D₄×C₂₉	—	4	1	no	—
82	C₂²×C₄×C₂₉	—	2	2	no	—
83	C₂²×C₄×C₂₉	—	4	1	no	—
84	C₂²×C₄×C₂₉	—	4	1	no	—
85	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
88	SD₁₆×C₂₉	—	8	4	no	—
89	M₁₆×C₂₉	—	2	2	no	—
90	M₁₆×C₂₉	—	2	2	no	—
91	M₁₆×C₂₉	—	4	2	no	—
92	C₂×C₈×C₂₉	—	2	2	no	—
93	C₂×C₈×C₂₉	—	2	2	no	—
94	C₂×C₈×C₂₉	—	2	2	no	—
95	C₂×C₈×C₂₉	—	4	2	no	—
96	C₂×C₈×C₂₉	—	4	1	no	—
97	C₄⋊C₄×C₂₉	—	2	2	no	—
98	C₄⋊C₄×C₂₉	—	2	2	no	—
99	C₄⋊C₄×C₂₉	—	2	1	no	—
100	C₄⋊C₄×C₂₉	—	2	1	no	—
101	C₄⋊C₄×C₂₉	—	4	1	no	—
102	C₄⋊C₄×C₂₉	—	4	1	no	—
103	C₄⋊C₄×C₂₉	—	4	1	no	—
104	C₄⋊C₄×C₂₉	—	4	1	no	—
105	C₄⋊C₄×C₂₉	—	4	1	no	—
106	C₄⋊C₄×C₂₉	—	4	1	no	—
107	C₂²⋊C₄×C₂₉	—	4	1	no	—
108	C₂²⋊C₄×C₂₉	—	4	1	no	—
109	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
113	#10162	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₉	—	4	1	no	—
115	C₂³×C₂₉	#121	1	1	yes	characteristic
116	Q₈×C₂₉	—	4	4	no	—
117	Q₈×C₂₉	—	4	2	no	—
118	Q₈×C₂₉	—	4	4	no	—
119	Q₈×C₂₉	—	8	2	no	—
120	Q₈×C₂₉	—	8	2	no	—
121	D₄×C₂₉	—	4	2	no	—
122	D₄×C₂₉	—	4	2	no	—
123	D₄×C₂₉	—	4	2	no	—
124	D₄×C₂₉	—	8	2	no	—
125	C₂×C₄×C₂₉	—	2	2	no	—
126	C₂×C₄×C₂₉	—	2	2	no	—
127	C₂×C₄×C₂₉	—	4	1	no	—
128	C₂×C₄×C₂₉	—	4	2	no	—
129	C₂×C₄×C₂₉	—	4	2	no	—
130	C₂×C₄×C₂₉	—	8	2	no	—
131	C₂×C₄×C₂₉	—	8	1	no	—
132	C₂×C₄×C₂₉	—	8	1	no	—
133	C₂×C₄×C₂₉	—	8	2	no	—
134	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
137	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₂₉	#135	1	2	yes	—
141	C₂×C₄×C₂₉	#137	1	2	yes	—
142	C₈×C₂₉	—	4	2	no	—
143	C₈×C₂₉	—	4	2	no	—
144	C₈×C₂₉	—	4	2	no	—
145	C₈×C₂₉	—	8	1	no	—
146	#3206	C₂×C₂₉	1	1	yes	characteristic
147	#3141	C₂×C₂₉	1	1	yes	characteristic
148	C₂×#487	C₂×C₂₉	1	1	yes	characteristic, complemented
149	#1997	C₂×C₂₉	1	1	yes	characteristic, complemented
150	#1548	C₂×C₂₉	1	1	yes	characteristic, complemented
151	#1492	C₂×C₂₉	1	1	yes	characteristic
152	#1278	C₂×C₂₉	1	1	yes	characteristic, complemented
153	C₂²×C₂₉	—	2	1	no	—
154	C₂²×C₂₉	—	2	1	no	—
155	C₂²×C₂₉	—	8	1	no	—
156	C₂²×C₂₉	—	8	1	no	—
157	C₂²×C₂₉	—	8	1	no	—
158	C₂²×C₂₉	#392	1	1	yes	characteristic
159	C₂²×C₂₉	#464	1	2	yes	—
160	C₄×C₂₉	—	2	1	no	—
161	C₄×C₂₉	—	2	1	no	—
162	C₄×C₂₉	—	8	2	no	—
163	C₄×C₂₉	—	8	1	no	—
164	C₄×C₂₉	—	8	2	no	—
165	C₄×C₂₉	#452	1	2	yes	—
166	C₄×C₂₉	#455	1	2	yes	—
167	C₄×C₂₉	#471	1	2	yes	—
168	C₄×C₂₉	#495	1	1	yes	characteristic
169	C₄×C₂₉	#497	1	1	yes	characteristic
170	#532	C₂²×C₂₉	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
173	#492	—	2	1	no	—
174	#489	—	2	2	no	—
175	#487	—	2	2	no	—
176	#487	—	2	1	no	—
177	#487	—	2	1	no	—
178	#487	—	2	2	no	—
179	#486	—	2	1	no	—
180	#461	—	2	2	no	—
181	#448	—	2	2	no	—
182	#431	C₂²×C₂₉	1	1	yes	characteristic
183	#423	C₂²×C₂₉	1	1	yes	characteristic
184	#417	—	2	2	no	—
185	C₂×#103	—	2	2	no	—
186	C₄×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
187	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
188	#343	—	2	2	no	—
189	#333	—	2	2	no	—
190	D₄⋊C₈	—	2	2	no	—
191	C₂×C₂₉	—	2	2	no	—
192	C₂×C₂₉	—	16	1	no	—
193	C₂×C₂₉	#2004	1	1	yes	characteristic
194	C₂×C₂₉	#2015	1	1	yes	characteristic
195	C₂×C₂₉	#2019	1	1	yes	characteristic
196	C₂×#41	—	2	1	no	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×Q₈	D₄×C₂₉	1	2	yes	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×M₁₆	—	2	1	no	—
201	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
202	C₂²×C₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	D₄×C₂₉	1	2	yes	—
211	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
212	#127	D₄×C₂₉	1	1	yes	characteristic
213	#124	—	2	1	no	—
214	#121	—	2	1	no	—
215	#121	—	2	1	no	—
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	D₄×C₂₉	1	2	yes	—
226	#103	—	2	8	no	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	#102	—	2	4	no	—
230	C₂²⋊C₈	—	2	1	no	—
231	#97	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	D₄×C₂₉	1	2	yes	—
236	C₂₉	#10162	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	4	2	no	—
238	#41	—	4	2	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	8	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
255	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
256	SD₁₆	—	8	4	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	1	no	—
277	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
281	C₂³	—	4	1	no	—
282	C₂³	C₂₉×#121	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
302	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
303	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
304	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
305	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
306	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
307	C₂×C₄	C₂₉×#135	1	2	yes	—
308	C₂×C₄	C₂₉×#137	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	8	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	C₂₉×#392	1	1	yes	characteristic
319	C₂²	C₂₉×#464	1	2	yes	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	C₂₉×#452	1	2	yes	—
326	C₄	C₂₉×#455	1	2	yes	—
327	C₄	C₂₉×#471	1	2	yes	—
328	C₄	C₂₉×#495	1	1	yes	characteristic
329	C₄	C₂₉×#497	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	16	1	no	—
332	C₂	C₂₉×#2004	1	1	yes	characteristic
333	C₂	C₂₉×#2015	1	1	yes	characteristic
334	C₂	C₂₉×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
prime radical	C₂₉×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₉×#487	C₂	semidirect
4. C₂₉×#1997	C₂	semidirect
5. C₂₉×#1548	C₂	semidirect
7. C₂₉×#1278	C₂	semidirect
113. #10162	C₂₉	direct (N × H)
148. C₂×#487	C₂×C₂₉	semidirect
149. #1997	C₂×C₂₉	semidirect
150. #1548	C₂×C₂₉	semidirect
152. #1278	C₂×C₂₉	semidirect
236. C₂₉	#10162	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×#3206	2
C₂₉×#3141	2
C₂×C₂₉×#487	2
C₂₉×#1997	2
C₂₉×#1548	2
C₂₉×#1492	2
C₂₉×#1278	2
#10162	29

Subgroups — C29×#10162 #20305062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#10162 #20305062