Subgroups — C₂₉×#7854 #20302754

Order 7424.

714 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#431	C₂	1	2	yes	maximal
2	C₄×C₂₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1274	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₉×#117	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
6	C₂₉×#431	—	2	2	no	—
7	C₂₉×#431	—	2	2	no	—
8	C₂₉×#431	C₄	1	4	yes	complemented
9	C₂₉×#431	C₂²	1	4	yes	—
10	C₂×C₂₉×C₄⋊C₈	—	2	2	no	—
11	C₂×C₂₉×C₂²⋊C₈	—	2	2	no	—
12	C₂×C₄×C₈×C₂₉	C₄	1	4	yes	complemented
13	C₄×C₄⋊C₄×C₂₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₂₉	—	2	2	no	—
15	C₂₉×#376	C₂²	1	2	yes	—
16	C₄³×C₂₉	C₄	1	2	yes	—
17	C₂³×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₉	D₄	1	2	yes	—
19	C₂²×C₈×C₂₉	—	2	4	no	—
20	C₂₉×#117	—	2	4	no	—
21	C₂₉×#117	C₈	1	8	yes	complemented
22	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
23	C₂×C₂²⋊C₄×C₂₉	—	2	2	no	—
24	C₂×C₄²×C₂₉	—	2	4	no	—
25	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₉	D₄	1	2	yes	—
30	C₂×C₄²×C₂₉	D₄	1	2	yes	—
31	C₂×C₄²×C₂₉	Q₈	1	2	yes	—
32	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
33	C₂₉×C₄⋊C₈	—	2	4	no	—
34	C₂₉×C₄⋊C₈	—	2	4	no	—
35	C₂₉×C₂²⋊C₈	—	2	4	no	—
36	C₂₉×C₂²⋊C₈	—	2	4	no	—
37	C₄×C₈×C₂₉	—	2	4	no	—
38	C₄×C₈×C₂₉	C₂×C₄	1	8	yes	—
39	C₂₉×#95	—	2	4	no	—
40	C₂⁴×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₉	—	2	4	no	—
42	C₂²×C₄×C₂₉	—	4	4	no	—
43	C₂²×C₄×C₂₉	C₄²	1	2	yes	—
44	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₉	C₂×C₈	1	4	yes	—
52	C₂²×C₄×C₂₉	M₁₆	1	4	yes	—
53	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₂₉	—	2	8	no	—
56	C₂×C₈×C₂₉	—	4	8	no	—
57	C₄⋊C₄×C₂₉	—	2	8	no	—
58	C₂²⋊C₄×C₂₉	—	2	8	no	—
59	C₄²×C₂₉	—	2	4	no	—
60	C₄²×C₂₉	—	4	8	no	—
61	C₄²×C₂₉	C₄²	1	1	yes	characteristic
62	C₄²×C₂₉	C₄²	1	1	yes	characteristic
63	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
66	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
67	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
68	C₄²×C₂₉	C₂×C₈	1	8	yes	complemented
69	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
70	C₄²×C₂₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₂₉	C₂×Q₈	1	1	yes	characteristic
72	#7854	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₉	—	4	2	no	—
74	C₂³×C₂₉	#95	1	2	yes	—
75	C₂³×C₂₉	C₂²⋊C₈	1	4	yes	—
76	C₂³×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₉	—	2	4	no	—
78	C₂×C₄×C₂₉	—	2	4	no	—
79	C₂×C₄×C₂₉	—	2	4	no	—
80	C₂×C₄×C₂₉	—	2	8	no	—
81	C₂×C₄×C₂₉	—	4	2	no	—
82	C₂×C₄×C₂₉	—	4	2	no	—
83	C₂×C₄×C₂₉	—	4	2	no	—
84	C₂×C₄×C₂₉	—	4	8	no	—
85	C₂×C₄×C₂₉	#95	1	2	yes	—
86	C₂×C₄×C₂₉	#95	1	2	yes	—
87	C₂×C₄×C₂₉	#95	1	2	yes	—
88	C₂×C₄×C₂₉	C₄×C₈	1	4	yes	—
89	C₂×C₄×C₂₉	#97	1	4	yes	—
90	C₂×C₄×C₂₉	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₂₉	C₄⋊C₈	1	8	yes	—
92	C₂×C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₂₉	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₂₉	C₂×M₁₆	1	2	yes	—
98	C₈×C₂₉	—	4	8	no	—
99	C₂×#431	C₂×C₂₉	1	2	yes	—
100	C₄×#117	C₂×C₂₉	1	1	yes	characteristic
101	#1274	C₂×C₂₉	1	4	yes	complemented
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	2	no	—
104	C₂²×C₂₉	—	2	2	no	—
105	C₂²×C₂₉	—	4	4	no	—
106	C₂²×C₂₉	#336	1	4	yes	—
107	C₂²×C₂₉	C₂×#95	1	1	yes	characteristic
108	C₂²×C₂₉	C₂×C₂²⋊C₈	1	2	yes	—
109	C₄×C₂₉	—	2	2	no	—
110	C₄×C₂₉	—	2	2	no	—
111	C₄×C₂₉	—	2	2	no	—
112	C₄×C₂₉	—	4	8	no	—
113	C₄×C₂₉	#336	1	4	yes	—
114	C₄×C₂₉	H(3,ℤ/4)	1	4	yes	—
115	C₄×C₂₉	C₂×C₄×C₈	1	1	yes	characteristic
116	C₄×C₂₉	C₂×#97	1	1	yes	characteristic
117	C₄×C₂₉	C₂×C₄⋊C₈	1	2	yes	—
118	C₂×#117	C₄×C₂₉	1	2	yes	—
119	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
120	#431	—	2	2	no	—
121	#431	—	2	2	no	—
122	#431	C₄×C₂₉	1	4	yes	complemented
123	#431	C₂²×C₂₉	1	4	yes	—
124	C₂×C₄⋊C₈	—	2	2	no	—
125	C₂×C₂²⋊C₈	—	2	2	no	—
126	C₂×C₄×C₈	C₄×C₂₉	1	4	yes	complemented
127	C₄×C₄⋊C₄	—	2	2	no	—
128	C₄×C₂²⋊C₄	—	2	2	no	—
129	#376	C₂²×C₂₉	1	2	yes	—
130	C₄³	C₄×C₂₉	1	2	yes	—
131	C₂×C₂₉	—	2	2	no	—
132	C₂×C₂₉	—	2	4	no	—
133	C₂×C₂₉	C₂×#336	1	1	yes	characteristic
134	C₂×C₂₉	C₂×H(3,ℤ/4)	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
136	C₂³×C₄	D₄×C₂₉	1	2	yes	—
137	C₂²×C₈	—	2	4	no	—
138	#117	—	2	4	no	—
139	#117	C₈×C₂₉	1	8	yes	complemented
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₂²⋊C₄	—	2	2	no	—
142	C₂×C₄²	—	2	4	no	—
143	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
144	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
147	C₂×C₄²	D₄×C₂₉	1	2	yes	—
148	C₂×C₄²	D₄×C₂₉	1	2	yes	—
149	C₂×C₄²	Q₈×C₂₉	1	2	yes	—
150	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	2	4	no	—
153	C₂²⋊C₈	—	2	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₄×C₈	—	2	4	no	—
156	C₄×C₈	C₂×C₄×C₂₉	1	8	yes	—
157	#95	—	2	4	no	—
158	C₂₉	#7854	1	1	yes	characteristic, direct product factor, Sylow
159	C₂⁴	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₄²×C₂₉	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
167	C₂²×C₄	C₄⋊C₄×C₂₉	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₂₉	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₂₉	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₂₉	1	4	yes	—
171	C₂²×C₄	M₁₆×C₂₉	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	4	8	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	4	8	no	—
180	C₄²	C₄²×C₂₉	1	1	yes	characteristic
181	C₄²	C₄²×C₂₉	1	1	yes	characteristic
182	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
183	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
184	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
185	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
186	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
187	C₄²	C₂×C₈×C₂₉	1	8	yes	complemented
188	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
189	C₄²	C₂×D₄×C₂₉	1	1	yes	characteristic
190	C₄²	C₂×Q₈×C₂₉	1	1	yes	characteristic
191	C₂³	—	4	2	no	—
192	C₂³	C₂₉×#95	1	2	yes	—
193	C₂³	C₂₉×C₂²⋊C₈	1	4	yes	—
194	C₂³	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	C₂₉×#95	1	2	yes	—
204	C₂×C₄	C₂₉×#95	1	2	yes	—
205	C₂×C₄	C₂₉×#95	1	2	yes	—
206	C₂×C₄	C₄×C₈×C₂₉	1	4	yes	—
207	C₂×C₄	C₂₉×#97	1	4	yes	—
208	C₂×C₄	C₂₉×C₂²⋊C₈	1	4	yes	—
209	C₂×C₄	C₂₉×C₄⋊C₈	1	8	yes	—
210	C₂×C₄	C₂×C₄²×C₂₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₈×C₂₉	1	2	yes	—
215	C₂×C₄	C₂×M₁₆×C₂₉	1	2	yes	—
216	C₈	—	4	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	C₂₉×#336	1	4	yes	—
222	C₂²	C₂×C₂₉×#95	1	1	yes	characteristic
223	C₂²	C₂×C₂₉×C₂²⋊C₈	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	8	no	—
228	C₄	C₂₉×#336	1	4	yes	—
229	C₄	C₂₉×H(3,ℤ/4)	1	4	yes	—
230	C₄	C₂×C₄×C₈×C₂₉	1	1	yes	characteristic
231	C₄	C₂×C₂₉×#97	1	1	yes	characteristic
232	C₄	C₂×C₂₉×C₄⋊C₈	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	—	2	4	no	—
235	C₂	C₂×C₂₉×#336	1	1	yes	characteristic
236	C₂	C₂×C₂₉×H(3,ℤ/4)	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×#95
prime radical	C₂⁴×C₂₉	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₉×#1274	C₂	semidirect
8. C₂₉×#431	C₄	semidirect
12. C₂×C₄×C₈×C₂₉	C₄	semidirect
21. C₂₉×#117	C₈	semidirect
68. C₄²×C₂₉	C₂×C₈	semidirect
72. #7854	C₂₉	direct (N × H)
101. #1274	C₂×C₂₉	semidirect
122. #431	C₄×C₂₉	semidirect
126. C₂×C₄×C₈	C₄×C₂₉	semidirect
139. #117	C₈×C₂₉	semidirect
158. C₂₉	#7854	direct (N × H)
187. C₄²	C₂×C₈×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#431	2
C₄×C₂₉×#117	2
C₂₉×#1274	2
#7854	29

Subgroups — C29×#7854 #20302754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#7854 #20302754