Subgroups — C₄×C₂₉×#339 #20305600

Order 7424.

1214 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₉×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₉×#339	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₂₉	C₂²	1	2	yes	—
6	C₂×C₂₉×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₉	—	2	2	no	—
8	C₂²×C₄²×C₂₉	C₂²	1	2	yes	—
9	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
10	C₄×M₁₆×C₂₉	—	2	2	no	—
11	C₄×M₁₆×C₂₉	C₄	1	4	yes	complemented
12	C₂×C₂₉×#97	—	2	1	no	—
13	C₂×C₄×C₈×C₂₉	—	2	1	no	—
14	C₄×C₄⋊C₄×C₂₉	C₄	1	4	yes	complemented
15	C₄×C₂²⋊C₄×C₂₉	—	2	2	no	—
16	C₂₉×#376	—	2	1	no	—
17	C₄³×C₂₉	—	2	2	no	—
18	C₄³×C₂₉	—	2	1	no	—
19	C₄³×C₂₉	C₄	1	4	yes	complemented
20	C₂₉×#339	C₄	1	16	yes	direct product factor
21	C₂³×C₄×C₂₉	—	2	2	no	—
22	C₂³×C₄×C₂₉	D₄	1	2	yes	—
23	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₂₉	—	2	2	no	—
25	C₂×M₁₆×C₂₉	—	2	4	no	—
26	C₂×M₁₆×C₂₉	C₂×C₄	1	8	yes	—
27	C₂×M₁₆×C₂₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₂₉	—	2	2	no	—
29	C₂₉×#117	—	2	4	no	—
30	C₂₉×#117	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊C₄×C₂₉	—	2	2	no	—
33	C₂×C₄²×C₂₉	—	2	16	no	—
34	C₂×C₄²×C₂₉	—	2	8	no	—
35	C₂×C₄²×C₂₉	—	2	2	no	—
36	C₂×C₄²×C₂₉	—	2	2	no	—
37	C₂×C₄²×C₂₉	—	2	4	no	—
38	C₂×C₄²×C₂₉	—	2	4	no	—
39	C₂×C₄²×C₂₉	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₂₉	D₄	1	2	yes	—
45	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₉	Q₈	1	2	yes	—
48	C₂₉×#97	—	2	4	no	—
49	C₄×C₈×C₂₉	—	2	4	no	—
50	C₂₉×#95	—	2	4	no	—
51	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	—	2	16	no	—
53	C₂²×C₄×C₂₉	—	2	4	no	—
54	C₂²×C₄×C₂₉	—	2	8	no	—
55	C₂²×C₄×C₂₉	—	2	8	no	—
56	C₂²×C₄×C₂₉	—	2	4	no	—
57	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₂₉	#41	1	4	yes	—
69	M₁₆×C₂₉	—	2	4	no	—
70	M₁₆×C₂₉	—	2	8	no	—
71	M₁₆×C₂₉	C₄²	1	8	yes	complemented
72	C₂×C₈×C₂₉	—	2	4	no	—
73	C₂×C₈×C₂₉	—	2	8	no	—
74	C₄⋊C₄×C₂₉	—	2	4	no	—
75	C₄⋊C₄×C₂₉	C₄²	1	8	yes	complemented
76	C₂²⋊C₄×C₂₉	—	2	8	no	—
77	C₄²×C₂₉	—	2	32	no	—
78	C₄²×C₂₉	—	2	4	no	—
79	C₄²×C₂₉	—	2	8	no	—
80	C₄²×C₂₉	—	2	8	no	—
81	C₄²×C₂₉	—	4	8	no	—
82	C₄²×C₂₉	C₄²	1	4	yes	—
83	C₄²×C₂₉	C₄²	1	8	yes	complemented
84	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₂₉	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₉	C₄⋊C₄	1	4	yes	—
88	C₄²×C₂₉	C₄⋊C₄	1	2	yes	—
89	C₄×#339	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₉	—	2	4	no	—
91	C₂³×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₂₉	C₄×D₄	1	4	yes	—
95	C₂×C₄×C₂₉	—	2	16	no	—
96	C₂×C₄×C₂₉	—	2	32	no	—
97	C₂×C₄×C₂₉	—	2	8	no	—
98	C₂×C₄×C₂₉	—	2	4	no	—
99	C₂×C₄×C₂₉	—	2	8	no	—
100	C₂×C₄×C₂₉	—	2	8	no	—
101	C₂×C₄×C₂₉	—	2	8	no	—
102	C₂×C₄×C₂₉	—	4	8	no	—
103	C₂×C₄×C₂₉	#95	1	4	yes	—
104	C₂×C₄×C₂₉	#95	1	4	yes	—
105	C₂×C₄×C₂₉	#104	1	8	yes	—
106	C₂×C₄×C₂₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₂₉	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₂₉	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₉	#117	1	2	yes	—
114	C₂×C₄×C₂₉	#117	1	2	yes	—
115	C₂×C₄×C₂₉	#117	1	2	yes	—
116	C₂×C₄×C₂₉	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₂₉	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₂₉	C₄×Q₈	1	4	yes	—
119	C₈×C₂₉	—	2	8	no	—
120	C₂×C₄×M₁₆	C₂×C₂₉	1	1	yes	characteristic
121	C₄×#117	C₂×C₂₉	1	1	yes	characteristic
122	C₂×C₄³	C₂×C₂₉	1	1	yes	characteristic
123	C₂×#339	C₂×C₂₉	1	4	yes	—
124	C₂²×C₂₉	—	2	2	no	—
125	C₂²×C₂₉	—	2	2	no	—
126	C₂²×C₂₉	—	2	8	no	—
127	C₂²×C₂₉	C₂×#95	1	1	yes	characteristic
128	C₂²×C₂₉	C₄×C₂²⋊C₄	1	2	yes	—
129	C₂²×C₂₉	C₄×C₂²⋊C₄	1	2	yes	—
130	C₂²×C₂₉	C₄×C₄⋊C₄	1	2	yes	—
131	C₂²×C₂₉	C₂×#104	1	4	yes	—
132	C₄×C₂₉	—	2	16	no	—
133	C₄×C₂₉	—	2	4	no	—
134	C₄×C₂₉	—	4	8	no	—
135	C₄×C₂₉	#339	1	8	yes	direct product factor
136	C₄×C₂₉	C₄³	1	1	yes	characteristic
137	C₄×C₂₉	#376	1	1	yes	characteristic
138	C₄×C₂₉	C₄×C₂²⋊C₄	1	2	yes	—
139	C₄×C₂₉	C₄×C₄⋊C₄	1	2	yes	—
140	C₄×C₂₉	C₄×C₄⋊C₄	1	2	yes	—
141	C₂²×M₁₆	C₂²×C₂₉	1	2	yes	—
142	C₂×#117	C₂²×C₂₉	1	2	yes	—
143	C₂²×C₄²	—	2	2	no	—
144	C₂²×C₄²	C₂²×C₂₉	1	2	yes	—
145	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
146	C₄×M₁₆	—	2	2	no	—
147	C₄×M₁₆	C₄×C₂₉	1	4	yes	complemented
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₄×C₄⋊C₄	C₄×C₂₉	1	4	yes	complemented
151	C₄×C₂²⋊C₄	—	2	2	no	—
152	#376	—	2	1	no	—
153	C₄³	—	2	2	no	—
154	C₄³	—	2	1	no	—
155	C₄³	C₄×C₂₉	1	4	yes	complemented
156	#339	C₄×C₂₉	1	16	yes	direct product factor
157	C₂×C₂₉	—	2	2	no	—
158	C₂×C₂₉	—	2	2	no	—
159	C₂×C₂₉	C₄×#95	1	1	yes	characteristic
160	C₂×C₂₉	C₂×#339	1	2	yes	—
161	C₂×C₂₉	C₄×#104	1	4	yes	—
162	C₂³×C₄	—	2	2	no	—
163	C₂³×C₄	D₄×C₂₉	1	2	yes	—
164	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	C₂×C₄×C₂₉	1	8	yes	—
168	C₂×M₁₆	C₂×C₄×C₂₉	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	#117	—	2	4	no	—
171	#117	C₂×C₄×C₂₉	1	8	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
173	C₂×C₂²⋊C₄	—	2	2	no	—
174	C₂×C₄²	—	2	16	no	—
175	C₂×C₄²	—	2	8	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	—	2	4	no	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	C₂×C₄×C₂₉	1	8	yes	—
181	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
185	C₂×C₄²	D₄×C₂₉	1	2	yes	—
186	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
188	C₂×C₄²	Q₈×C₂₉	1	2	yes	—
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	—	2	4	no	—
192	C₂₉	C₄×#339	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	16	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	8	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₂₉	1	2	yes	—
202	C₂²×C₄	C₄⋊C₄×C₂₉	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₂₉	1	1	yes	characteristic
210	C₂²×C₄	#41×C₂₉	1	4	yes	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	8	no	—
213	M₁₆	C₄²×C₂₉	1	8	yes	complemented
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₄²×C₂₉	1	8	yes	complemented
218	C₂²⋊C₄	—	2	8	no	—
219	C₄²	—	2	32	no	—
220	C₄²	—	2	4	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	4	8	no	—
224	C₄²	C₄²×C₂₉	1	4	yes	—
225	C₄²	C₄²×C₂₉	1	8	yes	complemented
226	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
227	C₄²	C₂²⋊C₄×C₂₉	1	2	yes	—
228	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
229	C₄²	C₄⋊C₄×C₂₉	1	4	yes	—
230	C₄²	C₄⋊C₄×C₂₉	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
233	C₂³	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
234	C₂³	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
235	C₂³	C₄×D₄×C₂₉	1	4	yes	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	32	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	C₂₉×#95	1	4	yes	—
245	C₂×C₄	C₂₉×#95	1	4	yes	—
246	C₂×C₄	C₂₉×#104	1	8	yes	—
247	C₂×C₄	C₂×C₄²×C₂₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄²×C₂₉	1	2	yes	—
249	C₂×C₄	C₂×C₄²×C₂₉	1	2	yes	—
250	C₂×C₄	C₂×C₄²×C₂₉	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	1	yes	characteristic
254	C₂×C₄	C₂₉×#117	1	2	yes	—
255	C₂×C₄	C₂₉×#117	1	2	yes	—
256	C₂×C₄	C₂₉×#117	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₂₉	1	4	yes	—
258	C₂×C₄	C₄×D₄×C₂₉	1	4	yes	—
259	C₂×C₄	C₄×Q₈×C₂₉	1	4	yes	—
260	C₈	—	2	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	C₂×C₂₉×#95	1	1	yes	characteristic
265	C₂²	C₄×C₂²⋊C₄×C₂₉	1	2	yes	—
266	C₂²	C₄×C₂²⋊C₄×C₂₉	1	2	yes	—
267	C₂²	C₄×C₄⋊C₄×C₂₉	1	2	yes	—
268	C₂²	C₂×C₂₉×#104	1	4	yes	—
269	C₄	—	2	16	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	C₂₉×#339	1	8	yes	direct product factor
273	C₄	C₄³×C₂₉	1	1	yes	characteristic
274	C₄	C₂₉×#376	1	1	yes	characteristic
275	C₄	C₄×C₂²⋊C₄×C₂₉	1	2	yes	—
276	C₄	C₄×C₄⋊C₄×C₂₉	1	2	yes	—
277	C₄	C₄×C₄⋊C₄×C₂₉	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	C₄×C₂₉×#95	1	1	yes	characteristic
281	C₂	C₂×C₂₉×#339	1	2	yes	—
282	C₂	C₄×C₂₉×#104	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₂₉	D₄
commutator	C₄	C₄³×C₂₉
Frattini, cosocle	C₂³×C₄	C₂³×C₂₉
socle	C₂³×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₉ ◁ C₂²×C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₄×M₁₆×C₂₉	C₄	semidirect
14. C₄×C₄⋊C₄×C₂₉	C₄	semidirect
19. C₄³×C₂₉	C₄	semidirect
20. C₂₉×#339	C₄	direct (N × H)
71. M₁₆×C₂₉	C₄²	semidirect
75. C₄⋊C₄×C₂₉	C₄²	semidirect
83. C₄²×C₂₉	C₄²	semidirect
89. C₄×#339	C₂₉	direct (N × H)
135. C₄×C₂₉	#339	direct (N × H)
147. C₄×M₁₆	C₄×C₂₉	semidirect
150. C₄×C₄⋊C₄	C₄×C₂₉	semidirect
155. C₄³	C₄×C₂₉	semidirect
156. #339	C₄×C₂₉	direct (N × H)
192. C₂₉	C₄×#339	direct (N × H)
213. M₁₆	C₄²×C₂₉	semidirect
217. C₄⋊C₄	C₄²×C₂₉	semidirect
225. C₄²	C₄²×C₂₉	semidirect
272. C₄	C₂₉×#339	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×C₄×M₁₆×C₂₉	2
C₄×C₂₉×#117	2
C₂×C₄³×C₂₉	2
C₂×C₂₉×#339	2
C₄×#339	29

Subgroups — C4×C29×#339 #20305600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₉×#339 #20305600