Subgroups — C₂×C₂₉×#1728 #20315233

Order 7424.

2782 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₂₉	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#1728	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₉×#336	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₂₉	C₄	1	2	yes	—
6	C₂⁴×C₄×C₂₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₉	—	2	8	no	—
8	C₂²×M₁₆×C₂₉	—	2	8	no	—
9	C₂²×M₁₆×C₂₉	C₄	1	16	yes	complemented
10	C₂²×M₁₆×C₂₉	C₂²	1	8	yes	—
11	C₂³×C₈×C₂₉	C₂²	1	4	yes	—
12	C₂²×C₄²×C₂₉	—	2	8	no	—
13	C₂²×C₄²×C₂₉	C₄	1	4	yes	—
14	C₂²×C₄²×C₂₉	C₄	1	2	yes	—
15	C₂²×C₄²×C₂₉	C₂²	1	4	yes	—
16	C₂²×C₄²×C₂₉	C₂²	1	2	yes	—
17	C₂₉×#336	C₂²	1	16	yes	complemented
18	C₂⁵×C₂₉	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₉	—	2	8	no	—
20	C₂³×C₄×C₂₉	—	2	16	no	—
21	C₂³×C₄×C₂₉	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₂₉	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₉	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₂₉	D₄	1	4	yes	—
27	C₂³×C₄×C₂₉	D₄	1	8	yes	—
28	C₂³×C₄×C₂₉	Q₈	1	4	yes	—
29	C₂³×C₄×C₂₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₂₉	—	2	64	no	—
32	C₂×M₁₆×C₂₉	—	2	16	no	—
33	C₂×M₁₆×C₂₉	—	2	16	no	—
34	C₂×M₁₆×C₂₉	C₂×C₄	1	32	yes	complemented
35	C₂²×C₈×C₂₉	—	2	8	no	—
36	C₂²×C₈×C₂₉	—	2	8	no	—
37	C₂²×C₈×C₂₉	C₂×C₄	1	16	yes	complemented
38	C₂²×C₈×C₂₉	C₂³	1	8	yes	—
39	C₂×C₄²×C₂₉	—	2	16	no	—
40	C₂×C₄²×C₂₉	—	2	8	no	—
41	C₂×C₄²×C₂₉	—	2	8	no	—
42	C₂×C₄²×C₂₉	—	2	16	no	—
43	C₂×C₄²×C₂₉	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₂₉	C₂³	1	4	yes	—
46	C₂⁴×C₂₉	—	2	8	no	—
47	C₂⁴×C₂₉	C₄²	1	4	yes	—
48	C₂⁴×C₂₉	C₂²⋊C₄	1	4	yes	—
49	C₂⁴×C₂₉	C₄⋊C₄	1	4	yes	—
50	C₂⁴×C₂₉	C₂²×C₄	1	2	yes	—
51	C₂⁴×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	—	2	64	no	—
53	C₂²×C₄×C₂₉	—	2	16	no	—
54	C₂²×C₄×C₂₉	—	2	8	no	—
55	C₂²×C₄×C₂₉	—	2	8	no	—
56	C₂²×C₄×C₂₉	—	2	32	no	—
57	C₂²×C₄×C₂₉	C₄²	1	8	yes	—
58	C₂²×C₄×C₂₉	C₄²	1	4	yes	—
59	C₂²×C₄×C₂₉	C₂²⋊C₄	1	8	yes	—
60	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₂₉	C₂²⋊C₄	1	32	yes	—
62	C₂²×C₄×C₂₉	C₄⋊C₄	1	8	yes	—
63	C₂²×C₄×C₂₉	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₂₉	C₄⋊C₄	1	32	yes	—
65	C₂²×C₄×C₂₉	M₁₆	1	16	yes	complemented
66	C₂²×C₄×C₂₉	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₂₉	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₂₉	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₂₉	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₂₉	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₂₉	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₂₉	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₂₉	—	2	64	no	—
79	C₂×C₈×C₂₉	—	2	32	no	—
80	C₂×C₈×C₂₉	—	2	8	no	—
81	C₂×C₈×C₂₉	—	2	8	no	—
82	C₂×C₈×C₂₉	C₂²×C₄	1	16	yes	complemented
83	C₄²×C₂₉	—	2	16	no	—
84	C₄²×C₂₉	—	2	8	no	—
85	C₄²×C₂₉	—	2	8	no	—
86	C₂×#1728	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₉	—	2	8	no	—
88	C₂³×C₂₉	—	2	16	no	—
89	C₂³×C₂₉	—	2	16	no	—
90	C₂³×C₂₉	—	2	16	no	—
91	C₂³×C₂₉	#95	1	16	yes	—
92	C₂³×C₂₉	C₂×C₄²	1	4	yes	—
93	C₂³×C₂₉	C₂×C₄²	1	2	yes	—
94	C₂³×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
95	C₂³×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂³×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
97	C₂³×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂³×C₂₉	C₂×M₁₆	1	8	yes	—
99	C₂³×C₂₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₂₉	—	2	64	no	—
101	C₂×C₄×C₂₉	—	2	8	no	—
102	C₂×C₄×C₂₉	—	2	8	no	—
103	C₂×C₄×C₂₉	—	2	32	no	—
104	C₂×C₄×C₂₉	#95	1	32	yes	—
105	C₂×C₄×C₂₉	#95	1	16	yes	—
106	C₂×C₄×C₂₉	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₂₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	8	yes	—
109	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	16	yes	—
110	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
111	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	16	yes	—
113	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₂₉	C₂×M₁₆	1	16	yes	complemented
117	C₂×C₄×C₂₉	C₂³×C₄	1	2	yes	—
118	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
121	C₈×C₂₉	—	2	16	no	—
122	C₂³×M₁₆	C₂×C₂₉	1	2	yes	—
123	C₂³×C₄²	C₂×C₂₉	1	1	yes	characteristic
124	#1728	C₂×C₂₉	1	8	yes	direct product factor
125	C₂×#336	C₂×C₂₉	1	4	yes	complemented
126	C₂²×C₂₉	—	2	8	no	—
127	C₂²×C₂₉	—	2	8	no	—
128	C₂²×C₂₉	—	2	8	no	—
129	C₂²×C₂₉	—	2	32	no	—
130	C₂²×C₂₉	C₂×#95	1	8	yes	—
131	C₂²×C₂₉	C₂×#95	1	4	yes	—
132	C₂²×C₂₉	C₄×M₁₆	1	8	yes	—
133	C₂²×C₂₉	#407	1	8	yes	—
134	C₂²×C₂₉	#423	1	8	yes	—
135	C₂²×C₂₉	C₂²×C₄²	1	1	yes	characteristic
136	C₂²×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₂₉	C₂²×M₁₆	1	4	yes	—
139	C₄×C₂₉	—	2	16	no	—
140	C₄×C₂₉	C₂×#95	1	4	yes	—
141	C₄×C₂₉	C₂×#95	1	8	yes	—
142	C₄×C₂₉	C₂²×C₂²⋊C₄	1	2	yes	—
143	C₄×C₂₉	C₂²×C₄⋊C₄	1	2	yes	—
144	C₂⁴×C₄	C₄×C₂₉	1	2	yes	—
145	C₂⁴×C₄	C₂²×C₂₉	1	1	yes	characteristic
146	C₂²×M₁₆	—	2	8	no	—
147	C₂²×M₁₆	—	2	8	no	—
148	C₂²×M₁₆	C₄×C₂₉	1	16	yes	complemented
149	C₂²×M₁₆	C₂²×C₂₉	1	8	yes	—
150	C₂³×C₈	C₂²×C₂₉	1	4	yes	—
151	C₂²×C₄²	—	2	8	no	—
152	C₂²×C₄²	C₄×C₂₉	1	4	yes	—
153	C₂²×C₄²	C₄×C₂₉	1	2	yes	—
154	C₂²×C₄²	C₂²×C₂₉	1	2	yes	—
155	C₂²×C₄²	C₂²×C₂₉	1	4	yes	—
156	#336	C₂²×C₂₉	1	16	yes	complemented
157	C₂×C₂₉	—	2	8	no	—
158	C₂×C₂₉	#1728	1	8	yes	direct product factor
159	C₂×C₂₉	C₂²×#95	1	1	yes	characteristic
160	C₂×C₂₉	C₂×C₄×M₁₆	1	2	yes	—
161	C₂×C₂₉	C₂×#407	1	2	yes	—
162	C₂×C₂₉	C₂×#423	1	2	yes	—
163	C₂⁵	C₂×C₄×C₂₉	1	1	yes	characteristic
164	C₂³×C₄	—	2	8	no	—
165	C₂³×C₄	—	2	16	no	—
166	C₂³×C₄	C₂×C₄×C₂₉	1	8	yes	—
167	C₂³×C₄	C₂×C₄×C₂₉	1	4	yes	—
168	C₂³×C₄	C₂×C₄×C₂₉	1	2	yes	—
169	C₂³×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₂₉	1	8	yes	—
171	C₂³×C₄	D₄×C₂₉	1	4	yes	—
172	C₂³×C₄	D₄×C₂₉	1	8	yes	—
173	C₂³×C₄	Q₈×C₂₉	1	4	yes	—
174	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₂₉	1	2	yes	—
176	C₂×M₁₆	—	2	64	no	—
177	C₂×M₁₆	—	2	16	no	—
178	C₂×M₁₆	—	2	16	no	—
179	C₂×M₁₆	C₂×C₄×C₂₉	1	32	yes	complemented
180	C₂²×C₈	—	2	8	no	—
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	C₂×C₄×C₂₉	1	16	yes	complemented
183	C₂²×C₈	C₂³×C₂₉	1	8	yes	—
184	C₂×C₄²	—	2	16	no	—
185	C₂×C₄²	—	2	8	no	—
186	C₂×C₄²	—	2	8	no	—
187	C₂×C₄²	—	2	16	no	—
188	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
189	C₂×C₄²	C₂×C₄×C₂₉	1	8	yes	—
190	C₂×C₄²	C₂³×C₂₉	1	4	yes	—
191	C₂₉	C₂×#1728	1	1	yes	characteristic, direct product factor, Sylow
192	C₂⁴	—	2	8	no	—
193	C₂⁴	C₄²×C₂₉	1	4	yes	—
194	C₂⁴	C₂²⋊C₄×C₂₉	1	4	yes	—
195	C₂⁴	C₄⋊C₄×C₂₉	1	4	yes	—
196	C₂⁴	C₂²×C₄×C₂₉	1	2	yes	—
197	C₂⁴	C₂²×C₄×C₂₉	1	1	yes	characteristic
198	C₂²×C₄	—	2	64	no	—
199	C₂²×C₄	—	2	16	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	32	no	—
203	C₂²×C₄	C₄²×C₂₉	1	8	yes	—
204	C₂²×C₄	C₄²×C₂₉	1	4	yes	—
205	C₂²×C₄	C₂²⋊C₄×C₂₉	1	32	yes	—
206	C₂²×C₄	C₂²⋊C₄×C₂₉	1	8	yes	—
207	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
208	C₂²×C₄	C₄⋊C₄×C₂₉	1	32	yes	—
209	C₂²×C₄	C₄⋊C₄×C₂₉	1	8	yes	—
210	C₂²×C₄	C₄⋊C₄×C₂₉	1	4	yes	—
211	C₂²×C₄	M₁₆×C₂₉	1	16	yes	complemented
212	C₂²×C₄	C₂²×C₄×C₂₉	1	8	yes	—
213	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
215	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₂₉	1	8	yes	—
217	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₂₉	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₂₉	1	4	yes	—
220	C₂²×C₄	C₂×D₄×C₂₉	1	8	yes	—
221	C₂²×C₄	C₂×Q₈×C₂₉	1	4	yes	—
222	C₂²×C₄	C₂×Q₈×C₂₉	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
224	M₁₆	—	2	64	no	—
225	C₂×C₈	—	2	32	no	—
226	C₂×C₈	—	2	8	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	C₂²×C₄×C₂₉	1	16	yes	complemented
229	C₄²	—	2	8	no	—
230	C₄²	—	2	16	no	—
231	C₄²	—	2	8	no	—
232	C₂³	—	2	8	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	16	no	—
235	C₂³	—	2	16	no	—
236	C₂³	C₂₉×#95	1	16	yes	—
237	C₂³	C₂×C₄²×C₂₉	1	4	yes	—
238	C₂³	C₂×C₄²×C₂₉	1	2	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
240	C₂³	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
241	C₂³	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
242	C₂³	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
243	C₂³	C₂×M₁₆×C₂₉	1	8	yes	—
244	C₂³	C₂³×C₄×C₂₉	1	1	yes	characteristic
245	C₂×C₄	—	2	64	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	32	no	—
249	C₂×C₄	C₂₉×#95	1	32	yes	—
250	C₂×C₄	C₂₉×#95	1	16	yes	—
251	C₂×C₄	C₂×C₄²×C₂₉	1	2	yes	—
252	C₂×C₄	C₂×C₄²×C₂₉	1	4	yes	—
253	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	8	yes	—
254	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	16	yes	—
255	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	16	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	8	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	4	yes	—
261	C₂×C₄	C₂×M₁₆×C₂₉	1	16	yes	complemented
262	C₂×C₄	C₂³×C₄×C₂₉	1	2	yes	—
263	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
265	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
266	C₈	—	2	16	no	—
267	C₂²	—	2	32	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	2	8	no	—
271	C₂²	C₂×C₂₉×#95	1	8	yes	—
272	C₂²	C₂×C₂₉×#95	1	4	yes	—
273	C₂²	C₄×M₁₆×C₂₉	1	8	yes	—
274	C₂²	C₂₉×#407	1	8	yes	—
275	C₂²	C₂₉×#423	1	8	yes	—
276	C₂²	C₂²×C₄²×C₂₉	1	1	yes	characteristic
277	C₂²	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
278	C₂²	C₂²×C₄⋊C₄×C₂₉	1	1	yes	characteristic
279	C₂²	C₂²×M₁₆×C₂₉	1	4	yes	—
280	C₄	—	2	16	no	—
281	C₄	C₂×C₂₉×#95	1	4	yes	—
282	C₄	C₂×C₂₉×#95	1	8	yes	—
283	C₄	C₂²×C₂²⋊C₄×C₂₉	1	2	yes	—
284	C₄	C₂²×C₄⋊C₄×C₂₉	1	2	yes	—
285	C₂	—	2	8	no	—
286	C₂	C₂₉×#1728	1	8	yes	direct product factor
287	C₂	C₂²×C₂₉×#95	1	1	yes	characteristic
288	C₂	C₂×C₄×M₁₆×C₂₉	1	2	yes	—
289	C₂	C₂×C₂₉×#407	1	2	yes	—
290	C₂	C₂×C₂₉×#423	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₉×#1728	C₂	direct (N × H)
4. C₂×C₂₉×#336	C₂	semidirect
9. C₂²×M₁₆×C₂₉	C₄	semidirect
17. C₂₉×#336	C₂²	semidirect
34. C₂×M₁₆×C₂₉	C₂×C₄	semidirect
37. C₂²×C₈×C₂₉	C₂×C₄	semidirect
65. C₂²×C₄×C₂₉	M₁₆	semidirect
82. C₂×C₈×C₂₉	C₂²×C₄	semidirect
86. C₂×#1728	C₂₉	direct (N × H)
116. C₂×C₄×C₂₉	C₂×M₁₆	semidirect
124. #1728	C₂×C₂₉	direct (N × H)
125. C₂×#336	C₂×C₂₉	semidirect
148. C₂²×M₁₆	C₄×C₂₉	semidirect
156. #336	C₂²×C₂₉	semidirect
158. C₂×C₂₉	#1728	direct (N × H)
179. C₂×M₁₆	C₂×C₄×C₂₉	semidirect
182. C₂²×C₈	C₂×C₄×C₂₉	semidirect
191. C₂₉	C₂×#1728	direct (N × H)
211. C₂²×C₄	M₁₆×C₂₉	semidirect
228. C₂×C₈	C₂²×C₄×C₂₉	semidirect
261. C₂×C₄	C₂×M₁₆×C₂₉	semidirect
286. C₂	C₂₉×#1728	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₂₉	2
C₂³×C₄²×C₂₉	2
C₂₉×#1728	2
C₂×C₂₉×#336	2
C₂×#1728	29

Subgroups — C2×C29×#1728 #20315233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#1728 #20315233