Subgroups — C₂₉×#10286 #20305186

Order 7424.

542 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3227	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#3226	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1962	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#1546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1276	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×#500	—	2	1	no	—
12	C₂₉×#498	—	2	1	no	—
13	C₂₉×#486	—	2	1	no	—
14	C₂₉×#486	—	2	1	no	—
15	C₂₉×#481	—	2	2	no	—
16	C₂₉×#480	—	2	2	no	—
17	C₂₉×#476	—	2	2	no	—
18	C₂₉×#474	—	2	2	no	—
19	C₂₉×#431	C₂²	1	1	yes	characteristic
20	C₂₉×#431	C₂²	1	1	yes	characteristic
21	C₂₉×#423	C₂²	1	1	yes	characteristic
22	C₂×C₂₉×#104	—	2	2	no	—
23	C₂₉×#391	—	2	2	no	—
24	C₂₉×#391	—	2	2	no	—
25	C₂₉×#382	—	2	2	no	—
26	C₄³×C₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×#332	—	2	2	no	—
28	C₂₉×#332	—	2	2	no	—
29	C₂×#41×C₂₉	—	2	1	no	—
30	C₂²×Q₈×C₂₉	—	2	1	no	—
31	C₂²×Q₈×C₂₉	—	2	1	no	—
32	C₂×M₁₆×C₂₉	—	2	1	no	—
33	C₂₉×C₄⋊Q₈	—	2	1	no	—
34	C₂₉×C₄⋊Q₈	—	2	1	no	—
35	C₂₉×C₄⋊Q₈	—	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	D₄	1	1	yes	characteristic
42	C₂₉×#127	D₄	1	1	yes	characteristic
43	C₂₉×#124	—	2	1	no	—
44	C₂₉×#121	—	2	1	no	—
45	C₂₉×#121	—	2	1	no	—
46	C₄×D₄×C₂₉	—	2	1	no	—
47	C₄×D₄×C₂₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
52	C₂×C₄²×C₂₉	—	2	1	no	—
53	C₂×C₄²×C₂₉	—	2	1	no	—
54	C₂×C₄²×C₂₉	D₄	1	2	yes	—
55	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
56	C₂₉×#107	—	2	4	no	—
57	C₂₉×#106	—	2	4	no	—
58	C₂₉×C₄⋊C₈	—	2	1	no	—
59	C₂₉×C₄⋊C₈	—	2	1	no	—
60	C₂₉×C₄⋊C₈	D₄	1	2	yes	—
61	C₂₉×C₄⋊C₈	D₄	1	2	yes	—
62	C₂₉×#104	—	4	4	no	—
63	C₂₉×#102	—	2	4	no	—
64	C₂₉×#102	—	2	4	no	—
65	C₂₉×C₂²⋊C₈	—	2	1	no	—
66	C₂₉×C₂²⋊C₈	—	2	1	no	—
67	C₄×C₈×C₂₉	—	2	1	no	—
68	C₄×C₈×C₂₉	—	2	1	no	—
69	C₂₉×#95	—	2	8	no	—
70	#41×C₂₉	—	4	2	no	—
71	#41×C₂₉	—	4	2	no	—
72	C₂×Q₈×C₂₉	—	2	1	no	—
73	C₂×Q₈×C₂₉	—	4	4	no	—
74	C₂×Q₈×C₂₉	—	4	1	no	—
75	C₂×Q₈×C₂₉	—	4	1	no	—
76	C₂×Q₈×C₂₉	—	4	1	no	—
77	C₂×Q₈×C₂₉	—	4	1	no	—
78	C₂×Q₈×C₂₉	—	4	2	no	—
79	C₂×Q₈×C₂₉	—	4	2	no	—
80	C₂×D₄×C₂₉	—	2	1	no	—
81	C₂×D₄×C₂₉	—	2	1	no	—
82	C₂×D₄×C₂₉	—	2	1	no	—
83	C₂×D₄×C₂₉	—	4	1	no	—
84	C₂²×C₄×C₂₉	—	2	2	no	—
85	C₂²×C₄×C₂₉	—	2	2	no	—
86	C₂²×C₄×C₂₉	—	2	2	no	—
87	C₂²×C₄×C₂₉	—	4	1	no	—
88	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
91	M₁₆×C₂₉	—	4	2	no	—
92	C₂×C₈×C₂₉	—	2	2	no	—
93	C₂×C₈×C₂₉	—	2	2	no	—
94	C₂×C₈×C₂₉	—	4	1	no	—
95	C₄⋊C₄×C₂₉	—	2	2	no	—
96	C₄⋊C₄×C₂₉	—	2	2	no	—
97	C₄⋊C₄×C₂₉	—	2	1	no	—
98	C₄⋊C₄×C₂₉	—	2	1	no	—
99	C₄⋊C₄×C₂₉	—	2	2	no	—
100	C₄⋊C₄×C₂₉	—	2	2	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	C₂²⋊C₄×C₂₉	—	4	1	no	—
106	C₂²⋊C₄×C₂₉	—	4	1	no	—
107	C₄²×C₂₉	—	2	4	no	—
108	C₄²×C₂₉	—	2	2	no	—
109	C₄²×C₂₉	—	2	2	no	—
110	C₄²×C₂₉	—	4	2	no	—
111	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₂₉	#41	1	1	yes	characteristic
115	#10286	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₂₉	—	4	1	no	—
117	C₂³×C₂₉	#120	1	1	yes	characteristic
118	Q₈×C₂₉	—	4	2	no	—
119	Q₈×C₂₉	—	8	2	no	—
120	Q₈×C₂₉	—	8	2	no	—
121	Q₈×C₂₉	—	8	2	no	—
122	Q₈×C₂₉	—	8	1	no	—
123	Q₈×C₂₉	—	8	1	no	—
124	D₄×C₂₉	—	4	2	no	—
125	D₄×C₂₉	—	4	2	no	—
126	D₄×C₂₉	—	4	2	no	—
127	D₄×C₂₉	—	8	2	no	—
128	C₂×C₄×C₂₉	—	2	2	no	—
129	C₂×C₄×C₂₉	—	2	2	no	—
130	C₂×C₄×C₂₉	—	2	4	no	—
131	C₂×C₄×C₂₉	—	4	2	no	—
132	C₂×C₄×C₂₉	—	4	2	no	—
133	C₂×C₄×C₂₉	—	4	4	no	—
134	C₂×C₄×C₂₉	—	4	1	no	—
135	C₂×C₄×C₂₉	—	4	2	no	—
136	C₂×C₄×C₂₉	—	4	2	no	—
137	C₂×C₄×C₂₉	—	4	4	no	—
138	C₂×C₄×C₂₉	—	8	1	no	—
139	C₂×C₄×C₂₉	—	8	1	no	—
140	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₂₉	#127	1	1	yes	characteristic
146	C₂×C₄×C₂₉	#135	1	2	yes	—
147	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₂₉	—	4	2	no	—
149	C₈×C₂₉	—	4	2	no	—
150	C₈×C₂₉	—	8	1	no	—
151	#3227	C₂×C₂₉	1	1	yes	characteristic
152	#3226	C₂×C₂₉	1	1	yes	characteristic
153	#2862	C₂×C₂₉	1	1	yes	characteristic, complemented
154	#1962	C₂×C₂₉	1	1	yes	characteristic
155	#1549	C₂×C₂₉	1	1	yes	characteristic, complemented
156	#1546	C₂×C₂₉	1	1	yes	characteristic, complemented
157	#1276	C₂×C₂₉	1	1	yes	characteristic, complemented
158	C₂²×C₂₉	—	2	1	no	—
159	C₂²×C₂₉	—	2	1	no	—
160	C₂²×C₂₉	—	8	1	no	—
161	C₂²×C₂₉	—	8	1	no	—
162	C₂²×C₂₉	—	8	1	no	—
163	C₂²×C₂₉	#392	1	1	yes	characteristic
164	C₂²×C₂₉	#449	1	2	yes	—
165	C₄×C₂₉	—	2	1	no	—
166	C₄×C₂₉	—	2	1	no	—
167	C₄×C₂₉	—	4	4	no	—
168	C₄×C₂₉	—	8	2	no	—
169	C₄×C₂₉	—	8	2	no	—
170	C₄×C₂₉	—	8	1	no	—
171	C₄×C₂₉	#454	1	2	yes	—
172	C₄×C₂₉	#456	1	2	yes	—
173	C₄×C₂₉	#463	1	2	yes	—
174	C₄×C₂₉	#495	1	1	yes	characteristic
175	C₄×C₂₉	#496	1	1	yes	characteristic
176	#532	C₂²×C₂₉	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
179	#500	—	2	1	no	—
180	#498	—	2	1	no	—
181	#486	—	2	1	no	—
182	#486	—	2	1	no	—
183	#481	—	2	2	no	—
184	#480	—	2	2	no	—
185	#476	—	2	2	no	—
186	#474	—	2	2	no	—
187	#431	C₂²×C₂₉	1	1	yes	characteristic
188	#431	C₂²×C₂₉	1	1	yes	characteristic
189	#423	C₂²×C₂₉	1	1	yes	characteristic
190	C₂×#104	—	2	2	no	—
191	#391	—	2	2	no	—
192	#391	—	2	2	no	—
193	#382	—	2	2	no	—
194	C₄³	C₂²×C₂₉	1	1	yes	characteristic
195	#332	—	2	2	no	—
196	#332	—	2	2	no	—
197	C₂×C₂₉	—	2	2	no	—
198	C₂×C₂₉	—	16	1	no	—
199	C₂×C₂₉	#2003	1	1	yes	characteristic
200	C₂×C₂₉	#2005	1	1	yes	characteristic
201	C₂×C₂₉	#2012	1	1	yes	characteristic
202	C₂×#41	—	2	1	no	—
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂×M₁₆	—	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	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
215	#127	D₄×C₂₉	1	1	yes	characteristic
216	#124	—	2	1	no	—
217	#121	—	2	1	no	—
218	#121	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
224	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	—	2	1	no	—
227	C₂×C₄²	D₄×C₂₉	1	2	yes	—
228	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
229	#107	—	2	4	no	—
230	#106	—	2	4	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	D₄×C₂₉	1	2	yes	—
234	C₄⋊C₈	D₄×C₂₉	1	2	yes	—
235	#104	—	4	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	C₂²⋊C₈	—	2	1	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	#95	—	2	8	no	—
243	C₂₉	#10286	1	1	yes	characteristic, direct product factor, Sylow
244	#41	—	4	2	no	—
245	#41	—	4	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
265	M₁₆	—	4	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
288	C₄²	#41×C₂₉	1	1	yes	characteristic
289	C₂³	—	4	1	no	—
290	C₂³	C₂₉×#120	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	8	2	no	—
295	Q₈	—	8	1	no	—
296	Q₈	—	8	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
314	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
315	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
316	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
317	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
318	C₂×C₄	C₂₉×#127	1	1	yes	characteristic
319	C₂×C₄	C₂₉×#135	1	2	yes	—
320	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	2	yes	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₂₉×#392	1	1	yes	characteristic
330	C₂²	C₂₉×#449	1	2	yes	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	4	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	C₂₉×#454	1	2	yes	—
338	C₄	C₂₉×#456	1	2	yes	—
339	C₄	C₂₉×#463	1	2	yes	—
340	C₄	C₂₉×#495	1	1	yes	characteristic
341	C₄	C₂₉×#496	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	16	1	no	—
344	C₂	C₂₉×#2003	1	1	yes	characteristic
345	C₂	C₂₉×#2005	1	1	yes	characteristic
346	C₂	C₂₉×#2012	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. C₂₉×#2862	C₂	semidirect
5. C₂₉×#1549	C₂	semidirect
6. C₂₉×#1546	C₂	semidirect
7. C₂₉×#1276	C₂	semidirect
115. #10286	C₂₉	direct (N × H)
153. #2862	C₂×C₂₉	semidirect
155. #1549	C₂×C₂₉	semidirect
156. #1546	C₂×C₂₉	semidirect
157. #1276	C₂×C₂₉	semidirect
243. C₂₉	#10286	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×#3227	2
C₂₉×#3226	2
C₂₉×#2862	2
C₂₉×#1962	2
C₂₉×#1549	2
C₂₉×#1546	2
C₂₉×#1276	2
#10286	29

Subgroups — C29×#10286 #20305186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#10286 #20305186