Subgroups — C₄×C₂₉×#493 #20316587

Order 7424.

1422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₂₉	C₂	1	4	yes	maximal, complemented
3	C₄×C₂₉×#127	C₂	1	2	yes	maximal, complemented
4	C₂₉×#1970	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1957	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₂₉	C₂²	1	4	yes	—
9	C₂×C₂₉×#127	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₂₉	—	2	4	no	—
11	C₂×C₄×D₄×C₂₉	C₂²	1	4	yes	—
12	C₂₉×#493	C₄	1	8	yes	direct product factor
13	C₄×D₈×C₂₉	—	2	16	no	—
14	C₂×C₂₉×#107	C₂²	1	4	yes	—
15	C₂×C₂₉×#102	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
20	C₂₉×#390	—	2	4	no	—
21	C₂₉×#390	C₂²	1	4	yes	—
22	C₂₉×#384	C₂²	1	2	yes	—
23	C₄³×C₂₉	C₂²	1	1	yes	characteristic
24	C₂²×D₄×C₂₉	—	2	4	no	—
25	C₂²×D₄×C₂₉	C₂³	1	4	yes	—
26	C₂³×C₄×C₂₉	—	2	4	no	—
27	C₂×D₈×C₂₉	—	2	8	no	—
28	C₂×D₈×C₂₉	—	2	8	no	—
29	C₂×D₈×C₂₉	C₂×C₄	1	16	yes	complemented
30	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
32	C₂₉×#127	—	2	4	no	—
33	C₂₉×#127	C₂×C₄	1	8	yes	complemented
34	C₄×D₄×C₂₉	—	2	16	no	—
35	C₄×D₄×C₂₉	—	4	16	no	—
36	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
38	C₂×C₂²⋊C₄×C₂₉	—	2	8	no	—
39	C₂×C₂²⋊C₄×C₂₉	—	2	4	no	—
40	C₂×C₄²×C₂₉	D₄	1	2	yes	—
41	C₂×C₄²×C₂₉	D₄	1	2	yes	—
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₂₉×#107	—	2	8	no	—
46	C₂₉×#102	—	2	16	no	—
47	C₄×C₈×C₂₉	—	2	4	no	—
48	C₄×C₈×C₂₉	—	2	2	no	—
49	C₄×C₈×C₂₉	C₂×C₄	1	4	yes	complemented
50	C₄×C₈×C₂₉	D₄	1	8	yes	complemented
51	C₂⁴×C₂₉	—	2	4	no	—
52	C₂×D₄×C₂₉	—	2	16	no	—
53	C₂×D₄×C₂₉	—	2	8	no	—
54	C₂×D₄×C₂₉	—	2	8	no	—
55	C₂×D₄×C₂₉	—	2	4	no	—
56	C₂×D₄×C₂₉	—	4	8	no	—
57	C₂×D₄×C₂₉	—	4	8	no	—
58	C₂×D₄×C₂₉	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₂₉	—	2	4	no	—
60	C₂²×C₄×C₂₉	—	4	8	no	—
61	C₂²×C₄×C₂₉	—	4	16	no	—
62	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
67	D₈×C₂₉	—	2	32	no	—
68	C₂×C₈×C₂₉	—	2	2	no	—
69	C₂×C₈×C₂₉	—	2	2	no	—
70	C₂×C₈×C₂₉	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₂₉	#41	1	4	yes	—
74	C₄⋊C₄×C₂₉	—	2	8	no	—
75	C₄⋊C₄×C₂₉	—	4	8	no	—
76	C₂²⋊C₄×C₂₉	—	4	8	no	—
77	C₂²⋊C₄×C₂₉	—	4	16	no	—
78	C₄²×C₂₉	—	2	4	no	—
79	C₄²×C₂₉	—	2	2	no	—
80	C₄²×C₂₉	D₈	1	8	yes	complemented
81	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
82	C₄²×C₂₉	C₂×D₄	1	4	yes	—
83	C₄²×C₂₉	#41	1	2	yes	—
84	C₄×#493	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₂₉	—	2	8	no	—
86	C₂³×C₂₉	—	4	4	no	—
87	C₂³×C₂₉	—	4	4	no	—
88	C₂³×C₂₉	—	4	8	no	—
89	C₂³×C₂₉	—	4	8	no	—
90	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
91	D₄×C₂₉	—	2	16	no	—
92	D₄×C₂₉	—	4	8	no	—
93	D₄×C₂₉	—	4	32	no	—
94	C₂×C₄×C₂₉	—	2	2	no	—
95	C₂×C₄×C₂₉	—	2	2	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	4	4	no	—
98	C₂×C₄×C₂₉	—	4	8	no	—
99	C₂×C₄×C₂₉	—	8	16	no	—
100	C₂×C₄×C₂₉	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₂₉	#127	1	2	yes	—
102	C₂×C₄×C₂₉	#127	1	2	yes	—
103	C₂×C₄×C₂₉	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₂₉	C₂×D₈	1	4	yes	—
105	C₂×C₄×C₂₉	C₂×D₈	1	4	yes	—
106	C₂×C₄×C₂₉	#135	1	4	yes	—
107	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
110	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
111	C₈×C₂₉	—	2	4	no	—
112	C₈×C₂₉	C₄×D₄	1	8	yes	complemented
113	C₂×#493	C₂×C₂₉	1	1	yes	characteristic
114	C₂×C₄×D₈	C₂×C₂₉	1	4	yes	complemented
115	C₄×#127	C₂×C₂₉	1	2	yes	complemented
116	#1970	C₂×C₂₉	1	4	yes	complemented
117	#1957	C₂×C₂₉	1	2	yes	complemented
118	#1845	C₂×C₂₉	1	1	yes	characteristic, complemented
119	C₄²×C₈	C₂×C₂₉	1	1	yes	characteristic, complemented
120	C₂²×C₂₉	—	4	8	no	—
121	C₂²×C₂₉	—	4	16	no	—
122	C₂²×C₂₉	—	8	4	no	—
123	C₂²×C₂₉	—	8	4	no	—
124	C₂²×C₂₉	—	8	8	no	—
125	C₂²×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₂₉	C₂×#127	1	1	yes	characteristic
127	C₂²×C₂₉	#532	1	1	yes	characteristic
128	C₂²×C₂₉	C₂²×D₈	1	2	yes	—
129	C₂²×C₂₉	C₂×#135	1	2	yes	—
130	C₄×C₂₉	—	2	2	no	—
131	C₄×C₂₉	—	2	4	no	—
132	C₄×C₂₉	—	8	4	no	—
133	C₄×C₂₉	C₄×D₈	1	8	yes	complemented
134	C₄×C₂₉	#493	1	4	yes	direct product factor
135	C₄×C₂₉	C₂×C₄×D₄	1	2	yes	—
136	C₄×C₂₉	#532	1	2	yes	—
137	C₂²×D₈	C₂²×C₂₉	1	4	yes	—
138	C₂×#127	C₂²×C₂₉	1	2	yes	—
139	C₂×C₄×D₄	—	2	4	no	—
140	C₂×C₄×D₄	C₂²×C₂₉	1	4	yes	—
141	#493	C₄×C₂₉	1	8	yes	direct product factor
142	C₄×D₈	—	2	16	no	—
143	C₂×#107	C₂²×C₂₉	1	4	yes	—
144	C₂×#102	C₂²×C₂₉	1	8	yes	—
145	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
149	#390	—	2	4	no	—
150	#390	C₂²×C₂₉	1	4	yes	—
151	#384	C₂²×C₂₉	1	2	yes	—
152	C₄³	C₂²×C₂₉	1	1	yes	characteristic
153	C₂×C₂₉	—	8	8	no	—
154	C₂×C₂₉	C₄×#127	1	1	yes	characteristic
155	C₂×C₂₉	C₂×C₄×D₈	1	2	yes	—
156	C₂×C₂₉	C₂×#493	1	1	yes	characteristic
157	C₂×C₂₉	#3147	1	1	yes	characteristic
158	C₂×C₂₉	#3169	1	2	yes	—
159	C₂²×D₄	—	2	4	no	—
160	C₂²×D₄	C₂³×C₂₉	1	4	yes	—
161	C₂³×C₄	—	2	4	no	—
162	C₂×D₈	—	2	8	no	—
163	C₂×D₈	—	2	8	no	—
164	C₂×D₈	C₂×C₄×C₂₉	1	16	yes	complemented
165	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
166	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
167	#127	—	2	4	no	—
168	#127	C₂×C₄×C₂₉	1	8	yes	complemented
169	C₄×D₄	—	2	16	no	—
170	C₄×D₄	—	4	16	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	8	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₄²	D₄×C₂₉	1	2	yes	—
176	C₂×C₄²	D₄×C₂₉	1	2	yes	—
177	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
180	#107	—	2	8	no	—
181	#102	—	2	16	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	C₂×C₄×C₂₉	1	4	yes	complemented
185	C₄×C₈	D₄×C₂₉	1	8	yes	complemented
186	C₂₉	C₄×#493	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	4	no	—
188	C₂×D₄	—	2	16	no	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	8	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	4	8	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂²×C₄×C₂₉	1	8	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	8	no	—
197	C₂²×C₄	—	4	16	no	—
198	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
203	D₈	—	2	32	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²×C₄×C₂₉	1	4	yes	—
207	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
208	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
209	C₂×C₈	#41×C₂₉	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₂²⋊C₄	—	4	16	no	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	2	no	—
216	C₄²	D₈×C₂₉	1	8	yes	complemented
217	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
218	C₄²	C₂×D₄×C₂₉	1	4	yes	—
219	C₄²	#41×C₂₉	1	2	yes	—
220	C₂³	—	2	8	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	8	no	—
224	C₂³	—	4	8	no	—
225	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
226	D₄	—	2	16	no	—
227	D₄	—	4	32	no	—
228	D₄	—	4	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	8	16	no	—
235	C₂×C₄	C₄×D₄×C₂₉	1	4	yes	—
236	C₂×C₄	C₂₉×#127	1	2	yes	—
237	C₂×C₄	C₂₉×#127	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₂₉	1	4	yes	—
239	C₂×C₄	C₂×D₈×C₂₉	1	4	yes	—
240	C₂×C₄	C₂×D₈×C₂₉	1	4	yes	—
241	C₂×C₄	C₂₉×#135	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
244	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
245	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
246	C₈	—	2	4	no	—
247	C₈	C₄×D₄×C₂₉	1	8	yes	complemented
248	C₂²	—	4	16	no	—
249	C₂²	—	4	8	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	8	no	—
253	C₂²	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
254	C₂²	C₂×C₂₉×#127	1	1	yes	characteristic
255	C₂²	C₂₉×#532	1	1	yes	characteristic
256	C₂²	C₂²×D₈×C₂₉	1	2	yes	—
257	C₂²	C₂×C₂₉×#135	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	8	4	no	—
261	C₄	C₄×D₈×C₂₉	1	8	yes	complemented
262	C₄	C₂₉×#493	1	4	yes	direct product factor
263	C₄	C₂×C₄×D₄×C₂₉	1	2	yes	—
264	C₄	C₂₉×#532	1	2	yes	—
265	C₂	—	8	8	no	—
266	C₂	C₄×C₂₉×#127	1	1	yes	characteristic
267	C₂	C₂×C₄×D₈×C₂₉	1	2	yes	—
268	C₂	C₂×C₂₉×#493	1	1	yes	characteristic
269	C₂	C₂₉×#3147	1	1	yes	characteristic
270	C₂	C₂₉×#3169	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle	C₂³×C₂₉	C₂²×D₄
prime radical	C₂×C₂₉×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₉ ◁ C₄³×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₂₉	C₂	semidirect
3. C₄×C₂₉×#127	C₂	semidirect
4. C₂₉×#1970	C₂	semidirect
5. C₂₉×#1957	C₂	semidirect
6. C₂₉×#1845	C₂	semidirect
7. C₄²×C₈×C₂₉	C₂	semidirect
12. C₂₉×#493	C₄	direct (N × H)
29. C₂×D₈×C₂₉	C₂×C₄	semidirect
33. C₂₉×#127	C₂×C₄	semidirect
49. C₄×C₈×C₂₉	C₂×C₄	semidirect
50. C₄×C₈×C₂₉	D₄	semidirect
80. C₄²×C₂₉	D₈	semidirect
84. C₄×#493	C₂₉	direct (N × H)
112. C₈×C₂₉	C₄×D₄	semidirect
114. C₂×C₄×D₈	C₂×C₂₉	semidirect
115. C₄×#127	C₂×C₂₉	semidirect
116. #1970	C₂×C₂₉	semidirect
117. #1957	C₂×C₂₉	semidirect
118. #1845	C₂×C₂₉	semidirect
119. C₄²×C₈	C₂×C₂₉	semidirect
133. C₄×C₂₉	C₄×D₈	semidirect
134. C₄×C₂₉	#493	direct (N × H)
141. #493	C₄×C₂₉	direct (N × H)
164. C₂×D₈	C₂×C₄×C₂₉	semidirect
168. #127	C₂×C₄×C₂₉	semidirect
184. C₄×C₈	C₂×C₄×C₂₉	semidirect
185. C₄×C₈	D₄×C₂₉	semidirect
186. C₂₉	C₄×#493	direct (N × H)
216. C₄²	D₈×C₂₉	semidirect
247. C₈	C₄×D₄×C₂₉	semidirect
261. C₄	C₄×D₈×C₂₉	semidirect
262. C₄	C₂₉×#493	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×#493	2
C₂×C₄×D₈×C₂₉	2
C₄×C₂₉×#127	2
C₂₉×#1970	2
C₂₉×#1957	2
C₂₉×#1845	2
C₄²×C₈×C₂₉	2
C₄×#493	29

Subgroups — C4×C29×#493 #20316587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₉×#493 #20316587