Subgroups — C₂₉×#3455 #16817169

Order 3712.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#542	C₂	1	4	yes	maximal, complemented
2	C₂₉×#541	C₂	1	2	yes	maximal, complemented
3	C₂₉×#540	C₂	1	2	yes	maximal, complemented
4	C₂₉×#539	C₂	1	4	yes	maximal, complemented
5	C₂₉×#538	C₂	1	4	yes	maximal, complemented
6	C₂₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#529	C₂	1	8	yes	maximal, complemented
9	C₂₉×#525	C₂	1	2	yes	maximal, complemented
10	C₄×#41×C₂₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
16	C₂₉×#127	C₂²	1	1	yes	characteristic, complemented
17	C₂₉×#126	C₂²	1	8	yes	—
18	C₂₉×#126	C₂²	1	8	yes	complemented
19	C₂₉×#125	C₂²	1	8	yes	complemented
20	C₂₉×#125	C₂²	1	2	yes	complemented
21	C₂₉×#124	C₂²	1	8	yes	—
22	C₂₉×#124	C₂²	1	2	yes	complemented
23	C₂₉×#123	C₂²	1	4	yes	—
24	C₂₉×#123	C₂²	1	8	yes	—
25	C₂₉×#123	C₂²	1	8	yes	complemented
26	C₂₉×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂₉×C₂²⋊Q₈	C₂²	1	4	yes	—
28	C₂₉×C₂²⋊Q₈	C₂²	1	8	yes	complemented
29	C₂₉×#121	C₂²	1	4	yes	—
30	C₂₉×#121	C₂²	1	8	yes	—
31	C₂₉×#121	C₂²	1	4	yes	—
32	C₂₉×#120	C₂²	1	4	yes	—
33	C₄×Q₈×C₂₉	C₂²	1	4	yes	complemented
34	C₄×Q₈×C₂₉	C₂²	1	4	yes	complemented
35	C₄×D₄×C₂₉	C₂²	1	8	yes	—
36	C₄×D₄×C₂₉	C₂²	1	8	yes	—
37	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
38	C₄×D₄×C₂₉	C₂²	1	4	yes	complemented
39	C₂₉×#117	C₂²	1	4	yes	—
40	C₂₉×#117	C₂²	1	2	yes	—
41	C₂₉×#117	C₂²	1	2	yes	—
42	C₂₉×#117	C₂²	1	2	yes	—
43	C₂₉×#117	C₂²	1	4	yes	complemented
44	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
45	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
46	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
47	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₂₉	C₂²	1	4	yes	complemented
49	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
50	#41×C₂₉	—	2	8	no	—
51	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
52	C₂×Q₈×C₂₉	C₂³	1	4	yes	—
53	C₂×D₄×C₂₉	C₂³	1	4	yes	—
54	C₂×D₄×C₂₉	C₂³	1	2	yes	—
55	C₂×D₄×C₂₉	C₂³	1	8	yes	—
56	C₂×D₄×C₂₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₂₉	—	2	4	no	—
58	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
64	C₂²×C₄×C₂₉	C₂³	1	8	yes	—
65	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
67	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
68	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
69	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
70	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
71	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
72	C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
73	C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₂₉	C₂³	1	4	yes	—
75	C₂²⋊C₄×C₂₉	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₂₉	C₂³	1	4	yes	—
77	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
78	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
80	C₂²⋊C₄×C₂₉	C₂³	1	8	yes	—
81	C₄²×C₂₉	C₂³	1	2	yes	—
82	C₄²×C₂₉	C₂³	1	1	yes	characteristic
83	C₄²×C₂₉	C₂³	1	1	yes	characteristic
84	C₄²×C₂₉	C₂³	1	8	yes	—
85	C₄²×C₂₉	C₂³	1	2	yes	—
86	C₄²×C₂₉	C₂³	1	2	yes	—
87	C₄²×C₂₉	C₂³	1	4	yes	—
88	C₂³×C₂₉	—	2	2	no	—
89	C₂³×C₂₉	—	4	2	no	—
90	C₂³×C₂₉	C₂⁴	1	2	yes	—
91	C₂³×C₂₉	C₂⁴	1	4	yes	—
92	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
93	Q₈×C₂₉	—	2	4	no	—
94	Q₈×C₂₉	—	4	4	no	—
95	D₄×C₂₉	—	2	8	no	—
96	D₄×C₂₉	—	2	4	no	—
97	D₄×C₂₉	—	4	4	no	—
98	D₄×C₂₉	—	4	8	no	—
99	C₂×C₄×C₂₉	—	2	2	no	—
100	C₂×C₄×C₂₉	—	2	4	no	—
101	C₂×C₄×C₂₉	—	2	4	no	—
102	C₂×C₄×C₂₉	—	2	8	no	—
103	C₂×C₄×C₂₉	—	4	2	no	—
104	C₂×C₄×C₂₉	—	4	8	no	—
105	C₂×C₄×C₂₉	—	4	2	no	—
106	C₂×C₄×C₂₉	#41	1	4	yes	—
107	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₂₉	C₂⁴	1	4	yes	—
110	C₂×C₄×C₂₉	C₂⁴	1	8	yes	—
111	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
112	C₂×C₄×C₂₉	C₂⁴	1	2	yes	—
113	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
115	#3455	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₂₉	—	2	2	no	—
117	C₂²×C₂₉	—	2	2	no	—
118	C₂²×C₂₉	—	2	4	no	—
119	C₂²×C₂₉	—	2	8	no	—
120	C₂²×C₂₉	—	2	4	no	—
121	C₂²×C₂₉	—	4	4	no	—
122	C₂²×C₂₉	C₂×#41	1	2	yes	—
123	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
124	C₄×C₂₉	—	2	2	no	—
125	C₄×C₂₉	—	2	4	no	—
126	C₄×C₂₉	—	2	4	no	—
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	2	8	no	—
129	C₄×C₂₉	—	2	2	no	—
130	C₄×C₂₉	C₂×#41	1	2	yes	—
131	C₄×C₂₉	C₂×#41	1	2	yes	—
132	#542	C₂×C₂₉	1	4	yes	complemented
133	#541	C₂×C₂₉	1	2	yes	complemented
134	#540	C₂×C₂₉	1	2	yes	complemented
135	#539	C₂×C₂₉	1	4	yes	complemented
136	#538	C₂×C₂₉	1	4	yes	complemented
137	#533	C₂×C₂₉	1	1	yes	characteristic, complemented
138	#532	C₂×C₂₉	1	1	yes	characteristic, complemented
139	#529	C₂×C₂₉	1	8	yes	complemented
140	#525	C₂×C₂₉	1	2	yes	complemented
141	C₄×#41	C₂×C₂₉	1	2	yes	complemented
142	C₂×#117	C₂×C₂₉	1	1	yes	characteristic, complemented
143	C₂×C₂₉	—	2	2	no	—
144	C₂×C₂₉	—	4	4	no	—
145	C₂×C₂₉	—	4	2	no	—
146	C₂×C₂₉	C₂²×#41	1	1	yes	characteristic
147	C₂×C₂₉	#585	1	2	yes	—
148	C₂×#41	C₂²×C₂₉	1	2	yes	—
149	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
150	C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic, complemented
151	C₄⋊Q₈	C₂²×C₂₉	1	2	yes	complemented
152	#127	C₂²×C₂₉	1	1	yes	characteristic, complemented
153	#126	C₂²×C₂₉	1	8	yes	—
154	#126	C₂²×C₂₉	1	8	yes	complemented
155	#125	C₂²×C₂₉	1	8	yes	complemented
156	#125	C₂²×C₂₉	1	2	yes	complemented
157	#124	C₂²×C₂₉	1	8	yes	—
158	#124	C₂²×C₂₉	1	2	yes	complemented
159	#123	C₂²×C₂₉	1	4	yes	—
160	#123	C₂²×C₂₉	1	8	yes	—
161	#123	C₂²×C₂₉	1	8	yes	complemented
162	C₂²⋊Q₈	C₂²×C₂₉	1	4	yes	—
163	C₂²⋊Q₈	C₂²×C₂₉	1	4	yes	—
164	C₂²⋊Q₈	C₂²×C₂₉	1	8	yes	complemented
165	#121	C₂²×C₂₉	1	4	yes	—
166	#121	C₂²×C₂₉	1	8	yes	—
167	#121	C₂²×C₂₉	1	4	yes	—
168	#120	C₂²×C₂₉	1	4	yes	—
169	C₄×Q₈	C₂²×C₂₉	1	4	yes	complemented
170	C₄×Q₈	C₂²×C₂₉	1	4	yes	complemented
171	C₄×D₄	C₂²×C₂₉	1	8	yes	—
172	C₄×D₄	C₂²×C₂₉	1	8	yes	—
173	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
174	C₄×D₄	C₂²×C₂₉	1	4	yes	complemented
175	#117	C₂²×C₂₉	1	4	yes	—
176	#117	C₂²×C₂₉	1	2	yes	—
177	#117	C₂²×C₂₉	1	2	yes	—
178	#117	C₂²×C₂₉	1	2	yes	—
179	#117	C₂²×C₂₉	1	4	yes	complemented
180	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
182	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
184	C₂×C₄²	C₂²×C₂₉	1	4	yes	complemented
185	C₂₉	#3455	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₂³×C₂₉	1	1	yes	characteristic
187	#41	—	2	8	no	—
188	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
189	C₂×Q₈	C₂³×C₂₉	1	4	yes	—
190	C₂×D₄	C₂³×C₂₉	1	4	yes	—
191	C₂×D₄	C₂³×C₂₉	1	2	yes	—
192	C₂×D₄	C₂³×C₂₉	1	8	yes	—
193	C₂×D₄	C₂³×C₂₉	1	4	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂³×C₂₉	1	4	yes	—
196	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
198	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
199	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
200	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
201	C₂²×C₄	C₂³×C₂₉	1	8	yes	—
202	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
204	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
205	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
206	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
207	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
208	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
209	C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
210	C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₂₉	1	4	yes	—
212	C₂²⋊C₄	C₂³×C₂₉	1	4	yes	—
213	C₂²⋊C₄	C₂³×C₂₉	1	4	yes	—
214	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
216	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
217	C₂²⋊C₄	C₂³×C₂₉	1	8	yes	—
218	C₄²	C₂³×C₂₉	1	2	yes	—
219	C₄²	C₂³×C₂₉	1	2	yes	—
220	C₄²	C₂³×C₂₉	1	2	yes	—
221	C₄²	C₂³×C₂₉	1	1	yes	characteristic
222	C₄²	C₂³×C₂₉	1	1	yes	characteristic
223	C₄²	C₂³×C₂₉	1	4	yes	—
224	C₄²	C₂³×C₂₉	1	8	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂⁴×C₂₉	1	2	yes	—
228	C₂³	C₂⁴×C₂₉	1	4	yes	—
229	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	4	4	no	—
232	D₄	—	2	8	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	#41×C₂₉	1	4	yes	—
244	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
245	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
246	C₂×C₄	C₂⁴×C₂₉	1	4	yes	—
247	C₂×C₄	C₂⁴×C₂₉	1	8	yes	—
248	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
249	C₂×C₄	C₂⁴×C₂₉	1	2	yes	—
250	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
251	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	4	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₂×#41×C₂₉	1	2	yes	—
259	C₂²	C₂⁵×C₂₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	4	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	8	no	—
265	C₄	—	2	2	no	—
266	C₄	C₂×#41×C₂₉	1	2	yes	—
267	C₄	C₂×#41×C₂₉	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	4	4	no	—
271	C₂	C₂²×#41×C₂₉	1	1	yes	characteristic
272	C₂	C₂₉×#585	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₉
socle	C₂²×C₂₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#542	C₂	semidirect
2. C₂₉×#541	C₂	semidirect
3. C₂₉×#540	C₂	semidirect
4. C₂₉×#539	C₂	semidirect
5. C₂₉×#538	C₂	semidirect
6. C₂₉×#533	C₂	semidirect
7. C₂₉×#532	C₂	semidirect
8. C₂₉×#529	C₂	semidirect
9. C₂₉×#525	C₂	semidirect
10. C₄×#41×C₂₉	C₂	semidirect
11. C₂×C₂₉×#117	C₂	semidirect
14. C₂₉×C₄⋊Q₈	C₂²	semidirect
15. C₂₉×C₄⋊Q₈	C₂²	semidirect
16. C₂₉×#127	C₂²	semidirect
18. C₂₉×#126	C₂²	semidirect
19. C₂₉×#125	C₂²	semidirect
20. C₂₉×#125	C₂²	semidirect
22. C₂₉×#124	C₂²	semidirect
25. C₂₉×#123	C₂²	semidirect
28. C₂₉×C₂²⋊Q₈	C₂²	semidirect
33. C₄×Q₈×C₂₉	C₂²	semidirect
34. C₄×Q₈×C₂₉	C₂²	semidirect
37. C₄×D₄×C₂₉	C₂²	semidirect
38. C₄×D₄×C₂₉	C₂²	semidirect
43. C₂₉×#117	C₂²	semidirect
48. C₂×C₄²×C₂₉	C₂²	semidirect
115. #3455	C₂₉	direct (N × H)
132. #542	C₂×C₂₉	semidirect
133. #541	C₂×C₂₉	semidirect
134. #540	C₂×C₂₉	semidirect
135. #539	C₂×C₂₉	semidirect
136. #538	C₂×C₂₉	semidirect
137. #533	C₂×C₂₉	semidirect
138. #532	C₂×C₂₉	semidirect
139. #529	C₂×C₂₉	semidirect
140. #525	C₂×C₂₉	semidirect
141. C₄×#41	C₂×C₂₉	semidirect
142. C₂×#117	C₂×C₂₉	semidirect
150. C₄⋊Q₈	C₂²×C₂₉	semidirect
151. C₄⋊Q₈	C₂²×C₂₉	semidirect
152. #127	C₂²×C₂₉	semidirect
154. #126	C₂²×C₂₉	semidirect
155. #125	C₂²×C₂₉	semidirect
156. #125	C₂²×C₂₉	semidirect
158. #124	C₂²×C₂₉	semidirect
161. #123	C₂²×C₂₉	semidirect
164. C₂²⋊Q₈	C₂²×C₂₉	semidirect
169. C₄×Q₈	C₂²×C₂₉	semidirect
170. C₄×Q₈	C₂²×C₂₉	semidirect
173. C₄×D₄	C₂²×C₂₉	semidirect
174. C₄×D₄	C₂²×C₂₉	semidirect
179. #117	C₂²×C₂₉	semidirect
184. C₂×C₄²	C₂²×C₂₉	semidirect
185. C₂₉	#3455	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×#542	2
C₂₉×#541	2
C₂₉×#540	2
C₂₉×#539	2
C₂₉×#538	2
C₂₉×#533	2
C₂₉×#532	2
C₂₉×#529	2
C₂₉×#525	2
C₄×#41×C₂₉	2
C₂×C₂₉×#117	2
#3455	29

Subgroups — C29×#3455 #16817169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3455 #16817169