Subgroups — C₂₉×#13398 #20308298

Order 7424.

458 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₉×#361	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#1811	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×#121	—	2	1	no	—
11	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
13	C₂₉×#469	—	2	2	no	—
14	C₂₉×#469	—	2	2	no	—
15	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×#108	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
18	C₂×C₂₉×#102	—	2	1	no	—
19	C₂₉×#407	—	2	1	no	—
20	C₂₉×#365	—	2	2	no	—
21	C₂₉×#361	—	2	2	no	—
22	C₂₉×#361	—	2	2	no	—
23	C₂₉×#349	—	2	2	no	—
24	C₂₉×#341	—	2	1	no	—
25	C₂²×D₄×C₂₉	—	2	1	no	—
26	C₂²×D₄×C₂₉	—	4	1	no	—
27	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
28	C₂×D₈×C₂₉	—	2	1	no	—
29	C₂×D₈×C₂₉	—	2	1	no	—
30	C₂×D₈×C₂₉	—	2	1	no	—
31	C₂×D₈×C₂₉	—	2	1	no	—
32	C₂×D₈×C₂₉	—	2	1	no	—
33	C₂×D₈×C₂₉	D₄	1	2	yes	—
34	C₂×M₁₆×C₂₉	—	2	1	no	—
35	C₂×M₁₆×C₂₉	—	2	1	no	—
36	C₂×M₁₆×C₂₉	—	2	1	no	—
37	C₂×M₁₆×C₂₉	—	2	1	no	—
38	C₂×M₁₆×C₂₉	—	2	1	no	—
39	C₂×M₁₆×C₂₉	—	2	1	no	—
40	C₂×M₁₆×C₂₉	—	2	1	no	—
41	C₂×M₁₆×C₂₉	—	2	1	no	—
42	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
44	C₂₉×#121	—	2	4	no	—
45	C₂₉×#121	—	4	2	no	—
46	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₂₉	—	4	1	no	—
48	C₂₉×M₃₂	—	2	1	no	—
49	C₂₉×M₃₂	—	2	1	no	—
50	C₂₉×M₃₂	—	2	1	no	—
51	C₂₉×M₃₂	D₄	1	2	yes	complemented
52	C₂×C₁₆×C₂₉	—	2	1	no	—
53	C₂×C₁₆×C₂₉	—	2	1	no	—
54	C₂₉×#108	—	2	1	no	—
55	C₂₉×#108	—	2	1	no	—
56	C₂₉×#108	—	2	1	no	—
57	C₂₉×#108	—	2	1	no	—
58	C₂₉×#106	—	2	1	no	—
59	C₂₉×#106	D₄	1	2	yes	—
60	C₂₉×#102	—	2	4	no	—
61	C₂₉×C₂²⋊C₈	—	4	2	no	—
62	C₂⁴×C₂₉	—	4	1	no	—
63	C₂⁴×C₂₉	SD₁₆	1	1	yes	characteristic
64	C₂×D₄×C₂₉	—	2	2	no	—
65	C₂×D₄×C₂₉	—	4	4	no	—
66	C₂×D₄×C₂₉	—	4	1	no	—
67	C₂×D₄×C₂₉	—	4	1	no	—
68	C₂×D₄×C₂₉	—	4	1	no	—
69	C₂×D₄×C₂₉	—	4	1	no	—
70	C₂×D₄×C₂₉	—	4	1	no	—
71	C₂×D₄×C₂₉	—	8	4	no	—
72	C₂²×C₄×C₂₉	—	2	1	no	—
73	C₂²×C₄×C₂₉	—	2	1	no	—
74	C₂²×C₄×C₂₉	—	2	1	no	—
75	C₂²×C₄×C₂₉	—	2	1	no	—
76	C₂²×C₄×C₂₉	—	2	1	no	—
77	C₂²×C₄×C₂₉	—	2	1	no	—
78	C₂²×C₄×C₂₉	—	4	1	no	—
79	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
81	D₈×C₂₉	—	2	2	no	—
82	D₈×C₂₉	—	2	2	no	—
83	D₈×C₂₉	—	4	1	no	—
84	D₈×C₂₉	—	4	1	no	—
85	M₁₆×C₂₉	—	4	1	no	—
86	M₁₆×C₂₉	—	4	1	no	—
87	M₁₆×C₂₉	—	4	1	no	—
88	M₁₆×C₂₉	—	4	1	no	—
89	M₁₆×C₂₉	—	4	1	no	—
90	M₁₆×C₂₉	—	4	1	no	—
91	M₁₆×C₂₉	—	4	1	no	—
92	M₁₆×C₂₉	—	4	1	no	—
93	C₂×C₈×C₂₉	—	2	1	no	—
94	C₂×C₈×C₂₉	—	2	1	no	—
95	C₂×C₈×C₂₉	—	2	1	no	—
96	C₂×C₈×C₂₉	—	4	1	no	—
97	C₂×C₈×C₂₉	—	4	1	no	—
98	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
102	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
103	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₂₉	—	2	2	no	—
105	C₄⋊C₄×C₂₉	—	4	1	no	—
106	C₂²⋊C₄×C₂₉	—	4	4	no	—
107	C₁₆×C₂₉	—	2	2	no	—
108	C₁₆×C₂₉	—	4	1	no	—
109	#13398	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₉	—	2	1	no	—
111	C₂³×C₂₉	—	2	1	no	—
112	C₂³×C₂₉	—	2	1	no	—
113	C₂³×C₂₉	—	4	2	no	—
114	C₂³×C₂₉	—	4	2	no	—
115	C₂³×C₂₉	—	8	2	no	—
116	C₂³×C₂₉	—	8	2	no	—
117	C₂³×C₂₉	—	8	1	no	—
118	C₂³×C₂₉	—	8	1	no	—
119	C₂³×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
120	D₄×C₂₉	—	4	2	no	—
121	D₄×C₂₉	—	4	2	no	—
122	D₄×C₂₉	—	8	8	no	—
123	D₄×C₂₉	—	8	1	no	—
124	D₄×C₂₉	—	8	1	no	—
125	C₂×C₄×C₂₉	—	2	1	no	—
126	C₂×C₄×C₂₉	—	2	1	no	—
127	C₂×C₄×C₂₉	—	2	1	no	—
128	C₂×C₄×C₂₉	—	4	1	no	—
129	C₂×C₄×C₂₉	—	4	1	no	—
130	C₂×C₄×C₂₉	—	4	1	no	—
131	C₂×C₄×C₂₉	—	4	1	no	—
132	C₂×C₄×C₂₉	—	4	2	no	—
133	C₂×C₄×C₂₉	—	8	2	no	—
134	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
137	C₂×C₄×C₂₉	#135	1	1	yes	characteristic
138	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₈×C₂₉	—	4	1	no	—
141	C₈×C₂₉	—	8	1	no	—
142	C₈×C₂₉	—	8	1	no	—
143	C₈×C₂₉	#121	1	1	yes	characteristic
144	C₈×C₂₉	#121	1	1	yes	characteristic
145	C₈×C₂₉	#123	1	1	yes	characteristic
146	C₈×C₂₉	#124	1	1	yes	characteristic
147	C₂×#469	C₂×C₂₉	1	1	yes	characteristic
148	C₂×#365	C₂×C₂₉	1	1	yes	characteristic, complemented
149	C₂×#361	C₂×C₂₉	1	1	yes	characteristic, complemented
150	C₂×#361	C₂×C₂₉	1	1	yes	characteristic, complemented
151	C₂×#349	C₂×C₂₉	1	1	yes	characteristic, complemented
152	#1811	C₂×C₂₉	1	1	yes	characteristic, complemented
153	#1383	C₂×C₂₉	1	1	yes	characteristic, complemented
154	C₂²×C₂₉	—	2	2	no	—
155	C₂²×C₂₉	—	4	2	no	—
156	C₂²×C₂₉	—	4	2	no	—
157	C₂²×C₂₉	—	4	1	no	—
158	C₂²×C₂₉	—	8	1	no	—
159	C₂²×C₂₉	—	8	2	no	—
160	C₂²×C₂₉	—	8	2	no	—
161	C₂²×C₂₉	—	8	2	no	—
162	C₂²×C₂₉	—	16	2	no	—
163	C₂²×C₂₉	—	16	2	no	—
164	C₂²×C₂₉	#450	1	1	yes	characteristic
165	C₂²×C₂₉	#465	1	1	yes	characteristic
166	C₂²×C₂₉	#481	1	1	yes	characteristic
167	C₄×C₂₉	—	4	1	no	—
168	C₄×C₂₉	—	8	2	no	—
169	C₄×C₂₉	#394	1	1	yes	characteristic
170	C₄×C₂₉	#449	1	1	yes	characteristic
171	C₄×C₂₉	#469	1	1	yes	characteristic
172	C₄×C₂₉	#482	1	1	yes	characteristic
173	C₂²×D₈	C₂²×C₂₉	1	1	yes	characteristic
174	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
175	C₂×#121	—	2	1	no	—
176	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
177	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
178	#469	—	2	2	no	—
179	#469	—	2	2	no	—
180	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
181	C₂×#108	C₂²×C₂₉	1	1	yes	characteristic
182	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
183	C₂×#102	—	2	1	no	—
184	#407	—	2	1	no	—
185	#365	—	2	2	no	—
186	#361	—	2	2	no	—
187	#361	—	2	2	no	—
188	#349	—	2	2	no	—
189	#341	—	2	1	no	—
190	C₂×C₂₉	—	2	1	no	—
191	C₂×C₂₉	—	2	1	no	—
192	C₂×C₂₉	—	8	1	no	—
193	C₂×C₂₉	—	16	2	no	—
194	C₂×C₂₉	#2034	1	1	yes	characteristic
195	C₂×C₂₉	#2238	1	2	yes	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	—	4	1	no	—
198	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	D₄×C₂₉	1	2	yes	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
215	#121	—	2	4	no	—
216	#121	—	4	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	4	1	no	—
219	M₃₂	—	2	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	D₄×C₂₉	1	2	yes	complemented
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	—	2	1	no	—
227	#108	—	2	1	no	—
228	#108	—	2	1	no	—
229	#106	—	2	1	no	—
230	#106	D₄×C₂₉	1	2	yes	—
231	#102	—	2	4	no	—
232	C₂²⋊C₈	—	4	2	no	—
233	C₂₉	#13398	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	—	4	1	no	—
235	C₂⁴	SD₁₆×C₂₉	1	1	yes	characteristic
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	8	4	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	1	no	—
256	D₈	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
273	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
274	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
275	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₁₆	—	2	2	no	—
280	C₁₆	—	4	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	8	2	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	8	8	no	—
294	D₄	—	8	1	no	—
295	D₄	—	8	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
306	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
307	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
308	C₂×C₄	C₂₉×#135	1	1	yes	characteristic
309	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	—	4	1	no	—
312	C₈	—	8	1	no	—
313	C₈	—	8	1	no	—
314	C₈	C₂₉×#121	1	1	yes	characteristic
315	C₈	C₂₉×#121	1	1	yes	characteristic
316	C₈	C₂₉×#123	1	1	yes	characteristic
317	C₈	C₂₉×#124	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	16	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	C₂₉×#450	1	1	yes	characteristic
329	C₂²	C₂₉×#465	1	1	yes	characteristic
330	C₂²	C₂₉×#481	1	1	yes	characteristic
331	C₄	—	4	1	no	—
332	C₄	—	8	2	no	—
333	C₄	C₂₉×#394	1	1	yes	characteristic
334	C₄	C₂₉×#449	1	1	yes	characteristic
335	C₄	C₂₉×#469	1	1	yes	characteristic
336	C₄	C₂₉×#482	1	1	yes	characteristic
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	8	1	no	—
340	C₂	—	16	2	no	—
341	C₂	C₂₉×#2034	1	1	yes	characteristic
342	C₂	C₂₉×#2238	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₉	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
prime radical	C₂×C₂₉×#469	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×M₁₆×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#365	C₂	semidirect
3. C₂×C₂₉×#361	C₂	semidirect
4. C₂×C₂₉×#361	C₂	semidirect
5. C₂×C₂₉×#349	C₂	semidirect
6. C₂₉×#1811	C₂	semidirect
7. C₂₉×#1383	C₂	semidirect
51. C₂₉×M₃₂	D₄	semidirect
109. #13398	C₂₉	direct (N × H)
148. C₂×#365	C₂×C₂₉	semidirect
149. C₂×#361	C₂×C₂₉	semidirect
150. C₂×#361	C₂×C₂₉	semidirect
151. C₂×#349	C₂×C₂₉	semidirect
152. #1811	C₂×C₂₉	semidirect
153. #1383	C₂×C₂₉	semidirect
222. M₃₂	D₄×C₂₉	semidirect
233. C₂₉	#13398	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×#469	2
C₂×C₂₉×#365	2
C₂×C₂₉×#361	2
C₂×C₂₉×#349	2
C₂₉×#1811	2
C₂₉×#1383	2
#13398	29

Subgroups — C29×#13398 #20308298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#13398 #20308298