Subgroups — C₂₉×#28813 #20323713

Order 7424.

822 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×#500	C₂	1	2	yes	maximal
2	C₂₉×#2551	C₂	1	2	yes	maximal
3	C₂₉×#2057	C₂	1	4	yes	maximal
4	C₂₉×#1947	C₂	1	2	yes	maximal
5	C₂₉×#1942	C₂	1	2	yes	maximal
6	C₂₉×#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₉×#107	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₉×C₄⋊Q₈	—	2	2	no	—
10	C₂×C₂₉×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₂₉×#500	—	2	8	no	—
12	C₂×C₂₉×#107	C₂²	1	2	yes	—
13	C₂×C₂₉×#107	C₂²	1	4	yes	—
14	C₂×C₂₉×#107	C₂²	1	2	yes	—
15	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×#395	—	2	4	no	—
21	C₂₉×#389	—	2	2	no	—
22	C₂₉×#389	C₂²	1	4	yes	—
23	C₂₉×#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
26	C₂₉×#340	C₂²	1	4	yes	—
27	C₂₉×#340	C₂²	1	2	yes	—
28	C₂₉×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₂₉	—	2	4	no	—
30	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
32	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
33	C₂₉×C₄⋊Q₈	—	2	8	no	—
34	C₂₉×C₄⋊Q₈	—	4	4	no	—
35	C₂₉×C₄⋊Q₈	—	4	8	no	—
36	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
45	C₂×C₄²×C₂₉	—	2	2	no	—
46	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
48	C₂₉×#107	—	2	4	no	—
49	C₂₉×#107	—	2	8	no	—
50	C₂₉×#107	—	4	2	no	—
51	C₂₉×#107	D₄	1	4	yes	—
52	C₂₉×#107	Q₈	1	4	yes	complemented
53	C₂₉×C₄⋊C₈	Q₈	1	4	yes	complemented
54	C₄×C₈×C₂₉	—	2	4	no	—
55	C₄×C₈×C₂₉	Q₈	1	4	yes	complemented
56	C₂₉×#95	—	2	2	no	—
57	C₂₉×#95	—	2	4	no	—
58	C₂₉×#95	D₄	1	2	yes	—
59	C₂×Q₈×C₂₉	—	4	8	no	—
60	C₂×Q₈×C₂₉	—	4	8	no	—
61	C₂×Q₈×C₂₉	—	8	4	no	—
62	C₂²×C₄×C₂₉	—	2	2	no	—
63	C₂²×C₄×C₂₉	—	2	2	no	—
64	C₂²×C₄×C₂₉	—	2	4	no	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₂₉	—	2	2	no	—
71	C₂×C₈×C₂₉	—	2	1	no	—
72	C₂×C₈×C₂₉	—	2	1	no	—
73	C₂×C₈×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
79	C₄⋊C₄×C₂₉	—	2	2	no	—
80	C₄⋊C₄×C₂₉	—	2	2	no	—
81	C₄⋊C₄×C₂₉	—	2	4	no	—
82	C₄⋊C₄×C₂₉	—	2	4	no	—
83	C₄⋊C₄×C₂₉	—	4	8	no	—
84	C₄⋊C₄×C₂₉	—	4	8	no	—
85	C₄⋊C₄×C₂₉	—	4	4	no	—
86	C₄⋊C₄×C₂₉	—	8	2	no	—
87	C₄⋊C₄×C₂₉	D₈	1	4	yes	—
88	C₄⋊C₄×C₂₉	Q₁₆	1	4	yes	—
89	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₂₉	C₂×Q₈	1	4	yes	—
91	C₄²×C₂₉	—	2	2	no	—
92	C₄²×C₂₉	—	2	2	no	—
93	C₄²×C₂₉	—	4	4	no	—
94	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
95	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
96	#28813	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₂₉	—	8	16	no	—
99	C₂×C₄×C₂₉	—	2	2	no	—
100	C₂×C₄×C₂₉	—	2	2	no	—
101	C₂×C₄×C₂₉	—	2	1	no	—
102	C₂×C₄×C₂₉	—	2	2	no	—
103	C₂×C₄×C₂₉	—	2	1	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	—	2	4	no	—
106	C₂×C₄×C₂₉	—	2	4	no	—
107	C₂×C₄×C₂₉	—	4	2	no	—
108	C₂×C₄×C₂₉	—	4	2	no	—
109	C₂×C₄×C₂₉	—	4	2	no	—
110	C₂×C₄×C₂₉	—	4	2	no	—
111	C₂×C₄×C₂₉	—	4	4	no	—
112	C₂×C₄×C₂₉	—	4	4	no	—
113	C₂×C₄×C₂₉	—	4	4	no	—
114	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₂₉	C₂×D₈	1	4	yes	—
117	C₂×C₄×C₂₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₂₉	C₂×Q₁₆	1	4	yes	—
119	C₂×C₄×C₂₉	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
126	C₈×C₂₉	—	2	2	no	—
127	C₈×C₂₉	—	2	4	no	—
128	C₈×C₂₉	C₄⋊Q₈	1	4	yes	complemented
129	C₂×#500	C₂×C₂₉	1	2	yes	—
130	#2551	C₂×C₂₉	1	2	yes	—
131	#2057	C₂×C₂₉	1	4	yes	—
132	#1947	C₂×C₂₉	1	2	yes	—
133	#1942	C₂×C₂₉	1	2	yes	—
134	#1845	C₂×C₂₉	1	1	yes	characteristic
135	C₄×#107	C₂×C₂₉	1	1	yes	characteristic
136	C₈×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
137	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₂²×C₂₉	D₄²	1	1	yes	characteristic
139	C₂²×C₂₉	D₄×Q₈	1	1	yes	characteristic
140	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
141	C₂²×C₂₉	C₂²×Q₁₆	1	1	yes	characteristic
142	C₂²×C₂₉	#576	1	1	yes	characteristic
143	C₂²×C₂₉	#578	1	1	yes	characteristic
144	C₄×C₂₉	—	2	2	no	—
145	C₄×C₂₉	—	2	4	no	—
146	C₄×C₂₉	—	2	4	no	—
147	C₄×C₂₉	—	4	4	no	—
148	C₄×C₂₉	—	4	4	no	—
149	C₄×C₂₉	—	8	4	no	—
150	C₄×C₂₉	#500	1	4	yes	complemented
151	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₄×C₂₉	#544	1	1	yes	characteristic
153	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
154	C₄×C₂₉	Q₈²	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	—	2	2	no	—
156	C₂×C₄⋊Q₈	C₂²×C₂₉	1	4	yes	—
157	#500	—	2	8	no	—
158	C₂×#107	C₂²×C₂₉	1	4	yes	—
159	C₂×#107	C₂²×C₂₉	1	2	yes	—
160	C₂×#107	C₂²×C₂₉	1	2	yes	—
161	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
162	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
165	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
166	#395	—	2	4	no	—
167	#389	—	2	2	no	—
168	#389	C₂²×C₂₉	1	4	yes	—
169	#384	C₂²×C₂₉	1	2	yes	—
170	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
171	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
172	#340	C₂²×C₂₉	1	2	yes	—
173	#340	C₂²×C₂₉	1	4	yes	—
174	#336	C₂²×C₂₉	1	2	yes	—
175	C₂×C₂₉	#2551	1	1	yes	characteristic
176	C₂×C₂₉	C₂×#500	1	1	yes	characteristic
177	C₂×C₂₉	#3164	1	1	yes	characteristic
178	C₂×C₂₉	D₄×D₈	1	1	yes	characteristic
179	C₂×C₂₉	D₄×Q₁₆	1	1	yes	characteristic
180	C₂×C₂₉	Q₈×D₈	1	1	yes	characteristic
181	C₂×C₂₉	Q₈×Q₁₆	1	1	yes	characteristic
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
185	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
186	C₄⋊Q₈	—	2	8	no	—
187	C₄⋊Q₈	—	4	4	no	—
188	C₄⋊Q₈	—	4	8	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
194	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
200	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	—	2	8	no	—
203	#107	—	4	2	no	—
204	#107	D₄×C₂₉	1	4	yes	—
205	#107	Q₈×C₂₉	1	4	yes	complemented
206	C₄⋊C₈	Q₈×C₂₉	1	4	yes	complemented
207	C₄×C₈	—	2	4	no	—
208	C₄×C₈	Q₈×C₂₉	1	4	yes	complemented
209	#95	—	2	2	no	—
210	#95	—	2	4	no	—
211	#95	D₄×C₂₉	1	2	yes	—
212	C₂₉	#28813	1	1	yes	characteristic, direct product factor, Sylow
213	C₂×Q₈	—	4	8	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
230	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
231	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	8	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	D₈×C₂₉	1	4	yes	—
242	C₄⋊C₄	Q₁₆×C₂₉	1	4	yes	—
243	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
244	C₄⋊C₄	C₂×Q₈×C₂₉	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
249	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
250	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
251	Q₈	—	8	16	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
268	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₂₉	1	4	yes	—
270	C₂×C₄	C₂×D₈×C₂₉	1	2	yes	—
271	C₂×C₄	C₂×Q₁₆×C₂₉	1	4	yes	—
272	C₂×C₄	C₂×Q₁₆×C₂₉	1	2	yes	—
273	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
277	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	C₂₉×C₄⋊Q₈	1	4	yes	complemented
282	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
283	C₂²	D₄²×C₂₉	1	1	yes	characteristic
284	C₂²	D₄×Q₈×C₂₉	1	1	yes	characteristic
285	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
286	C₂²	C₂²×Q₁₆×C₂₉	1	1	yes	characteristic
287	C₂²	C₂₉×#576	1	1	yes	characteristic
288	C₂²	C₂₉×#578	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	2	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₂₉×#500	1	4	yes	complemented
296	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
297	C₄	C₂₉×#544	1	1	yes	characteristic
298	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
299	C₄	Q₈²×C₂₉	1	1	yes	characteristic
300	C₂	C₂₉×#2551	1	1	yes	characteristic
301	C₂	C₂×C₂₉×#500	1	1	yes	characteristic
302	C₂	C₂₉×#3164	1	1	yes	characteristic
303	C₂	D₄×D₈×C₂₉	1	1	yes	characteristic
304	C₂	D₄×Q₁₆×C₂₉	1	1	yes	characteristic
305	C₂	Q₈×D₈×C₂₉	1	1	yes	characteristic
306	C₂	Q₈×Q₁₆×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₄×C₄⋊C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
52. C₂₉×#107	Q₈	semidirect
53. C₂₉×C₄⋊C₈	Q₈	semidirect
55. C₄×C₈×C₂₉	Q₈	semidirect
96. #28813	C₂₉	direct (N × H)
128. C₈×C₂₉	C₄⋊Q₈	semidirect
150. C₄×C₂₉	#500	semidirect
205. #107	Q₈×C₂₉	semidirect
206. C₄⋊C₈	Q₈×C₂₉	semidirect
208. C₄×C₈	Q₈×C₂₉	semidirect
212. C₂₉	#28813	direct (N × H)
281. C₈	C₂₉×C₄⋊Q₈	semidirect
295. C₄	C₂₉×#500	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#500	2
C₂₉×#2551	2
C₂₉×#2057	2
C₂₉×#1947	2
C₂₉×#1942	2
C₂₉×#1845	2
C₄×C₂₉×#107	2
C₈×C₄⋊C₄×C₂₉	2
#28813	29

Subgroups — C29×#28813 #20323713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28813 #20323713