Subgroups — C₂×C₂₉×#1987 #20316208

Order 7424.

1510 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂₉×#1987	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₉×#340	C₂	1	4	yes	maximal
6	C₂³×Q₈×C₂₉	—	2	1	no	—
7	C₂³×C₈×C₂₉	C₂²	1	2	yes	—
8	C₂×C₂₉×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₂₉×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₂₉×C₄⋊Q₈	C₄	1	8	yes	complemented
11	C₂×C₂₉×C₄⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₂₉	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₂₉	—	2	1	no	—
16	C₂²×C₄²×C₂₉	C₂²	1	1	yes	characteristic
17	C₂×C₂₉×C₄⋊C₈	—	2	4	no	—
18	C₂×C₂₉×C₄⋊C₈	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₂₉	—	2	2	no	—
20	C₄×C₄⋊C₄×C₂₉	C₂²	1	4	yes	—
21	C₂×C₂₉×#95	—	2	1	no	—
22	C₂₉×#340	C₂²	1	16	yes	—
23	C₂²×Q₈×C₂₉	—	2	8	no	—
24	C₂²×Q₈×C₂₉	—	2	4	no	—
25	C₂²×Q₈×C₂₉	—	4	8	no	—
26	C₂³×C₄×C₂₉	—	2	2	no	—
27	C₂³×C₄×C₂₉	—	2	2	no	—
28	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₂₉	—	2	4	no	—
32	C₂²×C₈×C₂₉	—	2	8	no	—
33	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
34	C₂₉×C₄⋊Q₈	—	2	16	no	—
35	C₂₉×C₄⋊Q₈	—	2	8	no	—
36	C₂₉×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
37	C₂×C₄⋊C₄×C₂₉	—	2	8	no	—
38	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
41	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₂₉	Q₈	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
47	C₂×C₄²×C₂₉	—	2	4	no	—
48	C₂×C₄²×C₂₉	—	4	4	no	—
49	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₂₉	D₄	1	2	yes	—
51	C₂×C₄²×C₂₉	D₄	1	4	yes	—
52	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
53	C₂₉×C₄⋊C₈	—	2	8	no	—
54	C₂₉×#95	—	2	4	no	—
55	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₂₉	—	2	16	no	—
57	C₂×Q₈×C₂₉	—	4	16	no	—
58	C₂×Q₈×C₂₉	—	4	32	no	—
59	C₂²×C₄×C₂₉	—	2	4	no	—
60	C₂²×C₄×C₂₉	—	2	8	no	—
61	C₂²×C₄×C₂₉	—	2	8	no	—
62	C₂²×C₄×C₂₉	—	2	4	no	—
63	C₂²×C₄×C₂₉	—	2	4	no	—
64	C₂²×C₄×C₂₉	—	4	4	no	—
65	C₂²×C₄×C₂₉	—	4	4	no	—
66	C₂²×C₄×C₂₉	—	4	8	no	—
67	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₂₉	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₂₉	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₂₉	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₂₉	#41	1	4	yes	—
78	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₂₉	—	2	4	no	—
80	C₂×C₈×C₂₉	—	2	8	no	—
81	C₂×C₈×C₂₉	—	4	8	no	—
82	C₄⋊C₄×C₂₉	—	2	16	no	—
83	C₄⋊C₄×C₂₉	—	2	8	no	—
84	C₄⋊C₄×C₂₉	—	2	8	no	—
85	C₄⋊C₄×C₂₉	—	2	4	no	—
86	C₄⋊C₄×C₂₉	—	2	4	no	—
87	C₄⋊C₄×C₂₉	C₂²×C₄	1	8	yes	—
88	C₄⋊C₄×C₂₉	C₂×Q₈	1	8	yes	complemented
89	C₄²×C₂₉	—	4	8	no	—
90	C₄²×C₂₉	C₂²⋊C₄	1	8	yes	—
91	C₄²×C₂₉	C₂²×C₄	1	4	yes	—
92	C₄²×C₂₉	C₂×D₄	1	4	yes	—
93	C₂×#1987	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂³×C₂₉	C₂²⋊Q₈	1	4	yes	—
96	C₂³×C₂₉	C₂×SD₁₆	1	2	yes	—
97	C₂³×C₂₉	C₂×Q₁₆	1	2	yes	—
98	C₂³×C₂₉	AGL(1,ℤ/8)	1	4	yes	—
99	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
100	Q₈×C₂₉	—	4	32	no	—
101	C₂×C₄×C₂₉	—	2	4	no	—
102	C₂×C₄×C₂₉	—	2	8	no	—
103	C₂×C₄×C₂₉	—	2	8	no	—
104	C₂×C₄×C₂₉	—	2	8	no	—
105	C₂×C₄×C₂₉	—	4	4	no	—
106	C₂×C₄×C₂₉	—	4	4	no	—
107	C₂×C₄×C₂₉	—	4	8	no	—
108	C₂×C₄×C₂₉	—	4	16	no	—
109	C₂×C₄×C₂₉	—	4	16	no	—
110	C₂×C₄×C₂₉	#103	1	16	yes	—
111	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
114	C₂×C₄×C₂₉	C₄×Q₈	1	8	yes	—
115	C₂×C₄×C₂₉	C₂²⋊Q₈	1	4	yes	—
116	C₂×C₄×C₂₉	#124	1	4	yes	—
117	C₂×C₄×C₂₉	C₄⋊Q₈	1	4	yes	—
118	C₂×C₄×C₂₉	C₂×SD₁₆	1	4	yes	—
119	C₂×C₄×C₂₉	C₂×Q₁₆	1	4	yes	—
120	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₉	C₂²×Q₈	1	2	yes	—
123	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
124	C₈×C₂₉	—	4	4	no	—
125	C₂²×C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic
126	C₂²×C₄⋊C₈	C₂×C₂₉	1	1	yes	characteristic
127	C₂×C₄×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
128	#1987	C₂×C₂₉	1	8	yes	direct product factor
129	C₂×#340	C₂×C₂₉	1	4	yes	—
130	C₂²×C₂₉	#391	1	4	yes	—
131	C₂²×C₂₉	C₂×#103	1	4	yes	—
132	C₂²×C₂₉	#418	1	4	yes	—
133	C₂²×C₂₉	#476	1	8	yes	—
134	C₂²×C₂₉	#477	1	8	yes	—
135	C₂²×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₂₉	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₂²×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
138	C₂²×C₂₉	C₂²×Q₁₆	1	1	yes	characteristic
139	C₂²×C₂₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
140	C₄×C₂₉	—	2	4	no	—
141	C₄×C₂₉	—	4	8	no	—
142	C₄×C₂₉	—	4	8	no	—
143	C₄×C₂₉	#391	1	4	yes	—
144	C₄×C₂₉	C₂×#103	1	8	yes	—
145	C₄×C₂₉	C₂×C₄×Q₈	1	2	yes	—
146	C₄×C₂₉	C₂×#124	1	1	yes	characteristic
147	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂³×Q₈	—	2	1	no	—
149	C₂³×C₈	C₂²×C₂₉	1	2	yes	—
150	C₂×C₄⋊Q₈	—	2	2	no	—
151	C₂×C₄⋊Q₈	—	2	4	no	—
152	C₂×C₄⋊Q₈	C₄×C₂₉	1	8	yes	complemented
153	C₂×C₄⋊Q₈	C₂²×C₂₉	1	4	yes	—
154	C₂²×C₄⋊C₄	—	2	1	no	—
155	C₂²×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
156	C₂²×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
157	C₂²×C₄²	—	2	1	no	—
158	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
159	C₂×C₄⋊C₈	—	2	4	no	—
160	C₂×C₄⋊C₈	C₂²×C₂₉	1	4	yes	—
161	C₄×C₄⋊C₄	—	2	2	no	—
162	C₄×C₄⋊C₄	C₂²×C₂₉	1	4	yes	—
163	C₂×#95	—	2	1	no	—
164	#340	C₂²×C₂₉	1	16	yes	—
165	C₂×C₂₉	#1987	1	8	yes	direct product factor
166	C₂×C₂₉	C₂×#391	1	1	yes	characteristic
167	C₂×C₂₉	C₂²×#103	1	1	yes	characteristic
168	C₂×C₂₉	C₂×#418	1	1	yes	characteristic
169	C₂×C₂₉	C₂×#476	1	2	yes	—
170	C₂×C₂₉	C₂×#477	1	2	yes	—
171	C₂²×Q₈	—	2	8	no	—
172	C₂²×Q₈	—	2	4	no	—
173	C₂²×Q₈	—	4	8	no	—
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	8	no	—
181	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
182	C₄⋊Q₈	—	2	16	no	—
183	C₄⋊Q₈	—	2	8	no	—
184	C₄⋊Q₈	C₂×C₄×C₂₉	1	16	yes	complemented
185	C₂×C₄⋊C₄	—	2	8	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	4	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	8	yes	—
192	C₂×C₄⋊C₄	Q₈×C₂₉	1	8	yes	complemented
193	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
195	C₂×C₄²	—	2	4	no	—
196	C₂×C₄²	—	4	4	no	—
197	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
198	C₂×C₄²	D₄×C₂₉	1	2	yes	—
199	C₂×C₄²	D₄×C₂₉	1	4	yes	—
200	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
201	C₄⋊C₈	—	2	8	no	—
202	#95	—	2	4	no	—
203	C₂₉	C₂×#1987	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
205	C₂×Q₈	—	2	16	no	—
206	C₂×Q₈	—	4	16	no	—
207	C₂×Q₈	—	4	32	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	8	no	—
215	C₂²×C₄	—	4	4	no	—
216	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
218	C₂²×C₄	SD₁₆×C₂₉	1	4	yes	—
219	C₂²×C₄	Q₁₆×C₂₉	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
225	C₂²×C₄	C₂×Q₈×C₂₉	1	4	yes	—
226	C₂²×C₄	#41×C₂₉	1	4	yes	—
227	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	8	no	—
230	C₂×C₈	—	4	8	no	—
231	C₄⋊C₄	—	2	16	no	—
232	C₄⋊C₄	—	2	8	no	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	C₂²×C₄×C₂₉	1	8	yes	—
237	C₄⋊C₄	C₂×Q₈×C₂₉	1	8	yes	complemented
238	C₄²	—	4	8	no	—
239	C₄²	C₂²⋊C₄×C₂₉	1	8	yes	—
240	C₄²	C₂²×C₄×C₂₉	1	4	yes	—
241	C₄²	C₂×D₄×C₂₉	1	4	yes	—
242	C₂³	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
243	C₂³	C₂₉×C₂²⋊Q₈	1	4	yes	—
244	C₂³	C₂×SD₁₆×C₂₉	1	2	yes	—
245	C₂³	C₂×Q₁₆×C₂₉	1	2	yes	—
246	C₂³	C₂₉×AGL(1,ℤ/8)	1	4	yes	—
247	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
248	Q₈	—	4	32	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	4	16	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	16	no	—
258	C₂×C₄	C₂₉×#103	1	16	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
262	C₂×C₄	C₄×Q₈×C₂₉	1	8	yes	—
263	C₂×C₄	C₂₉×C₂²⋊Q₈	1	4	yes	—
264	C₂×C₄	C₂₉×#124	1	4	yes	—
265	C₂×C₄	C₂₉×C₄⋊Q₈	1	4	yes	—
266	C₂×C₄	C₂×SD₁₆×C₂₉	1	4	yes	—
267	C₂×C₄	C₂×Q₁₆×C₂₉	1	4	yes	—
268	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₂₉	1	2	yes	—
271	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
272	C₈	—	4	4	no	—
273	C₂²	C₂₉×#391	1	4	yes	—
274	C₂²	C₂×C₂₉×#103	1	4	yes	—
275	C₂²	C₂₉×#418	1	4	yes	—
276	C₂²	C₂₉×#476	1	8	yes	—
277	C₂²	C₂₉×#477	1	8	yes	—
278	C₂²	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
279	C₂²	C₂×C₂₉×C₂²⋊Q₈	1	2	yes	—
280	C₂²	C₂²×SD₁₆×C₂₉	1	1	yes	characteristic
281	C₂²	C₂²×Q₁₆×C₂₉	1	1	yes	characteristic
282	C₂²	C₂×C₂₉×AGL(1,ℤ/8)	1	2	yes	—
283	C₄	—	2	4	no	—
284	C₄	—	4	8	no	—
285	C₄	—	4	8	no	—
286	C₄	C₂₉×#391	1	4	yes	—
287	C₄	C₂×C₂₉×#103	1	8	yes	—
288	C₄	C₂×C₄×Q₈×C₂₉	1	2	yes	—
289	C₄	C₂×C₂₉×#124	1	1	yes	characteristic
290	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
291	C₂	C₂₉×#1987	1	8	yes	direct product factor
292	C₂	C₂×C₂₉×#391	1	1	yes	characteristic
293	C₂	C₂²×C₂₉×#103	1	1	yes	characteristic
294	C₂	C₂×C₂₉×#418	1	1	yes	characteristic
295	C₂	C₂×C₂₉×#476	1	2	yes	—
296	C₂	C₂×C₂₉×#477	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₉×#1987	C₂	direct (N × H)
10. C₂×C₂₉×C₄⋊Q₈	C₄	semidirect
36. C₂₉×C₄⋊Q₈	C₂×C₄	semidirect
44. C₂×C₄⋊C₄×C₂₉	Q₈	semidirect
88. C₄⋊C₄×C₂₉	C₂×Q₈	semidirect
93. C₂×#1987	C₂₉	direct (N × H)
128. #1987	C₂×C₂₉	direct (N × H)
152. C₂×C₄⋊Q₈	C₄×C₂₉	semidirect
165. C₂×C₂₉	#1987	direct (N × H)
184. C₄⋊Q₈	C₂×C₄×C₂₉	semidirect
192. C₂×C₄⋊C₄	Q₈×C₂₉	semidirect
203. C₂₉	C₂×#1987	direct (N × H)
237. C₄⋊C₄	C₂×Q₈×C₂₉	semidirect
291. C₂	C₂₉×#1987	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₉×C₄⋊Q₈	2
C₂²×C₂₉×C₄⋊C₈	2
C₂×C₄×C₄⋊C₄×C₂₉	2
C₂₉×#1987	2
C₂×C₂₉×#340	2
C₂×#1987	29

Subgroups — C2×C29×#1987 #20316208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#1987 #20316208