Subgroups — C₂²×C₂₉×#360 #20328286

Order 7424.

1214 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₉×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₉×#360	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#41×C₂₉	—	2	1	no	—
6	C₂×C₂₉×#135	—	2	3	no	—
7	C₂×C₂₉×#135	—	2	3	no	—
8	C₂×C₂₉×#135	C₄	1	6	yes	complemented
9	C₂×C₂₉×#135	C₂²	1	6	yes	—
10	C₂²×Q₁₆×C₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂²×SD₁₆×C₂₉	—	2	1	no	—
12	C₂²×D₈×C₂₉	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₂₉	—	2	1	no	—
15	C₂×C₂₉×M₃₂	—	2	3	no	—
16	C₂×C₂₉×M₃₂	C₂²	1	6	yes	complemented
17	C₂²×C₁₆×C₂₉	—	2	1	no	—
18	C₂×C₂₉×#106	—	2	3	no	—
19	C₂×C₂₉×#106	C₂²	1	6	yes	complemented
20	C₂₉×#360	C₂²	1	16	yes	direct product factor
21	C₂×#41×C₂₉	—	2	6	no	—
22	C₂×#41×C₂₉	—	2	6	no	—
23	C₂×#41×C₂₉	—	4	3	no	—
24	C₂×#41×C₂₉	—	4	3	no	—
25	C₂²×Q₈×C₂₉	—	2	1	no	—
26	C₂²×D₄×C₂₉	—	2	1	no	—
27	C₂²×D₄×C₂₉	—	4	1	no	—
28	C₂³×C₄×C₂₉	—	4	1	no	—
29	C₂³×C₄×C₂₉	—	4	1	no	—
30	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
31	C₂₉×#135	—	2	12	no	—
32	C₂₉×#135	—	2	6	no	—
33	C₂₉×#135	—	2	6	no	—
34	C₂₉×#135	C₂×C₄	1	12	yes	complemented
35	C₂₉×#135	C₂³	1	4	yes	—
36	C₂×Q₁₆×C₂₉	—	2	3	no	—
37	C₂×Q₁₆×C₂₉	C₂×C₄	1	6	yes	complemented
38	C₂×SD₁₆×C₂₉	—	2	6	no	—
39	C₂×SD₁₆×C₂₉	—	2	6	no	—
40	C₂×D₈×C₂₉	—	2	3	no	—
41	C₂×D₈×C₂₉	C₂×C₄	1	6	yes	complemented
42	C₂²×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₂₉	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₂₉	D₄	1	3	yes	—
45	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₂₉	D₄	1	3	yes	—
47	C₂²×C₈×C₂₉	C₂³	1	3	yes	—
48	C₂×C₄⋊C₄×C₂₉	—	2	6	no	—
49	C₂×C₄⋊C₄×C₂₉	—	4	3	no	—
50	C₂₉×M₃₂	—	2	6	no	—
51	C₂₉×M₃₂	C₂³	1	4	yes	complemented
52	C₂×C₁₆×C₂₉	—	2	6	no	—
53	C₂₉×#106	—	2	6	no	—
54	C₂₉×#106	C₂³	1	4	yes	complemented
55	C₂⁴×C₂₉	—	4	1	no	—
56	C₂⁴×C₂₉	D₈	1	1	yes	characteristic
57	#41×C₂₉	—	2	12	no	—
58	#41×C₂₉	—	2	4	no	—
59	#41×C₂₉	—	4	12	no	—
60	#41×C₂₉	—	4	6	no	—
61	#41×C₂₉	—	4	6	no	—
62	C₂×Q₈×C₂₉	—	2	6	no	—
63	C₂×Q₈×C₂₉	—	4	3	no	—
64	C₂×D₄×C₂₉	—	2	6	no	—
65	C₂×D₄×C₂₉	—	4	6	no	—
66	C₂×D₄×C₂₉	—	4	6	no	—
67	C₂×D₄×C₂₉	—	4	3	no	—
68	C₂²×C₄×C₂₉	—	4	1	no	—
69	C₂²×C₄×C₂₉	—	4	6	no	—
70	C₂²×C₄×C₂₉	—	4	6	no	—
71	C₂²×C₄×C₂₉	—	4	6	no	—
72	C₂²×C₄×C₂₉	—	8	4	no	—
73	C₂²×C₄×C₂₉	C₂²⋊C₄	1	3	yes	—
74	C₂²×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₉	D₈	1	3	yes	—
76	C₂²×C₄×C₂₉	SD₁₆	1	3	yes	—
77	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₂₉	C₂×D₄	1	3	yes	—
79	Q₁₆×C₂₉	—	2	6	no	—
80	Q₁₆×C₂₉	C₂²×C₄	1	4	yes	complemented
81	SD₁₆×C₂₉	—	2	12	no	—
82	SD₁₆×C₂₉	—	2	4	no	—
83	D₈×C₂₉	—	2	6	no	—
84	D₈×C₂₉	C₂²×C₄	1	4	yes	complemented
85	C₂×C₈×C₂₉	C₂²⋊C₄	1	3	yes	—
86	C₂×C₈×C₂₉	C₂²⋊C₄	1	3	yes	—
87	C₂×C₈×C₂₉	C₂²⋊C₄	1	6	yes	—
88	C₂×C₈×C₂₉	C₂²×C₄	1	3	yes	—
89	C₂×C₈×C₂₉	C₂²×C₄	1	3	yes	—
90	C₂×C₈×C₂₉	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₂₉	C₂×D₄	1	3	yes	—
92	C₂×C₈×C₂₉	C₂×D₄	1	3	yes	—
93	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₂₉	—	2	4	no	—
95	C₄⋊C₄×C₂₉	—	4	6	no	—
96	C₁₆×C₂₉	—	2	4	no	—
97	C₂²×#360	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₂₉	—	2	4	no	—
99	C₂³×C₂₉	—	4	6	no	—
100	C₂³×C₂₉	—	8	4	no	—
101	C₂³×C₂₉	#102	1	3	yes	—
102	C₂³×C₂₉	#102	1	1	yes	characteristic
103	C₂³×C₂₉	C₂×D₈	1	3	yes	—
104	Q₈×C₂₉	—	2	4	no	—
105	Q₈×C₂₉	—	4	6	no	—
106	D₄×C₂₉	—	2	4	no	—
107	D₄×C₂₉	—	4	4	no	—
108	D₄×C₂₉	—	4	12	no	—
109	D₄×C₂₉	—	4	6	no	—
110	C₂×C₄×C₂₉	—	4	6	no	—
111	C₂×C₄×C₂₉	—	4	4	no	—
112	C₂×C₄×C₂₉	—	4	12	no	—
113	C₂×C₄×C₂₉	—	4	4	no	—
114	C₂×C₄×C₂₉	—	8	12	no	—
115	C₂×C₄×C₂₉	#102	1	3	yes	—
116	C₂×C₄×C₂₉	#102	1	3	yes	—
117	C₂×C₄×C₂₉	#102	1	6	yes	—
118	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	3	yes	—
119	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	3	yes	—
120	C₂×C₄×C₂₉	C₂×D₈	1	3	yes	—
121	C₂×C₄×C₂₉	C₂×SD₁₆	1	3	yes	—
122	C₂×C₄×C₂₉	C₂×SD₁₆	1	3	yes	—
123	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₂₉	C₂×C₂²⋊C₄	1	3	yes	—
125	C₈×C₂₉	C₂×C₂²⋊C₄	1	3	yes	—
126	C₈×C₂₉	C₂³×C₄	1	1	yes	characteristic
127	C₈×C₂₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²×#135	C₂×C₂₉	1	1	yes	characteristic
129	C₂²×M₃₂	C₂×C₂₉	1	1	yes	characteristic, complemented
130	C₂²×#106	C₂×C₂₉	1	1	yes	characteristic, complemented
131	C₂×#360	C₂×C₂₉	1	12	yes	direct product factor
132	C₂²×C₂₉	—	2	6	no	—
133	C₂²×C₂₉	—	2	6	no	—
134	C₂²×C₂₉	—	4	4	no	—
135	C₂²×C₂₉	—	8	12	no	—
136	C₂²×C₂₉	#360	1	4	yes	direct product factor
137	C₂²×C₂₉	C₂×#102	1	3	yes	—
138	C₂²×C₂₉	C₂×#102	1	3	yes	—
139	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₂₉	—	4	4	no	—
141	C₄×C₂₉	—	8	4	no	—
142	C₄×C₂₉	C₂×#102	1	3	yes	—
143	C₄×C₂₉	C₂×#102	1	3	yes	—
144	C₄×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₄×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
146	C₂²×#41	—	2	1	no	—
147	C₂×#135	—	2	3	no	—
148	C₂×#135	—	2	3	no	—
149	C₂×#135	C₄×C₂₉	1	6	yes	complemented
150	C₂×#135	C₂²×C₂₉	1	6	yes	—
151	C₂²×Q₁₆	C₄×C₂₉	1	1	yes	characteristic, complemented
152	C₂²×SD₁₆	—	2	1	no	—
153	C₂²×D₈	C₄×C₂₉	1	1	yes	characteristic, complemented
154	C₂³×C₈	C₂²×C₂₉	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	—	2	1	no	—
156	C₂×M₃₂	—	2	3	no	—
157	C₂×M₃₂	C₂²×C₂₉	1	6	yes	complemented
158	C₂²×C₁₆	—	2	1	no	—
159	C₂×#106	—	2	3	no	—
160	C₂×#106	C₂²×C₂₉	1	6	yes	complemented
161	#360	C₂²×C₂₉	1	16	yes	direct product factor
162	C₂×C₂₉	—	2	1	no	—
163	C₂×C₂₉	—	2	3	no	—
164	C₂×C₂₉	—	8	4	no	—
165	C₂×C₂₉	C₂×#360	1	6	yes	direct product factor
166	C₂×C₂₉	C₂²×#102	1	1	yes	characteristic
167	C₂×#41	—	2	6	no	—
168	C₂×#41	—	2	6	no	—
169	C₂×#41	—	4	3	no	—
170	C₂×#41	—	4	3	no	—
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×D₄	—	2	1	no	—
173	C₂²×D₄	—	4	1	no	—
174	C₂³×C₄	—	4	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
177	#135	—	2	12	no	—
178	#135	—	2	6	no	—
179	#135	—	2	6	no	—
180	#135	C₂×C₄×C₂₉	1	12	yes	complemented
181	#135	C₂³×C₂₉	1	4	yes	—
182	C₂×Q₁₆	—	2	3	no	—
183	C₂×Q₁₆	C₂×C₄×C₂₉	1	6	yes	complemented
184	C₂×SD₁₆	—	2	6	no	—
185	C₂×SD₁₆	—	2	6	no	—
186	C₂×D₈	—	2	3	no	—
187	C₂×D₈	C₂×C₄×C₂₉	1	6	yes	complemented
188	C₂²×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
189	C₂²×C₈	C₂×C₄×C₂₉	1	3	yes	—
190	C₂²×C₈	D₄×C₂₉	1	3	yes	—
191	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
192	C₂²×C₈	D₄×C₂₉	1	3	yes	—
193	C₂²×C₈	C₂³×C₂₉	1	3	yes	—
194	C₂×C₄⋊C₄	—	2	6	no	—
195	C₂×C₄⋊C₄	—	4	3	no	—
196	M₃₂	—	2	6	no	—
197	M₃₂	C₂³×C₂₉	1	4	yes	complemented
198	C₂×C₁₆	—	2	6	no	—
199	#106	—	2	6	no	—
200	#106	C₂³×C₂₉	1	4	yes	complemented
201	C₂₉	C₂²×#360	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	4	1	no	—
203	C₂⁴	D₈×C₂₉	1	1	yes	characteristic
204	#41	—	2	12	no	—
205	#41	—	2	4	no	—
206	#41	—	4	6	no	—
207	#41	—	4	6	no	—
208	#41	—	4	12	no	—
209	C₂×Q₈	—	2	6	no	—
210	C₂×Q₈	—	4	3	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	4	6	no	—
213	C₂×D₄	—	4	6	no	—
214	C₂×D₄	—	4	3	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	6	no	—
217	C₂²×C₄	—	4	6	no	—
218	C₂²×C₄	—	4	6	no	—
219	C₂²×C₄	—	8	4	no	—
220	C₂²×C₄	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₂₉	1	3	yes	—
222	C₂²×C₄	D₈×C₂₉	1	3	yes	—
223	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
224	C₂²×C₄	SD₁₆×C₂₉	1	3	yes	—
225	C₂²×C₄	C₂×D₄×C₂₉	1	3	yes	—
226	Q₁₆	—	2	6	no	—
227	Q₁₆	C₂²×C₄×C₂₉	1	4	yes	complemented
228	SD₁₆	—	2	12	no	—
229	SD₁₆	—	2	4	no	—
230	D₈	—	2	6	no	—
231	D₈	C₂²×C₄×C₂₉	1	4	yes	complemented
232	C₂×C₈	C₂²⋊C₄×C₂₉	1	3	yes	—
233	C₂×C₈	C₂²⋊C₄×C₂₉	1	3	yes	—
234	C₂×C₈	C₂²⋊C₄×C₂₉	1	6	yes	—
235	C₂×C₈	C₂²×C₄×C₂₉	1	3	yes	—
236	C₂×C₈	C₂²×C₄×C₂₉	1	3	yes	—
237	C₂×C₈	C₂×D₄×C₂₉	1	3	yes	—
238	C₂×C₈	C₂×D₄×C₂₉	1	3	yes	—
239	C₂×C₈	C₂×D₄×C₂₉	1	3	yes	—
240	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	6	no	—
243	C₁₆	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	6	no	—
246	C₂³	—	8	4	no	—
247	C₂³	C₂₉×#102	1	3	yes	—
248	C₂³	C₂₉×#102	1	1	yes	characteristic
249	C₂³	C₂×D₈×C₂₉	1	3	yes	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	6	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	12	no	—
255	D₄	—	4	6	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	12	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	12	no	—
261	C₂×C₄	C₂₉×#102	1	3	yes	—
262	C₂×C₄	C₂₉×#102	1	3	yes	—
263	C₂×C₄	C₂₉×#102	1	6	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	3	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	3	yes	—
266	C₂×C₄	C₂×D₈×C₂₉	1	3	yes	—
267	C₂×C₄	C₂×SD₁₆×C₂₉	1	3	yes	—
268	C₂×C₄	C₂×SD₁₆×C₂₉	1	3	yes	—
269	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
270	C₈	C₂×C₂²⋊C₄×C₂₉	1	3	yes	—
271	C₈	C₂×C₂²⋊C₄×C₂₉	1	3	yes	—
272	C₈	C₂³×C₄×C₂₉	1	1	yes	characteristic
273	C₈	C₂²×D₄×C₂₉	1	1	yes	characteristic
274	C₂²	—	2	6	no	—
275	C₂²	—	2	6	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	12	no	—
278	C₂²	C₂₉×#360	1	4	yes	direct product factor
279	C₂²	C₂×C₂₉×#102	1	3	yes	—
280	C₂²	C₂×C₂₉×#102	1	3	yes	—
281	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
282	C₄	—	4	4	no	—
283	C₄	—	8	4	no	—
284	C₄	C₂×C₂₉×#102	1	3	yes	—
285	C₄	C₂×C₂₉×#102	1	3	yes	—
286	C₄	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
287	C₄	C₂²×SD₁₆×C₂₉	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	3	no	—
290	C₂	—	8	4	no	—
291	C₂	C₂×C₂₉×#360	1	6	yes	direct product factor
292	C₂	C₂²×C₂₉×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₉	#102
commutator	C₈	C₂³×C₄×C₂₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
prime radical	C₂²×C₂₉×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂³×C₄×C₂₉ ◁ C₂³×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₉×M₃₂	C₂	semidirect
3. C₂²×C₂₉×#106	C₂	semidirect
4. C₂×C₂₉×#360	C₂	direct (N × H)
8. C₂×C₂₉×#135	C₄	semidirect
10. C₂²×Q₁₆×C₂₉	C₄	semidirect
12. C₂²×D₈×C₂₉	C₄	semidirect
16. C₂×C₂₉×M₃₂	C₂²	semidirect
19. C₂×C₂₉×#106	C₂²	semidirect
20. C₂₉×#360	C₂²	direct (N × H)
34. C₂₉×#135	C₂×C₄	semidirect
37. C₂×Q₁₆×C₂₉	C₂×C₄	semidirect
41. C₂×D₈×C₂₉	C₂×C₄	semidirect
51. C₂₉×M₃₂	C₂³	semidirect
54. C₂₉×#106	C₂³	semidirect
80. Q₁₆×C₂₉	C₂²×C₄	semidirect
84. D₈×C₂₉	C₂²×C₄	semidirect
97. C₂²×#360	C₂₉	direct (N × H)
129. C₂²×M₃₂	C₂×C₂₉	semidirect
130. C₂²×#106	C₂×C₂₉	semidirect
131. C₂×#360	C₂×C₂₉	direct (N × H)
136. C₂²×C₂₉	#360	direct (N × H)
149. C₂×#135	C₄×C₂₉	semidirect
151. C₂²×Q₁₆	C₄×C₂₉	semidirect
153. C₂²×D₈	C₄×C₂₉	semidirect
157. C₂×M₃₂	C₂²×C₂₉	semidirect
160. C₂×#106	C₂²×C₂₉	semidirect
161. #360	C₂²×C₂₉	direct (N × H)
165. C₂×C₂₉	C₂×#360	direct (N × H)
180. #135	C₂×C₄×C₂₉	semidirect
183. C₂×Q₁₆	C₂×C₄×C₂₉	semidirect
187. C₂×D₈	C₂×C₄×C₂₉	semidirect
197. M₃₂	C₂³×C₂₉	semidirect
200. #106	C₂³×C₂₉	semidirect
201. C₂₉	C₂²×#360	direct (N × H)
227. Q₁₆	C₂²×C₄×C₂₉	semidirect
231. D₈	C₂²×C₄×C₂₉	semidirect
278. C₂²	C₂₉×#360	direct (N × H)
291. C₂	C₂×C₂₉×#360	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×#135	2
C₂²×C₂₉×M₃₂	2
C₂²×C₂₉×#106	2
C₂×C₂₉×#360	2
C₂²×#360	29

Subgroups — C22×C29×#360 #20328286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₉×#360 #20328286