Subgroups — C₂₉×#21718 #20316618

Order 7424.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2399	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1985	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1985	C₂	1	1	yes	maximal, characteristic
6	C₂₉×#1939	C₂	1	4	yes	maximal
7	C₂₉×#1843	C₂	1	1	yes	maximal, characteristic
8	C₂₉×#1771	C₂	1	4	yes	maximal
9	C₂₉×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#446	—	2	4	no	—
13	C₈×Q₈×C₂₉	—	2	4	no	—
14	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₉×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₂₉×#97	C₂²	1	2	yes	—
20	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
21	C₂×C₂₉×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
25	C₂₉×#398	C₄	1	2	yes	—
26	C₂₉×#398	C₄	1	2	yes	—
27	C₂₉×#395	C₄	1	2	yes	—
28	C₂₉×#395	C₄	1	2	yes	—
29	C₄×C₄⋊C₄×C₂₉	C₂²	1	4	yes	—
30	C₂₉×#376	C₂²	1	1	yes	characteristic
31	C₂₉×#336	C₂²	1	4	yes	—
32	C₂₉×#336	C₂²	1	4	yes	—
33	C₂²×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₂₉	—	2	2	no	—
41	C₄×Q₈×C₂₉	—	2	2	no	—
42	C₄×Q₈×C₂₉	—	2	2	no	—
43	C₄×Q₈×C₂₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
52	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
54	C₂₉×C₄⋊C₈	—	2	4	no	—
55	C₂₉×C₄⋊C₈	—	2	4	no	—
56	C₂₉×C₄⋊C₈	—	2	2	no	—
57	C₂₉×C₄⋊C₈	—	2	2	no	—
58	C₂₉×C₄⋊C₈	—	2	8	no	—
59	C₂₉×#97	—	2	4	no	—
60	C₂₉×#97	—	2	2	no	—
61	C₂₉×#97	Q₈	1	8	yes	complemented
62	C₄×C₈×C₂₉	—	2	4	no	—
63	C₄×C₈×C₂₉	—	2	4	no	—
64	C₄×C₈×C₂₉	D₄	1	4	yes	—
65	C₂₉×#95	C₂×C₄	1	2	yes	—
66	C₂₉×#95	C₂×C₄	1	2	yes	—
67	C₂₉×#95	C₂×C₄	1	4	yes	—
68	C₂₉×#95	C₂×C₄	1	4	yes	—
69	C₂×Q₈×C₂₉	—	2	2	no	—
70	C₂×Q₈×C₂₉	—	2	2	no	—
71	C₂×Q₈×C₂₉	—	2	4	no	—
72	C₂×Q₈×C₂₉	—	4	2	no	—
73	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₂₉	—	2	2	no	—
82	C₂×C₈×C₂₉	—	2	2	no	—
83	C₂×C₈×C₂₉	—	2	2	no	—
84	C₂×C₈×C₂₉	—	2	2	no	—
85	C₂×C₈×C₂₉	—	2	2	no	—
86	C₂×C₈×C₂₉	—	2	4	no	—
87	C₂×C₈×C₂₉	—	2	4	no	—
88	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
91	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
92	C₄⋊C₄×C₂₉	—	2	4	no	—
93	C₄⋊C₄×C₂₉	—	2	2	no	—
94	C₄⋊C₄×C₂₉	—	2	2	no	—
95	C₄⋊C₄×C₂₉	—	2	4	no	—
96	C₄⋊C₄×C₂₉	—	2	8	no	—
97	C₄⋊C₄×C₂₉	—	4	4	no	—
98	C₄²×C₂₉	—	2	2	no	—
99	C₄²×C₂₉	—	2	2	no	—
100	C₄²×C₂₉	—	2	2	no	—
101	C₄²×C₂₉	—	2	2	no	—
102	C₄²×C₂₉	C₂×D₄	1	2	yes	—
103	C₄²×C₂₉	C₂×Q₈	1	4	yes	—
104	C₄²×C₂₉	#41	1	2	yes	—
105	C₄²×C₂₉	#41	1	4	yes	—
106	#21718	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₉	C₂³×C₄	1	1	yes	characteristic
108	Q₈×C₂₉	—	2	8	no	—
109	Q₈×C₂₉	—	4	4	no	—
110	C₂×C₄×C₂₉	—	2	2	no	—
111	C₂×C₄×C₂₉	—	2	4	no	—
112	C₂×C₄×C₂₉	—	2	2	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	2	4	no	—
115	C₂×C₄×C₂₉	—	2	2	no	—
116	C₂×C₄×C₂₉	—	2	2	no	—
117	C₂×C₄×C₂₉	—	2	2	no	—
118	C₂×C₄×C₂₉	—	2	2	no	—
119	C₂×C₄×C₂₉	—	2	4	no	—
120	C₂×C₄×C₂₉	—	2	2	no	—
121	C₂×C₄×C₂₉	—	2	4	no	—
122	C₂×C₄×C₂₉	—	2	2	no	—
123	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₂₉	C₄×Q₈	1	8	yes	—
126	C₂×C₄×C₂₉	#131	1	4	yes	—
127	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂₉	C₂²×Q₈	1	2	yes	—
129	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
130	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₂₉	—	2	2	no	—
132	C₈×C₂₉	—	2	4	no	—
133	C₈×C₂₉	—	4	2	no	—
134	C₈×C₂₉	—	4	2	no	—
135	C₈×C₂₉	C₄⋊Q₈	1	4	yes	—
136	C₂×#446	C₂×C₂₉	1	1	yes	characteristic
137	C₂×C₈×Q₈	C₂×C₂₉	1	1	yes	characteristic
138	#2399	C₂×C₂₉	1	1	yes	characteristic
139	#1985	C₂×C₂₉	1	1	yes	characteristic
140	#1985	C₂×C₂₉	1	1	yes	characteristic
141	#1939	C₂×C₂₉	1	4	yes	—
142	#1843	C₂×C₂₉	1	1	yes	characteristic
143	#1771	C₂×C₂₉	1	4	yes	—
144	#1726	C₂×C₂₉	1	1	yes	characteristic
145	C₂²×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₂₉	C₂×C₄×Q₈	1	2	yes	—
147	C₂²×C₂₉	C₂×#131	1	2	yes	—
148	C₂²×C₂₉	#568	1	2	yes	—
149	C₄×C₂₉	—	2	2	no	—
150	C₄×C₂₉	—	2	4	no	—
151	C₄×C₂₉	—	2	2	no	—
152	C₄×C₂₉	—	2	8	no	—
153	C₄×C₂₉	—	4	2	no	—
154	C₄×C₂₉	—	4	2	no	—
155	C₄×C₂₉	—	4	2	no	—
156	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₄×C₂₉	#532	1	1	yes	characteristic
158	C₄×C₂₉	#533	1	2	yes	—
159	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
160	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
161	#446	—	2	4	no	—
162	C₈×Q₈	—	2	4	no	—
163	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₂₉	1	4	yes	—
166	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
168	C₂×#97	C₂²×C₂₉	1	2	yes	—
169	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
170	C₂×#97	C₂²×C₂₉	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
173	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
174	#398	C₄×C₂₉	1	2	yes	—
175	#398	C₄×C₂₉	1	2	yes	—
176	#395	C₄×C₂₉	1	2	yes	—
177	#395	C₄×C₂₉	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₂₉	1	4	yes	—
179	#376	C₂²×C₂₉	1	1	yes	characteristic
180	#336	C₂²×C₂₉	1	4	yes	—
181	#336	C₂²×C₂₉	1	4	yes	—
182	C₂×C₂₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
183	C₂×C₂₉	#2929	1	1	yes	characteristic
184	C₂×C₂₉	#2930	1	1	yes	characteristic
185	C₂×C₂₉	#2962	1	4	yes	—
186	C₂²×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
187	C₂²×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
189	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
190	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
192	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
193	C₄×Q₈	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
203	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
204	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
205	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	8	no	—
212	#97	—	2	4	no	—
213	#97	—	2	2	no	—
214	#97	Q₈×C₂₉	1	8	yes	complemented
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	D₄×C₂₉	1	4	yes	—
218	#95	C₂×C₄×C₂₉	1	2	yes	—
219	#95	C₂×C₄×C₂₉	1	2	yes	—
220	#95	C₂×C₄×C₂₉	1	4	yes	—
221	#95	C₂×C₄×C₂₉	1	4	yes	—
222	C₂₉	#21718	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
245	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂×D₄×C₂₉	1	2	yes	—
257	C₄²	C₂×Q₈×C₂₉	1	4	yes	—
258	C₄²	#41×C₂₉	1	2	yes	—
259	C₄²	#41×C₂₉	1	4	yes	—
260	C₂³	C₂³×C₄×C₂₉	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	Q₈	—	4	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
278	C₂×C₄	C₄×Q₈×C₂₉	1	8	yes	—
279	C₂×C₄	C₂₉×#131	1	4	yes	—
280	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×Q₈×C₂₉	1	2	yes	—
282	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
283	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	C₂₉×C₄⋊Q₈	1	4	yes	—
289	C₂²	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
290	C₂²	C₂×C₄×Q₈×C₂₉	1	2	yes	—
291	C₂²	C₂×C₂₉×#131	1	2	yes	—
292	C₂²	C₂₉×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
301	C₄	C₂₉×#532	1	1	yes	characteristic
302	C₄	C₂₉×#533	1	2	yes	—
303	C₂	C₄×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
304	C₂	C₂₉×#2929	1	1	yes	characteristic
305	C₂	C₂₉×#2930	1	1	yes	characteristic
306	C₂	C₂₉×#2962	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle, prime radical	C₂³×C₂₉	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
61. C₂₉×#97	Q₈	semidirect
106. #21718	C₂₉	direct (N × H)
214. #97	Q₈×C₂₉	semidirect
222. C₂₉	#21718	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×#446	2
C₂×C₈×Q₈×C₂₉	2
C₂₉×#2399	2
C₂₉×#1985	2
C₂₉×#1939	2
C₂₉×#1843	2
C₂₉×#1771	2
C₂₉×#1726	2
#21718	29

Subgroups — C29×#21718 #20316618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#21718 #20316618