Subgroups — C₂₉×#28737 #20323637

Order 7424.

814 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×C₂₉×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2568	C₂	1	2	yes	maximal
4	C₂₉×#2056	C₂	1	4	yes	maximal
5	C₂₉×#1981	C₂	1	2	yes	maximal
6	C₂₉×#1969	C₂	1	4	yes	maximal
7	C₄×C₂₉×#107	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₂₉	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₂₉	C₂²	1	2	yes	—
10	C₂×C₂₉×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
12	C₂₉×#494	—	2	4	no	—
13	C₈×Q₈×C₂₉	—	2	4	no	—
14	C₂×C₂₉×#107	C₂²	1	2	yes	—
15	C₂×C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₉×#103	C₂²	1	8	yes	—
18	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
20	C₂₉×#400	—	2	4	no	—
21	C₂₉×#391	—	2	2	no	—
22	C₂₉×#391	C₂²	1	8	yes	—
23	C₂₉×#382	—	2	2	no	—
24	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
25	C₂₉×#340	C₂²	1	2	yes	—
26	C₂²×Q₈×C₂₉	D₄	1	1	yes	characteristic
27	C₂²×Q₈×C₂₉	C₂³	1	4	yes	—
28	C₂×Q₁₆×C₂₉	—	2	4	no	—
29	C₂×Q₁₆×C₂₉	—	2	4	no	—
30	C₂×Q₁₆×C₂₉	—	2	8	no	—
31	C₂×Q₁₆×C₂₉	—	4	4	no	—
32	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	C₂₉×C₄⋊Q₈	—	2	4	no	—
36	C₂₉×C₄⋊Q₈	—	2	4	no	—
37	C₄×Q₈×C₂₉	—	2	2	no	—
38	C₄×Q₈×C₂₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₉	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
44	C₂×C₄²×C₂₉	—	2	2	no	—
45	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
47	C₂₉×#107	—	2	4	no	—
48	C₂₉×C₄⋊C₈	—	2	4	no	—
49	C₂₉×C₄⋊C₈	—	2	2	no	—
50	C₂₉×#103	—	4	8	no	—
51	C₄×C₈×C₂₉	—	2	4	no	—
52	C₄×C₈×C₂₉	D₄	1	4	yes	—
53	C₂₉×#95	—	2	2	no	—
54	C₂₉×#95	—	2	8	no	—
55	C₂₉×#95	—	2	8	no	—
56	C₂×Q₈×C₂₉	—	2	2	no	—
57	C₂×Q₈×C₂₉	—	2	2	no	—
58	C₂×Q₈×C₂₉	—	2	4	no	—
59	C₂×Q₈×C₂₉	—	2	4	no	—
60	C₂×Q₈×C₂₉	—	2	8	no	—
61	C₂×Q₈×C₂₉	—	2	8	no	—
62	C₂×Q₈×C₂₉	Q₁₆	1	4	yes	complemented
63	C₂²×C₄×C₂₉	—	2	4	no	—
64	C₂²×C₄×C₂₉	—	2	4	no	—
65	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
70	Q₁₆×C₂₉	—	4	8	no	—
71	Q₁₆×C₂₉	—	4	8	no	—
72	C₂×C₈×C₂₉	—	2	2	no	—
73	C₂×C₈×C₂₉	—	2	2	no	—
74	C₂×C₈×C₂₉	—	2	4	no	—
75	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₂₉	#41	1	4	yes	—
78	C₄⋊C₄×C₂₉	—	2	4	no	—
79	C₄⋊C₄×C₂₉	—	2	2	no	—
80	C₄⋊C₄×C₂₉	—	2	4	no	—
81	C₄⋊C₄×C₂₉	—	2	4	no	—
82	C₄⋊C₄×C₂₉	—	4	4	no	—
83	C₄⋊C₄×C₂₉	—	4	4	no	—
84	C₄²×C₂₉	—	2	2	no	—
85	C₄²×C₂₉	—	2	2	no	—
86	C₄²×C₂₉	—	4	4	no	—
87	C₄²×C₂₉	C₂×D₄	1	2	yes	—
88	C₄²×C₂₉	#41	1	2	yes	—
89	#28737	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₂₉	—	2	8	no	—
92	Q₈×C₂₉	—	4	4	no	—
93	Q₈×C₂₉	—	4	4	no	—
94	Q₈×C₂₉	—	4	8	no	—
95	C₂×C₄×C₂₉	—	2	2	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	2	2	no	—
98	C₂×C₄×C₂₉	—	2	2	no	—
99	C₂×C₄×C₂₉	—	2	4	no	—
100	C₂×C₄×C₂₉	—	4	4	no	—
101	C₂×C₄×C₂₉	—	4	4	no	—
102	C₂×C₄×C₂₉	—	4	8	no	—
103	C₂×C₄×C₂₉	—	4	4	no	—
104	C₂×C₄×C₂₉	—	4	4	no	—
105	C₂×C₄×C₂₉	#121	1	2	yes	—
106	C₂×C₄×C₂₉	#121	1	2	yes	—
107	C₂×C₄×C₂₉	C₂×Q₁₆	1	4	yes	—
108	C₂×C₄×C₂₉	C₂×Q₁₆	1	2	yes	—
109	C₂×C₄×C₂₉	#135	1	2	yes	—
110	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
112	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₂₉	—	2	2	no	—
115	C₈×C₂₉	—	4	2	no	—
116	C₈×C₂₉	#124	1	4	yes	—
117	C₂×#494	C₂×C₂₉	1	1	yes	characteristic
118	C₂×C₈×Q₈	C₂×C₂₉	1	1	yes	characteristic
119	#2568	C₂×C₂₉	1	2	yes	—
120	#2056	C₂×C₂₉	1	4	yes	—
121	#1981	C₂×C₂₉	1	2	yes	—
122	#1969	C₂×C₂₉	1	4	yes	—
123	C₄×#107	C₂×C₂₉	1	1	yes	characteristic
124	C₂²×C₂₉	C₂×#121	1	1	yes	characteristic
125	C₂²×C₂₉	#550	1	2	yes	—
126	C₂²×C₂₉	C₂²×Q₁₆	1	1	yes	characteristic
127	C₂²×C₂₉	C₂×#135	1	1	yes	characteristic
128	C₂²×C₂₉	#578	1	2	yes	—
129	C₄×C₂₉	—	2	4	no	—
130	C₄×C₂₉	—	2	2	no	—
131	C₄×C₂₉	—	4	2	no	—
132	C₄×C₂₉	—	8	4	no	—
133	C₄×C₂₉	—	8	4	no	—
134	C₄×C₂₉	#467	1	4	yes	—
135	C₄×C₂₉	C₂×#124	1	1	yes	characteristic
136	C₄×C₂₉	#535	1	1	yes	characteristic
137	C₄×C₂₉	#559	1	2	yes	—
138	C₂²×Q₁₆	C₂²×C₂₉	1	2	yes	—
139	C₂²×Q₁₆	C₂²×C₂₉	1	2	yes	—
140	C₂×C₄⋊Q₈	C₂²×C₂₉	1	2	yes	—
141	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
142	#494	—	2	4	no	—
143	C₈×Q₈	—	2	4	no	—
144	C₂×#107	C₂²×C₂₉	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
146	C₂×C₄⋊C₈	C₂²×C₂₉	1	1	yes	characteristic
147	C₂×#103	C₂²×C₂₉	1	8	yes	—
148	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
150	#400	—	2	4	no	—
151	#391	—	2	2	no	—
152	#391	C₂²×C₂₉	1	8	yes	—
153	#382	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
155	#340	C₂²×C₂₉	1	2	yes	—
156	C₂×C₂₉	#2548	1	1	yes	characteristic
157	C₂×C₂₉	C₂×#467	1	1	yes	characteristic
158	C₂×C₂₉	#3059	1	1	yes	characteristic
159	C₂×C₂₉	#3393	1	2	yes	—
160	C₂×C₂₉	#3395	1	2	yes	—
161	C₂²×Q₈	D₄×C₂₉	1	1	yes	characteristic
162	C₂²×Q₈	C₂³×C₂₉	1	4	yes	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	2	8	no	—
166	C₂×Q₁₆	—	4	4	no	—
167	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
168	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	—	2	4	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	D₄×C₂₉	1	1	yes	characteristic
177	C₂×C₄⋊C₄	D₄×C₂₉	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
181	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
182	#107	—	2	4	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	2	no	—
185	#103	—	4	8	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	D₄×C₂₉	1	4	yes	—
188	#95	—	2	2	no	—
189	#95	—	2	8	no	—
190	#95	—	2	8	no	—
191	C₂₉	#28737	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	8	no	—
198	C₂×Q₈	Q₁₆×C₂₉	1	4	yes	complemented
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
205	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
206	Q₁₆	—	4	8	no	—
207	Q₁₆	—	4	8	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
213	C₂×C₈	#41×C₂₉	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₂×D₄×C₂₉	1	2	yes	—
224	C₄²	#41×C₂₉	1	2	yes	—
225	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	Q₈	—	4	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂₉×#121	1	2	yes	—
241	C₂×C₄	C₂₉×#121	1	2	yes	—
242	C₂×C₄	C₂×Q₁₆×C₂₉	1	4	yes	—
243	C₂×C₄	C₂×Q₁₆×C₂₉	1	2	yes	—
244	C₂×C₄	C₂₉×#135	1	2	yes	—
245	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
246	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
247	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
248	C₂×C₄	C₂₉×H(5,𝔽₂)	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	4	2	no	—
251	C₈	C₂₉×#124	1	4	yes	—
252	C₂²	C₂×C₂₉×#121	1	1	yes	characteristic
253	C₂²	C₂₉×#550	1	2	yes	—
254	C₂²	C₂²×Q₁₆×C₂₉	1	1	yes	characteristic
255	C₂²	C₂×C₂₉×#135	1	1	yes	characteristic
256	C₂²	C₂₉×#578	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	2	no	—
260	C₄	—	8	4	no	—
261	C₄	—	8	4	no	—
262	C₄	C₂₉×#467	1	4	yes	—
263	C₄	C₂×C₂₉×#124	1	1	yes	characteristic
264	C₄	C₂₉×#535	1	1	yes	characteristic
265	C₄	C₂₉×#559	1	2	yes	—
266	C₂	C₂₉×#2548	1	1	yes	characteristic
267	C₂	C₂×C₂₉×#467	1	1	yes	characteristic
268	C₂	C₂₉×#3059	1	1	yes	characteristic
269	C₂	C₂₉×#3393	1	2	yes	—
270	C₂	C₂₉×#3395	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₉ ◁ C₂×C₄×Q₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
62. C₂×Q₈×C₂₉	Q₁₆	semidirect
89. #28737	C₂₉	direct (N × H)
191. C₂₉	#28737	direct (N × H)
198. C₂×Q₈	Q₁₆×C₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₉×#494	2
C₂×C₈×Q₈×C₂₉	2
C₂₉×#2568	2
C₂₉×#2056	2
C₂₉×#1981	2
C₂₉×#1969	2
C₄×C₂₉×#107	2
#28737	29

Subgroups — C29×#28737 #20323637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#28737 #20323637