Subgroups — C₂×C₂₉×#1290 #20302885

Order 7424.

602 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#432	C₂	1	2	yes	maximal
2	C₂²×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1290	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₂₉	C₄	1	2	yes	—
5	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
6	C₂₉×#432	—	2	2	no	—
7	C₂₉×#432	—	2	2	no	—
8	C₂₉×#432	C₄	1	4	yes	—
9	C₂₉×#432	C₂²	1	4	yes	—
10	C₂×C₂₉×C₄⋊C₈	—	2	2	no	—
11	C₂×C₂₉×C₄⋊C₈	—	2	2	no	—
12	C₂×C₂₉×C₄⋊C₈	—	2	2	no	—
13	C₂×C₂₉×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₉×C₂²⋊C₈	—	2	2	no	—
16	C₂×C₂₉×#97	C₄	1	4	yes	—
17	C₂³×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₉	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₂₉	—	2	2	no	—
21	C₂²×C₈×C₂₉	—	2	2	no	—
22	C₂²×C₈×C₂₉	—	2	4	no	—
23	C₂²×C₈×C₂₉	—	4	4	no	—
24	C₂²×C₈×C₂₉	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₂₉	D₄	1	2	yes	—
30	C₂×C₄²×C₂₉	D₄	1	2	yes	—
31	C₂×C₄²×C₂₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
33	C₂₉×C₄⋊C₈	—	2	4	no	—
34	C₂₉×C₄⋊C₈	—	2	4	no	—
35	C₂₉×C₄⋊C₈	—	2	4	no	—
36	C₂₉×C₄⋊C₈	—	2	4	no	—
37	C₂₉×C₂²⋊C₈	—	2	4	no	—
38	C₂₉×C₂²⋊C₈	—	2	4	no	—
39	C₂₉×#97	—	2	4	no	—
40	C₂₉×#97	C₂×C₄	1	8	yes	—
41	C₂⁴×C₂₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	—	2	4	no	—
43	C₂²×C₄×C₂₉	—	2	2	no	—
44	C₂²×C₄×C₂₉	—	2	2	no	—
45	C₂²×C₄×C₂₉	—	4	2	no	—
46	C₂²×C₄×C₂₉	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₂₉	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₂₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₂₉	—	2	4	no	—
61	C₂×C₈×C₂₉	—	2	8	no	—
62	C₂×C₈×C₂₉	—	4	8	no	—
63	C₂×C₈×C₂₉	—	4	4	no	—
64	C₂×C₈×C₂₉	—	4	4	no	—
65	C₂×C₈×C₂₉	—	4	4	no	—
66	C₄²×C₂₉	C₄²	1	2	yes	—
67	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
71	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
73	C₄²×C₂₉	C₂×D₄	1	2	yes	—
74	C₂×#1290	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₂₉	—	2	2	no	—
76	C₂³×C₂₉	—	2	2	no	—
77	C₂³×C₂₉	#95	1	1	yes	characteristic
78	C₂³×C₂₉	#95	1	1	yes	characteristic
79	C₂³×C₂₉	#108	1	4	yes	—
80	C₂³×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₉	—	2	4	no	—
82	C₂×C₄×C₂₉	—	2	4	no	—
83	C₂×C₄×C₂₉	—	2	4	no	—
84	C₂×C₄×C₂₉	—	2	2	no	—
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	—	2	2	no	—
87	C₂×C₄×C₂₉	—	2	2	no	—
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	1	no	—
90	C₂×C₄×C₂₉	—	2	1	no	—
91	C₂×C₄×C₂₉	—	2	1	no	—
92	C₂×C₄×C₂₉	—	2	1	no	—
93	C₂×C₄×C₂₉	—	4	2	no	—
94	C₂×C₄×C₂₉	—	4	1	no	—
95	C₂×C₄×C₂₉	—	4	1	no	—
96	C₂×C₄×C₂₉	#95	1	2	yes	—
97	C₂×C₄×C₂₉	#95	1	2	yes	—
98	C₂×C₄×C₂₉	#95	1	1	yes	characteristic
99	C₂×C₄×C₂₉	#95	1	1	yes	characteristic
100	C₂×C₄×C₂₉	#100	1	2	yes	—
101	C₂×C₄×C₂₉	#101	1	2	yes	—
102	C₂×C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₂₉	—	4	8	no	—
107	C₈×C₂₉	—	4	4	no	—
108	C₂×#432	C₂×C₂₉	1	2	yes	—
109	C₂²×C₄⋊C₈	C₂×C₂₉	1	1	yes	characteristic
110	#1290	C₂×C₂₉	1	4	yes	direct product factor
111	C₂²×C₂₉	—	2	8	no	—
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	H(3,ℤ/4)	1	2	yes	—
117	C₂²×C₂₉	#341	1	2	yes	—
118	C₂²×C₂₉	#341	1	2	yes	—
119	C₂²×C₂₉	#343	1	2	yes	—
120	C₂²×C₂₉	C₂×#95	1	1	yes	characteristic
121	C₂²×C₂₉	C₂×#108	1	2	yes	—
122	C₄×C₂₉	—	2	2	no	—
123	C₄×C₂₉	—	2	2	no	—
124	C₄×C₂₉	—	2	2	no	—
125	C₄×C₂₉	—	2	2	no	—
126	C₄×C₂₉	—	2	1	no	—
127	C₄×C₂₉	—	2	1	no	—
128	C₄×C₂₉	#343	1	2	yes	—
129	C₄×C₂₉	C₂×#100	1	1	yes	characteristic
130	C₄×C₂₉	C₂×#101	1	1	yes	characteristic
131	C₂³×C₈	C₄×C₂₉	1	2	yes	—
132	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
133	#432	—	2	2	no	—
134	#432	—	2	2	no	—
135	#432	C₄×C₂₉	1	4	yes	—
136	#432	C₂²×C₂₉	1	4	yes	—
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	—	2	2	no	—
139	C₂×C₄⋊C₈	—	2	2	no	—
140	C₂×C₄⋊C₈	C₄×C₂₉	1	2	yes	—
141	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
142	C₂×C₂²⋊C₈	—	2	2	no	—
143	C₂×#97	C₄×C₂₉	1	4	yes	—
144	C₂×C₂₉	—	2	2	no	—
145	C₂×C₂₉	—	2	2	no	—
146	C₂×C₂₉	#1290	1	4	yes	direct product factor
147	C₂×C₂₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic
148	C₂×C₂₉	C₂×#341	1	1	yes	characteristic
149	C₂×C₂₉	C₂×#343	1	1	yes	characteristic
150	C₂³×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
151	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
152	C₂³×C₄	Q₈×C₂₉	1	1	yes	characteristic
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	4	4	no	—
157	C₂²×C₈	C₂×C₄×C₂₉	1	4	yes	—
158	C₂×C₄²	C₂×C₄×C₂₉	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
161	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
162	C₂×C₄²	D₄×C₂₉	1	2	yes	—
163	C₂×C₄²	D₄×C₂₉	1	2	yes	—
164	C₂×C₄²	Q₈×C₂₉	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	4	no	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₂²⋊C₈	—	2	4	no	—
171	C₂²⋊C₈	—	2	4	no	—
172	#97	—	2	4	no	—
173	#97	C₂×C₄×C₂₉	1	8	yes	—
174	C₂₉	C₂×#1290	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁴	C₄⋊C₄×C₂₉	1	1	yes	characteristic
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	C₄²×C₂₉	1	1	yes	characteristic
181	C₂²×C₄	C₄²×C₂₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
183	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
185	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊C₄×C₂₉	1	2	yes	—
187	C₂²×C₄	C₄⋊C₄×C₂₉	1	2	yes	—
188	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₂₉	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₂₉	1	1	yes	characteristic
194	C₂×C₈	—	2	8	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₂×C₈	—	4	8	no	—
198	C₂×C₈	—	4	4	no	—
199	C₂×C₈	—	4	4	no	—
200	C₄²	C₄²×C₂₉	1	2	yes	—
201	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
202	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
203	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
204	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
205	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
206	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
207	C₄²	C₂×D₄×C₂₉	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂₉×#95	1	1	yes	characteristic
211	C₂³	C₂₉×#95	1	1	yes	characteristic
212	C₂³	C₂₉×#108	1	4	yes	—
213	C₂³	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	C₂₉×#95	1	2	yes	—
230	C₂×C₄	C₂₉×#95	1	2	yes	—
231	C₂×C₄	C₂₉×#95	1	1	yes	characteristic
232	C₂×C₄	C₂₉×#95	1	1	yes	characteristic
233	C₂×C₄	C₂₉×#100	1	2	yes	—
234	C₂×C₄	C₂₉×#101	1	2	yes	—
235	C₂×C₄	C₂×C₄²×C₂₉	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
239	C₈	—	4	8	no	—
240	C₈	—	4	4	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	8	no	—
246	C₂²	C₂₉×H(3,ℤ/4)	1	2	yes	—
247	C₂²	C₂₉×#341	1	2	yes	—
248	C₂²	C₂₉×#341	1	2	yes	—
249	C₂²	C₂₉×#343	1	2	yes	—
250	C₂²	C₂×C₂₉×#95	1	1	yes	characteristic
251	C₂²	C₂×C₂₉×#108	1	2	yes	—
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	C₂₉×#343	1	2	yes	—
259	C₄	C₂×C₂₉×#100	1	1	yes	characteristic
260	C₄	C₂×C₂₉×#101	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	C₂₉×#1290	1	4	yes	direct product factor
264	C₂	C₂×C₂₉×H(3,ℤ/4)	1	1	yes	characteristic
265	C₂	C₂×C₂₉×#341	1	1	yes	characteristic
266	C₂	C₂×C₂₉×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₉	#95
commutator	C₂×C₄	C₂×C₄²×C₂₉
Frattini, cosocle	C₂×C₄²	C₂³×C₂₉
prime radical	C₂⁴×C₂₉	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂²×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₉×#1290	C₂	direct (N × H)
74. C₂×#1290	C₂₉	direct (N × H)
110. #1290	C₂×C₂₉	direct (N × H)
146. C₂×C₂₉	#1290	direct (N × H)
174. C₂₉	C₂×#1290	direct (N × H)
263. C₂	C₂₉×#1290	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×#432	2
C₂²×C₂₉×C₄⋊C₈	2
C₂₉×#1290	2
C₂×#1290	29

Subgroups — C2×C29×#1290 #20302885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#1290 #20302885