Subgroups — C₂×C₂₉×#1988 #20316209

Order 7424.

1510 conjugacy classes of subgroups.

The subgroup lattice has 306 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×#1988	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	4	no	—
26	C₂²×Q₈×C₂₉	—	4	4	no	—
27	C₂³×C₄×C₂₉	—	2	2	no	—
28	C₂³×C₄×C₂₉	—	2	2	no	—
29	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₂₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₂₉	—	2	4	no	—
33	C₂²×C₈×C₂₉	—	2	8	no	—
34	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
35	C₂₉×C₄⋊Q₈	—	2	16	no	—
36	C₂₉×C₄⋊Q₈	—	2	8	no	—
37	C₂₉×C₄⋊Q₈	C₂×C₄	1	16	yes	complemented
38	C₂×C₄⋊C₄×C₂₉	—	2	8	no	—
39	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₂₉	Q₈	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₂₉	—	2	4	no	—
49	C₂×C₄²×C₂₉	—	4	4	no	—
50	C₂×C₄²×C₂₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₂₉	D₄	1	2	yes	—
52	C₂×C₄²×C₂₉	D₄	1	4	yes	—
53	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
54	C₂₉×C₄⋊C₈	—	2	8	no	—
55	C₂₉×#95	—	2	4	no	—
56	C₂⁴×C₂₉	C₂×D₄	1	1	yes	characteristic
57	C₂×Q₈×C₂₉	—	2	16	no	—
58	C₂×Q₈×C₂₉	—	4	32	no	—
59	C₂×Q₈×C₂₉	—	4	8	no	—
60	C₂×Q₈×C₂₉	—	4	8	no	—
61	C₂²×C₄×C₂₉	—	2	4	no	—
62	C₂²×C₄×C₂₉	—	2	8	no	—
63	C₂²×C₄×C₂₉	—	2	8	no	—
64	C₂²×C₄×C₂₉	—	2	4	no	—
65	C₂²×C₄×C₂₉	—	2	4	no	—
66	C₂²×C₄×C₂₉	—	4	4	no	—
67	C₂²×C₄×C₂₉	—	4	4	no	—
68	C₂²×C₄×C₂₉	—	4	4	no	—
69	C₂²×C₄×C₂₉	—	4	4	no	—
70	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₂₉	D₈	1	4	yes	—
73	C₂²×C₄×C₂₉	SD₁₆	1	4	yes	—
74	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₂₉	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₂₉	#41	1	4	yes	—
81	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₂₉	—	2	4	no	—
83	C₂×C₈×C₂₉	—	2	8	no	—
84	C₂×C₈×C₂₉	—	4	8	no	—
85	C₄⋊C₄×C₂₉	—	2	16	no	—
86	C₄⋊C₄×C₂₉	—	2	8	no	—
87	C₄⋊C₄×C₂₉	—	2	8	no	—
88	C₄⋊C₄×C₂₉	—	2	4	no	—
89	C₄⋊C₄×C₂₉	—	2	4	no	—
90	C₄⋊C₄×C₂₉	C₂²×C₄	1	8	yes	—
91	C₄⋊C₄×C₂₉	C₂×Q₈	1	8	yes	complemented
92	C₄²×C₂₉	—	4	8	no	—
93	C₄²×C₂₉	C₂²⋊C₄	1	8	yes	—
94	C₄²×C₂₉	C₂²×C₄	1	4	yes	—
95	C₄²×C₂₉	C₂×D₄	1	4	yes	—
96	C₂×#1988	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₂₉	C₂²⋊Q₈	1	4	yes	—
99	C₂³×C₂₉	C₂×D₈	1	2	yes	—
100	C₂³×C₂₉	C₂×SD₁₆	1	2	yes	—
101	C₂³×C₂₉	#137	1	4	yes	—
102	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
103	Q₈×C₂₉	—	4	32	no	—
104	C₂×C₄×C₂₉	—	2	4	no	—
105	C₂×C₄×C₂₉	—	2	8	no	—
106	C₂×C₄×C₂₉	—	2	8	no	—
107	C₂×C₄×C₂₉	—	2	8	no	—
108	C₂×C₄×C₂₉	—	4	4	no	—
109	C₂×C₄×C₂₉	—	4	4	no	—
110	C₂×C₄×C₂₉	—	4	4	no	—
111	C₂×C₄×C₂₉	—	4	4	no	—
112	C₂×C₄×C₂₉	—	4	16	no	—
113	C₂×C₄×C₂₉	—	4	16	no	—
114	C₂×C₄×C₂₉	#102	1	16	yes	—
115	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
116	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	4	yes	—
118	C₂×C₄×C₂₉	C₄×Q₈	1	8	yes	—
119	C₂×C₄×C₂₉	C₂²⋊Q₈	1	4	yes	—
120	C₂×C₄×C₂₉	#124	1	4	yes	—
121	C₂×C₄×C₂₉	C₄⋊Q₈	1	4	yes	—
122	C₂×C₄×C₂₉	C₂×D₈	1	4	yes	—
123	C₂×C₄×C₂₉	C₂×SD₁₆	1	4	yes	—
124	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₉	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
128	C₈×C₂₉	—	4	4	no	—
129	C₂²×C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic
130	C₂²×C₄⋊C₈	C₂×C₂₉	1	1	yes	characteristic
131	C₂×C₄×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
132	#1988	C₂×C₂₉	1	8	yes	direct product factor
133	C₂×#340	C₂×C₂₉	1	4	yes	—
134	C₂²×C₂₉	#391	1	4	yes	—
135	C₂²×C₂₉	C₂×#102	1	4	yes	—
136	C₂²×C₂₉	#419	1	4	yes	—
137	C₂²×C₂₉	#474	1	8	yes	—
138	C₂²×C₂₉	#475	1	8	yes	—
139	C₂²×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₂₉	C₂×C₂²⋊Q₈	1	2	yes	—
141	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
142	C₂²×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
143	C₂²×C₂₉	C₂×#137	1	2	yes	—
144	C₄×C₂₉	—	2	4	no	—
145	C₄×C₂₉	—	4	8	no	—
146	C₄×C₂₉	—	4	8	no	—
147	C₄×C₂₉	#391	1	4	yes	—
148	C₄×C₂₉	C₂×#102	1	8	yes	—
149	C₄×C₂₉	C₂×C₄×Q₈	1	2	yes	—
150	C₄×C₂₉	C₂×#124	1	1	yes	characteristic
151	C₄×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₂³×Q₈	—	2	1	no	—
153	C₂³×C₈	C₂²×C₂₉	1	2	yes	—
154	C₂×C₄⋊Q₈	—	2	2	no	—
155	C₂×C₄⋊Q₈	—	2	4	no	—
156	C₂×C₄⋊Q₈	C₄×C₂₉	1	8	yes	complemented
157	C₂×C₄⋊Q₈	C₂²×C₂₉	1	4	yes	—
158	C₂²×C₄⋊C₄	—	2	1	no	—
159	C₂²×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
160	C₂²×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
161	C₂²×C₄²	—	2	1	no	—
162	C₂²×C₄²	C₂²×C₂₉	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	2	4	no	—
164	C₂×C₄⋊C₈	C₂²×C₂₉	1	4	yes	—
165	C₄×C₄⋊C₄	—	2	2	no	—
166	C₄×C₄⋊C₄	C₂²×C₂₉	1	4	yes	—
167	C₂×#95	—	2	1	no	—
168	#340	C₂²×C₂₉	1	16	yes	—
169	C₂×C₂₉	#1988	1	8	yes	direct product factor
170	C₂×C₂₉	C₂×#391	1	1	yes	characteristic
171	C₂×C₂₉	C₂²×#102	1	1	yes	characteristic
172	C₂×C₂₉	C₂×#419	1	1	yes	characteristic
173	C₂×C₂₉	C₂×#474	1	2	yes	—
174	C₂×C₂₉	C₂×#475	1	2	yes	—
175	C₂²×Q₈	—	2	8	no	—
176	C₂²×Q₈	—	2	4	no	—
177	C₂²×Q₈	—	4	4	no	—
178	C₂²×Q₈	—	4	4	no	—
179	C₂³×C₄	—	2	2	no	—
180	C₂³×C₄	—	2	2	no	—
181	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
182	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₂₉	1	1	yes	characteristic
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	2	8	no	—
186	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
187	C₄⋊Q₈	—	2	16	no	—
188	C₄⋊Q₈	—	2	8	no	—
189	C₄⋊Q₈	C₂×C₄×C₂₉	1	16	yes	complemented
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	8	yes	—
197	C₂×C₄⋊C₄	Q₈×C₂₉	1	8	yes	complemented
198	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	—	4	4	no	—
202	C₂×C₄²	C₂×C₄×C₂₉	1	4	yes	—
203	C₂×C₄²	D₄×C₂₉	1	2	yes	—
204	C₂×C₄²	D₄×C₂₉	1	4	yes	—
205	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
206	C₄⋊C₈	—	2	8	no	—
207	#95	—	2	4	no	—
208	C₂₉	C₂×#1988	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	C₂×D₄×C₂₉	1	1	yes	characteristic
210	C₂×Q₈	—	2	16	no	—
211	C₂×Q₈	—	4	8	no	—
212	C₂×Q₈	—	4	8	no	—
213	C₂×Q₈	—	4	32	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
224	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
225	C₂²×C₄	D₈×C₂₉	1	4	yes	—
226	C₂²×C₄	SD₁₆×C₂₉	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
232	C₂²×C₄	C₂×Q₈×C₂₉	1	4	yes	—
233	C₂²×C₄	#41×C₂₉	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	4	8	no	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₂²×C₄×C₂₉	1	8	yes	—
244	C₄⋊C₄	C₂×Q₈×C₂₉	1	8	yes	complemented
245	C₄²	—	4	8	no	—
246	C₄²	C₂²⋊C₄×C₂₉	1	8	yes	—
247	C₄²	C₂²×C₄×C₂₉	1	4	yes	—
248	C₄²	C₂×D₄×C₂₉	1	4	yes	—
249	C₂³	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
250	C₂³	C₂₉×C₂²⋊Q₈	1	4	yes	—
251	C₂³	C₂×D₈×C₂₉	1	2	yes	—
252	C₂³	C₂×SD₁₆×C₂₉	1	2	yes	—
253	C₂³	C₂₉×#137	1	4	yes	—
254	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
255	Q₈	—	4	32	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	16	no	—
266	C₂×C₄	C₂₉×#102	1	16	yes	—
267	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
268	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	4	yes	—
270	C₂×C₄	C₄×Q₈×C₂₉	1	8	yes	—
271	C₂×C₄	C₂₉×C₂²⋊Q₈	1	4	yes	—
272	C₂×C₄	C₂₉×#124	1	4	yes	—
273	C₂×C₄	C₂₉×C₄⋊Q₈	1	4	yes	—
274	C₂×C₄	C₂×D₈×C₂₉	1	4	yes	—
275	C₂×C₄	C₂×SD₁₆×C₂₉	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₂₉	1	2	yes	—
279	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
280	C₈	—	4	4	no	—
281	C₂²	C₂₉×#391	1	4	yes	—
282	C₂²	C₂×C₂₉×#102	1	4	yes	—
283	C₂²	C₂₉×#419	1	4	yes	—
284	C₂²	C₂₉×#474	1	8	yes	—
285	C₂²	C₂₉×#475	1	8	yes	—
286	C₂²	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
287	C₂²	C₂×C₂₉×C₂²⋊Q₈	1	2	yes	—
288	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
289	C₂²	C₂²×SD₁₆×C₂₉	1	1	yes	characteristic
290	C₂²	C₂×C₂₉×#137	1	2	yes	—
291	C₄	—	2	4	no	—
292	C₄	—	4	8	no	—
293	C₄	—	4	8	no	—
294	C₄	C₂₉×#391	1	4	yes	—
295	C₄	C₂×C₂₉×#102	1	8	yes	—
296	C₄	C₂×C₄×Q₈×C₂₉	1	2	yes	—
297	C₄	C₂×C₂₉×#124	1	1	yes	characteristic
298	C₄	C₂×C₂₉×C₄⋊Q₈	1	1	yes	characteristic
299	C₂	C₂₉×#1988	1	8	yes	direct product factor
300	C₂	C₂×C₂₉×#391	1	1	yes	characteristic
301	C₂	C₂²×C₂₉×#102	1	1	yes	characteristic
302	C₂	C₂×C₂₉×#419	1	1	yes	characteristic
303	C₂	C₂×C₂₉×#474	1	2	yes	—
304	C₂	C₂×C₂₉×#475	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×#1988	C₂	direct (N × H)
10. C₂×C₂₉×C₄⋊Q₈	C₄	semidirect
37. C₂₉×C₄⋊Q₈	C₂×C₄	semidirect
45. C₂×C₄⋊C₄×C₂₉	Q₈	semidirect
91. C₄⋊C₄×C₂₉	C₂×Q₈	semidirect
96. C₂×#1988	C₂₉	direct (N × H)
132. #1988	C₂×C₂₉	direct (N × H)
156. C₂×C₄⋊Q₈	C₄×C₂₉	semidirect
169. C₂×C₂₉	#1988	direct (N × H)
189. C₄⋊Q₈	C₂×C₄×C₂₉	semidirect
197. C₂×C₄⋊C₄	Q₈×C₂₉	semidirect
208. C₂₉	C₂×#1988	direct (N × H)
244. C₄⋊C₄	C₂×Q₈×C₂₉	semidirect
299. C₂	C₂₉×#1988	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×#1988	2
C₂×C₂₉×#340	2
C₂×#1988	29

Subgroups — C2×C29×#1988 #20316209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#1988 #20316209