Subgroups — C₂₉×#28809 #20323709

Order 7424.

822 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#498	C₂	1	2	yes	maximal
2	C₂₉×#2551	C₂	1	2	yes	maximal
3	C₂₉×#2058	C₂	1	4	yes	maximal
4	C₂₉×#1948	C₂	1	2	yes	maximal
5	C₂₉×#1941	C₂	1	2	yes	maximal
6	C₂₉×#1844	C₂	1	1	yes	maximal, characteristic
7	C₄×C₂₉×#106	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×#498	—	2	8	no	—
12	C₂×C₂₉×#106	C₂²	1	2	yes	—
13	C₂×C₂₉×#106	C₂²	1	4	yes	—
14	C₂×C₂₉×#106	C₂²	1	2	yes	—
15	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×#106	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	2	yes	—
41	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₉	—	2	2	no	—
45	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
47	C₂₉×#106	—	2	4	no	—
48	C₂₉×#106	—	2	8	no	—
49	C₂₉×#106	—	4	2	no	—
50	C₂₉×#106	D₄	1	4	yes	—
51	C₂₉×#106	Q₈	1	4	yes	complemented
52	C₂₉×C₄⋊C₈	Q₈	1	4	yes	complemented
53	C₄×C₈×C₂₉	—	2	4	no	—
54	C₄×C₈×C₂₉	Q₈	1	4	yes	complemented
55	C₂₉×#95	—	2	2	no	—
56	C₂₉×#95	—	2	4	no	—
57	C₂₉×#95	D₄	1	2	yes	—
58	C₂×Q₈×C₂₉	—	4	8	no	—
59	C₂×Q₈×C₂₉	—	4	8	no	—
60	C₂×Q₈×C₂₉	—	8	4	no	—
61	C₂²×C₄×C₂₉	—	2	2	no	—
62	C₂²×C₄×C₂₉	—	2	2	no	—
63	C₂²×C₄×C₂₉	—	2	4	no	—
64	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₉	—	2	2	no	—
69	C₂×C₈×C₂₉	—	2	1	no	—
70	C₂×C₈×C₂₉	—	2	1	no	—
71	C₂×C₈×C₂₉	—	2	2	no	—
72	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₂₉	C₂×Q₈	1	2	yes	—
77	C₄⋊C₄×C₂₉	—	2	2	no	—
78	C₄⋊C₄×C₂₉	—	2	2	no	—
79	C₄⋊C₄×C₂₉	—	2	4	no	—
80	C₄⋊C₄×C₂₉	—	2	4	no	—
81	C₄⋊C₄×C₂₉	—	4	8	no	—
82	C₄⋊C₄×C₂₉	—	4	8	no	—
83	C₄⋊C₄×C₂₉	—	4	4	no	—
84	C₄⋊C₄×C₂₉	—	8	2	no	—
85	C₄⋊C₄×C₂₉	SD₁₆	1	8	yes	—
86	C₄⋊C₄×C₂₉	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₂₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₂₉	—	2	2	no	—
89	C₄²×C₂₉	—	2	2	no	—
90	C₄²×C₂₉	—	4	4	no	—
91	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
92	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
93	#28809	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₂₉	—	8	16	no	—
96	C₂×C₄×C₂₉	—	2	4	no	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	2	4	no	—
99	C₂×C₄×C₂₉	—	2	4	no	—
100	C₂×C₄×C₂₉	—	2	4	no	—
101	C₂×C₄×C₂₉	—	4	2	no	—
102	C₂×C₄×C₂₉	—	4	2	no	—
103	C₂×C₄×C₂₉	—	4	2	no	—
104	C₂×C₄×C₂₉	—	4	2	no	—
105	C₂×C₄×C₂₉	—	4	4	no	—
106	C₂×C₄×C₂₉	—	4	4	no	—
107	C₂×C₄×C₂₉	—	4	4	no	—
108	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
110	C₂×C₄×C₂₉	C₂×SD₁₆	1	8	yes	—
111	C₂×C₄×C₂₉	C₂×SD₁₆	1	4	yes	—
112	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
118	C₈×C₂₉	—	2	2	no	—
119	C₈×C₂₉	—	2	4	no	—
120	C₈×C₂₉	C₄⋊Q₈	1	4	yes	complemented
121	C₂×#498	C₂×C₂₉	1	2	yes	—
122	#2551	C₂×C₂₉	1	2	yes	—
123	#2058	C₂×C₂₉	1	4	yes	—
124	#1948	C₂×C₂₉	1	2	yes	—
125	#1941	C₂×C₂₉	1	2	yes	—
126	#1844	C₂×C₂₉	1	1	yes	characteristic
127	C₄×#106	C₂×C₂₉	1	1	yes	characteristic
128	C₈×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
129	C₂²×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₂₉	D₄²	1	1	yes	characteristic
131	C₂²×C₂₉	D₄×Q₈	1	1	yes	characteristic
132	C₂²×C₂₉	C₂²×SD₁₆	1	2	yes	—
133	C₂²×C₂₉	#577	1	2	yes	—
134	C₄×C₂₉	—	2	2	no	—
135	C₄×C₂₉	—	2	8	no	—
136	C₄×C₂₉	—	4	4	no	—
137	C₄×C₂₉	—	4	4	no	—
138	C₄×C₂₉	—	8	4	no	—
139	C₄×C₂₉	#498	1	4	yes	complemented
140	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₄×C₂₉	#544	1	1	yes	characteristic
142	C₄×C₂₉	D₄×Q₈	1	1	yes	characteristic
143	C₄×C₂₉	Q₈²	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	—	2	2	no	—
145	C₂×C₄⋊Q₈	C₂²×C₂₉	1	4	yes	—
146	#498	—	2	8	no	—
147	C₂×#106	C₂²×C₂₉	1	4	yes	—
148	C₂×#106	C₂²×C₂₉	1	2	yes	—
149	C₂×#106	C₂²×C₂₉	1	2	yes	—
150	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
151	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
155	#395	—	2	4	no	—
156	#389	—	2	2	no	—
157	#389	C₂²×C₂₉	1	4	yes	—
158	#384	C₂²×C₂₉	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
161	#340	C₂²×C₂₉	1	2	yes	—
162	#340	C₂²×C₂₉	1	4	yes	—
163	#336	C₂²×C₂₉	1	2	yes	—
164	C₂×C₂₉	#2551	1	1	yes	characteristic
165	C₂×C₂₉	C₂×#498	1	1	yes	characteristic
166	C₂×C₂₉	#3163	1	1	yes	characteristic
167	C₂×C₂₉	D₄×SD₁₆	1	2	yes	—
168	C₂×C₂₉	Q₈×SD₁₆	1	2	yes	—
169	C₂²×Q₈	—	2	4	no	—
170	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
172	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	8	no	—
174	C₄⋊Q₈	—	4	4	no	—
175	C₄⋊Q₈	—	4	8	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
182	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
186	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
187	#106	—	2	4	no	—
188	#106	—	2	8	no	—
189	#106	—	4	2	no	—
190	#106	D₄×C₂₉	1	4	yes	—
191	#106	Q₈×C₂₉	1	4	yes	complemented
192	C₄⋊C₈	Q₈×C₂₉	1	4	yes	complemented
193	C₄×C₈	—	2	4	no	—
194	C₄×C₈	Q₈×C₂₉	1	4	yes	complemented
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	#95	D₄×C₂₉	1	2	yes	—
198	C₂₉	#28809	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×Q₈	—	4	8	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	8	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₉	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
214	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
215	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
216	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
217	C₂×C₈	C₂×Q₈×C₂₉	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	8	2	no	—
226	C₄⋊C₄	SD₁₆×C₂₉	1	8	yes	—
227	C₄⋊C₄	C₂×D₄×C₂₉	1	4	yes	—
228	C₄⋊C₄	C₂×Q₈×C₂₉	1	4	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	4	no	—
232	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
233	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
234	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
235	Q₈	—	8	16	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
249	C₂×C₄	C₂₉×C₄⋊Q₈	1	2	yes	—
250	C₂×C₄	C₂×SD₁₆×C₂₉	1	8	yes	—
251	C₂×C₄	C₂×SD₁₆×C₂₉	1	4	yes	—
252	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
256	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₄	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	4	no	—
260	C₈	C₂₉×C₄⋊Q₈	1	4	yes	complemented
261	C₂²	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
262	C₂²	D₄²×C₂₉	1	1	yes	characteristic
263	C₂²	D₄×Q₈×C₂₉	1	1	yes	characteristic
264	C₂²	C₂²×SD₁₆×C₂₉	1	2	yes	—
265	C₂²	C₂₉×#577	1	2	yes	—
266	C₄	—	2	8	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	4	no	—
271	C₄	C₂₉×#498	1	4	yes	complemented
272	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
273	C₄	C₂₉×#544	1	1	yes	characteristic
274	C₄	D₄×Q₈×C₂₉	1	1	yes	characteristic
275	C₄	Q₈²×C₂₉	1	1	yes	characteristic
276	C₂	C₂₉×#2551	1	1	yes	characteristic
277	C₂	C₂×C₂₉×#498	1	1	yes	characteristic
278	C₂	C₂₉×#3163	1	1	yes	characteristic
279	C₂	D₄×SD₁₆×C₂₉	1	2	yes	—
280	C₂	Q₈×SD₁₆×C₂₉	1	2	yes	—

	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
51. C₂₉×#106	Q₈	semidirect
52. C₂₉×C₄⋊C₈	Q₈	semidirect
54. C₄×C₈×C₂₉	Q₈	semidirect
93. #28809	C₂₉	direct (N × H)
120. C₈×C₂₉	C₄⋊Q₈	semidirect
139. C₄×C₂₉	#498	semidirect
191. #106	Q₈×C₂₉	semidirect
192. C₄⋊C₈	Q₈×C₂₉	semidirect
194. C₄×C₈	Q₈×C₂₉	semidirect
198. C₂₉	#28809	direct (N × H)
260. C₈	C₂₉×C₄⋊Q₈	semidirect
271. C₄	C₂₉×#498	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#498	2
C₂₉×#2551	2
C₂₉×#2058	2
C₂₉×#1948	2
C₂₉×#1941	2
C₂₉×#1844	2
C₄×C₂₉×#106	2
C₈×C₄⋊C₄×C₂₉	2
#28809	29

Subgroups — C29×#28809 #20323709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28809 #20323709