Subgroups — C₂₉×#2318 #16816032

Order 3712.

568 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#388	C₂	1	2	yes	maximal, complemented
3	C₂₉×#387	C₂	1	2	yes	maximal, complemented
4	C₂₉×#386	C₂	1	2	yes	maximal
5	C₂₉×#385	C₂	1	2	yes	maximal, complemented
6	C₂₉×#383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₉×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₉	C₄	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×#95	C₂²	1	1	yes	characteristic
28	C₂₉×#95	C₂²	1	1	yes	characteristic
29	C₂₉×#95	C₂²	1	2	yes	—
30	C₂₉×#95	C₂²	1	1	yes	characteristic
31	C₂₉×#95	C₂²	1	1	yes	characteristic
32	C₂₉×#95	C₂²	1	2	yes	—
33	C₂₉×#95	C₂²	1	2	yes	—
34	C₂₉×#95	C₂²	1	2	yes	—
35	C₂₉×#95	C₂²	1	2	yes	—
36	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
37	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₂₉	—	2	4	no	—
39	C₂×D₄×C₂₉	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₂₉	—	2	4	no	—
41	C₂²×C₄×C₂₉	—	4	2	no	—
42	C₂²×C₄×C₂₉	—	4	2	no	—
43	C₂²×C₄×C₂₉	—	4	2	no	—
44	C₂²×C₄×C₂₉	—	4	2	no	—
45	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₂₉	C₂×C₄	1	4	yes	complemented
47	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₉	—	4	2	no	—
57	C₄⋊C₄×C₂₉	—	4	2	no	—
58	C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₂₉	—	2	4	no	—
60	C₂²⋊C₄×C₂₉	—	4	2	no	—
61	C₂²⋊C₄×C₂₉	—	4	2	no	—
62	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
63	C₂²⋊C₄×C₂₉	C₂×C₄	1	4	yes	complemented
64	C₄²×C₂₉	—	4	2	no	—
65	C₄²×C₂₉	C₂×C₄	1	4	yes	complemented
66	C₂³×C₂₉	—	2	2	no	—
67	C₂³×C₂₉	—	2	1	no	—
68	C₂³×C₂₉	—	2	1	no	—
69	C₂³×C₂₉	—	2	2	no	—
70	C₂³×C₂₉	—	2	1	no	—
71	C₂³×C₂₉	—	2	2	no	—
72	C₂³×C₂₉	—	2	1	no	—
73	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₉	#41	1	4	yes	—
76	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
77	D₄×C₂₉	—	2	8	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	2	no	—
80	C₂×C₄×C₂₉	—	2	1	no	—
81	C₂×C₄×C₂₉	—	2	2	no	—
82	C₂×C₄×C₂₉	—	2	2	no	—
83	C₂×C₄×C₂₉	—	2	1	no	—
84	C₂×C₄×C₂₉	—	2	2	no	—
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	—	2	2	no	—
87	C₂×C₄×C₂₉	—	2	2	no	—
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	2	no	—
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	—	2	2	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	—	4	4	no	—
97	C₂×C₄×C₂₉	—	4	4	no	—
98	C₂×C₄×C₂₉	—	4	8	no	—
99	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
104	#2318	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₂₉	—	2	2	no	—
106	C₂²×C₂₉	—	2	2	no	—
107	C₂²×C₂₉	—	2	4	no	—
108	C₂²×C₂₉	—	2	4	no	—
109	C₂²×C₂₉	—	2	2	no	—
110	C₂²×C₂₉	—	2	2	no	—
111	C₂²×C₂₉	—	2	2	no	—
112	C₂²×C₂₉	—	4	2	no	—
113	C₂²×C₂₉	—	4	2	no	—
114	C₂²×C₂₉	#117	1	4	yes	—
115	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₂₉	C₂×#41	1	2	yes	—
117	C₂²×C₂₉	H(5,𝔽₂)	1	2	yes	—
118	C₂²×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
120	C₄×C₂₉	—	2	2	no	—
121	C₄×C₂₉	—	2	2	no	—
122	C₄×C₂₉	—	2	2	no	—
123	C₄×C₂₉	—	2	2	no	—
124	C₄×C₂₉	—	2	2	no	—
125	C₄×C₂₉	—	4	2	no	—
126	C₄×C₂₉	—	4	2	no	—
127	C₄×C₂₉	—	4	2	no	—
128	C₄×C₂₉	—	4	2	no	—
129	C₂×C₄×D₄	C₂×C₂₉	1	1	yes	characteristic
130	#388	C₂×C₂₉	1	2	yes	complemented
131	#387	C₂×C₂₉	1	2	yes	complemented
132	#386	C₂×C₂₉	1	2	yes	—
133	#385	C₂×C₂₉	1	2	yes	complemented
134	#383	C₂×C₂₉	1	1	yes	characteristic, complemented
135	#382	C₂×C₂₉	1	1	yes	characteristic, complemented
136	#381	C₂×C₂₉	1	1	yes	characteristic, complemented
137	#380	C₂×C₂₉	1	1	yes	characteristic, complemented
138	C₄×C₂²⋊C₄	C₂×C₂₉	1	1	yes	characteristic, complemented
139	C₂×#95	C₂×C₂₉	1	1	yes	characteristic, complemented
140	C₂×C₂₉	—	2	4	no	—
141	C₂×C₂₉	—	4	2	no	—
142	C₂×C₂₉	C₂×#117	1	1	yes	characteristic
143	C₂×C₂₉	#518	1	1	yes	characteristic
144	C₂×C₂₉	#520	1	1	yes	characteristic
145	C₂×C₂₉	#538	1	2	yes	—
146	C₂×C₂₉	#539	1	2	yes	—
147	C₂²×D₄	C₂²×C₂₉	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₂₉	1	2	yes	—
149	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₂₉	1	1	yes	characteristic
151	C₄×D₄	C₄×C₂₉	1	8	yes	complemented
152	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
153	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
155	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
156	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₉	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
159	C₂×C₂²⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
160	C₂×C₄²	C₂²×C₂₉	1	2	yes	—
161	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
162	#95	C₂²×C₂₉	1	1	yes	characteristic
163	#95	C₂²×C₂₉	1	1	yes	characteristic
164	#95	C₂²×C₂₉	1	2	yes	—
165	#95	C₂²×C₂₉	1	1	yes	characteristic
166	#95	C₂²×C₂₉	1	1	yes	characteristic
167	#95	C₂²×C₂₉	1	2	yes	—
168	#95	C₂²×C₂₉	1	2	yes	—
169	#95	C₂²×C₂₉	1	2	yes	—
170	#95	C₂²×C₂₉	1	2	yes	—
171	C₂₉	#2318	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₂³×C₂₉	1	1	yes	characteristic
173	C₂⁴	C₂³×C₂₉	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂×C₄×C₂₉	1	4	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₄×C₂₉	1	4	yes	—
182	C₂²×C₄	C₂×C₄×C₂₉	1	4	yes	complemented
183	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
185	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
186	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	complemented
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
199	C₂²⋊C₄	C₂×C₄×C₂₉	1	4	yes	complemented
200	C₄²	—	4	2	no	—
201	C₄²	C₂×C₄×C₂₉	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂²×C₄×C₂₉	1	2	yes	—
210	C₂³	C₂²×C₄×C₂₉	1	2	yes	—
211	C₂³	#41×C₂₉	1	4	yes	—
212	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
213	D₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
238	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
239	C₂×C₄	C₂²×C₄×C₂₉	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	C₂₉×#117	1	4	yes	—
250	C₂²	C₂³×C₄×C₂₉	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₉	1	2	yes	—
252	C₂²	C₂₉×H(5,𝔽₂)	1	2	yes	—
253	C₂²	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₂	—	2	4	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×C₂₉×#117	1	1	yes	characteristic
267	C₂	C₂₉×#518	1	1	yes	characteristic
268	C₂	C₂₉×#520	1	1	yes	characteristic
269	C₂	C₂₉×#538	1	2	yes	—
270	C₂	C₂₉×#539	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂₉×#388	C₂	semidirect
3. C₂₉×#387	C₂	semidirect
5. C₂₉×#385	C₂	semidirect
6. C₂₉×#383	C₂	semidirect
7. C₂₉×#382	C₂	semidirect
8. C₂₉×#381	C₂	semidirect
9. C₂₉×#380	C₂	semidirect
10. C₄×C₂²⋊C₄×C₂₉	C₂	semidirect
11. C₂×C₂₉×#95	C₂	semidirect
16. C₄×D₄×C₂₉	C₄	semidirect
46. C₂²×C₄×C₂₉	C₂×C₄	semidirect
58. C₄⋊C₄×C₂₉	C₂×C₄	semidirect
63. C₂²⋊C₄×C₂₉	C₂×C₄	semidirect
65. C₄²×C₂₉	C₂×C₄	semidirect
104. #2318	C₂₉	direct (N × H)
130. #388	C₂×C₂₉	semidirect
131. #387	C₂×C₂₉	semidirect
133. #385	C₂×C₂₉	semidirect
134. #383	C₂×C₂₉	semidirect
135. #382	C₂×C₂₉	semidirect
136. #381	C₂×C₂₉	semidirect
137. #380	C₂×C₂₉	semidirect
138. C₄×C₂²⋊C₄	C₂×C₂₉	semidirect
139. C₂×#95	C₂×C₂₉	semidirect
151. C₄×D₄	C₄×C₂₉	semidirect
171. C₂₉	#2318	direct (N × H)
182. C₂²×C₄	C₂×C₄×C₂₉	semidirect
194. C₄⋊C₄	C₂×C₄×C₂₉	semidirect
199. C₂²⋊C₄	C₂×C₄×C₂₉	semidirect
201. C₄²	C₂×C₄×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₉	2
C₂₉×#388	2
C₂₉×#387	2
C₂₉×#386	2
C₂₉×#385	2
C₂₉×#383	2
C₂₉×#382	2
C₂₉×#381	2
C₂₉×#380	2
C₄×C₂²⋊C₄×C₂₉	2
C₂×C₂₉×#95	2
#2318	29

Subgroups — C29×#2318 #16816032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#2318 #16816032