Subgroups — C₂₉×#3500 #16817214

Order 3712.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#548	C₂	1	4	yes	maximal, complemented
4	C₂₉×#546	C₂	1	12	yes	maximal, complemented
5	C₂₉×#536	C₂	1	3	yes	maximal, complemented
6	C₂₉×#535	C₂	1	3	yes	maximal, complemented
7	C₂₉×#532	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₉×#124	C₂	1	3	yes	maximal, complemented
9	C₄×#41×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×2₋¹⁺⁴	—	2	8	no	—
11	C₂₉×H(5,𝔽₂)	—	2	8	no	—
12	C₂×#41×C₂₉	C₂²	1	3	yes	—
13	C₂×#41×C₂₉	C₂²	1	2	yes	—
14	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
15	C₂×#41×C₂₉	C₂²	1	3	yes	complemented
16	C₂×#41×C₂₉	C₂²	1	6	yes	complemented
17	C₂²×Q₈×C₂₉	C₂²	1	3	yes	complemented
18	C₂²×Q₈×C₂₉	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₂₉	C₂²	1	3	yes	—
20	C₂²×D₄×C₂₉	C₂²	1	6	yes	—
21	C₂₉×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₂₉×#127	C₂²	1	3	yes	complemented
23	C₂₉×#124	C₂²	1	3	yes	complemented
24	C₂₉×#124	C₂²	1	6	yes	complemented
25	C₂₉×#124	C₂²	1	12	yes	complemented
26	C₂₉×#124	C₂²	1	1	yes	characteristic, complemented
27	C₂₉×#123	C₂²	1	12	yes	complemented
28	C₂₉×C₂²⋊Q₈	C₂²	1	12	yes	complemented
29	C₂₉×#121	C₂²	1	12	yes	complemented
30	C₂₉×#121	C₂²	1	12	yes	complemented
31	C₂₉×#120	C₂²	1	12	yes	complemented
32	C₄×Q₈×C₂₉	C₂²	1	2	yes	complemented
33	C₄×D₄×C₂₉	C₂²	1	6	yes	complemented
34	C₄×D₄×C₂₉	C₂²	1	12	yes	complemented
35	C₂₉×#117	C₂²	1	3	yes	complemented
36	C₂×C₂²⋊C₄×C₂₉	C₂²	1	12	yes	complemented
37	C₂×C₄²×C₂₉	C₂²	1	3	yes	complemented
38	C₂⁴×C₂₉	C₂³	1	6	yes	—
39	#41×C₂₉	—	2	8	no	—
40	#41×C₂₉	—	2	24	no	—
41	#41×C₂₉	—	2	12	no	—
42	#41×C₂₉	—	2	12	no	—
43	#41×C₂₉	D₄	1	8	yes	complemented
44	C₂×Q₈×C₂₉	—	2	8	no	—
45	C₂×Q₈×C₂₉	—	2	12	no	—
46	C₂×Q₈×C₂₉	C₂³	1	6	yes	—
47	C₂×Q₈×C₂₉	C₂³	1	3	yes	—
48	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₂₉	—	2	24	no	—
51	C₂×D₄×C₂₉	—	2	12	no	—
52	C₂×D₄×C₂₉	C₂³	1	6	yes	—
53	C₂×D₄×C₂₉	C₂³	1	6	yes	—
54	C₂×D₄×C₂₉	C₂³	1	6	yes	—
55	C₂×D₄×C₂₉	C₂³	1	3	yes	—
56	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₂₉	C₂³	1	12	yes	—
58	C₂×D₄×C₂₉	C₂³	1	3	yes	—
59	C₂²×C₄×C₂₉	C₂³	1	6	yes	—
60	C₂²×C₄×C₂₉	C₂³	1	6	yes	—
61	C₂²×C₄×C₂₉	C₂³	1	6	yes	—
62	C₂²×C₄×C₂₉	C₂³	1	6	yes	—
63	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
64	C₄⋊C₄×C₂₉	C₂³	1	6	yes	—
65	C₄⋊C₄×C₂₉	C₂³	1	12	yes	—
66	C₂²⋊C₄×C₂₉	C₂³	1	12	yes	—
67	C₂²⋊C₄×C₂₉	C₂³	1	24	yes	—
68	C₂²⋊C₄×C₂₉	C₂³	1	6	yes	—
69	C₂²⋊C₄×C₂₉	C₂³	1	4	yes	—
70	C₄²×C₂₉	C₂³	1	1	yes	characteristic
71	C₄²×C₂₉	C₂³	1	3	yes	—
72	C₄²×C₂₉	C₂³	1	6	yes	—
73	C₂³×C₂₉	—	2	12	no	—
74	C₂³×C₂₉	—	4	12	no	—
75	C₂³×C₂₉	C₂⁴	1	6	yes	—
76	C₂³×C₂₉	C₂⁴	1	2	yes	—
77	C₂³×C₂₉	C₂⁴	1	3	yes	—
78	Q₈×C₂₉	—	2	12	no	—
79	Q₈×C₂₉	—	2	6	no	—
80	Q₈×C₂₉	—	2	2	no	—
81	Q₈×C₂₉	C₂×D₄	1	4	yes	—
82	D₄×C₂₉	—	2	12	no	—
83	D₄×C₂₉	—	2	12	no	—
84	D₄×C₂₉	—	2	12	no	—
85	D₄×C₂₉	—	2	6	no	—
86	D₄×C₂₉	—	2	2	no	—
87	D₄×C₂₉	—	4	12	no	—
88	D₄×C₂₉	C₂×D₄	1	12	yes	—
89	C₂×C₄×C₂₉	—	2	12	no	—
90	C₂×C₄×C₂₉	—	2	12	no	—
91	C₂×C₄×C₂₉	—	2	12	no	—
92	C₂×C₄×C₂₉	—	4	6	no	—
93	C₂×C₄×C₂₉	C₂×D₄	1	12	yes	—
94	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
95	C₂×C₄×C₂₉	C₂⁴	1	6	yes	—
96	C₂×C₄×C₂₉	C₂⁴	1	6	yes	—
97	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
98	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₂₉	C₂⁴	1	3	yes	—
100	#3500	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₂₉	—	2	6	no	—
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	6	no	—
104	C₂²×C₂₉	—	2	4	no	—
105	C₂²×C₂₉	—	2	12	no	—
106	C₂²×C₂₉	—	8	12	no	—
107	C₂²×C₂₉	C₂²×D₄	1	6	yes	—
108	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
109	C₄×C₂₉	—	2	2	no	—
110	C₄×C₂₉	—	2	6	no	—
111	C₄×C₂₉	—	2	2	no	—
112	C₄×C₂₉	—	2	6	no	—
113	C₄×C₂₉	C₂²×D₄	1	6	yes	—
114	C₄×C₂₉	C₂²×D₄	1	2	yes	—
115	C₂×2₋¹⁺⁴	C₂×C₂₉	1	1	yes	characteristic, complemented
116	C₂×H(5,𝔽₂)	C₂×C₂₉	1	1	yes	characteristic, complemented
117	#548	C₂×C₂₉	1	4	yes	complemented
118	#546	C₂×C₂₉	1	12	yes	complemented
119	#536	C₂×C₂₉	1	3	yes	complemented
120	#535	C₂×C₂₉	1	3	yes	complemented
121	#532	C₂×C₂₉	1	3	yes	complemented
122	C₂×#124	C₂×C₂₉	1	3	yes	complemented
123	C₄×#41	C₂×C₂₉	1	1	yes	characteristic, complemented
124	C₂×C₂₉	—	2	6	no	—
125	C₂×C₂₉	—	4	2	no	—
126	C₂×C₂₉	—	4	6	no	—
127	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
128	C₂×C₂₉	#585	1	2	yes	—
129	2₋¹⁺⁴	—	2	8	no	—
130	H(5,𝔽₂)	—	2	8	no	—
131	C₂×#41	C₂²×C₂₉	1	3	yes	—
132	C₂×#41	C₂²×C₂₉	1	2	yes	—
133	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
134	C₂×#41	C₂²×C₂₉	1	3	yes	complemented
135	C₂×#41	C₂²×C₂₉	1	6	yes	complemented
136	C₂²×Q₈	C₂²×C₂₉	1	3	yes	complemented
137	C₂²×Q₈	C₂²×C₂₉	1	2	yes	complemented
138	C₂²×D₄	C₂²×C₂₉	1	3	yes	—
139	C₂²×D₄	C₂²×C₂₉	1	6	yes	—
140	C₄⋊Q₈	C₂²×C₂₉	1	3	yes	complemented
141	#127	C₂²×C₂₉	1	3	yes	complemented
142	#124	C₂²×C₂₉	1	3	yes	complemented
143	#124	C₂²×C₂₉	1	6	yes	complemented
144	#124	C₂²×C₂₉	1	12	yes	complemented
145	#124	C₂²×C₂₉	1	1	yes	characteristic, complemented
146	#123	C₂²×C₂₉	1	12	yes	complemented
147	C₂²⋊Q₈	C₂²×C₂₉	1	12	yes	complemented
148	#121	C₂²×C₂₉	1	12	yes	complemented
149	#121	C₂²×C₂₉	1	12	yes	complemented
150	#120	C₂²×C₂₉	1	12	yes	complemented
151	C₄×Q₈	C₂²×C₂₉	1	2	yes	complemented
152	C₄×D₄	C₂²×C₂₉	1	6	yes	complemented
153	C₄×D₄	C₂²×C₂₉	1	12	yes	complemented
154	#117	C₂²×C₂₉	1	3	yes	complemented
155	C₂×C₂²⋊C₄	C₂²×C₂₉	1	12	yes	complemented
156	C₂×C₄²	C₂²×C₂₉	1	3	yes	complemented
157	C₂₉	#3500	1	1	yes	characteristic, direct product factor, Sylow
158	C₂⁴	C₂³×C₂₉	1	6	yes	—
159	#41	—	2	8	no	—
160	#41	—	2	24	no	—
161	#41	—	2	12	no	—
162	#41	—	2	12	no	—
163	#41	D₄×C₂₉	1	8	yes	complemented
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	12	no	—
166	C₂×Q₈	C₂³×C₂₉	1	6	yes	—
167	C₂×Q₈	C₂³×C₂₉	1	3	yes	—
168	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
170	C₂×D₄	—	2	24	no	—
171	C₂×D₄	—	2	12	no	—
172	C₂×D₄	C₂³×C₂₉	1	6	yes	—
173	C₂×D₄	C₂³×C₂₉	1	6	yes	—
174	C₂×D₄	C₂³×C₂₉	1	6	yes	—
175	C₂×D₄	C₂³×C₂₉	1	3	yes	—
176	C₂×D₄	C₂³×C₂₉	1	1	yes	characteristic
177	C₂×D₄	C₂³×C₂₉	1	12	yes	—
178	C₂×D₄	C₂³×C₂₉	1	3	yes	—
179	C₂²×C₄	C₂³×C₂₉	1	6	yes	—
180	C₂²×C₄	C₂³×C₂₉	1	6	yes	—
181	C₂²×C₄	C₂³×C₂₉	1	6	yes	—
182	C₂²×C₄	C₂³×C₂₉	1	6	yes	—
183	C₂²×C₄	C₂³×C₂₉	1	3	yes	—
184	C₄⋊C₄	C₂³×C₂₉	1	6	yes	—
185	C₄⋊C₄	C₂³×C₂₉	1	12	yes	—
186	C₂²⋊C₄	C₂³×C₂₉	1	12	yes	—
187	C₂²⋊C₄	C₂³×C₂₉	1	24	yes	—
188	C₂²⋊C₄	C₂³×C₂₉	1	6	yes	—
189	C₂²⋊C₄	C₂³×C₂₉	1	4	yes	—
190	C₄²	C₂³×C₂₉	1	1	yes	characteristic
191	C₄²	C₂³×C₂₉	1	3	yes	—
192	C₄²	C₂³×C₂₉	1	6	yes	—
193	C₂³	—	2	12	no	—
194	C₂³	—	4	12	no	—
195	C₂³	C₂⁴×C₂₉	1	6	yes	—
196	C₂³	C₂⁴×C₂₉	1	2	yes	—
197	C₂³	C₂⁴×C₂₉	1	3	yes	—
198	Q₈	—	2	12	no	—
199	Q₈	—	2	6	no	—
200	Q₈	—	2	2	no	—
201	Q₈	C₂×D₄×C₂₉	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	12	no	—
204	D₄	—	2	12	no	—
205	D₄	—	2	6	no	—
206	D₄	—	2	2	no	—
207	D₄	—	4	12	no	—
208	D₄	C₂×D₄×C₂₉	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	4	6	no	—
213	C₂×C₄	C₂×D₄×C₂₉	1	12	yes	—
214	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
215	C₂×C₄	C₂⁴×C₂₉	1	6	yes	—
216	C₂×C₄	C₂⁴×C₂₉	1	6	yes	—
217	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
218	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₂₉	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	12	no	—
225	C₂²	—	8	12	no	—
226	C₂²	C₂²×D₄×C₂₉	1	6	yes	—
227	C₂²	C₂⁵×C₂₉	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	6	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	6	no	—
232	C₄	C₂²×D₄×C₂₉	1	6	yes	—
233	C₄	C₂²×D₄×C₂₉	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	6	no	—
236	C₂	—	4	2	no	—
237	C₂	C₂³×D₄×C₂₉	1	1	yes	characteristic
238	C₂	C₂₉×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×2₋¹⁺⁴	C₂	semidirect
2. C₂×C₂₉×H(5,𝔽₂)	C₂	semidirect
3. C₂₉×#548	C₂	semidirect
4. C₂₉×#546	C₂	semidirect
5. C₂₉×#536	C₂	semidirect
6. C₂₉×#535	C₂	semidirect
7. C₂₉×#532	C₂	semidirect
8. C₂×C₂₉×#124	C₂	semidirect
9. C₄×#41×C₂₉	C₂	semidirect
15. C₂×#41×C₂₉	C₂²	semidirect
16. C₂×#41×C₂₉	C₂²	semidirect
17. C₂²×Q₈×C₂₉	C₂²	semidirect
18. C₂²×Q₈×C₂₉	C₂²	semidirect
21. C₂₉×C₄⋊Q₈	C₂²	semidirect
22. C₂₉×#127	C₂²	semidirect
23. C₂₉×#124	C₂²	semidirect
24. C₂₉×#124	C₂²	semidirect
25. C₂₉×#124	C₂²	semidirect
26. C₂₉×#124	C₂²	semidirect
27. C₂₉×#123	C₂²	semidirect
28. C₂₉×C₂²⋊Q₈	C₂²	semidirect
29. C₂₉×#121	C₂²	semidirect
30. C₂₉×#121	C₂²	semidirect
31. C₂₉×#120	C₂²	semidirect
32. C₄×Q₈×C₂₉	C₂²	semidirect
33. C₄×D₄×C₂₉	C₂²	semidirect
34. C₄×D₄×C₂₉	C₂²	semidirect
35. C₂₉×#117	C₂²	semidirect
36. C₂×C₂²⋊C₄×C₂₉	C₂²	semidirect
37. C₂×C₄²×C₂₉	C₂²	semidirect
43. #41×C₂₉	D₄	semidirect
100. #3500	C₂₉	direct (N × H)
115. C₂×2₋¹⁺⁴	C₂×C₂₉	semidirect
116. C₂×H(5,𝔽₂)	C₂×C₂₉	semidirect
117. #548	C₂×C₂₉	semidirect
118. #546	C₂×C₂₉	semidirect
119. #536	C₂×C₂₉	semidirect
120. #535	C₂×C₂₉	semidirect
121. #532	C₂×C₂₉	semidirect
122. C₂×#124	C₂×C₂₉	semidirect
123. C₄×#41	C₂×C₂₉	semidirect
134. C₂×#41	C₂²×C₂₉	semidirect
135. C₂×#41	C₂²×C₂₉	semidirect
136. C₂²×Q₈	C₂²×C₂₉	semidirect
137. C₂²×Q₈	C₂²×C₂₉	semidirect
140. C₄⋊Q₈	C₂²×C₂₉	semidirect
141. #127	C₂²×C₂₉	semidirect
142. #124	C₂²×C₂₉	semidirect
143. #124	C₂²×C₂₉	semidirect
144. #124	C₂²×C₂₉	semidirect
145. #124	C₂²×C₂₉	semidirect
146. #123	C₂²×C₂₉	semidirect
147. C₂²⋊Q₈	C₂²×C₂₉	semidirect
148. #121	C₂²×C₂₉	semidirect
149. #121	C₂²×C₂₉	semidirect
150. #120	C₂²×C₂₉	semidirect
151. C₄×Q₈	C₂²×C₂₉	semidirect
152. C₄×D₄	C₂²×C₂₉	semidirect
153. C₄×D₄	C₂²×C₂₉	semidirect
154. #117	C₂²×C₂₉	semidirect
155. C₂×C₂²⋊C₄	C₂²×C₂₉	semidirect
156. C₂×C₄²	C₂²×C₂₉	semidirect
157. C₂₉	#3500	direct (N × H)
163. #41	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×C₂₉×2₋¹⁺⁴	2
C₂×C₂₉×H(5,𝔽₂)	2
C₂₉×#548	2
C₂₉×#546	2
C₂₉×#536	2
C₂₉×#535	2
C₂₉×#532	2
C₂×C₂₉×#124	2
C₄×#41×C₂₉	2
#3500	29

Subgroups — C29×#3500 #16817214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3500 #16817214