Subgroups — C₂×C₂₉×#1735 #20315245

Order 7424.

2398 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₉×#117	C₂	1	2	yes	maximal
3	C₂₉×#1735	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₉×#340	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₂₉	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₉	—	2	8	no	—
7	C₂³×C₈×C₂₉	C₄	1	8	yes	complemented
8	C₂³×C₈×C₂₉	C₂²	1	2	yes	—
9	C₂³×C₈×C₂₉	C₂²	1	4	yes	—
10	C₂×C₂₉×#117	—	2	8	no	—
11	C₂×C₂₉×#117	—	2	4	no	—
12	C₂×C₂₉×#117	C₄	1	16	yes	complemented
13	C₂×C₂₉×#117	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₂₉	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₂₉	—	2	2	no	—
16	C₂²×C₄²×C₂₉	—	2	2	no	—
17	C₂₉×#340	C₂²	1	16	yes	complemented
18	C₂⁵×C₂₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₉	—	2	4	no	—
20	C₂³×C₄×C₂₉	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₂₉	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₂₉	D₄	1	2	yes	—
23	C₂³×C₄×C₂₉	D₄	1	8	yes	—
24	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₉	Q₈	1	4	yes	—
26	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
27	C₂³×C₄×C₂₉	C₂³	1	2	yes	—
28	C₂²×C₈×C₂₉	—	2	32	no	—
29	C₂²×C₈×C₂₉	—	2	8	no	—
30	C₂²×C₈×C₂₉	—	2	16	no	—
31	C₂²×C₈×C₂₉	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₂₉	C₂×C₄	1	16	yes	complemented
33	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
34	C₂₉×#117	—	2	32	no	—
35	C₂₉×#117	—	2	16	no	—
36	C₂₉×#117	C₂×C₄	1	32	yes	complemented
37	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₉	—	2	8	no	—
39	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	16	yes	—
40	C₂×C₄⋊C₄×C₂₉	C₂³	1	8	yes	—
41	C₂×C₂²⋊C₄×C₂₉	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₂₉	—	2	8	no	—
43	C₂×C₄²×C₂₉	—	2	16	no	—
44	C₂×C₄²×C₂₉	—	2	8	no	—
45	C₂⁴×C₂₉	—	2	8	no	—
46	C₂⁴×C₂₉	C₂²⋊C₄	1	8	yes	—
47	C₂⁴×C₂₉	C₄⋊C₄	1	4	yes	—
48	C₂⁴×C₂₉	C₂×D₄	1	2	yes	—
49	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₉	—	2	16	no	—
51	C₂²×C₄×C₂₉	—	2	8	no	—
52	C₂²×C₄×C₂₉	—	4	16	no	—
53	C₂²×C₄×C₂₉	C₄²	1	16	yes	—
54	C₂²×C₄×C₂₉	C₂²⋊C₄	1	8	yes	—
55	C₂²×C₄×C₂₉	C₂²⋊C₄	1	16	yes	—
56	C₂²×C₄×C₂₉	C₂²⋊C₄	1	16	yes	—
57	C₂²×C₄×C₂₉	C₄⋊C₄	1	8	yes	—
58	C₂²×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₂₉	C₄⋊C₄	1	32	yes	—
60	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₂₉	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₂₉	C₂×D₄	1	8	yes	—
68	C₂²×C₄×C₂₉	C₂×Q₈	1	4	yes	—
69	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₂₉	—	2	16	no	—
72	C₂×C₈×C₂₉	—	2	32	no	—
73	C₂×C₈×C₂₉	—	2	8	no	—
74	C₂×C₈×C₂₉	C₂²×C₄	1	8	yes	—
75	C₄⋊C₄×C₂₉	—	2	16	no	—
76	C₄⋊C₄×C₂₉	—	2	8	no	—
77	C₄⋊C₄×C₂₉	C₂²×C₄	1	16	yes	—
78	C₂²⋊C₄×C₂₉	—	2	32	no	—
79	C₄²×C₂₉	—	2	32	no	—
80	C₂×#1735	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₉	—	2	8	no	—
82	C₂³×C₂₉	—	2	32	no	—
83	C₂³×C₂₉	—	2	16	no	—
84	C₂³×C₂₉	#95	1	16	yes	—
85	C₂³×C₂₉	C₂×C₂²⋊C₄	1	8	yes	—
86	C₂³×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
87	C₂³×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
88	C₂³×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₂₉	#135	1	8	yes	—
90	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₉	—	2	16	no	—
92	C₂×C₄×C₂₉	—	4	16	no	—
93	C₂×C₄×C₂₉	—	4	32	no	—
94	C₂×C₄×C₂₉	#95	1	32	yes	—
95	C₂×C₄×C₂₉	#95	1	16	yes	—
96	C₂×C₄×C₂₉	C₂×C₄²	1	8	yes	—
97	C₂×C₄×C₂₉	C₂×C₄²	1	4	yes	—
98	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	8	yes	—
99	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	8	yes	—
101	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	16	yes	—
103	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	8	yes	—
104	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂³×C₄	1	2	yes	—
108	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
110	C₈×C₂₉	—	2	8	no	—
111	C₂⁴×C₈	C₂×C₂₉	1	1	yes	characteristic
112	C₂²×#117	C₂×C₂₉	1	2	yes	—
113	#1735	C₂×C₂₉	1	8	yes	direct product factor
114	C₂×#340	C₂×C₂₉	1	4	yes	complemented
115	C₂²×C₂₉	—	2	16	no	—
116	C₂²×C₂₉	—	2	8	no	—
117	C₂²×C₂₉	—	2	32	no	—
118	C₂²×C₂₉	C₂×#95	1	8	yes	—
119	C₂²×C₂₉	C₂×#95	1	4	yes	—
120	C₂²×C₂₉	#416	1	16	yes	—
121	C₂²×C₂₉	#427	1	8	yes	—
122	C₂²×C₂₉	C₂²×C₂²⋊C₄	1	2	yes	—
123	C₂²×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₂₉	C₂×#135	1	4	yes	—
125	C₄×C₂₉	—	4	16	no	—
126	C₄×C₂₉	C₂×#95	1	8	yes	—
127	C₄×C₂₉	C₂×#95	1	4	yes	—
128	C₄×C₂₉	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₂₉	C₂²×C₄⋊C₄	1	2	yes	—
131	C₂⁴×C₄	C₂²×C₂₉	1	1	yes	characteristic
132	C₂³×C₈	—	2	8	no	—
133	C₂³×C₈	C₄×C₂₉	1	8	yes	complemented
134	C₂³×C₈	C₂²×C₂₉	1	2	yes	—
135	C₂³×C₈	C₂²×C₂₉	1	4	yes	—
136	C₂×#117	—	2	8	no	—
137	C₂×#117	—	2	4	no	—
138	C₂×#117	C₄×C₂₉	1	16	yes	complemented
139	C₂×#117	C₂²×C₂₉	1	8	yes	—
140	C₂²×C₄⋊C₄	C₂²×C₂₉	1	4	yes	—
141	C₂²×C₂²⋊C₄	—	2	2	no	—
142	C₂²×C₄²	—	2	2	no	—
143	#340	C₂²×C₂₉	1	16	yes	complemented
144	C₂×C₂₉	—	2	8	no	—
145	C₂×C₂₉	#1735	1	8	yes	direct product factor
146	C₂×C₂₉	C₂²×#95	1	1	yes	characteristic
147	C₂×C₂₉	C₂×#416	1	4	yes	—
148	C₂×C₂₉	C₂×#427	1	2	yes	—
149	C₂⁵	D₄×C₂₉	1	1	yes	characteristic
150	C₂³×C₄	—	2	4	no	—
151	C₂³×C₄	C₂×C₄×C₂₉	1	4	yes	—
152	C₂³×C₄	C₂×C₄×C₂₉	1	8	yes	—
153	C₂³×C₄	D₄×C₂₉	1	2	yes	—
154	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
155	C₂³×C₄	D₄×C₂₉	1	8	yes	—
156	C₂³×C₄	Q₈×C₂₉	1	4	yes	—
157	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
158	C₂³×C₄	C₂³×C₂₉	1	2	yes	—
159	C₂²×C₈	—	2	32	no	—
160	C₂²×C₈	—	2	8	no	—
161	C₂²×C₈	—	2	16	no	—
162	C₂²×C₈	C₂×C₄×C₂₉	1	8	yes	—
163	C₂²×C₈	C₂×C₄×C₂₉	1	16	yes	complemented
164	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
165	#117	—	2	32	no	—
166	#117	—	2	16	no	—
167	#117	C₂×C₄×C₂₉	1	32	yes	complemented
168	C₂×C₄⋊C₄	—	2	4	no	—
169	C₂×C₄⋊C₄	—	2	8	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	16	yes	—
171	C₂×C₄⋊C₄	C₂³×C₂₉	1	8	yes	—
172	C₂×C₂²⋊C₄	—	2	16	no	—
173	C₂×C₂²⋊C₄	—	2	8	no	—
174	C₂×C₄²	—	2	16	no	—
175	C₂×C₄²	—	2	8	no	—
176	C₂₉	C₂×#1735	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	—	2	8	no	—
178	C₂⁴	C₂²⋊C₄×C₂₉	1	8	yes	—
179	C₂⁴	C₄⋊C₄×C₂₉	1	4	yes	—
180	C₂⁴	C₂×D₄×C₂₉	1	2	yes	—
181	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	16	no	—
183	C₂²×C₄	—	2	8	no	—
184	C₂²×C₄	—	4	16	no	—
185	C₂²×C₄	C₄²×C₂₉	1	16	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₂₉	1	16	yes	—
187	C₂²×C₄	C₂²⋊C₄×C₂₉	1	8	yes	—
188	C₂²×C₄	C₂²⋊C₄×C₂₉	1	16	yes	—
189	C₂²×C₄	C₄⋊C₄×C₂₉	1	32	yes	—
190	C₂²×C₄	C₄⋊C₄×C₂₉	1	8	yes	—
191	C₂²×C₄	C₄⋊C₄×C₂₉	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
193	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
194	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
195	C₂²×C₄	C₂²×C₄×C₂₉	1	8	yes	—
196	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₂₉	1	4	yes	—
199	C₂²×C₄	C₂×D₄×C₂₉	1	8	yes	—
200	C₂²×C₄	C₂×Q₈×C₂₉	1	4	yes	—
201	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
203	C₂×C₈	—	2	16	no	—
204	C₂×C₈	—	2	32	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	C₂²×C₄×C₂₉	1	8	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	C₂²×C₄×C₂₉	1	16	yes	—
210	C₂²⋊C₄	—	2	32	no	—
211	C₄²	—	2	32	no	—
212	C₂³	—	2	16	no	—
213	C₂³	—	2	8	no	—
214	C₂³	—	2	32	no	—
215	C₂³	C₂₉×#95	1	16	yes	—
216	C₂³	C₂×C₂²⋊C₄×C₂₉	1	8	yes	—
217	C₂³	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
218	C₂³	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
219	C₂³	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
220	C₂³	C₂₉×#135	1	8	yes	—
221	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	—	4	32	no	—
225	C₂×C₄	C₂₉×#95	1	32	yes	—
226	C₂×C₄	C₂₉×#95	1	16	yes	—
227	C₂×C₄	C₂×C₄²×C₂₉	1	8	yes	—
228	C₂×C₄	C₂×C₄²×C₂₉	1	4	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	8	yes	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
231	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	8	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
233	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	16	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	8	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
237	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
238	C₂×C₄	C₂³×C₄×C₂₉	1	2	yes	—
239	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
240	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
241	C₈	—	2	8	no	—
242	C₂²	—	2	32	no	—
243	C₂²	—	2	16	no	—
244	C₂²	—	2	8	no	—
245	C₂²	C₂×C₂₉×#95	1	8	yes	—
246	C₂²	C₂×C₂₉×#95	1	4	yes	—
247	C₂²	C₂₉×#416	1	16	yes	—
248	C₂²	C₂₉×#427	1	8	yes	—
249	C₂²	C₂²×C₂²⋊C₄×C₂₉	1	2	yes	—
250	C₂²	C₂²×C₄⋊C₄×C₂₉	1	1	yes	characteristic
251	C₂²	C₂×C₂₉×#135	1	4	yes	—
252	C₄	—	4	16	no	—
253	C₄	C₂×C₂₉×#95	1	8	yes	—
254	C₄	C₂×C₂₉×#95	1	4	yes	—
255	C₄	C₂²×C₄²×C₂₉	1	1	yes	characteristic
256	C₄	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
257	C₄	C₂²×C₄⋊C₄×C₂₉	1	2	yes	—
258	C₂	—	2	8	no	—
259	C₂	C₂₉×#1735	1	8	yes	direct product factor
260	C₂	C₂²×C₂₉×#95	1	1	yes	characteristic
261	C₂	C₂×C₂₉×#416	1	4	yes	—
262	C₂	C₂×C₂₉×#427	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₉	D₄
commutator	C₄	C₂²×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle	C₂⁴×C₂₉	C₂×D₄
prime radical	C₂⁵×C₂₉	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₉ ◁ C₂⁴×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₉×#1735	C₂	direct (N × H)
4. C₂×C₂₉×#340	C₂	semidirect
7. C₂³×C₈×C₂₉	C₄	semidirect
12. C₂×C₂₉×#117	C₄	semidirect
17. C₂₉×#340	C₂²	semidirect
32. C₂²×C₈×C₂₉	C₂×C₄	semidirect
36. C₂₉×#117	C₂×C₄	semidirect
80. C₂×#1735	C₂₉	direct (N × H)
113. #1735	C₂×C₂₉	direct (N × H)
114. C₂×#340	C₂×C₂₉	semidirect
133. C₂³×C₈	C₄×C₂₉	semidirect
138. C₂×#117	C₄×C₂₉	semidirect
143. #340	C₂²×C₂₉	semidirect
145. C₂×C₂₉	#1735	direct (N × H)
163. C₂²×C₈	C₂×C₄×C₂₉	semidirect
167. #117	C₂×C₄×C₂₉	semidirect
176. C₂₉	C₂×#1735	direct (N × H)
259. C₂	C₂₉×#1735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₂₉	2
C₂²×C₂₉×#117	2
C₂₉×#1735	2
C₂×C₂₉×#340	2
C₂×#1735	29

Subgroups — C2×C29×#1735 #20315245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#1735 #20315245