Subgroups — C₂₉×#3496 #16817210

Order 3712.

1660 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₂₉×#547	C₂	1	4	yes	maximal, complemented
3	C₂₉×#546	C₂	1	8	yes	maximal, complemented
4	D₄²×C₂₉	C₂	1	4	yes	maximal, complemented
5	C₂₉×#537	C₂	1	2	yes	maximal, complemented
6	C₂₉×#534	C₂	1	4	yes	maximal, complemented
7	C₂₉×#532	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₉×#121	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×H(5,𝔽₂)	—	2	16	no	—
11	C₂×#41×C₂₉	C₂²	1	4	yes	complemented
12	C₂×#41×C₂₉	C₂²	1	8	yes	complemented
13	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₂₉	C₂²	1	8	yes	complemented
15	C₂²×D₄×C₂₉	C₂²	1	8	yes	complemented
16	C₂²×D₄×C₂₉	C₂²	1	4	yes	complemented
17	C₂³×C₄×C₂₉	C₂²	1	2	yes	complemented
18	C₂₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₂₉×#127	C₂²	1	2	yes	complemented
20	C₂₉×#124	C₂²	1	4	yes	complemented
21	C₂₉×#123	C₂²	1	16	yes	complemented
22	C₂₉×C₂²⋊Q₈	C₂²	1	8	yes	complemented
23	C₂₉×#121	C₂²	1	4	yes	complemented
24	C₂₉×#121	C₂²	1	8	yes	complemented
25	C₂₉×#121	C₂²	1	8	yes	complemented
26	C₂₉×#121	C₂²	1	8	yes	complemented
27	C₂₉×#121	C₂²	1	4	yes	complemented
28	C₂₉×#121	C₂²	1	8	yes	complemented
29	C₂₉×#120	C₂²	1	16	yes	complemented
30	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
31	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
32	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
33	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
34	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄⋊C₄×C₂₉	C₂²	1	4	yes	complemented
36	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	complemented
37	C₂×C₂²⋊C₄×C₂₉	C₂²	1	8	yes	complemented
38	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic, complemented
39	C₂⁴×C₂₉	C₂³	1	8	yes	—
40	C₂⁴×C₂₉	C₂³	1	2	yes	—
41	#41×C₂₉	—	2	32	no	—
42	#41×C₂₉	—	2	16	no	—
43	C₂×Q₈×C₂₉	C₂³	1	4	yes	—
44	C₂×D₄×C₂₉	—	2	32	no	—
45	C₂×D₄×C₂₉	—	2	32	no	—
46	C₂×D₄×C₂₉	—	4	8	no	—
47	C₂×D₄×C₂₉	D₄	1	8	yes	complemented
48	C₂×D₄×C₂₉	C₂³	1	4	yes	—
49	C₂×D₄×C₂₉	C₂³	1	16	yes	—
50	C₂×D₄×C₂₉	C₂³	1	8	yes	—
51	C₂×D₄×C₂₉	C₂³	1	4	yes	—
52	C₂×D₄×C₂₉	C₂³	1	4	yes	—
53	C₂×D₄×C₂₉	C₂³	1	8	yes	—
54	C₂×D₄×C₂₉	C₂³	1	4	yes	—
55	C₂²×C₄×C₂₉	—	4	4	no	—
56	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
58	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
62	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
65	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
66	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
67	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
69	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
70	C₂²⋊C₄×C₂₉	C₂³	1	16	yes	—
71	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
72	C₄²×C₂₉	C₂³	1	2	yes	—
73	C₄²×C₂₉	C₂³	1	2	yes	—
74	C₂³×C₂₉	—	2	16	no	—
75	C₂³×C₂₉	—	4	2	no	—
76	C₂³×C₂₉	—	4	2	no	—
77	C₂³×C₂₉	—	4	8	no	—
78	C₂³×C₂₉	—	4	8	no	—
79	C₂³×C₂₉	C₂×D₄	1	8	yes	complemented
80	C₂³×C₂₉	C₂⁴	1	8	yes	—
81	C₂³×C₂₉	C₂⁴	1	4	yes	—
82	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
83	Q₈×C₂₉	—	2	8	no	—
84	D₄×C₂₉	—	2	32	no	—
85	D₄×C₂₉	—	2	16	no	—
86	D₄×C₂₉	—	2	8	no	—
87	D₄×C₂₉	—	2	8	no	—
88	D₄×C₂₉	—	4	8	no	—
89	D₄×C₂₉	—	4	4	no	—
90	D₄×C₂₉	C₂×D₄	1	16	yes	complemented
91	C₂×C₄×C₂₉	—	2	16	no	—
92	C₂×C₄×C₂₉	—	2	16	no	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	—	2	8	no	—
95	C₂×C₄×C₂₉	—	4	4	no	—
96	C₂×C₄×C₂₉	—	4	2	no	—
97	C₂×C₄×C₂₉	—	4	1	no	—
98	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	complemented
99	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
100	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
102	C₂×C₄×C₂₉	C₂⁴	1	8	yes	—
103	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
104	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
105	#3496	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₂₉	—	2	1	no	—
107	C₂²×C₂₉	—	2	4	no	—
108	C₂²×C₂₉	—	2	1	no	—
109	C₂²×C₂₉	—	2	4	no	—
110	C₂²×C₂₉	—	2	8	no	—
111	C₂²×C₂₉	—	2	8	no	—
112	C₂²×C₂₉	—	2	8	no	—
113	C₂²×C₂₉	—	4	8	no	—
114	C₂²×C₂₉	—	8	16	no	—
115	C₂²×C₂₉	C₂²×D₄	1	8	yes	—
116	C₂²×C₂₉	C₂²×D₄	1	2	yes	—
117	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
118	C₄×C₂₉	—	2	8	no	—
119	C₄×C₂₉	—	2	1	no	—
120	C₄×C₂₉	—	2	4	no	—
121	C₄×C₂₉	—	2	2	no	—
122	C₄×C₂₉	C₂²×D₄	1	4	yes	—
123	C₄×C₂₉	H(5,𝔽₂)	1	2	yes	complemented
124	C₂×H(5,𝔽₂)	C₂×C₂₉	1	2	yes	complemented
125	#547	C₂×C₂₉	1	4	yes	complemented
126	#546	C₂×C₂₉	1	8	yes	complemented
127	D₄²	C₂×C₂₉	1	4	yes	complemented
128	#537	C₂×C₂₉	1	2	yes	complemented
129	#534	C₂×C₂₉	1	4	yes	complemented
130	#532	C₂×C₂₉	1	2	yes	complemented
131	C₂×#121	C₂×C₂₉	1	4	yes	complemented
132	C₂×C₄×D₄	C₂×C₂₉	1	1	yes	characteristic, complemented
133	C₂×C₂₉	—	2	2	no	—
134	C₂×C₂₉	—	2	8	no	—
135	C₂×C₂₉	—	4	8	no	—
136	C₂×C₂₉	C₂³×D₄	1	1	yes	characteristic
137	C₂×C₂₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₂₉	#585	1	1	yes	characteristic
139	H(5,𝔽₂)	—	2	16	no	—
140	C₂×#41	C₂²×C₂₉	1	4	yes	complemented
141	C₂×#41	C₂²×C₂₉	1	8	yes	complemented
142	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
143	C₂²×D₄	C₂²×C₂₉	1	8	yes	complemented
144	C₂²×D₄	C₂²×C₂₉	1	8	yes	complemented
145	C₂²×D₄	C₂²×C₂₉	1	4	yes	complemented
146	C₂³×C₄	C₂²×C₂₉	1	2	yes	complemented
147	C₄⋊Q₈	C₂²×C₂₉	1	2	yes	complemented
148	#127	C₂²×C₂₉	1	2	yes	complemented
149	#124	C₂²×C₂₉	1	4	yes	complemented
150	#123	C₂²×C₂₉	1	16	yes	complemented
151	C₂²⋊Q₈	C₂²×C₂₉	1	8	yes	complemented
152	#121	C₂²×C₂₉	1	4	yes	complemented
153	#121	C₂²×C₂₉	1	8	yes	complemented
154	#121	C₂²×C₂₉	1	8	yes	complemented
155	#121	C₂²×C₂₉	1	8	yes	complemented
156	#121	C₂²×C₂₉	1	4	yes	complemented
157	#121	C₂²×C₂₉	1	8	yes	complemented
158	#120	C₂²×C₂₉	1	16	yes	complemented
159	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
160	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
161	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
162	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
163	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic, complemented
164	C₂×C₄⋊C₄	C₂²×C₂₉	1	4	yes	complemented
165	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	complemented
166	C₂×C₂²⋊C₄	C₂²×C₂₉	1	8	yes	complemented
167	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic, complemented
168	C₂₉	#3496	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	C₂³×C₂₉	1	8	yes	—
170	C₂⁴	C₂³×C₂₉	1	2	yes	—
171	#41	—	2	32	no	—
172	#41	—	2	16	no	—
173	C₂×Q₈	C₂³×C₂₉	1	4	yes	—
174	C₂×D₄	—	2	32	no	—
175	C₂×D₄	—	2	32	no	—
176	C₂×D₄	—	4	8	no	—
177	C₂×D₄	D₄×C₂₉	1	8	yes	complemented
178	C₂×D₄	C₂³×C₂₉	1	4	yes	—
179	C₂×D₄	C₂³×C₂₉	1	16	yes	—
180	C₂×D₄	C₂³×C₂₉	1	8	yes	—
181	C₂×D₄	C₂³×C₂₉	1	4	yes	—
182	C₂×D₄	C₂³×C₂₉	1	4	yes	—
183	C₂×D₄	C₂³×C₂₉	1	8	yes	—
184	C₂×D₄	C₂³×C₂₉	1	4	yes	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
187	C₂²×C₄	C₂³×C₂₉	1	8	yes	—
188	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
189	C₂²×C₄	C₂³×C₂₉	1	8	yes	—
190	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
192	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
193	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
194	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
195	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
196	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
197	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
198	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
199	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
200	C₂²⋊C₄	C₂³×C₂₉	1	16	yes	—
201	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
202	C₄²	C₂³×C₂₉	1	2	yes	—
203	C₄²	C₂³×C₂₉	1	2	yes	—
204	C₂³	—	2	16	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	2	no	—
207	C₂³	—	4	8	no	—
208	C₂³	—	4	8	no	—
209	C₂³	C₂×D₄×C₂₉	1	8	yes	complemented
210	C₂³	C₂⁴×C₂₉	1	8	yes	—
211	C₂³	C₂⁴×C₂₉	1	4	yes	—
212	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
213	Q₈	—	2	8	no	—
214	D₄	—	2	32	no	—
215	D₄	—	2	16	no	—
216	D₄	—	2	8	no	—
217	D₄	—	2	8	no	—
218	D₄	—	4	8	no	—
219	D₄	—	4	4	no	—
220	D₄	C₂×D₄×C₂₉	1	16	yes	complemented
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×D₄×C₂₉	1	4	yes	complemented
229	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
230	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
231	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
232	C₂×C₄	C₂⁴×C₂₉	1	8	yes	—
233	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
234	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
235	C₂²	—	2	1	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	8	no	—
243	C₂²	—	8	16	no	—
244	C₂²	C₂²×D₄×C₂₉	1	8	yes	—
245	C₂²	C₂²×D₄×C₂₉	1	2	yes	—
246	C₂²	C₂⁵×C₂₉	1	1	yes	characteristic
247	C₄	—	2	8	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	4	no	—
250	C₄	—	2	2	no	—
251	C₄	C₂²×D₄×C₂₉	1	4	yes	—
252	C₄	C₂₉×H(5,𝔽₂)	1	2	yes	complemented
253	C₂	—	2	2	no	—
254	C₂	—	2	8	no	—
255	C₂	—	4	8	no	—
256	C₂	C₂³×D₄×C₂₉	1	1	yes	characteristic
257	C₂	C₂×C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
258	C₂	C₂₉×#585	1	1	yes	characteristic

	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×H(5,𝔽₂)	C₂	semidirect
2. C₂₉×#547	C₂	semidirect
3. C₂₉×#546	C₂	semidirect
4. D₄²×C₂₉	C₂	semidirect
5. C₂₉×#537	C₂	semidirect
6. C₂₉×#534	C₂	semidirect
7. C₂₉×#532	C₂	semidirect
8. C₂×C₂₉×#121	C₂	semidirect
9. C₂×C₄×D₄×C₂₉	C₂	semidirect
11. C₂×#41×C₂₉	C₂²	semidirect
12. C₂×#41×C₂₉	C₂²	semidirect
14. C₂²×D₄×C₂₉	C₂²	semidirect
15. C₂²×D₄×C₂₉	C₂²	semidirect
16. C₂²×D₄×C₂₉	C₂²	semidirect
17. C₂³×C₄×C₂₉	C₂²	semidirect
18. C₂₉×C₄⋊Q₈	C₂²	semidirect
19. C₂₉×#127	C₂²	semidirect
20. C₂₉×#124	C₂²	semidirect
21. C₂₉×#123	C₂²	semidirect
22. C₂₉×C₂²⋊Q₈	C₂²	semidirect
23. C₂₉×#121	C₂²	semidirect
24. C₂₉×#121	C₂²	semidirect
25. C₂₉×#121	C₂²	semidirect
26. C₂₉×#121	C₂²	semidirect
27. C₂₉×#121	C₂²	semidirect
28. C₂₉×#121	C₂²	semidirect
29. C₂₉×#120	C₂²	semidirect
30. C₄×D₄×C₂₉	C₂²	semidirect
31. C₄×D₄×C₂₉	C₂²	semidirect
32. C₄×D₄×C₂₉	C₂²	semidirect
33. C₄×D₄×C₂₉	C₂²	semidirect
34. C₂×C₄⋊C₄×C₂₉	C₂²	semidirect
35. C₂×C₄⋊C₄×C₂₉	C₂²	semidirect
36. C₂×C₂²⋊C₄×C₂₉	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₂₉	C₂²	semidirect
38. C₂×C₄²×C₂₉	C₂²	semidirect
47. C₂×D₄×C₂₉	D₄	semidirect
79. C₂³×C₂₉	C₂×D₄	semidirect
90. D₄×C₂₉	C₂×D₄	semidirect
98. C₂×C₄×C₂₉	C₂×D₄	semidirect
105. #3496	C₂₉	direct (N × H)
123. C₄×C₂₉	H(5,𝔽₂)	semidirect
124. C₂×H(5,𝔽₂)	C₂×C₂₉	semidirect
125. #547	C₂×C₂₉	semidirect
126. #546	C₂×C₂₉	semidirect
127. D₄²	C₂×C₂₉	semidirect
128. #537	C₂×C₂₉	semidirect
129. #534	C₂×C₂₉	semidirect
130. #532	C₂×C₂₉	semidirect
131. C₂×#121	C₂×C₂₉	semidirect
132. C₂×C₄×D₄	C₂×C₂₉	semidirect
140. C₂×#41	C₂²×C₂₉	semidirect
141. C₂×#41	C₂²×C₂₉	semidirect
143. C₂²×D₄	C₂²×C₂₉	semidirect
144. C₂²×D₄	C₂²×C₂₉	semidirect
145. C₂²×D₄	C₂²×C₂₉	semidirect
146. C₂³×C₄	C₂²×C₂₉	semidirect
147. C₄⋊Q₈	C₂²×C₂₉	semidirect
148. #127	C₂²×C₂₉	semidirect
149. #124	C₂²×C₂₉	semidirect
150. #123	C₂²×C₂₉	semidirect
151. C₂²⋊Q₈	C₂²×C₂₉	semidirect
152. #121	C₂²×C₂₉	semidirect
153. #121	C₂²×C₂₉	semidirect
154. #121	C₂²×C₂₉	semidirect
155. #121	C₂²×C₂₉	semidirect
156. #121	C₂²×C₂₉	semidirect
157. #121	C₂²×C₂₉	semidirect
158. #120	C₂²×C₂₉	semidirect
159. C₄×D₄	C₂²×C₂₉	semidirect
160. C₄×D₄	C₂²×C₂₉	semidirect
161. C₄×D₄	C₂²×C₂₉	semidirect
162. C₄×D₄	C₂²×C₂₉	semidirect
163. C₂×C₄⋊C₄	C₂²×C₂₉	semidirect
164. C₂×C₄⋊C₄	C₂²×C₂₉	semidirect
165. C₂×C₂²⋊C₄	C₂²×C₂₉	semidirect
166. C₂×C₂²⋊C₄	C₂²×C₂₉	semidirect
167. C₂×C₄²	C₂²×C₂₉	semidirect
168. C₂₉	#3496	direct (N × H)
177. C₂×D₄	D₄×C₂₉	semidirect
209. C₂³	C₂×D₄×C₂₉	semidirect
220. D₄	C₂×D₄×C₂₉	semidirect
228. C₂×C₄	C₂×D₄×C₂₉	semidirect
252. C₄	C₂₉×H(5,𝔽₂)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₉×H(5,𝔽₂)	2
C₂₉×#547	2
C₂₉×#546	2
D₄²×C₂₉	2
C₂₉×#537	2
C₂₉×#534	2
C₂₉×#532	2
C₂×C₂₉×#121	2
C₂×C₄×D₄×C₂₉	2
#3496	29

Subgroups — C29×#3496 #16817210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3496 #16817210