Subgroups — C₂×C₂₉×#2218 #20328501

Order 7424.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₉×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₉×#466	C₂	1	2	yes	maximal, complemented
4	C₂₉×#2218	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₂₉×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₉×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₂₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₂₉×#121	—	2	2	no	—
10	C₂×C₂₉×D₁₆	—	2	4	no	—
11	C₂×C₂₉×D₁₆	C₂²	1	4	yes	complemented
12	C₂×C₂₉×M₃₂	—	2	4	no	—
13	C₂×C₂₉×M₃₂	C₂²	1	4	yes	complemented
14	C₂²×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
16	C₂₉×#466	—	2	4	no	—
17	C₂₉×#466	C₂²	1	8	yes	complemented
18	C₂×C₂₉×#107	C₂²	1	2	yes	—
19	C₂×C₂₉×#102	—	2	2	no	—
20	C₂₉×#367	C₂²	1	4	yes	complemented
21	C₂₉×#357	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₂₉	—	2	2	no	—
23	C₂²×D₄×C₂₉	—	4	2	no	—
24	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
25	C₂×D₈×C₂₉	—	2	4	no	—
26	C₂×D₈×C₂₉	—	2	2	no	—
27	C₂×D₈×C₂₉	—	2	4	no	—
28	C₂×D₈×C₂₉	C₂³	1	4	yes	—
29	C₂²×C₈×C₂₉	—	2	2	no	—
30	C₂²×C₈×C₂₉	—	2	2	no	—
31	C₂²×C₈×C₂₉	D₄	1	2	yes	—
32	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	C₂₉×#121	—	2	8	no	—
36	C₂₉×#121	—	4	4	no	—
37	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₂₉	—	4	2	no	—
39	C₂₉×D₁₆	—	2	8	no	—
40	C₂₉×M₃₂	—	2	8	no	—
41	C₂×C₁₆×C₂₉	—	2	2	no	—
42	C₂×C₁₆×C₂₉	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
44	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
45	C₂₉×#107	—	2	2	no	—
46	C₂₉×#107	C₂³	1	4	yes	—
47	C₂₉×#102	—	2	8	no	—
48	C₂⁴×C₂₉	—	4	2	no	—
49	C₂⁴×C₂₉	D₈	1	1	yes	characteristic
50	C₂×D₄×C₂₉	—	2	4	no	—
51	C₂×D₄×C₂₉	—	4	8	no	—
52	C₂×D₄×C₂₉	—	4	4	no	—
53	C₂×D₄×C₂₉	—	4	2	no	—
54	C₂×D₄×C₂₉	—	4	4	no	—
55	C₂×D₄×C₂₉	—	8	8	no	—
56	C₂²×C₄×C₂₉	—	2	2	no	—
57	C₂²×C₄×C₂₉	—	2	2	no	—
58	C₂²×C₄×C₂₉	—	4	2	no	—
59	C₂²×C₄×C₂₉	D₈	1	2	yes	—
60	C₂²×C₄×C₂₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
63	D₈×C₂₉	—	2	4	no	—
64	D₈×C₂₉	—	2	4	no	—
65	D₈×C₂₉	—	4	4	no	—
66	C₂×C₈×C₂₉	—	2	2	no	—
67	C₂×C₈×C₂₉	—	2	4	no	—
68	C₂×C₈×C₂₉	—	2	2	no	—
69	C₂×C₈×C₂₉	—	2	2	no	—
70	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₂₉	#41	1	2	yes	—
74	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₂₉	—	2	4	no	—
76	C₄⋊C₄×C₂₉	—	4	2	no	—
77	C₂²⋊C₄×C₂₉	—	4	8	no	—
78	C₁₆×C₂₉	—	2	2	no	—
79	C₁₆×C₂₉	C₂×D₄	1	4	yes	complemented
80	C₂×#2218	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₉	—	2	2	no	—
82	C₂³×C₂₉	—	2	4	no	—
83	C₂³×C₂₉	—	4	4	no	—
84	C₂³×C₂₉	—	8	4	no	—
85	C₂³×C₂₉	—	8	4	no	—
86	C₂³×C₂₉	—	8	4	no	—
87	C₂³×C₂₉	C₂×D₈	1	2	yes	—
88	C₂³×C₂₉	C₂×D₈	1	1	yes	characteristic
89	D₄×C₂₉	—	4	4	no	—
90	D₄×C₂₉	—	4	4	no	—
91	D₄×C₂₉	—	8	16	no	—
92	D₄×C₂₉	—	8	4	no	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	—	2	2	no	—
95	C₂×C₄×C₂₉	—	2	2	no	—
96	C₂×C₄×C₂₉	—	2	4	no	—
97	C₂×C₄×C₂₉	—	4	4	no	—
98	C₂×C₄×C₂₉	—	8	4	no	—
99	C₂×C₄×C₂₉	#121	1	2	yes	—
100	C₂×C₄×C₂₉	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₂₉	#135	1	2	yes	—
103	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
104	C₈×C₂₉	—	2	2	no	—
105	C₈×C₂₉	#121	1	2	yes	—
106	C₈×C₂₉	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₂₉	C₂×#41	1	1	yes	characteristic
108	C₂²×D₁₆	C₂×C₂₉	1	1	yes	characteristic, complemented
109	C₂²×M₃₂	C₂×C₂₉	1	1	yes	characteristic, complemented
110	C₂×#466	C₂×C₂₉	1	2	yes	complemented
111	#2218	C₂×C₂₉	1	8	yes	direct product factor
112	C₂×#367	C₂×C₂₉	1	1	yes	characteristic, complemented
113	C₂×#357	C₂×C₂₉	1	2	yes	complemented
114	C₂²×C₂₉	—	2	4	no	—
115	C₂²×C₂₉	—	2	2	no	—
116	C₂²×C₂₉	—	4	4	no	—
117	C₂²×C₂₉	—	8	4	no	—
118	C₂²×C₂₉	—	8	4	no	—
119	C₂²×C₂₉	—	8	4	no	—
120	C₂²×C₂₉	—	16	8	no	—
121	C₂²×C₂₉	#466	1	2	yes	—
122	C₂²×C₂₉	#509	1	4	yes	—
123	C₂²×C₂₉	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₂₉	—	2	2	no	—
125	C₄×C₂₉	—	8	4	no	—
126	C₄×C₂₉	#466	1	2	yes	—
127	C₄×C₂₉	C₂×#121	1	1	yes	characteristic
128	C₄×C₂₉	C₂×#135	1	1	yes	characteristic
129	C₂²×D₈	C₂²×C₂₉	1	2	yes	—
130	C₂³×C₈	C₂²×C₂₉	1	1	yes	characteristic
131	C₂×#121	—	2	2	no	—
132	C₂×D₁₆	—	2	4	no	—
133	C₂×D₁₆	C₂²×C₂₉	1	4	yes	complemented
134	C₂×M₃₂	—	2	4	no	—
135	C₂×M₃₂	C₂²×C₂₉	1	4	yes	complemented
136	C₂²×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
138	#466	—	2	4	no	—
139	#466	C₂²×C₂₉	1	8	yes	complemented
140	C₂×#107	C₂²×C₂₉	1	2	yes	—
141	C₂×#102	—	2	2	no	—
142	#367	C₂²×C₂₉	1	4	yes	complemented
143	#357	C₂²×C₂₉	1	8	yes	complemented
144	C₂×C₂₉	—	4	2	no	—
145	C₂×C₂₉	—	16	4	no	—
146	C₂×C₂₉	#2218	1	4	yes	direct product factor
147	C₂×C₂₉	C₂×#466	1	1	yes	characteristic
148	C₂×C₂₉	C₂×#509	1	2	yes	—
149	C₂²×D₄	—	2	2	no	—
150	C₂²×D₄	—	4	2	no	—
151	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
152	C₂×D₈	—	2	4	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	4	no	—
155	C₂×D₈	C₂³×C₂₉	1	4	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₂₉	1	2	yes	—
160	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
161	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
162	#121	—	2	8	no	—
163	#121	—	4	4	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₂²⋊C₄	—	4	2	no	—
166	D₁₆	—	2	8	no	—
167	M₃₂	—	2	8	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	D₄×C₂₉	1	4	yes	complemented
170	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
171	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
172	#107	—	2	2	no	—
173	#107	C₂³×C₂₉	1	4	yes	—
174	#102	—	2	8	no	—
175	C₂₉	C₂×#2218	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	—	4	2	no	—
177	C₂⁴	D₈×C₂₉	1	1	yes	characteristic
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	4	8	no	—
180	C₂×D₄	—	4	4	no	—
181	C₂×D₄	—	4	2	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂×D₄	—	8	8	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	D₈×C₂₉	1	1	yes	characteristic
188	C₂²×C₄	D₈×C₂₉	1	2	yes	—
189	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
191	D₈	—	2	4	no	—
192	D₈	—	2	4	no	—
193	D₈	—	4	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
201	C₂×C₈	#41×C₂₉	1	2	yes	—
202	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₂²⋊C₄	—	4	8	no	—
206	C₁₆	—	2	2	no	—
207	C₁₆	C₂×D₄×C₂₉	1	4	yes	complemented
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	4	4	no	—
211	C₂³	—	8	4	no	—
212	C₂³	—	8	4	no	—
213	C₂³	—	8	4	no	—
214	C₂³	C₂×D₈×C₂₉	1	2	yes	—
215	C₂³	C₂×D₈×C₂₉	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	D₄	—	4	4	no	—
218	D₄	—	8	16	no	—
219	D₄	—	8	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	C₂₉×#121	1	2	yes	—
227	C₂×C₄	C₂×D₈×C₂₉	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
229	C₂×C₄	C₂₉×#135	1	2	yes	—
230	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	C₂₉×#121	1	2	yes	—
233	C₈	C₂²×D₄×C₂₉	1	1	yes	characteristic
234	C₈	C₂×#41×C₂₉	1	1	yes	characteristic
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	4	no	—
241	C₂²	—	16	8	no	—
242	C₂²	C₂₉×#466	1	2	yes	—
243	C₂²	C₂₉×#509	1	4	yes	—
244	C₂²	C₂²×D₈×C₂₉	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₂₉×#466	1	2	yes	—
248	C₄	C₂×C₂₉×#121	1	1	yes	characteristic
249	C₄	C₂×C₂₉×#135	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	16	4	no	—
252	C₂	C₂₉×#2218	1	4	yes	direct product factor
253	C₂	C₂×C₂₉×#466	1	1	yes	characteristic
254	C₂	C₂×C₂₉×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
LCS	G ▷ C₂×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
1. C₂²×C₂₉×D₁₆	C₂	semidirect
2. C₂²×C₂₉×M₃₂	C₂	semidirect
3. C₂×C₂₉×#466	C₂	semidirect
4. C₂₉×#2218	C₂	direct (N × H)
5. C₂×C₂₉×#367	C₂	semidirect
6. C₂×C₂₉×#357	C₂	semidirect
11. C₂×C₂₉×D₁₆	C₂²	semidirect
13. C₂×C₂₉×M₃₂	C₂²	semidirect
17. C₂₉×#466	C₂²	semidirect
20. C₂₉×#367	C₂²	semidirect
21. C₂₉×#357	C₂²	semidirect
42. C₂×C₁₆×C₂₉	D₄	semidirect
79. C₁₆×C₂₉	C₂×D₄	semidirect
80. C₂×#2218	C₂₉	direct (N × H)
108. C₂²×D₁₆	C₂×C₂₉	semidirect
109. C₂²×M₃₂	C₂×C₂₉	semidirect
110. C₂×#466	C₂×C₂₉	semidirect
111. #2218	C₂×C₂₉	direct (N × H)
112. C₂×#367	C₂×C₂₉	semidirect
113. C₂×#357	C₂×C₂₉	semidirect
133. C₂×D₁₆	C₂²×C₂₉	semidirect
135. C₂×M₃₂	C₂²×C₂₉	semidirect
139. #466	C₂²×C₂₉	semidirect
142. #367	C₂²×C₂₉	semidirect
143. #357	C₂²×C₂₉	semidirect
146. C₂×C₂₉	#2218	direct (N × H)
169. C₂×C₁₆	D₄×C₂₉	semidirect
175. C₂₉	C₂×#2218	direct (N × H)
207. C₁₆	C₂×D₄×C₂₉	semidirect
252. C₂	C₂₉×#2218	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×D₁₆	2
C₂²×C₂₉×M₃₂	2
C₂×C₂₉×#466	2
C₂₉×#2218	2
C₂×C₂₉×#367	2
C₂×C₂₉×#357	2
C₂×#2218	29

Subgroups — C2×C29×#2218 #20328501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#2218 #20328501