Subgroups — C₄×C₂₉×#137 #16816659

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#137	C₂	1	1	yes	maximal, characteristic
2	C₄×#41×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#441	C₂	1	2	yes	maximal, complemented
5	C₂₉×#440	C₂	1	2	yes	maximal, complemented
6	C₄×Q₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
7	C₄×SD₁₆×C₂₉	C₂	1	2	yes	maximal, complemented
8	C₂₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×#417	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×#137	C₄	1	8	yes	direct product factor
15	C₂×Q₁₆×C₂₉	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₂₉	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₉	—	2	1	no	—
20	C₄×Q₈×C₂₉	—	2	1	no	—
21	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
22	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
24	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₉	—	2	1	no	—
26	C₄×D₄×C₂₉	—	2	1	no	—
27	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
29	C₂₉×#117	—	2	1	no	—
30	C₂₉×#117	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₂₉	—	2	1	no	—
34	C₂×C₄²×C₂₉	—	2	1	no	—
35	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
36	C₂₉×#107	C₂²	1	2	yes	—
37	C₂₉×#106	C₂²	1	2	yes	—
38	C₂₉×#103	C₂²	1	2	yes	—
39	C₂₉×#103	C₂²	1	2	yes	—
40	C₂₉×#103	C₂²	1	2	yes	—
41	C₂₉×#102	C₂²	1	2	yes	—
42	C₂₉×#97	C₂²	1	2	yes	—
43	C₄×C₈×C₂₉	C₂²	1	2	yes	—
44	#41×C₂₉	—	2	2	no	—
45	#41×C₂₉	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₂₉	—	2	1	no	—
47	C₂×Q₈×C₂₉	—	2	2	no	—
48	C₂×Q₈×C₂₉	C₂×C₄	1	4	yes	complemented
49	C₂×Q₈×C₂₉	C₂³	1	2	yes	—
50	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₂₉	—	2	1	no	—
52	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	—	2	1	no	—
54	C₂²×C₄×C₂₉	—	2	1	no	—
55	C₂²×C₄×C₂₉	—	2	1	no	—
56	C₂²×C₄×C₂₉	—	2	1	no	—
57	C₂²×C₄×C₂₉	—	2	1	no	—
58	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
61	Q₁₆×C₂₉	C₂×C₄	1	8	yes	complemented
62	SD₁₆×C₂₉	C₂×C₄	1	8	yes	complemented
63	M₁₆×C₂₉	—	2	2	no	—
64	M₁₆×C₂₉	C₂×C₄	1	4	yes	complemented
65	C₂×C₈×C₂₉	C₂³	1	2	yes	—
66	C₂×C₈×C₂₉	C₂³	1	2	yes	—
67	C₄⋊C₄×C₂₉	—	2	2	no	—
68	C₄⋊C₄×C₂₉	—	2	2	no	—
69	C₄⋊C₄×C₂₉	—	2	1	no	—
70	C₄⋊C₄×C₂₉	—	2	1	no	—
71	C₄⋊C₄×C₂₉	—	2	1	no	—
72	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
73	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₉	—	2	1	no	—
76	C₂²⋊C₄×C₂₉	—	2	1	no	—
77	C₂²⋊C₄×C₂₉	—	2	1	no	—
78	C₄²×C₂₉	—	2	1	no	—
79	C₄²×C₂₉	—	2	1	no	—
80	C₄²×C₂₉	—	2	2	no	—
81	C₄²×C₂₉	—	2	1	no	—
82	C₄²×C₂₉	—	2	2	no	—
83	C₄²×C₂₉	D₄	1	1	yes	characteristic
84	C₄²×C₂₉	D₄	1	1	yes	characteristic
85	C₄²×C₂₉	C₂³	1	1	yes	characteristic
86	C₄²×C₂₉	C₂³	1	1	yes	characteristic
87	C₂³×C₂₉	—	2	1	no	—
88	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₂₉	—	2	4	no	—
90	Q₈×C₂₉	—	2	2	no	—
91	Q₈×C₂₉	—	2	1	no	—
92	Q₈×C₂₉	C₂²×C₄	1	4	yes	—
93	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
94	D₄×C₂₉	—	2	4	no	—
95	D₄×C₂₉	—	2	1	no	—
96	D₄×C₂₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₉	—	2	4	no	—
98	C₂×C₄×C₂₉	—	2	1	no	—
99	C₂×C₄×C₂₉	—	2	2	no	—
100	C₂×C₄×C₂₉	—	2	4	no	—
101	C₂×C₄×C₂₉	—	2	2	no	—
102	C₂×C₄×C₂₉	—	2	1	no	—
103	C₂×C₄×C₂₉	—	2	1	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	—	4	1	no	—
106	C₂×C₄×C₂₉	—	4	2	no	—
107	C₂×C₄×C₂₉	—	4	2	no	—
108	C₂×C₄×C₂₉	—	4	2	no	—
109	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₉	#41	1	2	yes	—
116	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
117	C₈×C₂₉	—	2	2	no	—
118	C₈×C₂₉	C₂²×C₄	1	4	yes	—
119	C₄×#137	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₂₉	—	2	1	no	—
121	C₂²×C₂₉	—	2	1	no	—
122	C₂²×C₂₉	—	2	2	no	—
123	C₂²×C₂₉	—	4	1	no	—
124	C₂²×C₂₉	—	4	1	no	—
125	C₂²×C₂₉	C₄×D₄	1	2	yes	—
126	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	2	4	no	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	—	2	1	no	—
131	C₄×C₂₉	—	2	1	no	—
132	C₄×C₂₉	—	4	1	no	—
133	C₄×C₂₉	—	4	1	no	—
134	C₄×C₂₉	—	4	2	no	—
135	C₄×C₂₉	C₄×D₄	1	2	yes	—
136	C₄×C₂₉	#137	1	2	yes	direct product factor
137	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
138	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
139	C₂×#137	C₂×C₂₉	1	1	yes	characteristic
140	C₄×#41	C₂×C₂₉	1	1	yes	characteristic
141	C₂×C₄×Q₈	C₂×C₂₉	1	1	yes	characteristic, complemented
142	#441	C₂×C₂₉	1	2	yes	complemented
143	#440	C₂×C₂₉	1	2	yes	complemented
144	C₄×Q₁₆	C₂×C₂₉	1	2	yes	complemented
145	C₄×SD₁₆	C₂×C₂₉	1	2	yes	complemented
146	#428	C₂×C₂₉	1	1	yes	characteristic, complemented
147	#419	C₂×C₂₉	1	1	yes	characteristic, complemented
148	#417	C₂×C₂₉	1	1	yes	characteristic
149	C₄×M₁₆	C₂×C₂₉	1	1	yes	characteristic, complemented
150	C₂×C₂₉	—	2	2	no	—
151	C₂×C₂₉	—	4	2	no	—
152	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂×C₂₉	C₂×#137	1	1	yes	characteristic
154	C₂×C₂₉	#575	1	1	yes	characteristic
155	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
156	C₂²×Q₈	C₂²×C₂₉	1	1	yes	characteristic
157	#137	C₄×C₂₉	1	8	yes	direct product factor
158	C₂×Q₁₆	C₂²×C₂₉	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₂₉	1	2	yes	—
160	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
161	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
162	C₄×Q₈	—	2	1	no	—
163	C₄×Q₈	—	2	1	no	—
164	C₄×Q₈	C₂²×C₂₉	1	2	yes	—
165	C₄×Q₈	C₂²×C₂₉	1	2	yes	—
166	C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
167	C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂²×C₂₉	1	1	yes	characteristic
171	C₄×D₄	C₂²×C₂₉	1	1	yes	characteristic
172	#117	—	2	1	no	—
173	#117	C₂²×C₂₉	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	—	2	1	no	—
178	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
179	#107	C₂²×C₂₉	1	2	yes	—
180	#106	C₂²×C₂₉	1	2	yes	—
181	#103	C₂²×C₂₉	1	2	yes	—
182	#103	C₂²×C₂₉	1	2	yes	—
183	#103	C₂²×C₂₉	1	2	yes	—
184	#102	C₂²×C₂₉	1	2	yes	—
185	#97	C₂²×C₂₉	1	2	yes	—
186	C₄×C₈	C₂²×C₂₉	1	2	yes	—
187	C₂₉	C₄×#137	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	2	2	no	—
189	#41	C₂×C₄×C₂₉	1	4	yes	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂×C₄×C₂₉	1	4	yes	complemented
193	C₂×Q₈	C₂³×C₂₉	1	2	yes	—
194	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂³×C₂₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
203	C₂²×C₄	D₄×C₂₉	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
205	Q₁₆	C₂×C₄×C₂₉	1	8	yes	complemented
206	SD₁₆	C₂×C₄×C₂₉	1	8	yes	complemented
207	M₁₆	—	2	2	no	—
208	M₁₆	C₂×C₄×C₂₉	1	4	yes	complemented
209	C₂×C₈	C₂³×C₂₉	1	2	yes	—
210	C₂×C₈	C₂³×C₂₉	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
217	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
218	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	D₄×C₂₉	1	1	yes	characteristic
228	C₄²	D₄×C₂₉	1	1	yes	characteristic
229	C₄²	C₂³×C₂₉	1	1	yes	characteristic
230	C₄²	C₂³×C₂₉	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	1	no	—
236	Q₈	C₂²×C₄×C₂₉	1	4	yes	—
237	Q₈	C₂²×C₄×C₂₉	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂²×C₄×C₂₉	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
259	C₂×C₄	#41×C₂₉	1	2	yes	—
260	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂²×C₄×C₂₉	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₄×D₄×C₂₉	1	2	yes	—
269	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₄×D₄×C₂₉	1	2	yes	—
279	C₄	C₂₉×#137	1	2	yes	direct product factor
280	C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
281	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
285	C₂	C₂×C₂₉×#137	1	1	yes	characteristic
286	C₂	C₂₉×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×D₄
commutator	C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂×#41×C₂₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Q₈×C₂₉	C₂	semidirect
4. C₂₉×#441	C₂	semidirect
5. C₂₉×#440	C₂	semidirect
6. C₄×Q₁₆×C₂₉	C₂	semidirect
7. C₄×SD₁₆×C₂₉	C₂	semidirect
8. C₂₉×#428	C₂	semidirect
9. C₂₉×#419	C₂	semidirect
11. C₄×M₁₆×C₂₉	C₂	semidirect
14. C₂₉×#137	C₄	direct (N × H)
48. C₂×Q₈×C₂₉	C₂×C₄	semidirect
61. Q₁₆×C₂₉	C₂×C₄	semidirect
62. SD₁₆×C₂₉	C₂×C₄	semidirect
64. M₁₆×C₂₉	C₂×C₄	semidirect
119. C₄×#137	C₂₉	direct (N × H)
136. C₄×C₂₉	#137	direct (N × H)
141. C₂×C₄×Q₈	C₂×C₂₉	semidirect
142. #441	C₂×C₂₉	semidirect
143. #440	C₂×C₂₉	semidirect
144. C₄×Q₁₆	C₂×C₂₉	semidirect
145. C₄×SD₁₆	C₂×C₂₉	semidirect
146. #428	C₂×C₂₉	semidirect
147. #419	C₂×C₂₉	semidirect
149. C₄×M₁₆	C₂×C₂₉	semidirect
157. #137	C₄×C₂₉	direct (N × H)
187. C₂₉	C₄×#137	direct (N × H)
192. C₂×Q₈	C₂×C₄×C₂₉	semidirect
205. Q₁₆	C₂×C₄×C₂₉	semidirect
206. SD₁₆	C₂×C₄×C₂₉	semidirect
208. M₁₆	C₂×C₄×C₂₉	semidirect
279. C₄	C₂₉×#137	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#137	2
C₄×#41×C₂₉	2
C₂×C₄×Q₈×C₂₉	2
C₂₉×#441	2
C₂₉×#440	2
C₄×Q₁₆×C₂₉	2
C₄×SD₁₆×C₂₉	2
C₂₉×#428	2
C₂₉×#419	2
C₂₉×#417	2
C₄×M₁₆×C₂₉	2
C₄×#137	29

Subgroups — C4×C29×#137 #16816659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₉×#137 #16816659