Subgroups — C₂₉×#21708 #20316608

Order 7424.

1006 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#443	C₂	1	4	yes	maximal, complemented
3	C₂₉×#2931	C₂	1	2	yes	maximal, complemented
4	C₂₉×#1975	C₂	1	4	yes	maximal, complemented
5	C₂₉×#1963	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#135	C₂²	1	4	yes	—
9	C₂×C₂₉×#131	C₂²	1	4	yes	—
10	C₂₉×#532	C₂²	1	2	yes	—
11	C₂₉×#497	C₄	1	4	yes	complemented
12	C₂₉×#496	C₄	1	4	yes	complemented
13	C₂₉×#443	—	2	16	no	—
14	C₂₉×#436	—	2	4	no	—
15	C₈×D₄×C₂₉	—	2	4	no	—
16	C₂×C₂₉×#108	C₂²	1	4	yes	—
17	C₂₉×#423	C₂²	1	2	yes	—
18	C₂×C₂₉×#104	C₂²	1	8	yes	—
19	C₂₉×#408	C₂²	1	4	yes	—
20	C₂×C₂₉×#97	C₂²	1	2	yes	—
21	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×#376	C₂²	1	1	yes	characteristic
25	C₂×#41×C₂₉	C₂³	1	4	yes	—
26	C₂₉×#135	—	2	8	no	—
27	C₂₉×#135	—	2	8	no	—
28	C₂×Q₁₆×C₂₉	C₂×C₄	1	4	yes	—
29	C₂×SD₁₆×C₂₉	C₂×C₄	1	8	yes	—
30	C₂×D₈×C₂₉	C₂×C₄	1	4	yes	—
31	C₂₉×#131	—	2	16	no	—
32	C₂×M₁₆×C₂₉	—	2	4	no	—
33	C₂×M₁₆×C₂₉	C₂³	1	4	yes	—
34	C₂²×C₈×C₂₉	—	2	4	no	—
35	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
38	C₂₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₂₉×#127	C₂×C₄	1	2	yes	—
40	C₂₉×#124	C₂×C₄	1	4	yes	—
41	C₂₉×#121	—	2	4	no	—
42	C₄×D₄×C₂₉	—	2	4	no	—
43	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₂₉×#108	—	2	8	no	—
46	C₂₉×C₄⋊C₈	—	2	4	no	—
47	C₂₉×#104	—	2	16	no	—
48	C₂₉×C₂²⋊C₈	—	2	8	no	—
49	C₂₉×#97	—	2	4	no	—
50	C₂₉×#97	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₂₉	D₄	1	2	yes	—
52	C₄×C₈×C₂₉	D₄	1	2	yes	—
53	C₄×C₈×C₂₉	D₄	1	8	yes	complemented
54	C₂₉×#95	—	2	2	no	—
55	#41×C₂₉	—	2	8	no	—
56	#41×C₂₉	—	2	8	no	—
57	C₂×Q₈×C₂₉	C₂²×C₄	1	4	yes	—
58	C₂×D₄×C₂₉	—	2	4	no	—
59	C₂×D₄×C₂₉	—	2	4	no	—
60	C₂×D₄×C₂₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₂₉	—	2	2	no	—
62	C₂²×C₄×C₂₉	—	2	4	no	—
63	C₂²×C₄×C₂₉	#41	1	2	yes	—
64	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
65	Q₁₆×C₂₉	—	2	8	no	—
66	SD₁₆×C₂₉	—	2	16	no	—
67	D₈×C₂₉	—	2	8	no	—
68	M₁₆×C₂₉	—	2	8	no	—
69	M₁₆×C₂₉	—	4	8	no	—
70	C₂×C₈×C₂₉	—	2	2	no	—
71	C₂×C₈×C₂₉	—	2	4	no	—
72	C₂×C₈×C₂₉	—	4	8	no	—
73	C₂×C₈×C₂₉	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₉	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₂₉	#41	1	4	yes	—
80	C₄⋊C₄×C₂₉	—	2	4	no	—
81	C₂²⋊C₄×C₂₉	—	2	8	no	—
82	C₄²×C₂₉	—	2	2	no	—
83	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
84	C₄²×C₂₉	C₂×D₄	1	4	yes	—
85	C₄²×C₂₉	C₂×D₄	1	2	yes	—
86	#21708	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₉	—	2	4	no	—
88	C₂³×C₂₉	C₂×#41	1	1	yes	characteristic
89	Q₈×C₂₉	—	2	8	no	—
90	D₄×C₂₉	—	2	8	no	—
91	D₄×C₂₉	—	4	8	no	—
92	D₄×C₂₉	—	4	8	no	—
93	C₂×C₄×C₂₉	—	2	4	no	—
94	C₂×C₄×C₂₉	—	2	4	no	—
95	C₂×C₄×C₂₉	—	4	2	no	—
96	C₂×C₄×C₂₉	—	4	4	no	—
97	C₂×C₄×C₂₉	—	4	4	no	—
98	C₂×C₄×C₂₉	—	4	4	no	—
99	C₂×C₄×C₂₉	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
102	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
104	C₈×C₂₉	—	2	2	no	—
105	C₈×C₂₉	—	4	4	no	—
106	C₈×C₂₉	C₄×D₄	1	8	yes	—
107	C₈×C₂₉	#127	1	2	yes	—
108	C₈×C₂₉	#127	1	2	yes	—
109	#3150	C₂×C₂₉	1	1	yes	characteristic
110	C₂×#443	C₂×C₂₉	1	4	yes	complemented
111	#2931	C₂×C₂₉	1	2	yes	complemented
112	#1975	C₂×C₂₉	1	4	yes	complemented
113	#1963	C₂×C₂₉	1	2	yes	complemented
114	#1849	C₂×C₂₉	1	1	yes	characteristic, complemented
115	#1726	C₂×C₂₉	1	1	yes	characteristic, complemented
116	C₂²×C₂₉	—	2	2	no	—
117	C₂²×C₂₉	—	4	4	no	—
118	C₂²×C₂₉	—	4	8	no	—
119	C₂²×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₂₉	#532	1	2	yes	—
121	C₄×C₂₉	—	2	4	no	—
122	C₄×C₂₉	—	4	4	no	—
123	C₄×C₂₉	—	4	4	no	—
124	C₄×C₂₉	C₂×C₄×D₄	1	2	yes	—
125	C₄×C₂₉	C₂×#127	1	1	yes	characteristic
126	C₄×C₂₉	#532	1	1	yes	characteristic
127	C₂×#135	C₂²×C₂₉	1	4	yes	—
128	C₂×#131	C₂²×C₂₉	1	4	yes	—
129	#532	C₂²×C₂₉	1	2	yes	—
130	#497	C₄×C₂₉	1	4	yes	complemented
131	#496	C₄×C₂₉	1	4	yes	complemented
132	#443	—	2	16	no	—
133	#436	—	2	4	no	—
134	C₈×D₄	—	2	4	no	—
135	C₂×#108	C₂²×C₂₉	1	4	yes	—
136	#423	C₂²×C₂₉	1	2	yes	—
137	C₂×#104	C₂²×C₂₉	1	8	yes	—
138	#408	C₂²×C₂₉	1	4	yes	—
139	C₂×#97	C₂²×C₂₉	1	2	yes	—
140	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
143	#376	C₂²×C₂₉	1	1	yes	characteristic
144	C₂×C₂₉	—	2	2	no	—
145	C₂×C₂₉	—	8	4	no	—
146	C₂×C₂₉	C₄×#127	1	1	yes	characteristic
147	C₂×C₂₉	#2955	1	2	yes	—
148	C₂×#41	C₂³×C₂₉	1	4	yes	—
149	#135	—	2	8	no	—
150	#135	—	2	8	no	—
151	C₂×Q₁₆	C₂×C₄×C₂₉	1	4	yes	—
152	C₂×SD₁₆	C₂×C₄×C₂₉	1	8	yes	—
153	C₂×D₈	C₂×C₄×C₂₉	1	4	yes	—
154	#131	—	2	16	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	C₂³×C₂₉	1	4	yes	—
157	C₂²×C₈	—	2	4	no	—
158	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
160	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
161	C₄⋊Q₈	C₂×C₄×C₂₉	1	2	yes	—
162	#127	C₂×C₄×C₂₉	1	2	yes	—
163	#124	C₂×C₄×C₂₉	1	4	yes	—
164	#121	—	2	4	no	—
165	C₄×D₄	—	2	4	no	—
166	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
167	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
168	#108	—	2	8	no	—
169	C₄⋊C₈	—	2	4	no	—
170	#104	—	2	16	no	—
171	C₂²⋊C₈	—	2	8	no	—
172	#97	—	2	4	no	—
173	#97	C₂×C₄×C₂₉	1	4	yes	—
174	C₄×C₈	D₄×C₂₉	1	2	yes	—
175	C₄×C₈	D₄×C₂₉	1	2	yes	—
176	C₄×C₈	D₄×C₂₉	1	8	yes	complemented
177	#95	—	2	2	no	—
178	C₂₉	#21708	1	1	yes	characteristic, direct product factor, Sylow
179	#41	—	2	8	no	—
180	#41	—	2	8	no	—
181	C₂×Q₈	C₂²×C₄×C₂₉	1	4	yes	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	C₂²×C₄×C₂₉	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	#41×C₂₉	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
189	Q₁₆	—	2	8	no	—
190	SD₁₆	—	2	16	no	—
191	D₈	—	2	8	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	—	4	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×C₄×C₂₉	1	4	yes	—
198	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
200	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₂₉	1	4	yes	—
203	C₂×C₈	#41×C₂₉	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
208	C₄²	C₂×D₄×C₂₉	1	4	yes	—
209	C₄²	C₂×D₄×C₂₉	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	C₂×#41×C₂₉	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	D₄	—	2	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×D₄×C₂₉	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
225	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
226	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	4	4	no	—
229	C₈	C₄×D₄×C₂₉	1	8	yes	—
230	C₈	C₂₉×#127	1	2	yes	—
231	C₈	C₂₉×#127	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
236	C₂²	C₂₉×#532	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	C₂×C₄×D₄×C₂₉	1	2	yes	—
241	C₄	C₂×C₂₉×#127	1	1	yes	characteristic
242	C₄	C₂₉×#532	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	C₄×C₂₉×#127	1	1	yes	characteristic
246	C₂	C₂₉×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₂₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle	C₂²×C₂₉	C₂×C₄×D₄
prime radical	C₂₉×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#443	C₂	semidirect
3. C₂₉×#2931	C₂	semidirect
4. C₂₉×#1975	C₂	semidirect
5. C₂₉×#1963	C₂	semidirect
6. C₂₉×#1849	C₂	semidirect
7. C₂₉×#1726	C₂	semidirect
11. C₂₉×#497	C₄	semidirect
12. C₂₉×#496	C₄	semidirect
53. C₄×C₈×C₂₉	D₄	semidirect
86. #21708	C₂₉	direct (N × H)
110. C₂×#443	C₂×C₂₉	semidirect
111. #2931	C₂×C₂₉	semidirect
112. #1975	C₂×C₂₉	semidirect
113. #1963	C₂×C₂₉	semidirect
114. #1849	C₂×C₂₉	semidirect
115. #1726	C₂×C₂₉	semidirect
130. #497	C₄×C₂₉	semidirect
131. #496	C₄×C₂₉	semidirect
176. C₄×C₈	D₄×C₂₉	semidirect
178. C₂₉	#21708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₉×#3150	2
C₂×C₂₉×#443	2
C₂₉×#2931	2
C₂₉×#1975	2
C₂₉×#1963	2
C₂₉×#1849	2
C₂₉×#1726	2
#21708	29

Subgroups — C29×#21708 #20316608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#21708 #20316608