Subgroups — C₂₉×#3210 #16816924

Order 3712.

394 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#464	C₂	1	4	yes	maximal, complemented
8	C₂₉×#452	C₂	1	4	yes	maximal, complemented
9	C₂₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₉	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₂₉	C₂²	1	4	yes	—
12	C₂×SD₁₆×C₂₉	C₂²	1	4	yes	—
13	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₉×#127	C₂²	1	1	yes	characteristic
17	C₂₉×#125	—	2	1	no	—
18	C₂₉×#124	C₂²	1	2	yes	—
19	C₂₉×C₂²⋊Q₈	—	2	1	no	—
20	C₂₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂₉×#121	—	2	1	no	—
22	C₂₉×#121	—	2	1	no	—
23	C₄×Q₈×C₂₉	—	2	1	no	—
24	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
25	C₄×D₄×C₂₉	—	2	1	no	—
26	C₄×D₄×C₂₉	—	2	1	no	—
27	C₂₉×#117	—	2	1	no	—
28	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
29	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₉×#103	C₂²	1	4	yes	—
31	C₂₉×#102	C₂²	1	4	yes	—
32	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
35	#41×C₂₉	—	4	4	no	—
36	C₂×Q₈×C₂₉	—	2	1	no	—
37	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
38	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
39	C₂×D₄×C₂₉	—	2	2	no	—
40	C₂×D₄×C₂₉	—	2	1	no	—
41	C₂×D₄×C₂₉	—	2	1	no	—
42	C₂×D₄×C₂₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₉	—	2	2	no	—
44	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
47	Q₁₆×C₂₉	—	4	4	no	—
48	SD₁₆×C₂₉	—	4	4	no	—
49	C₂×C₈×C₂₉	C₂³	1	2	yes	—
50	C₂×C₈×C₂₉	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₉	—	2	1	no	—
52	C₄⋊C₄×C₂₉	—	2	1	no	—
53	C₄⋊C₄×C₂₉	—	2	1	no	—
54	C₄⋊C₄×C₂₉	—	2	1	no	—
55	C₄⋊C₄×C₂₉	—	2	1	no	—
56	C₄⋊C₄×C₂₉	—	2	2	no	—
57	C₄⋊C₄×C₂₉	—	2	2	no	—
58	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
59	C₂²⋊C₄×C₂₉	—	2	2	no	—
60	C₂²⋊C₄×C₂₉	—	2	2	no	—
61	C₂²⋊C₄×C₂₉	—	2	2	no	—
62	C₄²×C₂₉	—	2	2	no	—
63	C₄²×C₂₉	D₄	1	1	yes	characteristic
64	C₄²×C₂₉	D₄	1	1	yes	characteristic
65	C₄²×C₂₉	C₂³	1	1	yes	characteristic
66	C₄²×C₂₉	C₂³	1	1	yes	characteristic
67	C₂³×C₂₉	—	2	2	no	—
68	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₂₉	—	2	4	no	—
70	Q₈×C₂₉	—	2	4	no	—
71	Q₈×C₂₉	—	4	2	no	—
72	D₄×C₂₉	—	2	4	no	—
73	D₄×C₂₉	—	4	4	no	—
74	D₄×C₂₉	—	4	2	no	—
75	D₄×C₂₉	—	4	2	no	—
76	C₂×C₄×C₂₉	—	2	2	no	—
77	C₂×C₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	2	no	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	—	4	4	no	—
83	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
89	C₈×C₂₉	—	2	2	no	—
90	C₈×C₂₉	—	2	2	no	—
91	#3210	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₂₉	—	2	1	no	—
93	C₂²×C₂₉	—	2	1	no	—
94	C₂²×C₂₉	—	2	1	no	—
95	C₂²×C₂₉	—	4	2	no	—
96	C₂²×C₂₉	—	4	2	no	—
97	C₂²×C₂₉	—	4	2	no	—
98	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
99	C₄×C₂₉	—	2	1	no	—
100	C₄×C₂₉	—	2	1	no	—
101	C₄×C₂₉	—	2	1	no	—
102	C₄×C₂₉	—	4	2	no	—
103	C₄×C₂₉	—	4	2	no	—
104	C₄×C₂₉	—	4	2	no	—
105	C₄×C₂₉	#135	1	2	yes	—
106	C₄×C₂₉	#137	1	2	yes	—
107	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
108	#533	C₂×C₂₉	1	1	yes	characteristic, complemented
109	#532	C₂×C₂₉	1	1	yes	characteristic
110	#494	C₂×C₂₉	1	1	yes	characteristic, complemented
111	#492	C₂×C₂₉	1	1	yes	characteristic, complemented
112	#487	C₂×C₂₉	1	1	yes	characteristic, complemented
113	#486	C₂×C₂₉	1	1	yes	characteristic, complemented
114	#464	C₂×C₂₉	1	4	yes	complemented
115	#452	C₂×C₂₉	1	4	yes	complemented
116	#431	C₂×C₂₉	1	1	yes	characteristic, complemented
117	C₂×C₂₉	—	4	1	no	—
118	C₂×C₂₉	—	8	2	no	—
119	C₂×C₂₉	#535	1	1	yes	characteristic
120	C₂×C₂₉	C₂×#135	1	1	yes	characteristic
121	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
122	C₂×#41	C₂²×C₂₉	1	2	yes	—
123	C₂×Q₁₆	C₂²×C₂₉	1	4	yes	—
124	C₂×SD₁₆	C₂²×C₂₉	1	4	yes	—
125	C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
126	C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
128	#127	C₂²×C₂₉	1	1	yes	characteristic
129	#125	—	2	1	no	—
130	#124	C₂²×C₂₉	1	2	yes	—
131	C₂²⋊Q₈	—	2	1	no	—
132	C₂²⋊Q₈	C₂²×C₂₉	1	2	yes	—
133	#121	—	2	1	no	—
134	#121	—	2	1	no	—
135	C₄×Q₈	—	2	1	no	—
136	C₄×Q₈	C₂²×C₂₉	1	2	yes	—
137	C₄×D₄	—	2	1	no	—
138	C₄×D₄	—	2	1	no	—
139	#117	—	2	1	no	—
140	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
141	C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
142	#103	C₂²×C₂₉	1	4	yes	—
143	#102	C₂²×C₂₉	1	4	yes	—
144	C₂²⋊C₈	C₂²×C₂₉	1	2	yes	—
145	C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
147	C₂₉	#3210	1	1	yes	characteristic, direct product factor, Sylow
148	#41	—	4	4	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
151	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	C₂³×C₂₉	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
158	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
160	Q₁₆	—	4	4	no	—
161	SD₁₆	—	4	4	no	—
162	C₂×C₈	C₂³×C₂₉	1	2	yes	—
163	C₂×C₈	C₂³×C₂₉	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	D₄×C₂₉	1	1	yes	characteristic
177	C₄²	D₄×C₂₉	1	1	yes	characteristic
178	C₄²	C₂³×C₂₉	1	1	yes	characteristic
179	C₄²	C₂³×C₂₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
182	Q₈	—	2	4	no	—
183	Q₈	—	2	4	no	—
184	Q₈	—	4	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₂₉×#135	1	2	yes	—
218	C₄	C₂₉×#137	1	2	yes	—
219	C₄	C₂₉×H(5,𝔽₂)	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₂₉×#535	1	1	yes	characteristic
223	C₂	C₂×C₂₉×#135	1	1	yes	characteristic
224	C₂	C₂×C₂₉×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂₉×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#533	C₂	semidirect
3. C₂₉×#494	C₂	semidirect
4. C₂₉×#492	C₂	semidirect
5. C₂₉×#487	C₂	semidirect
6. C₂₉×#486	C₂	semidirect
7. C₂₉×#464	C₂	semidirect
8. C₂₉×#452	C₂	semidirect
9. C₂₉×#431	C₂	semidirect
91. #3210	C₂₉	direct (N × H)
108. #533	C₂×C₂₉	semidirect
110. #494	C₂×C₂₉	semidirect
111. #492	C₂×C₂₉	semidirect
112. #487	C₂×C₂₉	semidirect
113. #486	C₂×C₂₉	semidirect
114. #464	C₂×C₂₉	semidirect
115. #452	C₂×C₂₉	semidirect
116. #431	C₂×C₂₉	semidirect
147. C₂₉	#3210	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×#533	2
C₂₉×#532	2
C₂₉×#494	2
C₂₉×#492	2
C₂₉×#487	2
C₂₉×#486	2
C₂₉×#464	2
C₂₉×#452	2
C₂₉×#431	2
#3210	29

Subgroups — C29×#3210 #16816924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3210 #16816924