Subgroups — C₂₉×#3061 #16816775

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#537	C₂	1	2	yes	maximal, complemented
4	C₂₉×#468	C₂	1	4	yes	maximal, complemented
5	C₂₉×#467	C₂	1	2	yes	maximal, complemented
6	C₂₉×#466	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₉×#107	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#417	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₉	C₂²	1	2	yes	—
10	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
11	C₂₉×#135	—	2	4	no	—
12	C₂×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	—
14	C₂×D₈×C₂₉	C₂²	1	1	yes	characteristic
15	C₂₉×#131	—	2	4	no	—
16	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
19	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×C₂²⋊Q₈	—	2	2	no	—
21	C₂₉×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₂₉×#121	—	2	2	no	—
23	C₂₉×#121	—	2	4	no	—
24	C₂₉×#121	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
26	C₂₉×#107	C₂²	1	2	yes	—
27	C₂₉×#107	C₂²	1	2	yes	—
28	C₂₉×#103	C₂²	1	4	yes	—
29	C₂₉×#102	C₂²	1	4	yes	—
30	#41×C₂₉	—	2	2	no	—
31	#41×C₂₉	—	2	2	no	—
32	#41×C₂₉	—	2	4	no	—
33	#41×C₂₉	—	2	4	no	—
34	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
36	C₂×D₄×C₂₉	—	2	2	no	—
37	C₂×D₄×C₂₉	—	2	4	no	—
38	C₂×D₄×C₂₉	D₄	1	2	yes	—
39	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
40	C₂×D₄×C₂₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₉	—	2	2	no	—
42	C₂²×C₄×C₂₉	—	2	2	no	—
43	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
45	Q₁₆×C₂₉	—	2	2	no	—
46	SD₁₆×C₂₉	—	2	4	no	—
47	D₈×C₂₉	—	2	2	no	—
48	M₁₆×C₂₉	—	2	2	no	—
49	M₁₆×C₂₉	—	2	4	no	—
50	C₂×C₈×C₂₉	—	2	4	no	—
51	C₂×C₈×C₂₉	D₄	1	4	yes	complemented
52	C₂×C₈×C₂₉	C₂³	1	2	yes	—
53	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₉	—	2	2	no	—
56	C₄⋊C₄×C₂₉	—	2	4	no	—
57	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
58	C₂²⋊C₄×C₂₉	—	2	4	no	—
59	C₂²⋊C₄×C₂₉	—	2	4	no	—
60	C₂³×C₂₉	—	2	2	no	—
61	C₂³×C₂₉	C₂×D₄	1	2	yes	—
62	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₂₉	—	2	2	no	—
64	Q₈×C₂₉	—	2	2	no	—
65	Q₈×C₂₉	—	2	2	no	—
66	D₄×C₂₉	—	2	4	no	—
67	D₄×C₂₉	—	2	2	no	—
68	D₄×C₂₉	—	2	2	no	—
69	D₄×C₂₉	—	2	2	no	—
70	D₄×C₂₉	—	4	4	no	—
71	D₄×C₂₉	—	8	4	no	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	—	2	2	no	—
74	C₂×C₄×C₂₉	—	2	4	no	—
75	C₂×C₄×C₂₉	—	2	2	no	—
76	C₂×C₄×C₂₉	—	4	4	no	—
77	C₂×C₄×C₂₉	—	8	2	no	—
78	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₉	#41	1	2	yes	—
82	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₉	—	2	2	no	—
84	C₈×C₂₉	C₂×D₄	1	2	yes	—
85	C₈×C₂₉	C₂×D₄	1	2	yes	—
86	#3061	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₂₉	—	2	1	no	—
88	C₂²×C₂₉	—	2	1	no	—
89	C₂²×C₂₉	—	2	2	no	—
90	C₂²×C₂₉	—	2	2	no	—
91	C₂²×C₂₉	—	2	2	no	—
92	C₂²×C₂₉	—	4	2	no	—
93	C₂²×C₂₉	—	4	2	no	—
94	C₂²×C₂₉	—	4	2	no	—
95	C₂²×C₂₉	#121	1	2	yes	—
96	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₂₉	—	2	2	no	—
98	C₄×C₂₉	—	4	2	no	—
99	C₄×C₂₉	—	4	4	no	—
100	C₄×C₂₉	#121	1	2	yes	—
101	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
103	C₂×#135	C₂×C₂₉	1	1	yes	characteristic, complemented
104	C₂×#131	C₂×C₂₉	1	1	yes	characteristic, complemented
105	#537	C₂×C₂₉	1	2	yes	complemented
106	#468	C₂×C₂₉	1	4	yes	complemented
107	#467	C₂×C₂₉	1	2	yes	complemented
108	#466	C₂×C₂₉	1	2	yes	complemented
109	C₂×#107	C₂×C₂₉	1	1	yes	characteristic, complemented
110	#417	C₂×C₂₉	1	2	yes	complemented
111	C₂×C₂₉	—	2	2	no	—
112	C₂×C₂₉	—	4	2	no	—
113	C₂×C₂₉	—	8	2	no	—
114	C₂×C₂₉	C₂×#121	1	1	yes	characteristic
115	C₂×C₂₉	#576	1	1	yes	characteristic
116	C₂×C₂₉	#578	1	1	yes	characteristic
117	C₂×#41	C₂²×C₂₉	1	2	yes	—
118	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
119	#135	—	2	4	no	—
120	C₂×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
121	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	—
122	C₂×D₈	C₂²×C₂₉	1	1	yes	characteristic
123	#131	—	2	4	no	—
124	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
125	C₂×M₁₆	C₂²×C₂₉	1	2	yes	—
126	C₂²×C₈	C₂²×C₂₉	1	2	yes	—
127	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
128	C₂²⋊Q₈	—	2	2	no	—
129	C₂²⋊Q₈	C₂²×C₂₉	1	4	yes	—
130	#121	—	2	2	no	—
131	#121	—	2	4	no	—
132	#121	C₂²×C₂₉	1	4	yes	—
133	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
134	#107	C₂²×C₂₉	1	2	yes	—
135	#107	C₂²×C₂₉	1	2	yes	—
136	#103	C₂²×C₂₉	1	4	yes	—
137	#102	C₂²×C₂₉	1	4	yes	—
138	C₂₉	#3061	1	1	yes	characteristic, direct product factor, Sylow
139	#41	—	2	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	4	no	—
142	#41	—	2	4	no	—
143	C₂×Q₈	D₄×C₂₉	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	D₄×C₂₉	1	2	yes	—
148	C₂×D₄	D₄×C₂₉	1	1	yes	characteristic
149	C₂×D₄	C₂³×C₂₉	1	2	yes	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
153	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
154	Q₁₆	—	2	2	no	—
155	SD₁₆	—	2	4	no	—
156	D₈	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	D₄×C₂₉	1	4	yes	complemented
161	C₂×C₈	C₂³×C₂₉	1	2	yes	—
162	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₂×D₄×C₂₉	1	2	yes	—
171	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	2	2	no	—
175	D₄	—	2	4	no	—
176	D₄	—	2	2	no	—
177	D₄	—	2	2	no	—
178	D₄	—	2	2	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
188	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
190	C₂×C₄	#41×C₂₉	1	2	yes	—
191	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₂×D₄×C₂₉	1	2	yes	—
194	C₈	C₂×D₄×C₂₉	1	2	yes	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	C₂₉×#121	1	2	yes	—
204	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	4	no	—
208	C₄	C₂₉×#121	1	2	yes	—
209	C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
210	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	4	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₂×C₂₉×#121	1	1	yes	characteristic
215	C₂	C₂₉×#576	1	1	yes	characteristic
216	C₂	C₂₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×#135	C₂	semidirect
2. C₂×C₂₉×#131	C₂	semidirect
3. C₂₉×#537	C₂	semidirect
4. C₂₉×#468	C₂	semidirect
5. C₂₉×#467	C₂	semidirect
6. C₂₉×#466	C₂	semidirect
7. C₂×C₂₉×#107	C₂	semidirect
8. C₂₉×#417	C₂	semidirect
51. C₂×C₈×C₂₉	D₄	semidirect
86. #3061	C₂₉	direct (N × H)
103. C₂×#135	C₂×C₂₉	semidirect
104. C₂×#131	C₂×C₂₉	semidirect
105. #537	C₂×C₂₉	semidirect
106. #468	C₂×C₂₉	semidirect
107. #467	C₂×C₂₉	semidirect
108. #466	C₂×C₂₉	semidirect
109. C₂×#107	C₂×C₂₉	semidirect
110. #417	C₂×C₂₉	semidirect
138. C₂₉	#3061	direct (N × H)
160. C₂×C₈	D₄×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×#135	2
C₂×C₂₉×#131	2
C₂₉×#537	2
C₂₉×#468	2
C₂₉×#467	2
C₂₉×#466	2
C₂×C₂₉×#107	2
C₂₉×#417	2
#3061	29

Subgroups — C29×#3061 #16816775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3061 #16816775