Subgroups — C₂₉×#9904 #20304804

Order 7424.

438 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1842	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1296	C₂	1	4	yes	maximal
5	C₂×C₂₉×#125	—	2	1	no	—
6	C₂×C₂₉×C₄⋊C₈	—	2	2	no	—
7	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂₉×#97	—	2	1	no	—
10	C₂₉×#389	—	2	2	no	—
11	C₂₉×#389	C₄	1	4	yes	—
12	C₂₉×#384	C₂²	1	2	yes	—
13	C₂₉×#382	—	2	1	no	—
14	C₄³×C₂₉	C₂²	1	1	yes	characteristic
15	C₂₉×#336	—	2	1	no	—
16	C₂²×C₈×C₂₉	—	2	2	no	—
17	C₂²×C₈×C₂₉	—	2	2	no	—
18	C₂₉×#125	—	4	2	no	—
19	C₂₉×#125	—	4	2	no	—
20	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₉	D₄	1	2	yes	—
27	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
29	C₂₉×C₄⋊C₈	—	2	2	no	—
30	C₂₉×C₄⋊C₈	—	2	4	no	—
31	C₂₉×C₄⋊C₈	—	4	4	no	—
32	C₂₉×C₄⋊C₈	Q₈	1	4	yes	—
33	C₂₉×#97	—	4	2	no	—
34	C₂₉×#95	—	2	4	no	—
35	C₂²×C₄×C₂₉	—	2	4	no	—
36	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₉	—	2	4	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	4	no	—
46	C₄⋊C₄×C₂₉	—	4	4	no	—
47	C₄⋊C₄×C₂₉	—	4	8	no	—
48	C₄⋊C₄×C₂₉	—	4	2	no	—
49	C₄⋊C₄×C₂₉	—	4	2	no	—
50	C₄⋊C₄×C₂₉	—	4	2	no	—
51	C₄⋊C₄×C₂₉	—	4	2	no	—
52	C₄²×C₂₉	—	2	2	no	—
53	C₄²×C₂₉	—	2	2	no	—
54	C₄²×C₂₉	—	2	2	no	—
55	C₄²×C₂₉	—	4	2	no	—
56	C₄²×C₂₉	D₈	1	2	yes	—
57	C₄²×C₂₉	SD₁₆	1	2	yes	—
58	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
59	C₄²×C₂₉	#41	1	2	yes	—
60	#9904	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₉	—	2	2	no	—
63	C₂×C₄×C₂₉	—	2	2	no	—
64	C₂×C₄×C₂₉	—	2	2	no	—
65	C₂×C₄×C₂₉	—	2	2	no	—
66	C₂×C₄×C₂₉	—	2	2	no	—
67	C₂×C₄×C₂₉	—	2	2	no	—
68	C₂×C₄×C₂₉	—	4	4	no	—
69	C₂×C₄×C₂₉	—	4	4	no	—
70	C₂×C₄×C₂₉	—	4	4	no	—
71	C₂×C₄×C₂₉	#102	1	4	yes	—
72	C₂×C₄×C₂₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₂₉	#124	1	1	yes	characteristic
75	C₂×C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₂₉	#137	1	2	yes	—
80	C₂×C₄×C₂₉	#137	1	2	yes	—
81	C₈×C₂₉	—	4	4	no	—
82	C₈×C₂₉	—	8	2	no	—
83	#2859	C₂×C₂₉	1	1	yes	characteristic
84	#1842	C₂×C₂₉	1	1	yes	characteristic
85	#1767	C₂×C₂₉	1	1	yes	characteristic
86	#1296	C₂×C₂₉	1	4	yes	—
87	C₂²×C₂₉	#391	1	1	yes	characteristic
88	C₂²×C₂₉	#413	1	2	yes	—
89	C₂²×C₂₉	C₂×#102	1	1	yes	characteristic
90	C₂²×C₂₉	#417	1	2	yes	—
91	C₂²×C₂₉	#419	1	1	yes	characteristic
92	C₄×C₂₉	—	2	4	no	—
93	C₄×C₂₉	—	2	2	no	—
94	C₄×C₂₉	—	4	2	no	—
95	C₄×C₂₉	—	8	4	no	—
96	C₄×C₂₉	#474	1	2	yes	—
97	C₄×C₂₉	#475	1	2	yes	—
98	C₄×C₂₉	#490	1	1	yes	characteristic
99	C₄×C₂₉	#491	1	1	yes	characteristic
100	C₄×C₂₉	#501	1	2	yes	—
101	C₂×#125	—	2	1	no	—
102	C₂×C₄⋊C₈	—	2	2	no	—
103	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
104	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
105	C₂×#97	—	2	1	no	—
106	#389	—	2	2	no	—
107	#389	C₄×C₂₉	1	4	yes	—
108	#384	C₂²×C₂₉	1	2	yes	—
109	#382	—	2	1	no	—
110	C₄³	C₂²×C₂₉	1	1	yes	characteristic
111	#336	—	2	1	no	—
112	C₂×C₂₉	#1547	1	2	yes	—
113	C₂×C₂₉	#1559	1	2	yes	—
114	C₂×C₂₉	#1988	1	1	yes	characteristic
115	C₂×C₂₉	#1992	1	1	yes	characteristic
116	C₂×C₂₉	#1998	1	1	yes	characteristic
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	2	2	no	—
119	#125	—	4	2	no	—
120	#125	—	4	2	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
125	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
126	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
127	C₂×C₄²	D₄×C₂₉	1	2	yes	—
128	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
129	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	4	4	no	—
133	C₄⋊C₈	Q₈×C₂₉	1	4	yes	—
134	#97	—	4	2	no	—
135	#95	—	2	4	no	—
136	C₂₉	#9904	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
139	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
140	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	—	4	4	no	—
148	C₄⋊C₄	—	4	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	D₈×C₂₉	1	2	yes	—
159	C₄²	SD₁₆×C₂₉	1	2	yes	—
160	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
161	C₄²	#41×C₂₉	1	2	yes	—
162	C₂³	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	C₂₉×#102	1	4	yes	—
173	C₂×C₄	C₄×Q₈×C₂₉	1	2	yes	—
174	C₂×C₄	C₂₉×C₂²⋊Q₈	1	2	yes	—
175	C₂×C₄	C₂₉×#124	1	1	yes	characteristic
176	C₂×C₄	C₂₉×C₄⋊Q₈	1	1	yes	characteristic
177	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
179	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	2	yes	—
180	C₂×C₄	C₂₉×#137	1	2	yes	—
181	C₂×C₄	C₂₉×#137	1	2	yes	—
182	C₈	—	4	4	no	—
183	C₈	—	8	2	no	—
184	C₂²	C₂₉×#391	1	1	yes	characteristic
185	C₂²	C₂₉×#413	1	2	yes	—
186	C₂²	C₂×C₂₉×#102	1	1	yes	characteristic
187	C₂²	C₂₉×#417	1	2	yes	—
188	C₂²	C₂₉×#419	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	4	no	—
191	C₄	—	4	2	no	—
192	C₄	—	8	4	no	—
193	C₄	C₂₉×#474	1	2	yes	—
194	C₄	C₂₉×#475	1	2	yes	—
195	C₄	C₂₉×#490	1	1	yes	characteristic
196	C₄	C₂₉×#491	1	1	yes	characteristic
197	C₄	C₂₉×#501	1	2	yes	—
198	C₂	C₂₉×#1547	1	2	yes	—
199	C₂	C₂₉×#1559	1	2	yes	—
200	C₂	C₂₉×#1988	1	1	yes	characteristic
201	C₂	C₂₉×#1992	1	1	yes	characteristic
202	C₂	C₂₉×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₄³×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
60. #9904	C₂₉	direct (N × H)
136. C₂₉	#9904	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×#2859	2
C₂₉×#1842	2
C₂₉×#1767	2
C₂₉×#1296	2
#9904	29

Subgroups — C29×#9904 #20304804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#9904 #20304804