Subgroups — C₂₉×#12616 #20307516

Order 7424.

420 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	C₂²×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×#347	C₂	1	2	yes	maximal, complemented
4	C₂₉×#1752	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#1375	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₂₉×M₃₂	—	2	4	no	—
8	C₂×C₂₉×M₃₂	C₄	1	2	yes	complemented
9	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
10	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₂₉	C₄	1	2	yes	complemented
13	C₂×C₂₉×C₂²⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₂₉×#97	C₂²	1	2	yes	—
15	C₄×C₂²⋊C₄×C₂₉	C₄	1	1	yes	characteristic
16	C₂₉×C₂²⋊C₁₆	C₄	1	2	yes	complemented
17	C₂₉×C₂²⋊C₁₆	C₄	1	2	yes	complemented
18	C₂₉×#347	—	2	4	no	—
19	C₂₉×#336	C₄	1	2	yes	—
20	C₂³×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₂₉	—	2	2	no	—
22	C₂²×C₈×C₂₉	—	2	2	no	—
23	C₂²×C₈×C₂₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₂₉	—	2	1	no	—
31	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
32	C₂₉×M₃₂	—	2	2	no	—
33	C₂₉×M₃₂	—	2	2	no	—
34	C₂₉×M₃₂	—	2	2	no	—
35	C₂₉×M₃₂	—	2	4	no	—
36	C₂₉×M₃₂	D₄	1	4	yes	—
37	C₂×C₁₆×C₂₉	—	2	4	no	—
38	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₂₉	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₂₉	C₂×C₄	1	2	yes	—
41	C₂₉×C₂²⋊C₈	—	2	1	no	—
42	C₂₉×C₂²⋊C₈	—	2	1	no	—
43	C₂₉×#97	—	2	2	no	—
44	C₂₉×#97	—	2	2	no	—
45	C₂₉×#95	—	2	1	no	—
46	C₂⁴×C₂₉	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₂₉	—	2	2	no	—
48	C₂²×C₄×C₂₉	—	2	2	no	—
49	C₂²×C₄×C₂₉	—	2	2	no	—
50	C₂²×C₄×C₂₉	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₂₉	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₂₉	—	2	2	no	—
57	C₂×C₈×C₂₉	—	2	1	no	—
58	C₂×C₈×C₂₉	—	2	1	no	—
59	C₂×C₈×C₂₉	—	2	1	no	—
60	C₂×C₈×C₂₉	—	2	1	no	—
61	C₂×C₈×C₂₉	—	2	2	no	—
62	C₂×C₈×C₂₉	—	2	2	no	—
63	C₂×C₈×C₂₉	—	2	2	no	—
64	C₂×C₈×C₂₉	—	2	2	no	—
65	C₂×C₈×C₂₉	—	2	2	no	—
66	C₂×C₈×C₂₉	C₄²	1	1	yes	characteristic
67	C₂×C₈×C₂₉	C₄²	1	1	yes	characteristic
68	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₉	#41	1	2	yes	—
72	C₂²⋊C₄×C₂₉	—	4	2	no	—
73	C₄²×C₂₉	—	2	2	no	—
74	C₄²×C₂₉	—	2	2	no	—
75	C₁₆×C₂₉	—	2	2	no	—
76	C₁₆×C₂₉	—	2	4	no	—
77	C₁₆×C₂₉	C₂²⋊C₄	1	4	yes	complemented
78	#12616	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₉	—	2	2	no	—
80	C₂³×C₂₉	—	2	2	no	—
81	C₂³×C₂₉	—	4	1	no	—
82	C₂³×C₂₉	#97	1	1	yes	characteristic
83	C₂³×C₂₉	#97	1	1	yes	characteristic
84	C₂³×C₂₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	—	2	1	no	—
87	C₂×C₄×C₂₉	—	2	1	no	—
88	C₂×C₄×C₂₉	—	2	1	no	—
89	C₂×C₄×C₂₉	—	2	1	no	—
90	C₂×C₄×C₂₉	—	2	2	no	—
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	—	4	2	no	—
93	C₂×C₄×C₂₉	—	4	2	no	—
94	C₂×C₄×C₂₉	—	4	2	no	—
95	C₂×C₄×C₂₉	#97	1	1	yes	characteristic
96	C₂×C₄×C₂₉	#97	1	1	yes	characteristic
97	C₂×C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₂₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₂₉	#131	1	2	yes	—
101	C₈×C₂₉	—	2	1	no	—
102	C₈×C₂₉	—	2	1	no	—
103	C₈×C₂₉	—	4	2	no	—
104	C₈×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₂₉	#117	1	1	yes	characteristic
106	C₈×C₂₉	C₄×D₄	1	2	yes	—
107	C₂²×M₃₂	C₂×C₂₉	1	1	yes	characteristic
108	#2112	C₂×C₂₉	1	1	yes	characteristic
109	C₂×#347	C₂×C₂₉	1	2	yes	complemented
110	#1752	C₂×C₂₉	1	1	yes	characteristic
111	#1375	C₂×C₂₉	1	2	yes	complemented
112	C₂²×C₂₉	—	2	1	no	—
113	C₂²×C₂₉	—	2	1	no	—
114	C₂²×C₂₉	—	2	1	no	—
115	C₂²×C₂₉	—	2	1	no	—
116	C₂²×C₂₉	—	2	1	no	—
117	C₂²×C₂₉	—	2	1	no	—
118	C₂²×C₂₉	—	2	1	no	—
119	C₂²×C₂₉	—	4	4	no	—
120	C₂²×C₂₉	#347	1	2	yes	complemented
121	C₂²×C₂₉	C₂×#97	1	1	yes	characteristic
122	C₂²×C₂₉	#435	1	2	yes	—
123	C₄×C₂₉	—	2	1	no	—
124	C₄×C₂₉	—	2	1	no	—
125	C₄×C₂₉	—	8	2	no	—
126	C₄×C₂₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₂₉	#405	1	1	yes	characteristic
128	C₄×C₂₉	#435	1	2	yes	—
129	C₂³×C₈	C₂²×C₂₉	1	1	yes	characteristic
130	C₂×M₃₂	—	2	4	no	—
131	C₂×M₃₂	C₄×C₂₉	1	2	yes	complemented
132	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
133	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
134	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
135	C₂²×C₁₆	C₄×C₂₉	1	2	yes	complemented
136	C₂×C₂²⋊C₈	C₄×C₂₉	1	1	yes	characteristic
137	C₂×#97	C₂²×C₂₉	1	2	yes	—
138	C₄×C₂²⋊C₄	C₄×C₂₉	1	1	yes	characteristic
139	C₂²⋊C₁₆	C₄×C₂₉	1	2	yes	complemented
140	C₂²⋊C₁₆	C₄×C₂₉	1	2	yes	complemented
141	#347	—	2	4	no	—
142	#336	C₄×C₂₉	1	2	yes	—
143	C₂×C₂₉	—	2	2	no	—
144	C₂×C₂₉	—	2	2	no	—
145	C₂×C₂₉	—	4	1	no	—
146	C₂×C₂₉	#1752	1	1	yes	characteristic
147	C₂×C₂₉	C₂×#347	1	1	yes	characteristic
148	C₂×C₂₉	#2110	1	1	yes	characteristic
149	C₂³×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	C₂×C₄×C₂₉	1	2	yes	—
153	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
156	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
157	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	—	2	1	no	—
160	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	—	2	4	no	—
165	M₃₂	D₄×C₂₉	1	4	yes	—
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	C₂×C₄×C₂₉	1	4	yes	—
168	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
169	C₂×C₁₆	C₂×C₄×C₂₉	1	2	yes	—
170	C₂²⋊C₈	—	2	1	no	—
171	C₂²⋊C₈	—	2	1	no	—
172	#97	—	2	2	no	—
173	#97	—	2	2	no	—
174	#95	—	2	1	no	—
175	C₂₉	#12616	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	M₁₆×C₂₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₄²×C₂₉	1	1	yes	characteristic
181	C₂²×C₄	C₄²×C₂₉	1	1	yes	characteristic
182	C₂²×C₄	M₁₆×C₂₉	1	1	yes	characteristic
183	C₂²×C₄	M₁₆×C₂₉	1	1	yes	characteristic
184	C₂²×C₄	M₁₆×C₂₉	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₄²×C₂₉	1	1	yes	characteristic
197	C₂×C₈	C₄²×C₂₉	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
201	C₂×C₈	#41×C₂₉	1	2	yes	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₁₆	—	2	2	no	—
206	C₁₆	—	2	4	no	—
207	C₁₆	C₂²⋊C₄×C₂₉	1	4	yes	complemented
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	C₂₉×#97	1	1	yes	characteristic
212	C₂³	C₂₉×#97	1	1	yes	characteristic
213	C₂³	C₂×M₁₆×C₂₉	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂₉×#97	1	1	yes	characteristic
225	C₂×C₄	C₂₉×#97	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄²×C₂₉	1	1	yes	characteristic
227	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
228	C₂×C₄	C₂×M₁₆×C₂₉	1	1	yes	characteristic
229	C₂×C₄	C₂₉×#131	1	2	yes	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	4	2	no	—
233	C₈	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
234	C₈	C₂₉×#117	1	1	yes	characteristic
235	C₈	C₄×D₄×C₂₉	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₂₉×#347	1	2	yes	complemented
245	C₂²	C₂×C₂₉×#97	1	1	yes	characteristic
246	C₂²	C₂₉×#435	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	8	2	no	—
250	C₄	C₄×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
251	C₄	C₂₉×#405	1	1	yes	characteristic
252	C₄	C₂₉×#435	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	C₂₉×#1752	1	1	yes	characteristic
257	C₂	C₂×C₂₉×#347	1	1	yes	characteristic
258	C₂	C₂₉×#2110	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×#97
prime radical	C₂⁴×C₂₉	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₉×#347	C₂	semidirect
5. C₂₉×#1375	C₂	semidirect
8. C₂×C₂₉×M₃₂	C₄	semidirect
12. C₂²×C₁₆×C₂₉	C₄	semidirect
16. C₂₉×C₂²⋊C₁₆	C₄	semidirect
17. C₂₉×C₂²⋊C₁₆	C₄	semidirect
77. C₁₆×C₂₉	C₂²⋊C₄	semidirect
78. #12616	C₂₉	direct (N × H)
109. C₂×#347	C₂×C₂₉	semidirect
111. #1375	C₂×C₂₉	semidirect
120. C₂²×C₂₉	#347	semidirect
131. C₂×M₃₂	C₄×C₂₉	semidirect
135. C₂²×C₁₆	C₄×C₂₉	semidirect
139. C₂²⋊C₁₆	C₄×C₂₉	semidirect
140. C₂²⋊C₁₆	C₄×C₂₉	semidirect
175. C₂₉	#12616	direct (N × H)
207. C₁₆	C₂²⋊C₄×C₂₉	semidirect
244. C₂²	C₂₉×#347	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₉×M₃₂	2
C₂₉×#2112	2
C₂×C₂₉×#347	2
C₂₉×#1752	2
C₂₉×#1375	2
#12616	29

Subgroups — C29×#12616 #20307516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#12616 #20307516