Subgroups — C₂₉×#12674 #20307574

Order 7424.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×#346	C₂	1	1	yes	maximal, characteristic
4	C₄×C₂₉×#107	C₂	1	2	yes	maximal
5	C₂₉×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₂₉×#107	C₂²	1	2	yes	—
9	C₂×C₂₉×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₂₉	—	2	2	no	—
12	C₂₉×#367	C₄	1	4	yes	complemented
13	C₂₉×#366	C₄	1	4	yes	complemented
14	C₂₉×#346	C₄	1	4	yes	complemented
15	C₂₉×#340	—	2	2	no	—
16	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
20	C₂×C₄²×C₂₉	—	4	2	no	—
21	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₂₉	—	2	1	no	—
23	C₂×C₁₆×C₂₉	—	2	1	no	—
24	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
28	C₂₉×#107	—	2	2	no	—
29	C₂₉×#107	—	2	2	no	—
30	C₂₉×#107	C₂×C₄	1	4	yes	—
31	C₂₉×#107	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₂₉	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₂₉	Q₈	1	2	yes	—
34	C₂₉×#95	—	4	2	no	—
35	C₂²×C₄×C₂₉	—	4	2	no	—
36	C₂²×C₄×C₂₉	—	4	2	no	—
37	C₂²×C₄×C₂₉	D₈	1	2	yes	—
38	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₂₉	—	2	1	no	—
40	C₂×C₈×C₂₉	—	2	1	no	—
41	C₂×C₈×C₂₉	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₂₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
47	C₂×C₈×C₂₉	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₂₉	—	2	4	no	—
49	C₄⋊C₄×C₂₉	—	2	4	no	—
50	C₄⋊C₄×C₂₉	—	4	2	no	—
51	C₄⋊C₄×C₂₉	—	4	2	no	—
52	C₄²×C₂₉	—	8	4	no	—
53	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₂₉	Q₁₆	1	2	yes	—
55	C₁₆×C₂₉	—	2	2	no	—
56	C₁₆×C₂₉	C₄²	1	4	yes	complemented
57	#12674	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₂₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₂₉	—	2	2	no	—
60	C₂×C₄×C₂₉	—	2	2	no	—
61	C₂×C₄×C₂₉	—	4	4	no	—
62	C₂×C₄×C₂₉	—	4	4	no	—
63	C₂×C₄×C₂₉	—	8	2	no	—
64	C₂×C₄×C₂₉	—	8	2	no	—
65	C₂×C₄×C₂₉	—	8	2	no	—
66	C₂×C₄×C₂₉	—	8	2	no	—
67	C₂×C₄×C₂₉	#107	1	4	yes	—
68	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₂₉	#135	1	1	yes	characteristic
73	C₂×C₄×C₂₉	#135	1	1	yes	characteristic
74	C₈×C₂₉	—	2	2	no	—
75	C₈×C₂₉	C₂×C₄²	1	1	yes	characteristic
76	C₈×C₂₉	#117	1	1	yes	characteristic
77	C₈×C₂₉	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₂₉	C₄×Q₈	1	1	yes	characteristic
79	C₂×#367	C₂×C₂₉	1	1	yes	characteristic
80	C₂×#366	C₂×C₂₉	1	1	yes	characteristic
81	C₂×#346	C₂×C₂₉	1	1	yes	characteristic
82	C₄×#107	C₂×C₂₉	1	2	yes	—
83	#1380	C₂×C₂₉	1	2	yes	—
84	C₂²×C₂₉	C₂×#107	1	1	yes	characteristic
85	C₂²×C₂₉	C₄×D₈	1	1	yes	characteristic
86	C₂²×C₂₉	C₄×D₈	1	1	yes	characteristic
87	C₂²×C₂₉	#509	1	2	yes	—
88	C₂²×C₂₉	#510	1	2	yes	—
89	C₄×C₂₉	—	2	4	no	—
90	C₄×C₂₉	—	8	4	no	—
91	C₄×C₂₉	—	8	4	no	—
92	C₄×C₂₉	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₂₉	#427	1	1	yes	characteristic
94	C₄×C₂₉	C₄×Q₁₆	1	1	yes	characteristic
95	C₄×C₂₉	C₄×Q₁₆	1	1	yes	characteristic
96	C₂²×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
97	C₂²×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
98	C₂×#107	C₂²×C₂₉	1	2	yes	—
99	C₂×#107	C₂²×C₂₉	1	2	yes	—
100	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
101	C₄×C₄⋊C₄	—	2	2	no	—
102	#367	C₄×C₂₉	1	4	yes	complemented
103	#366	C₄×C₂₉	1	4	yes	complemented
104	#346	C₄×C₂₉	1	4	yes	complemented
105	#340	—	2	2	no	—
106	C₂×C₂₉	C₄×#107	1	1	yes	characteristic
107	C₂×C₂₉	#2159	1	2	yes	—
108	C₂×C₂₉	#2175	1	2	yes	—
109	C₂×C₂₉	#2176	1	1	yes	characteristic
110	C₂×C₂₉	#2177	1	1	yes	characteristic
111	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄²	—	4	2	no	—
116	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
120	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
121	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	C₂×C₄×C₂₉	1	4	yes	—
126	#107	C₂×C₄×C₂₉	1	4	yes	—
127	C₄×C₈	C₂×C₄×C₂₉	1	2	yes	—
128	C₄×C₈	Q₈×C₂₉	1	2	yes	—
129	#95	—	4	2	no	—
130	C₂₉	#12674	1	1	yes	characteristic, direct product factor, Sylow
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₂₉	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₄⋊C₄×C₂₉	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₂₉	1	1	yes	characteristic
142	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
143	C₂×C₈	#41×C₂₉	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	—	8	4	no	—
149	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
150	C₄²	Q₁₆×C₂₉	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄²×C₂₉	1	4	yes	complemented
153	C₂³	C₂×D₈×C₂₉	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	C₂₉×#107	1	4	yes	—
163	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₂₉	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₂₉	1	1	yes	characteristic
167	C₂×C₄	C₂₉×#135	1	1	yes	characteristic
168	C₂×C₄	C₂₉×#135	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄²×C₂₉	1	1	yes	characteristic
171	C₈	C₂₉×#117	1	1	yes	characteristic
172	C₈	C₄×Q₈×C₂₉	1	1	yes	characteristic
173	C₈	C₄×Q₈×C₂₉	1	1	yes	characteristic
174	C₂²	C₂×C₂₉×#107	1	1	yes	characteristic
175	C₂²	C₄×D₈×C₂₉	1	1	yes	characteristic
176	C₂²	C₄×D₈×C₂₉	1	1	yes	characteristic
177	C₂²	C₂₉×#509	1	2	yes	—
178	C₂²	C₂₉×#510	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₄×C₄⋊C₄×C₂₉	1	1	yes	characteristic
183	C₄	C₂₉×#427	1	1	yes	characteristic
184	C₄	C₄×Q₁₆×C₂₉	1	1	yes	characteristic
185	C₄	C₄×Q₁₆×C₂₉	1	1	yes	characteristic
186	C₂	C₄×C₂₉×#107	1	1	yes	characteristic
187	C₂	C₂₉×#2159	1	2	yes	—
188	C₂	C₂₉×#2175	1	2	yes	—
189	C₂	C₂₉×#2176	1	1	yes	characteristic
190	C₂	C₂₉×#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂×D₈
commutator	C₈	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂₉×#367	C₄	semidirect
13. C₂₉×#366	C₄	semidirect
14. C₂₉×#346	C₄	semidirect
56. C₁₆×C₂₉	C₄²	semidirect
57. #12674	C₂₉	direct (N × H)
102. #367	C₄×C₂₉	semidirect
103. #366	C₄×C₂₉	semidirect
104. #346	C₄×C₂₉	semidirect
130. C₂₉	#12674	direct (N × H)
152. C₁₆	C₄²×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×#367	2
C₂×C₂₉×#366	2
C₂×C₂₉×#346	2
C₄×C₂₉×#107	2
C₂₉×#1380	2
#12674	29

Subgroups — C29×#12674 #20307574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#12674 #20307574