Subgroups — C₂₉×#19975 #20314875

Order 7424.

758 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#3391	C₂	1	2	yes	maximal, complemented
2	C₂₉×#3287	C₂	1	2	yes	maximal, complemented
3	C₂₉×#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₉×#1668	C₂	1	2	yes	maximal, complemented
5	C₂₉×#1667	C₂	1	2	yes	maximal, complemented
6	C₂₉×#1621	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₉×#135	—	2	2	no	—
10	C₂₉×#550	C₂²	1	4	yes	—
11	C₂₉×#493	—	2	2	no	—
12	C₂₉×#493	C₂²	1	2	yes	—
13	C₂₉×#493	C₂²	1	2	yes	—
14	C₂₉×#493	C₂²	1	2	yes	—
15	C₂₉×#466	—	2	2	no	—
16	C₂₉×#460	C₂²	1	4	yes	—
17	C₂₉×#459	—	2	4	no	—
18	C₂₉×#459	C₂²	1	4	yes	—
19	C₂₉×#449	—	2	4	no	—
20	C₈×Q₈×C₂₉	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₂₉	C₂²	1	1	yes	characteristic
22	C₄×D₈×C₂₉	—	2	2	no	—
23	C₄×D₈×C₂₉	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₂₉	C₂²	1	2	yes	—
25	C₂₉×#335	C₂²	1	1	yes	characteristic
26	C₂₉×#335	C₂²	1	1	yes	characteristic
27	C₂₉×#331	C₂²	1	4	yes	—
28	C₂₉×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₂₉×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₂₉	C₂²	1	1	yes	characteristic
31	C₂×#41×C₂₉	—	2	4	no	—
32	C₂₉×#135	—	4	8	no	—
33	C₂×Q₁₆×C₂₉	D₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₂₉	—	2	4	no	—
35	C₂×D₈×C₂₉	—	2	4	no	—
36	C₂×D₈×C₂₉	—	2	2	no	—
37	C₂×D₈×C₂₉	—	2	2	no	—
38	C₂×D₈×C₂₉	—	2	2	no	—
39	C₂×D₈×C₂₉	—	2	2	no	—
40	C₂×D₈×C₂₉	—	4	2	no	—
41	C₂×D₈×C₂₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₉	—	2	2	no	—
43	C₂₉×#127	—	2	4	no	—
44	C₂₉×#127	C₂³	1	4	yes	—
45	C₂₉×#121	—	2	4	no	—
46	C₂₉×#121	—	4	4	no	—
47	C₄×Q₈×C₂₉	C₂³	1	2	yes	—
48	C₄×D₄×C₂₉	—	2	4	no	—
49	C₄×D₄×C₂₉	C₂³	1	2	yes	—
50	C₂₉×#107	D₄	1	1	yes	characteristic
51	C₂₉×#107	D₄	1	1	yes	characteristic
52	C₂₉×C₄⋊C₈	—	2	2	no	—
53	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
54	C₂₉×C₄⋊C₈	C₂³	1	2	yes	—
55	C₂₉×#103	D₄	1	2	yes	—
56	C₂₉×#102	—	2	4	no	—
57	C₂₉×#102	—	2	4	no	—
58	C₂₉×#102	D₄	1	2	yes	—
59	C₂₉×C₂²⋊C₈	—	2	2	no	—
60	C₄×C₈×C₂₉	—	2	2	no	—
61	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₂₉	C₂³	1	1	yes	characteristic
64	#41×C₂₉	—	4	8	no	—
65	#41×C₂₉	—	8	4	no	—
66	C₂×Q₈×C₂₉	C₂×D₄	1	2	yes	—
67	C₂×D₄×C₂₉	—	2	4	no	—
68	C₂×D₄×C₂₉	—	2	4	no	—
69	C₂×D₄×C₂₉	—	2	4	no	—
70	C₂×D₄×C₂₉	—	4	4	no	—
71	C₂×D₄×C₂₉	—	4	4	no	—
72	C₂×D₄×C₂₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₂₉	—	2	2	no	—
74	C₂²×C₄×C₂₉	—	4	4	no	—
75	Q₁₆×C₂₉	D₈	1	4	yes	complemented
76	SD₁₆×C₂₉	—	4	8	no	—
77	D₈×C₂₉	—	2	2	no	—
78	D₈×C₂₉	—	4	4	no	—
79	D₈×C₂₉	—	4	4	no	—
80	D₈×C₂₉	—	8	2	no	—
81	D₈×C₂₉	—	8	2	no	—
82	D₈×C₂₉	—	8	4	no	—
83	D₈×C₂₉	—	8	4	no	—
84	C₂×C₈×C₂₉	—	2	1	no	—
85	C₂×C₈×C₂₉	—	2	2	no	—
86	C₂×C₈×C₂₉	—	2	2	no	—
87	C₂×C₈×C₂₉	—	4	4	no	—
88	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₂₉	—	2	2	no	—
93	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₂₉	C₂×D₄	1	2	yes	—
95	C₂²⋊C₄×C₂₉	—	2	2	no	—
96	C₂²⋊C₄×C₂₉	—	4	4	no	—
97	C₄²×C₂₉	—	2	2	no	—
98	C₄²×C₂₉	C₂⁴	1	1	yes	characteristic
99	#19975	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₉	—	2	2	no	—
101	C₂³×C₂₉	—	4	4	no	—
102	Q₈×C₂₉	—	2	2	no	—
103	Q₈×C₂₉	C₂×D₈	1	4	yes	—
104	D₄×C₂₉	—	2	2	no	—
105	D₄×C₂₉	—	2	2	no	—
106	D₄×C₂₉	—	4	4	no	—
107	D₄×C₂₉	—	4	4	no	—
108	D₄×C₂₉	—	4	4	no	—
109	D₄×C₂₉	—	4	4	no	—
110	D₄×C₂₉	—	8	8	no	—
111	D₄×C₂₉	—	8	4	no	—
112	D₄×C₂₉	—	16	4	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	2	2	no	—
115	C₂×C₄×C₂₉	—	2	2	no	—
116	C₂×C₄×C₂₉	—	4	2	no	—
117	C₂×C₄×C₂₉	—	4	2	no	—
118	C₂×C₄×C₂₉	—	8	8	no	—
119	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
122	C₈×C₂₉	—	2	2	no	—
123	C₈×C₂₉	—	2	1	no	—
124	C₈×C₂₉	—	2	1	no	—
125	C₈×C₂₉	—	4	2	no	—
126	C₈×C₂₉	—	4	2	no	—
127	C₈×C₂₉	C₂×D₈	1	2	yes	—
128	C₈×C₂₉	#135	1	2	yes	complemented
129	#3391	C₂×C₂₉	1	2	yes	complemented
130	#3287	C₂×C₂₉	1	2	yes	complemented
131	#1712	C₂×C₂₉	1	1	yes	characteristic, complemented
132	#1668	C₂×C₂₉	1	2	yes	complemented
133	#1667	C₂×C₂₉	1	2	yes	complemented
134	#1621	C₂×C₂₉	1	4	yes	complemented
135	C₈×Q₁₆	C₂×C₂₉	1	1	yes	characteristic, complemented
136	C₈×D₈	C₂×C₂₉	1	1	yes	characteristic, complemented
137	C₂²×C₂₉	—	4	2	no	—
138	C₂²×C₂₉	—	4	2	no	—
139	C₂²×C₂₉	—	4	2	no	—
140	C₂²×C₂₉	—	8	4	no	—
141	C₂²×C₂₉	—	8	4	no	—
142	C₂²×C₂₉	—	8	4	no	—
143	C₂²×C₂₉	D₄²	1	1	yes	characteristic
144	C₄×C₂₉	—	2	4	no	—
145	C₄×C₂₉	—	2	1	no	—
146	C₄×C₂₉	—	4	2	no	—
147	C₄×C₂₉	—	4	2	no	—
148	C₄×C₂₉	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₂₉	C₂×#135	1	1	yes	characteristic
150	C₄×C₂₉	#576	1	1	yes	characteristic
151	C₄×C₂₉	#576	1	1	yes	characteristic
152	C₂×#135	—	2	2	no	—
153	#550	C₂²×C₂₉	1	4	yes	—
154	#493	—	2	2	no	—
155	#493	C₂²×C₂₉	1	2	yes	—
156	#493	C₂²×C₂₉	1	2	yes	—
157	#493	C₂²×C₂₉	1	2	yes	—
158	#466	—	2	2	no	—
159	#460	C₂²×C₂₉	1	4	yes	—
160	#459	—	2	4	no	—
161	#459	C₂²×C₂₉	1	4	yes	—
162	#449	—	2	4	no	—
163	C₈×Q₈	C₂²×C₂₉	1	2	yes	—
164	C₄×Q₁₆	C₂²×C₂₉	1	1	yes	characteristic
165	C₄×D₈	—	2	2	no	—
166	C₄×D₈	C₂²×C₂₉	1	1	yes	characteristic
167	C₈×D₄	C₂²×C₂₉	1	2	yes	—
168	#335	C₂²×C₂₉	1	1	yes	characteristic
169	#335	C₂²×C₂₉	1	1	yes	characteristic
170	#331	C₂²×C₂₉	1	4	yes	—
171	Q₈⋊C₈	C₂²×C₂₉	1	2	yes	—
172	D₄⋊C₈	C₂²×C₂₉	1	2	yes	—
173	C₈²	C₂²×C₂₉	1	1	yes	characteristic
174	C₂×C₂₉	—	8	2	no	—
175	C₂×C₂₉	—	16	4	no	—
176	C₂×C₂₉	D₄×D₈	1	1	yes	characteristic
177	C₂×C₂₉	#3287	1	1	yes	characteristic
178	C₂×C₂₉	#3292	1	1	yes	characteristic
179	C₂×#41	—	2	4	no	—
180	#135	—	4	8	no	—
181	C₂×Q₁₆	D₄×C₂₉	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	4	no	—
183	C₂×D₈	—	2	4	no	—
184	C₂×D₈	—	2	2	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	—	4	2	no	—
189	C₂×D₈	D₄×C₂₉	1	1	yes	characteristic
190	C₂²×C₈	—	2	2	no	—
191	#127	—	2	4	no	—
192	#127	C₂³×C₂₉	1	4	yes	—
193	#121	—	2	4	no	—
194	#121	—	4	4	no	—
195	C₄×Q₈	C₂³×C₂₉	1	2	yes	—
196	C₄×D₄	—	2	4	no	—
197	C₄×D₄	C₂³×C₂₉	1	2	yes	—
198	#107	D₄×C₂₉	1	1	yes	characteristic
199	#107	D₄×C₂₉	1	1	yes	characteristic
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
202	C₄⋊C₈	C₂³×C₂₉	1	2	yes	—
203	#103	D₄×C₂₉	1	2	yes	—
204	#102	—	2	4	no	—
205	#102	—	2	4	no	—
206	#102	D₄×C₂₉	1	2	yes	—
207	C₂²⋊C₈	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₂₉	1	1	yes	characteristic
212	C₂₉	#19975	1	1	yes	characteristic, direct product factor, Sylow
213	#41	—	4	8	no	—
214	#41	—	8	4	no	—
215	C₂×Q₈	C₂×D₄×C₂₉	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	C₂×D₄×C₂₉	1	2	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	4	no	—
224	Q₁₆	D₈×C₂₉	1	4	yes	complemented
225	SD₁₆	—	4	8	no	—
226	D₈	—	2	2	no	—
227	D₈	—	4	4	no	—
228	D₈	—	4	4	no	—
229	D₈	—	8	2	no	—
230	D₈	—	8	2	no	—
231	D₈	—	8	4	no	—
232	D₈	—	8	4	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
243	C₄⋊C₄	C₂×D₄×C₂₉	1	2	yes	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₄²	—	2	2	no	—
247	C₄²	C₂⁴×C₂₉	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	4	4	no	—
250	Q₈	—	2	2	no	—
251	Q₈	C₂×D₈×C₂₉	1	4	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	4	no	—
260	D₄	—	16	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	8	no	—
267	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
269	C₂×C₄	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	C₂×D₈×C₂₉	1	2	yes	—
276	C₈	C₂₉×#135	1	2	yes	complemented
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	—	8	4	no	—
283	C₂²	D₄²×C₂₉	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	C₂²×D₈×C₂₉	1	1	yes	characteristic
289	C₄	C₂×C₂₉×#135	1	1	yes	characteristic
290	C₄	C₂₉×#576	1	1	yes	characteristic
291	C₄	C₂₉×#576	1	1	yes	characteristic
292	C₂	—	8	2	no	—
293	C₂	—	16	4	no	—
294	C₂	D₄×D₈×C₂₉	1	1	yes	characteristic
295	C₂	C₂₉×#3287	1	1	yes	characteristic
296	C₂	C₂₉×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₄²×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₉×#3391	C₂	semidirect
2. C₂₉×#3287	C₂	semidirect
3. C₂₉×#1712	C₂	semidirect
4. C₂₉×#1668	C₂	semidirect
5. C₂₉×#1667	C₂	semidirect
6. C₂₉×#1621	C₂	semidirect
7. C₈×Q₁₆×C₂₉	C₂	semidirect
8. C₈×D₈×C₂₉	C₂	semidirect
75. Q₁₆×C₂₉	D₈	semidirect
99. #19975	C₂₉	direct (N × H)
128. C₈×C₂₉	#135	semidirect
129. #3391	C₂×C₂₉	semidirect
130. #3287	C₂×C₂₉	semidirect
131. #1712	C₂×C₂₉	semidirect
132. #1668	C₂×C₂₉	semidirect
133. #1667	C₂×C₂₉	semidirect
134. #1621	C₂×C₂₉	semidirect
135. C₈×Q₁₆	C₂×C₂₉	semidirect
136. C₈×D₈	C₂×C₂₉	semidirect
212. C₂₉	#19975	direct (N × H)
224. Q₁₆	D₈×C₂₉	semidirect
276. C₈	C₂₉×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₉×#3391	2
C₂₉×#3287	2
C₂₉×#1712	2
C₂₉×#1668	2
C₂₉×#1667	2
C₂₉×#1621	2
C₈×Q₁₆×C₂₉	2
C₈×D₈×C₂₉	2
#19975	29

Subgroups — C29×#19975 #20314875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#19975 #20314875