Subgroups — C₂₉×#7030 #20301930

Order 7424.

344 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#1831	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1744	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₉×C₄⋊C₈	C₄	1	2	yes	—
5	C₂×C₂₉×#97	—	2	1	no	—
6	C₂×C₂₉×#97	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₂₉	—	2	1	no	—
8	C₂×C₄×C₈×C₂₉	—	2	1	no	—
9	C₄³×C₂₉	C₂²	1	1	yes	characteristic
10	C₂₉×#336	—	2	1	no	—
11	C₂₉×#336	—	2	1	no	—
12	C₂₉×#336	—	2	1	no	—
13	C₂₉×#336	C₄	1	2	yes	—
14	C₂²×C₈×C₂₉	—	2	2	no	—
15	C₂²×C₈×C₂₉	—	2	2	no	—
16	C₂²×C₈×C₂₉	—	2	2	no	—
17	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₉	Q₈	1	1	yes	characteristic
24	C₂₉×C₄⋊C₈	—	4	2	no	—
25	C₂₉×#97	—	2	4	no	—
26	C₂₉×#97	—	2	2	no	—
27	C₂₉×#97	C₈	1	4	yes	complemented
28	C₄×C₈×C₂₉	—	2	4	no	—
29	C₄×C₈×C₂₉	—	2	4	no	—
30	C₂²×C₄×C₂₉	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₂₉	—	2	4	no	—
38	C₂×C₈×C₂₉	—	2	4	no	—
39	C₂×C₈×C₂₉	—	2	4	no	—
40	C₂×C₈×C₂₉	—	2	4	no	—
41	C₂×C₈×C₂₉	—	4	2	no	—
42	C₂×C₈×C₂₉	—	4	2	no	—
43	C₂×C₈×C₂₉	—	4	2	no	—
44	C₂×C₈×C₂₉	—	4	2	no	—
45	C₂×C₈×C₂₉	—	4	2	no	—
46	C₄²×C₂₉	—	2	1	no	—
47	C₄²×C₂₉	—	2	1	no	—
48	C₄²×C₂₉	—	2	2	no	—
49	C₄²×C₂₉	—	2	2	no	—
50	C₄²×C₂₉	—	2	2	no	—
51	C₄²×C₂₉	—	2	2	no	—
52	C₄²×C₂₉	—	4	1	no	—
53	C₄²×C₂₉	C₂×C₈	1	2	yes	—
54	C₄²×C₂₉	M₁₆	1	2	yes	—
55	#7030	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₂₉	#95	1	1	yes	characteristic
57	C₂×C₄×C₂₉	—	2	1	no	—
58	C₂×C₄×C₂₉	—	2	1	no	—
59	C₂×C₄×C₂₉	—	2	1	no	—
60	C₂×C₄×C₂₉	—	2	1	no	—
61	C₂×C₄×C₂₉	—	2	1	no	—
62	C₂×C₄×C₂₉	—	2	1	no	—
63	C₂×C₄×C₂₉	—	2	1	no	—
64	C₂×C₄×C₂₉	—	2	1	no	—
65	C₂×C₄×C₂₉	—	2	1	no	—
66	C₂×C₄×C₂₉	—	2	1	no	—
67	C₂×C₄×C₂₉	—	2	1	no	—
68	C₂×C₄×C₂₉	—	2	1	no	—
69	C₂×C₄×C₂₉	—	2	1	no	—
70	C₂×C₄×C₂₉	C₄×C₈	1	1	yes	characteristic
71	C₂×C₄×C₂₉	#97	1	1	yes	characteristic
72	C₂×C₄×C₂₉	C₂²⋊C₈	1	2	yes	—
73	C₂×C₄×C₂₉	#100	1	1	yes	characteristic
74	C₂×C₄×C₂₉	#101	1	1	yes	characteristic
75	C₂×C₄×C₂₉	#104	1	2	yes	—
76	C₂×C₄×C₂₉	#104	1	2	yes	—
77	C₂×C₄×C₂₉	#104	1	2	yes	—
78	C₂×C₄×C₂₉	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₂₉	#108	1	2	yes	—
80	C₈×C₂₉	—	2	8	no	—
81	C₈×C₂₉	—	4	4	no	—
82	C₈×C₂₉	—	8	2	no	—
83	#1831	C₂×C₂₉	1	1	yes	characteristic
84	#1744	C₂×C₂₉	1	1	yes	characteristic
85	C₄×#97	C₂×C₂₉	1	1	yes	characteristic
86	C₂²×C₂₉	#336	1	1	yes	characteristic
87	C₂²×C₂₉	#339	1	1	yes	characteristic
88	C₂²×C₂₉	#339	1	1	yes	characteristic
89	C₂²×C₂₉	#339	1	1	yes	characteristic
90	C₂²×C₂₉	#341	1	1	yes	characteristic
91	C₂²×C₂₉	#343	1	1	yes	characteristic
92	C₂²×C₂₉	#344	1	1	yes	characteristic
93	C₄×C₂₉	—	2	1	no	—
94	C₄×C₂₉	—	2	1	no	—
95	C₄×C₂₉	—	2	2	no	—
96	C₄×C₂₉	—	2	2	no	—
97	C₄×C₂₉	—	2	2	no	—
98	C₄×C₂₉	—	2	2	no	—
99	C₄×C₂₉	—	4	1	no	—
100	C₄×C₂₉	#329	1	2	yes	—
101	C₄×C₂₉	#330	1	2	yes	—
102	C₂×C₄⋊C₈	C₄×C₂₉	1	2	yes	—
103	C₂×#97	—	2	1	no	—
104	C₂×#97	C₄×C₂₉	1	2	yes	—
105	C₂×C₄×C₈	—	2	1	no	—
106	C₂×C₄×C₈	—	2	1	no	—
107	C₄³	C₂²×C₂₉	1	1	yes	characteristic
108	#336	—	2	1	no	—
109	#336	—	2	1	no	—
110	#336	—	2	1	no	—
111	#336	C₄×C₂₉	1	2	yes	—
112	C₂×C₂₉	#1276	1	1	yes	characteristic
113	C₂×C₂₉	#1278	1	1	yes	characteristic
114	C₂×C₂₉	#1283	1	1	yes	characteristic
115	C₂×C₂₉	#1295	1	1	yes	characteristic
116	C₂×C₂₉	#1299	1	1	yes	characteristic
117	C₂×C₂₉	#1300	1	1	yes	characteristic
118	C₂×C₂₉	#1302	1	1	yes	characteristic
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
123	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
124	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
125	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
126	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
127	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
128	C₂×C₄²	Q₈×C₂₉	1	1	yes	characteristic
129	C₄⋊C₈	—	4	2	no	—
130	#97	—	2	4	no	—
131	#97	—	2	2	no	—
132	#97	C₈×C₂₉	1	4	yes	complemented
133	C₄×C₈	—	2	4	no	—
134	C₄×C₈	—	2	4	no	—
135	C₂₉	#7030	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×C₄	C₄²×C₂₉	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
140	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
141	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
142	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	4	2	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	C₂×C₈×C₂₉	1	2	yes	—
160	C₄²	M₁₆×C₂₉	1	2	yes	—
161	C₂³	C₂₉×#95	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	C₄×C₈×C₂₉	1	1	yes	characteristic
176	C₂×C₄	C₂₉×#97	1	1	yes	characteristic
177	C₂×C₄	C₂₉×C₂²⋊C₈	1	2	yes	—
178	C₂×C₄	C₂₉×#100	1	1	yes	characteristic
179	C₂×C₄	C₂₉×#101	1	1	yes	characteristic
180	C₂×C₄	C₂₉×#104	1	2	yes	—
181	C₂×C₄	C₂₉×#104	1	2	yes	—
182	C₂×C₄	C₂₉×#104	1	2	yes	—
183	C₂×C₄	C₂₉×C₄⋊C₈	1	2	yes	—
184	C₂×C₄	C₂₉×#108	1	2	yes	—
185	C₈	—	2	8	no	—
186	C₈	—	4	4	no	—
187	C₈	—	8	2	no	—
188	C₂²	C₂₉×#336	1	1	yes	characteristic
189	C₂²	C₂₉×#339	1	1	yes	characteristic
190	C₂²	C₂₉×#339	1	1	yes	characteristic
191	C₂²	C₂₉×#339	1	1	yes	characteristic
192	C₂²	C₂₉×#341	1	1	yes	characteristic
193	C₂²	C₂₉×#343	1	1	yes	characteristic
194	C₂²	C₂₉×#344	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	C₂₉×#329	1	2	yes	—
203	C₄	C₂₉×#330	1	2	yes	—
204	C₂	C₂₉×#1276	1	1	yes	characteristic
205	C₂	C₂₉×#1278	1	1	yes	characteristic
206	C₂	C₂₉×#1283	1	1	yes	characteristic
207	C₂	C₂₉×#1295	1	1	yes	characteristic
208	C₂	C₂₉×#1299	1	1	yes	characteristic
209	C₂	C₂₉×#1300	1	1	yes	characteristic
210	C₂	C₂₉×#1302	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₉	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₂₉
Frattini, cosocle	C₄³	C₂²×C₂₉
socle, prime radical	C₂³×C₂₉	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
27. C₂₉×#97	C₈	semidirect
55. #7030	C₂₉	direct (N × H)
132. #97	C₈×C₂₉	semidirect
135. C₂₉	#7030	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₉×#1831	2
C₂₉×#1744	2
C₄×C₂₉×#97	2
#7030	29

Subgroups — C29×#7030 #20301930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#7030 #20301930