Subgroups — C₂₉×#33441 #20328341

Order 7424.

646 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	C₂×C₂₉×#504	C₂	1	2	yes	maximal, complemented
2	C₂₉×#2955	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2171	C₂	1	4	yes	maximal, complemented
4	C₂₉×#2170	C₂	1	4	yes	maximal, complemented
5	C₂₉×#2153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₉×#2117	C₂	1	2	yes	maximal, complemented
7	C₂₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#575	C₄	1	1	yes	characteristic
9	C₂×C₂₉×#131	C₂²	1	2	yes	—
10	C₂₉×#504	—	2	2	no	—
11	C₂₉×#504	—	2	2	no	—
12	C₂₉×#504	C₂²	1	4	yes	complemented
13	C₂₉×#504	C₂²	1	4	yes	complemented
14	C₂×C₂₉×M₃₂	—	2	2	no	—
15	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
16	C₂×C₂₉×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₂₉	—	2	2	no	—
18	C₂²×C₁₆×C₂₉	C₂²	1	1	yes	characteristic
19	C₂₉×#444	C₄	1	4	yes	—
20	C₂₉×#443	C₂²	1	4	yes	—
21	C₂₉×#430	C₄	1	1	yes	characteristic
22	C₂₉×#421	C₄	1	2	yes	—
23	C₂₉×#405	C₂²	1	1	yes	characteristic
24	C₂₉×#364	C₂²	1	4	yes	complemented
25	C₂₉×#350	C₂²	1	8	yes	complemented
26	C₂₉×#346	C₂²	1	2	yes	complemented
27	C₄×C₁₆×C₂₉	C₂²	1	2	yes	complemented
28	C₂×#41×C₂₉	C₂×C₄	1	2	yes	—
29	C₂₉×#137	C₈	1	4	yes	—
30	C₂₉×AGL(1,ℤ/8)	C₈	1	4	yes	—
31	C₂₉×#135	C₂×C₄	1	4	yes	—
32	C₂₉×#131	—	2	2	no	—
33	C₂₉×#131	—	2	2	no	—
34	C₂₉×#131	C₂×C₄	1	4	yes	—
35	C₂₉×#131	C₂³	1	4	yes	—
36	C₂×M₁₆×C₂₉	—	2	2	no	—
37	C₂×M₁₆×C₂₉	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₂₉	—	2	2	no	—
40	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
41	C₂₉×#117	C₂×C₄	1	1	yes	characteristic
42	C₂₉×M₃₂	—	2	2	no	—
43	C₂₉×M₃₂	—	2	2	no	—
44	C₂₉×M₃₂	—	2	4	no	—
45	C₂₉×M₃₂	—	2	4	no	—
46	C₂₉×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆×C₂₉	—	2	2	no	—
48	C₂×C₁₆×C₂₉	—	2	4	no	—
49	C₂×C₁₆×C₂₉	—	2	4	no	—
50	C₂×C₁₆×C₂₉	D₄	1	2	yes	—
51	C₂×C₁₆×C₂₉	D₄	1	2	yes	—
52	C₂×C₁₆×C₂₉	C₂³	1	2	yes	—
53	C₂×C₁₆×C₂₉	C₂³	1	1	yes	characteristic
54	C₂×C₁₆×C₂₉	C₂³	1	1	yes	characteristic
55	C₂₉×#108	C₂×C₄	1	4	yes	—
56	C₂₉×#104	C₂×C₄	1	8	yes	—
57	C₂₉×#97	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₂₉	C₂³	1	2	yes	—
59	#41×C₂₉	—	2	2	no	—
60	#41×C₂₉	—	2	2	no	—
61	#41×C₂₉	C₂×C₈	1	4	yes	—
62	#41×C₂₉	C₂²×C₄	1	4	yes	—
63	C₂×Q₈×C₂₉	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₂₉	—	2	2	no	—
65	C₂×D₄×C₂₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₂₉	—	2	2	no	—
67	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
68	Q₁₆×C₂₉	C₂×C₈	1	4	yes	—
69	SD₁₆×C₂₉	C₂×C₈	1	8	yes	—
70	D₈×C₂₉	C₂×C₈	1	4	yes	—
71	M₁₆×C₂₉	—	2	2	no	—
72	M₁₆×C₂₉	—	2	4	no	—
73	M₁₆×C₂₉	—	2	4	no	—
74	M₁₆×C₂₉	C₂×C₈	1	4	yes	—
75	M₁₆×C₂₉	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₂₉	—	2	2	no	—
77	C₂×C₈×C₂₉	—	2	4	no	—
78	C₂×C₈×C₂₉	—	2	4	no	—
79	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₂₉	#41	1	2	yes	—
83	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₂₉	—	2	1	no	—
85	C₂²⋊C₄×C₂₉	—	2	1	no	—
86	C₄²×C₂₉	C₂²×C₄	1	2	yes	—
87	C₁₆×C₂₉	—	2	4	no	—
88	C₁₆×C₂₉	—	2	2	no	—
89	C₁₆×C₂₉	C₂×D₄	1	2	yes	—
90	C₁₆×C₂₉	C₂×D₄	1	1	yes	characteristic
91	C₁₆×C₂₉	C₂×D₄	1	1	yes	characteristic
92	#33441	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₉	—	2	2	no	—
94	C₂³×C₂₉	#131	1	1	yes	characteristic
95	Q₈×C₂₉	—	2	2	no	—
96	Q₈×C₂₉	C₂²×C₈	1	4	yes	—
97	D₄×C₂₉	—	2	4	no	—
98	D₄×C₂₉	—	2	4	no	—
99	D₄×C₂₉	—	2	2	no	—
100	D₄×C₂₉	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₂₉	—	2	2	no	—
102	C₂×C₄×C₂₉	—	2	4	no	—
103	C₂×C₄×C₂₉	—	2	4	no	—
104	C₂×C₄×C₂₉	—	2	2	no	—
105	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₂₉	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₂₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₂₉	—	2	4	no	—
110	C₈×C₂₉	C₄×D₄	1	2	yes	—
111	C₈×C₂₉	C₂²×C₈	1	4	yes	—
112	C₈×C₂₉	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₂₉	C₂×#41	1	1	yes	characteristic
114	C₂×#504	C₂×C₂₉	1	2	yes	complemented
115	#2955	C₂×C₂₉	1	1	yes	characteristic
116	#2171	C₂×C₂₉	1	4	yes	complemented
117	#2170	C₂×C₂₉	1	4	yes	complemented
118	#2153	C₂×C₂₉	1	1	yes	characteristic, complemented
119	#2117	C₂×C₂₉	1	2	yes	complemented
120	#2108	C₂×C₂₉	1	1	yes	characteristic, complemented
121	C₂²×C₂₉	—	2	4	no	—
122	C₂²×C₂₉	—	4	1	no	—
123	C₂²×C₂₉	—	4	4	no	—
124	C₂²×C₂₉	C₈×D₄	1	2	yes	—
125	C₂²×C₂₉	C₂×#131	1	1	yes	characteristic
126	C₄×C₂₉	—	2	4	no	—
127	C₄×C₂₉	—	4	2	no	—
128	C₄×C₂₉	C₈×D₄	1	2	yes	—
129	C₄×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₉	C₂³×C₈	1	1	yes	characteristic
131	#575	C₄×C₂₉	1	1	yes	characteristic
132	C₂×#131	C₂²×C₂₉	1	2	yes	—
133	#504	—	2	2	no	—
134	#504	—	2	2	no	—
135	#504	C₂²×C₂₉	1	4	yes	complemented
136	#504	C₂²×C₂₉	1	4	yes	complemented
137	C₂×M₃₂	—	2	2	no	—
138	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
139	C₂×M₃₂	C₂²×C₂₉	1	1	yes	characteristic
140	C₂²×C₁₆	—	2	2	no	—
141	C₂²×C₁₆	C₂²×C₂₉	1	1	yes	characteristic
142	#444	C₄×C₂₉	1	4	yes	—
143	#443	C₂²×C₂₉	1	4	yes	—
144	#430	C₄×C₂₉	1	1	yes	characteristic
145	#421	C₄×C₂₉	1	2	yes	—
146	#405	C₂²×C₂₉	1	1	yes	characteristic
147	#364	C₂²×C₂₉	1	4	yes	complemented
148	#350	C₂²×C₂₉	1	8	yes	complemented
149	#346	C₂²×C₂₉	1	2	yes	complemented
150	C₄×C₁₆	C₂²×C₂₉	1	2	yes	complemented
151	C₂×C₂₉	—	2	1	no	—
152	C₂×C₂₉	—	2	2	no	—
153	C₂×C₂₉	—	4	4	no	—
154	C₂×C₂₉	C₂×C₈×D₄	1	1	yes	characteristic
155	C₂×#41	C₂×C₄×C₂₉	1	2	yes	—
156	#137	C₈×C₂₉	1	4	yes	—
157	AGL(1,ℤ/8)	C₈×C₂₉	1	4	yes	—
158	#135	C₂×C₄×C₂₉	1	4	yes	—
159	#131	—	2	2	no	—
160	#131	—	2	2	no	—
161	#131	C₂×C₄×C₂₉	1	4	yes	—
162	#131	C₂³×C₂₉	1	4	yes	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	C₂×C₄×C₂₉	1	2	yes	—
165	C₂×M₁₆	C₂×C₄×C₂₉	1	1	yes	characteristic
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
168	#117	C₂×C₄×C₂₉	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	2	no	—
171	M₃₂	—	2	4	no	—
172	M₃₂	—	2	4	no	—
173	M₃₂	C₂³×C₂₉	1	4	yes	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	4	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	D₄×C₂₉	1	2	yes	—
178	C₂×C₁₆	D₄×C₂₉	1	2	yes	—
179	C₂×C₁₆	C₂³×C₂₉	1	2	yes	—
180	C₂×C₁₆	C₂³×C₂₉	1	1	yes	characteristic
181	C₂×C₁₆	C₂³×C₂₉	1	1	yes	characteristic
182	#108	C₂×C₄×C₂₉	1	4	yes	—
183	#104	C₂×C₄×C₂₉	1	8	yes	—
184	#97	C₂×C₄×C₂₉	1	2	yes	—
185	C₄×C₈	C₂³×C₂₉	1	2	yes	—
186	C₂₉	#33441	1	1	yes	characteristic, direct product factor, Sylow
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×C₈×C₂₉	1	4	yes	—
190	#41	C₂²×C₄×C₂₉	1	4	yes	—
191	C₂×Q₈	C₂×C₈×C₂₉	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	C₂×C₈×C₂₉	1	2	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
196	Q₁₆	C₂×C₈×C₂₉	1	4	yes	—
197	SD₁₆	C₂×C₈×C₂₉	1	8	yes	—
198	D₈	C₂×C₈×C₂₉	1	4	yes	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	C₂×C₈×C₂₉	1	4	yes	—
203	M₁₆	C₂²×C₄×C₂₉	1	4	yes	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₂₉	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
210	C₂×C₈	#41×C₂₉	1	2	yes	—
211	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂²×C₄×C₂₉	1	2	yes	—
215	C₁₆	—	2	4	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×D₄×C₂₉	1	1	yes	characteristic
218	C₁₆	C₂×D₄×C₂₉	1	2	yes	—
219	C₁₆	C₂×D₄×C₂₉	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	C₂₉×#131	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	Q₈	C₂²×C₈×C₂₉	1	4	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂²×C₈×C₂₉	1	4	yes	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
233	C₂×C₄	C₂²×C₈×C₂₉	1	2	yes	—
234	C₂×C₄	C₂₉×#131	1	1	yes	characteristic
235	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
236	C₈	—	2	4	no	—
237	C₈	C₄×D₄×C₂₉	1	2	yes	—
238	C₈	C₂²×C₈×C₂₉	1	4	yes	—
239	C₈	C₂²×D₄×C₂₉	1	1	yes	characteristic
240	C₈	C₂×#41×C₂₉	1	1	yes	characteristic
241	C₂²	—	2	4	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	C₈×D₄×C₂₉	1	2	yes	—
245	C₂²	C₂×C₂₉×#131	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₈×D₄×C₂₉	1	2	yes	—
249	C₄	C₂×C₄×D₄×C₂₉	1	1	yes	characteristic
250	C₄	C₂³×C₈×C₂₉	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₈×D₄×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₉	C₂×D₄
commutator	C₄	C₂³×C₈×C₂₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
socle	C₂×C₂₉	C₂×C₈×D₄
prime radical	C₂₉×#575	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₉ ◁ C₂²×C₁₆×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₉×#504	C₂	semidirect
3. C₂₉×#2171	C₂	semidirect
4. C₂₉×#2170	C₂	semidirect
5. C₂₉×#2153	C₂	semidirect
6. C₂₉×#2117	C₂	semidirect
7. C₂₉×#2108	C₂	semidirect
12. C₂₉×#504	C₂²	semidirect
13. C₂₉×#504	C₂²	semidirect
24. C₂₉×#364	C₂²	semidirect
25. C₂₉×#350	C₂²	semidirect
26. C₂₉×#346	C₂²	semidirect
27. C₄×C₁₆×C₂₉	C₂²	semidirect
92. #33441	C₂₉	direct (N × H)
114. C₂×#504	C₂×C₂₉	semidirect
116. #2171	C₂×C₂₉	semidirect
117. #2170	C₂×C₂₉	semidirect
118. #2153	C₂×C₂₉	semidirect
119. #2117	C₂×C₂₉	semidirect
120. #2108	C₂×C₂₉	semidirect
135. #504	C₂²×C₂₉	semidirect
136. #504	C₂²×C₂₉	semidirect
147. #364	C₂²×C₂₉	semidirect
148. #350	C₂²×C₂₉	semidirect
149. #346	C₂²×C₂₉	semidirect
150. C₄×C₁₆	C₂²×C₂₉	semidirect
186. C₂₉	#33441	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#504	2
C₂₉×#2955	2
C₂₉×#2171	2
C₂₉×#2170	2
C₂₉×#2153	2
C₂₉×#2117	2
C₂₉×#2108	2
#33441	29

Subgroups — C29×#33441 #20328341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#33441 #20328341