Subgroups — C₂₉×#3099 #16816813

Order 3712.

398 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#537	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#485	C₂	1	2	yes	maximal, complemented
4	C₂₉×#483	C₂	1	2	yes	maximal, complemented
5	C₂₉×#482	C₂	1	2	yes	maximal, complemented
6	C₂₉×#481	C₂	1	2	yes	maximal, complemented
7	C₂₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₉×#417	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₉×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
16	C₂₉×C₂²⋊Q₈	—	2	1	no	—
17	C₂₉×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂₉×#121	—	2	1	no	—
19	C₂₉×#121	—	2	2	no	—
20	C₂₉×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₉	—	2	1	no	—
22	C₄×D₄×C₂₉	—	2	2	no	—
23	C₄×D₄×C₂₉	—	2	1	no	—
24	C₂₉×#117	C₂²	1	1	yes	characteristic
25	C₂₉×#117	C₂²	1	2	yes	—
26	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
29	C₂₉×#107	C₂²	1	2	yes	—
30	C₂₉×#106	C₂²	1	2	yes	—
31	C₂₉×#106	C₂²	1	2	yes	—
32	C₂₉×#106	C₂²	1	2	yes	—
33	C₂₉×#103	C₂²	1	1	yes	characteristic
34	C₂₉×#103	C₂²	1	2	yes	—
35	C₂₉×#103	C₂²	1	1	yes	characteristic
36	C₂₉×#102	C₂²	1	1	yes	characteristic
37	C₂₉×#102	C₂²	1	2	yes	—
38	C₂₉×#102	C₂²	1	1	yes	characteristic
39	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	—
41	#41×C₂₉	—	2	2	no	—
42	#41×C₂₉	—	2	2	no	—
43	#41×C₂₉	—	2	2	no	—
44	#41×C₂₉	—	2	2	no	—
45	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
46	C₂×Q₈×C₂₉	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₂₉	—	2	1	no	—
48	C₂×D₄×C₂₉	—	2	2	no	—
49	C₂×D₄×C₂₉	D₄	1	2	yes	—
50	C₂×D₄×C₂₉	D₄	1	1	yes	characteristic
51	C₂×D₄×C₂₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₉	—	2	2	no	—
53	C₂²×C₄×C₂₉	—	2	1	no	—
54	C₂²×C₄×C₂₉	—	2	1	no	—
55	C₂²×C₄×C₂₉	—	2	1	no	—
56	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₂₉	—	2	2	no	—
59	C₂×C₈×C₂₉	—	2	2	no	—
60	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₂₉	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₉	—	2	2	no	—
64	C₄⋊C₄×C₂₉	—	2	1	no	—
65	C₄⋊C₄×C₂₉	—	2	1	no	—
66	C₄⋊C₄×C₂₉	—	2	2	no	—
67	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
70	C₄⋊C₄×C₂₉	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₉	—	2	2	no	—
72	C₂²⋊C₄×C₂₉	—	2	1	no	—
73	C₂²⋊C₄×C₂₉	—	2	2	no	—
74	C₂²⋊C₄×C₂₉	—	2	2	no	—
75	C₄²×C₂₉	—	2	2	no	—
76	C₄²×C₂₉	—	2	1	no	—
77	C₂³×C₂₉	—	2	1	no	—
78	C₂³×C₂₉	C₂×D₄	1	2	yes	—
79	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₂₉	—	2	2	no	—
81	Q₈×C₂₉	—	2	1	no	—
82	Q₈×C₂₉	—	2	1	no	—
83	D₄×C₂₉	—	2	4	no	—
84	D₄×C₂₉	—	2	2	no	—
85	D₄×C₂₉	—	2	1	no	—
86	D₄×C₂₉	—	2	1	no	—
87	D₄×C₂₉	—	4	2	no	—
88	D₄×C₂₉	—	8	2	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	2	no	—
91	C₂×C₄×C₂₉	—	2	1	no	—
92	C₂×C₄×C₂₉	—	2	2	no	—
93	C₂×C₄×C₂₉	—	2	1	no	—
94	C₂×C₄×C₂₉	—	2	2	no	—
95	C₂×C₄×C₂₉	—	2	1	no	—
96	C₂×C₄×C₂₉	—	4	2	no	—
97	C₂×C₄×C₂₉	—	8	1	no	—
98	C₂×C₄×C₂₉	—	8	2	no	—
99	C₂×C₄×C₂₉	—	8	1	no	—
100	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₉	#41	1	2	yes	—
103	C₂×C₄×C₂₉	#41	1	2	yes	—
104	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
105	C₈×C₂₉	—	2	1	no	—
106	C₈×C₂₉	—	2	1	no	—
107	C₈×C₂₉	—	2	2	no	—
108	#3099	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₂₉	—	2	1	no	—
110	C₂²×C₂₉	—	2	1	no	—
111	C₂²×C₂₉	—	2	2	no	—
112	C₂²×C₂₉	—	2	2	no	—
113	C₂²×C₂₉	—	2	2	no	—
114	C₂²×C₂₉	—	4	1	no	—
115	C₂²×C₂₉	—	4	1	no	—
116	C₂²×C₂₉	—	4	1	no	—
117	C₂²×C₂₉	#123	1	2	yes	—
118	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₂₉	—	2	2	no	—
120	C₄×C₂₉	—	4	1	no	—
121	C₄×C₂₉	—	4	1	no	—
122	C₄×C₂₉	—	4	2	no	—
123	C₄×C₂₉	—	4	1	no	—
124	C₄×C₂₉	—	4	2	no	—
125	C₄×C₂₉	#123	1	2	yes	—
126	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
127	C₄×C₂₉	C₂×#41	1	1	yes	characteristic
128	#537	C₂×C₂₉	1	1	yes	characteristic
129	#520	C₂×C₂₉	1	1	yes	characteristic, complemented
130	#485	C₂×C₂₉	1	2	yes	complemented
131	#483	C₂×C₂₉	1	2	yes	complemented
132	#482	C₂×C₂₉	1	2	yes	complemented
133	#481	C₂×C₂₉	1	2	yes	complemented
134	#428	C₂×C₂₉	1	1	yes	characteristic, complemented
135	C₂×#106	C₂×C₂₉	1	1	yes	characteristic, complemented
136	#417	C₂×C₂₉	1	1	yes	characteristic, complemented
137	#416	C₂×C₂₉	1	1	yes	characteristic, complemented
138	#408	C₂×C₂₉	1	1	yes	characteristic, complemented
139	C₂×C₂₉	—	2	2	no	—
140	C₂×C₂₉	—	4	2	no	—
141	C₂×C₂₉	—	8	1	no	—
142	C₂×C₂₉	C₂×#123	1	1	yes	characteristic
143	C₂×C₂₉	#577	1	1	yes	characteristic
144	C₂×C₂₉	#577	1	1	yes	characteristic
145	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
146	C₂×#41	C₂²×C₂₉	1	1	yes	characteristic
147	C₂×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
148	C₂²×C₈	C₂²×C₂₉	1	1	yes	characteristic
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	C₂²×C₂₉	1	2	yes	—
151	#121	—	2	1	no	—
152	#121	—	2	2	no	—
153	#121	C₂²×C₂₉	1	2	yes	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	—	2	1	no	—
157	#117	C₂²×C₂₉	1	1	yes	characteristic
158	#117	C₂²×C₂₉	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
162	#107	C₂²×C₂₉	1	2	yes	—
163	#106	C₂²×C₂₉	1	2	yes	—
164	#106	C₂²×C₂₉	1	2	yes	—
165	#106	C₂²×C₂₉	1	2	yes	—
166	#103	C₂²×C₂₉	1	1	yes	characteristic
167	#103	C₂²×C₂₉	1	1	yes	characteristic
168	#103	C₂²×C₂₉	1	2	yes	—
169	#102	C₂²×C₂₉	1	1	yes	characteristic
170	#102	C₂²×C₂₉	1	1	yes	characteristic
171	#102	C₂²×C₂₉	1	2	yes	—
172	C₂²⋊C₈	C₂²×C₂₉	1	2	yes	—
173	C₂²⋊C₈	C₂²×C₂₉	1	2	yes	—
174	C₂₉	#3099	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	#41	—	2	2	no	—
178	#41	—	2	2	no	—
179	C₂×Q₈	D₄×C₂₉	1	1	yes	characteristic
180	C₂×Q₈	C₂³×C₂₉	1	1	yes	characteristic
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	D₄×C₂₉	1	2	yes	—
184	C₂×D₄	D₄×C₂₉	1	1	yes	characteristic
185	C₂×D₄	C₂³×C₂₉	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
191	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
192	M₁₆	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
195	C₂×C₈	C₂³×C₂₉	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₂₉	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
203	C₄⋊C₄	C₂³×C₂₉	1	1	yes	characteristic
204	C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₂₉	1	2	yes	—
213	C₂³	C₂×D₄×C₂₉	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	—	2	1	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	1	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	8	1	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
236	C₂×C₄	#41×C₂₉	1	2	yes	—
237	C₂×C₄	#41×C₂₉	1	2	yes	—
238	C₂×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	C₂₉×#123	1	2	yes	—
251	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	C₂₉×#123	1	2	yes	—
259	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
260	C₄	C₂×#41×C₂₉	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₂×C₂₉×#123	1	1	yes	characteristic
265	C₂	C₂₉×#577	1	1	yes	characteristic
266	C₂	C₂₉×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₉
prime radical	C₂₉×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂×#41×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₉×#520	C₂	semidirect
3. C₂₉×#485	C₂	semidirect
4. C₂₉×#483	C₂	semidirect
5. C₂₉×#482	C₂	semidirect
6. C₂₉×#481	C₂	semidirect
7. C₂₉×#428	C₂	semidirect
8. C₂×C₂₉×#106	C₂	semidirect
9. C₂₉×#417	C₂	semidirect
10. C₂₉×#416	C₂	semidirect
11. C₂₉×#408	C₂	semidirect
108. #3099	C₂₉	direct (N × H)
129. #520	C₂×C₂₉	semidirect
130. #485	C₂×C₂₉	semidirect
131. #483	C₂×C₂₉	semidirect
132. #482	C₂×C₂₉	semidirect
133. #481	C₂×C₂₉	semidirect
134. #428	C₂×C₂₉	semidirect
135. C₂×#106	C₂×C₂₉	semidirect
136. #417	C₂×C₂₉	semidirect
137. #416	C₂×C₂₉	semidirect
138. #408	C₂×C₂₉	semidirect
174. C₂₉	#3099	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×#537	2
C₂₉×#520	2
C₂₉×#485	2
C₂₉×#483	2
C₂₉×#482	2
C₂₉×#481	2
C₂₉×#428	2
C₂×C₂₉×#106	2
C₂₉×#417	2
C₂₉×#416	2
C₂₉×#408	2
#3099	29

Subgroups — C29×#3099 #16816813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#3099 #16816813