Subgroups — C₂₉×#9274 #20304174

Order 7424.

492 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3208	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#1965	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₉×#331	C₂	1	1	yes	maximal, characteristic
6	C₂₉×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#533	C₂²	1	1	yes	characteristic
9	C₂×C₂₉×#127	C₂²	1	1	yes	characteristic
10	C₂₉×#492	—	2	1	no	—
11	C₂₉×#486	—	2	1	no	—
12	C₂₉×#460	—	2	2	no	—
13	C₂₉×#450	C₄	1	4	yes	—
14	C₂₉×#446	—	2	1	no	—
15	C₂₉×#446	—	2	2	no	—
16	C₈×Q₈×C₂₉	—	2	1	no	—
17	C₂₉×#431	C₂²	1	1	yes	characteristic
18	C₂₉×#424	—	2	1	no	—
19	C₂₉×#423	C₂²	1	1	yes	characteristic
20	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₉×#100	—	2	1	no	—
23	C₂×C₂₉×#100	—	2	1	no	—
24	C₂₉×#405	—	2	1	no	—
25	C₄×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#333	—	2	1	no	—
27	C₂₉×#332	—	2	1	no	—
28	C₂₉×#331	—	2	1	no	—
29	C₂₉×#331	—	2	1	no	—
30	C₂²×D₄×C₂₉	—	2	1	no	—
31	C₂²×D₄×C₂₉	—	2	1	no	—
32	C₂²×D₄×C₂₉	C₂×C₄	1	2	yes	—
33	C₂×SD₁₆×C₂₉	—	2	4	no	—
34	C₂×M₁₆×C₂₉	—	2	1	no	—
35	C₂×M₁₆×C₂₉	D₄	1	1	yes	characteristic
36	C₂×M₁₆×C₂₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₉	—	2	1	no	—
38	C₂²×C₈×C₂₉	—	2	1	no	—
39	C₂₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
40	C₂₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₂₉×#127	—	2	1	no	—
42	C₂₉×#127	—	2	1	no	—
43	C₂₉×#127	—	2	1	no	—
44	C₂₉×#127	—	2	1	no	—
45	C₂₉×#125	—	2	1	no	—
46	C₂₉×C₂²⋊Q₈	—	2	1	no	—
47	C₂₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₂₉	—	2	1	no	—
49	C₄×Q₈×C₂₉	D₄	1	2	yes	—
50	C₂₉×#117	—	2	1	no	—
51	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
52	C₂₉×C₄⋊C₈	—	2	2	no	—
53	C₂₉×C₄⋊C₈	—	2	2	no	—
54	C₂₉×C₄⋊C₈	—	2	1	no	—
55	C₂₉×C₄⋊C₈	—	2	1	no	—
56	C₂₉×C₄⋊C₈	—	2	1	no	—
57	C₂₉×C₄⋊C₈	—	2	1	no	—
58	C₂₉×C₄⋊C₈	—	2	1	no	—
59	C₂₉×C₄⋊C₈	—	2	1	no	—
60	C₂₉×C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₂₉×C₄⋊C₈	D₄	1	1	yes	characteristic
62	C₂₉×#102	—	2	4	no	—
63	C₂₉×#100	—	4	2	no	—
64	C₂₉×#100	—	4	2	no	—
65	C₂₉×C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₂₉×#97	—	2	2	no	—
67	C₂₉×#97	—	2	1	no	—
68	C₄×C₈×C₂₉	—	2	2	no	—
69	C₄×C₈×C₂₉	—	2	1	no	—
70	C₄×C₈×C₂₉	—	2	1	no	—
71	C₂⁴×C₂₉	—	2	2	no	—
72	C₂×Q₈×C₂₉	—	2	1	no	—
73	C₂×Q₈×C₂₉	C₂²⋊C₄	1	2	yes	—
74	C₂×D₄×C₂₉	—	2	2	no	—
75	C₂×D₄×C₂₉	—	2	2	no	—
76	C₂×D₄×C₂₉	—	4	1	no	—
77	C₂×D₄×C₂₉	—	4	1	no	—
78	C₂×D₄×C₂₉	—	4	1	no	—
79	C₂×D₄×C₂₉	—	4	1	no	—
80	C₂×D₄×C₂₉	—	4	1	no	—
81	C₂×D₄×C₂₉	—	4	1	no	—
82	C₂×D₄×C₂₉	—	4	1	no	—
83	C₂×D₄×C₂₉	—	4	1	no	—
84	C₂×D₄×C₂₉	—	8	1	no	—
85	C₂×D₄×C₂₉	—	8	1	no	—
86	C₂×D₄×C₂₉	—	8	2	no	—
87	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₂₉	#41	1	1	yes	characteristic
90	SD₁₆×C₂₉	—	8	4	no	—
91	M₁₆×C₂₉	—	2	1	no	—
92	M₁₆×C₂₉	—	2	1	no	—
93	M₁₆×C₂₉	—	2	1	no	—
94	M₁₆×C₂₉	—	4	1	no	—
95	M₁₆×C₂₉	—	4	1	no	—
96	M₁₆×C₂₉	SD₁₆	1	2	yes	complemented
97	C₂×C₈×C₂₉	—	2	1	no	—
98	C₂×C₈×C₂₉	—	2	2	no	—
99	C₂×C₈×C₂₉	—	2	1	no	—
100	C₂×C₈×C₂₉	—	2	1	no	—
101	C₂×C₈×C₂₉	—	2	1	no	—
102	C₂×C₈×C₂₉	—	4	1	no	—
103	C₂×C₈×C₂₉	—	4	1	no	—
104	C₂×C₈×C₂₉	—	4	1	no	—
105	C₂×C₈×C₂₉	—	4	1	no	—
106	C₂×C₈×C₂₉	—	8	1	no	—
107	C₄⋊C₄×C₂₉	—	2	2	no	—
108	C₄⋊C₄×C₂₉	—	2	1	no	—
109	C₄⋊C₄×C₂₉	—	2	1	no	—
110	C₄⋊C₄×C₂₉	—	2	1	no	—
111	C₄⋊C₄×C₂₉	—	2	2	no	—
112	C₄⋊C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
113	C₂²⋊C₄×C₂₉	—	2	2	no	—
114	C₄²×C₂₉	—	2	2	no	—
115	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₂₉	#41	1	1	yes	characteristic
119	#9274	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₉	—	4	2	no	—
121	C₂³×C₂₉	—	4	2	no	—
122	C₂³×C₂₉	—	4	2	no	—
123	C₂³×C₂₉	—	8	2	no	—
124	C₂³×C₂₉	—	8	2	no	—
125	C₂³×C₂₉	C₄×D₄	1	1	yes	characteristic
126	Q₈×C₂₉	—	4	2	no	—
127	Q₈×C₂₉	—	4	1	no	—
128	Q₈×C₂₉	—	4	1	no	—
129	D₄×C₂₉	—	4	4	no	—
130	D₄×C₂₉	—	8	2	no	—
131	D₄×C₂₉	—	8	2	no	—
132	D₄×C₂₉	—	8	1	no	—
133	D₄×C₂₉	—	8	1	no	—
134	D₄×C₂₉	—	8	1	no	—
135	D₄×C₂₉	—	8	1	no	—
136	D₄×C₂₉	—	8	1	no	—
137	D₄×C₂₉	—	8	1	no	—
138	C₂×C₄×C₂₉	—	2	1	no	—
139	C₂×C₄×C₂₉	—	2	1	no	—
140	C₂×C₄×C₂₉	—	2	2	no	—
141	C₂×C₄×C₂₉	—	2	2	no	—
142	C₂×C₄×C₂₉	—	4	1	no	—
143	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₂₉	#120	1	1	yes	characteristic
146	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₂₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
149	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₂₉	#135	1	1	yes	characteristic
151	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₂₉	#137	1	1	yes	characteristic
153	C₈×C₂₉	—	2	2	no	—
154	C₈×C₂₉	—	4	2	no	—
155	C₈×C₂₉	—	4	1	no	—
156	C₈×C₂₉	—	4	1	no	—
157	C₈×C₂₉	—	4	1	no	—
158	C₈×C₂₉	—	8	1	no	—
159	C₈×C₂₉	—	8	1	no	—
160	#3208	C₂×C₂₉	1	1	yes	characteristic
161	#2962	C₂×C₂₉	1	1	yes	characteristic, complemented
162	#1965	C₂×C₂₉	1	1	yes	characteristic
163	#1550	C₂×C₂₉	1	1	yes	characteristic, complemented
164	C₂×#331	C₂×C₂₉	1	1	yes	characteristic
165	#1301	C₂×C₂₉	1	1	yes	characteristic, complemented
166	#1283	C₂×C₂₉	1	1	yes	characteristic, complemented
167	C₂²×C₂₉	—	2	1	no	—
168	C₂²×C₂₉	—	2	1	no	—
169	C₂²×C₂₉	—	8	2	no	—
170	C₂²×C₂₉	—	8	2	no	—
171	C₂²×C₂₉	—	8	2	no	—
172	C₂²×C₂₉	—	16	2	no	—
173	C₂²×C₂₉	—	16	2	no	—
174	C₂²×C₂₉	#386	1	1	yes	characteristic
175	C₂²×C₂₉	C₄×SD₁₆	1	1	yes	characteristic
176	C₂²×C₂₉	#440	1	1	yes	characteristic
177	C₄×C₂₉	—	2	1	no	—
178	C₄×C₂₉	—	2	1	no	—
179	C₄×C₂₉	—	2	1	no	—
180	C₄×C₂₉	—	4	2	no	—
181	C₄×C₂₉	—	4	2	no	—
182	C₄×C₂₉	#448	1	1	yes	characteristic
183	C₄×C₂₉	#449	1	1	yes	characteristic
184	C₄×C₂₉	#460	1	1	yes	characteristic
185	C₄×C₂₉	#464	1	1	yes	characteristic
186	C₄×C₂₉	#471	1	1	yes	characteristic
187	C₄×C₂₉	#472	1	1	yes	characteristic
188	#533	C₂²×C₂₉	1	1	yes	characteristic
189	C₂×#127	C₂²×C₂₉	1	1	yes	characteristic
190	#492	—	2	1	no	—
191	#486	—	2	1	no	—
192	#460	—	2	2	no	—
193	#450	C₄×C₂₉	1	4	yes	—
194	#446	—	2	2	no	—
195	#446	—	2	1	no	—
196	C₈×Q₈	—	2	1	no	—
197	#431	C₂²×C₂₉	1	1	yes	characteristic
198	#424	—	2	1	no	—
199	#423	C₂²×C₂₉	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
202	C₂×#100	—	2	1	no	—
203	C₂×#100	—	2	1	no	—
204	#405	—	2	1	no	—
205	C₄×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
206	#333	—	2	1	no	—
207	#332	—	2	1	no	—
208	#331	—	2	1	no	—
209	#331	—	2	1	no	—
210	C₂×C₂₉	—	2	1	no	—
211	C₂×C₂₉	—	2	1	no	—
212	C₂×C₂₉	—	16	2	no	—
213	C₂×C₂₉	#1880	1	1	yes	characteristic
214	C₂×C₂₉	#1907	1	1	yes	characteristic
215	C₂×C₂₉	#1914	1	1	yes	characteristic
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	—	2	1	no	—
218	C₂²×D₄	C₂×C₄×C₂₉	1	2	yes	—
219	C₂×SD₁₆	—	2	4	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	D₄×C₂₉	1	1	yes	characteristic
222	C₂×M₁₆	D₄×C₂₉	1	1	yes	characteristic
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
226	C₄⋊Q₈	D₄×C₂₉	1	1	yes	characteristic
227	#127	—	2	1	no	—
228	#127	—	2	1	no	—
229	#127	—	2	1	no	—
230	#127	—	2	1	no	—
231	#125	—	2	1	no	—
232	C₂²⋊Q₈	—	2	1	no	—
233	C₂²⋊Q₈	C₂×C₄×C₂₉	1	2	yes	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×Q₈	D₄×C₂₉	1	2	yes	—
236	#117	—	2	1	no	—
237	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	D₄×C₂₉	1	1	yes	characteristic
247	C₄⋊C₈	D₄×C₂₉	1	1	yes	characteristic
248	#102	—	2	4	no	—
249	#100	—	4	2	no	—
250	#100	—	4	2	no	—
251	C₂²⋊C₈	C₂×C₄×C₂₉	1	2	yes	—
252	#97	—	2	2	no	—
253	#97	—	2	1	no	—
254	C₄×C₈	—	2	2	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	2	1	no	—
257	C₂₉	#9274	1	1	yes	characteristic, direct product factor, Sylow
258	C₂⁴	—	2	2	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	C₂²⋊C₄×C₂₉	1	2	yes	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	8	1	no	—
272	C₂×D₄	—	8	1	no	—
273	C₂×D₄	—	8	2	no	—
274	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
276	C₂²×C₄	#41×C₂₉	1	1	yes	characteristic
277	SD₁₆	—	8	4	no	—
278	M₁₆	—	2	1	no	—
279	M₁₆	—	2	1	no	—
280	M₁₆	—	2	1	no	—
281	M₁₆	—	4	1	no	—
282	M₁₆	—	4	1	no	—
283	M₁₆	SD₁₆×C₂₉	1	2	yes	complemented
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	8	1	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
305	C₄²	#41×C₂₉	1	1	yes	characteristic
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	8	2	no	—
310	C₂³	—	8	2	no	—
311	C₂³	C₄×D₄×C₂₉	1	1	yes	characteristic
312	Q₈	—	4	2	no	—
313	Q₈	—	4	1	no	—
314	Q₈	—	4	1	no	—
315	D₄	—	4	4	no	—
316	D₄	—	8	2	no	—
317	D₄	—	8	2	no	—
318	D₄	—	8	1	no	—
319	D₄	—	8	1	no	—
320	D₄	—	8	1	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
330	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
331	C₂×C₄	C₂₉×#120	1	1	yes	characteristic
332	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
333	C₂×C₄	C₂₉×#121	1	1	yes	characteristic
334	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
335	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
336	C₂×C₄	C₂₉×#135	1	1	yes	characteristic
337	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
338	C₂×C₄	C₂₉×#137	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	4	2	no	—
341	C₈	—	4	1	no	—
342	C₈	—	4	1	no	—
343	C₈	—	4	1	no	—
344	C₈	—	8	1	no	—
345	C₈	—	8	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	16	2	no	—
352	C₂²	—	16	2	no	—
353	C₂²	C₂₉×#386	1	1	yes	characteristic
354	C₂²	C₄×SD₁₆×C₂₉	1	1	yes	characteristic
355	C₂²	C₂₉×#440	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	C₂₉×#448	1	1	yes	characteristic
362	C₄	C₂₉×#449	1	1	yes	characteristic
363	C₄	C₂₉×#460	1	1	yes	characteristic
364	C₄	C₂₉×#464	1	1	yes	characteristic
365	C₄	C₂₉×#471	1	1	yes	characteristic
366	C₄	C₂₉×#472	1	1	yes	characteristic
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	16	2	no	—
370	C₂	C₂₉×#1880	1	1	yes	characteristic
371	C₂	C₂₉×#1907	1	1	yes	characteristic
372	C₂	C₂₉×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
prime radical	C₂×C₂₉×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ 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
2. C₂₉×#2962	C₂	semidirect
4. C₂₉×#1550	C₂	semidirect
6. C₂₉×#1301	C₂	semidirect
7. C₂₉×#1283	C₂	semidirect
96. M₁₆×C₂₉	SD₁₆	semidirect
119. #9274	C₂₉	direct (N × H)
161. #2962	C₂×C₂₉	semidirect
163. #1550	C₂×C₂₉	semidirect
165. #1301	C₂×C₂₉	semidirect
166. #1283	C₂×C₂₉	semidirect
257. C₂₉	#9274	direct (N × H)
283. M₁₆	SD₁₆×C₂₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₉×#3208	2
C₂₉×#2962	2
C₂₉×#1965	2
C₂₉×#1550	2
C₂×C₂₉×#331	2
C₂₉×#1301	2
C₂₉×#1283	2
#9274	29

Subgroups — C29×#9274 #20304174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#9274 #20304174