Subgroups — C₂₉×#29150 #20324050

Order 7424.

710 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#2026	C₂	1	4	yes	maximal
4	C₂₉×#1960	C₂	1	2	yes	maximal
5	C₂₉×#1836	C₂	1	2	yes	maximal
6	C₂₉×#1803	C₂	1	4	yes	maximal
7	C₂₉×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₉×#125	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₉	C₂²	1	2	yes	—
11	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₉×#103	C₂²	1	8	yes	—
13	C₂×C₂₉×#97	C₂²	1	2	yes	—
14	C₂₉×#391	C₂²	1	2	yes	—
15	C₂₉×#389	—	2	2	no	—
16	C₂₉×#389	—	2	2	no	—
17	C₂₉×#389	—	2	2	no	—
18	C₂₉×#384	C₂²	1	1	yes	characteristic
19	C₂₉×#384	C₂²	1	2	yes	—
20	C₂₉×#382	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₉	—	2	2	no	—
22	C₄³×C₂₉	C₂²	1	1	yes	characteristic
23	C₂₉×#340	C₂²	1	8	yes	—
24	C₂₉×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₂₉	C₂³	1	2	yes	—
26	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
27	C₂₉×C₄⋊Q₈	—	2	2	no	—
28	C₂₉×C₄⋊Q₈	Q₈	1	4	yes	—
29	C₂₉×#125	—	4	4	no	—
30	C₄×Q₈×C₂₉	—	2	8	no	—
31	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₉	—	2	4	no	—
34	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₉	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₉	C₂³	1	4	yes	—
39	C₂×C₄²×C₂₉	—	2	2	no	—
40	C₂×C₄²×C₂₉	D₄	1	2	yes	—
41	C₂×C₄²×C₂₉	D₄	1	2	yes	—
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
45	C₂₉×C₄⋊C₈	—	2	4	no	—
46	C₂₉×#103	—	4	8	no	—
47	C₂₉×#97	—	2	4	no	—
48	C₂₉×#95	—	2	4	no	—
49	C₂₉×#95	—	2	4	no	—
50	C₂×Q₈×C₂₉	—	2	2	no	—
51	C₂×Q₈×C₂₉	—	2	4	no	—
52	C₂×Q₈×C₂₉	—	2	4	no	—
53	C₂×Q₈×C₂₉	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₂₉	—	2	2	no	—
55	C₂²×C₄×C₂₉	—	2	2	no	—
56	C₂²×C₄×C₂₉	—	2	4	no	—
57	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₂₉	—	2	4	no	—
63	C₂×C₈×C₂₉	—	2	4	no	—
64	C₂×C₈×C₂₉	—	2	4	no	—
65	C₄⋊C₄×C₂₉	—	2	8	no	—
66	C₄⋊C₄×C₂₉	—	2	4	no	—
67	C₄⋊C₄×C₂₉	—	2	8	no	—
68	C₄⋊C₄×C₂₉	—	4	4	no	—
69	C₄⋊C₄×C₂₉	—	4	4	no	—
70	C₄⋊C₄×C₂₉	—	4	8	no	—
71	C₄⋊C₄×C₂₉	—	4	4	no	—
72	C₄⋊C₄×C₂₉	—	4	4	no	—
73	C₄²×C₂₉	—	2	4	no	—
74	C₄²×C₂₉	—	2	2	no	—
75	C₄²×C₂₉	—	4	2	no	—
76	C₄²×C₂₉	—	4	4	no	—
77	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
78	C₄²×C₂₉	#41	1	2	yes	—
79	C₄²×C₂₉	#41	1	4	yes	—
80	#29150	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₉	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₂₉	—	2	8	no	—
83	Q₈×C₂₉	—	4	4	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	4	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	—	4	2	no	—
92	C₂×C₄×C₂₉	—	4	2	no	—
93	C₂×C₄×C₂₉	—	4	2	no	—
94	C₂×C₄×C₂₉	—	4	4	no	—
95	C₂×C₄×C₂₉	—	4	4	no	—
96	C₂×C₄×C₂₉	—	4	4	no	—
97	C₂×C₄×C₂₉	—	4	4	no	—
98	C₂×C₄×C₂₉	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₂₉	#124	1	2	yes	—
100	C₂×C₄×C₂₉	#124	1	2	yes	—
101	C₂×C₄×C₂₉	#137	1	4	yes	—
102	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
104	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₂₉	H(5,𝔽₂)	1	2	yes	—
106	C₈×C₂₉	—	4	4	no	—
107	#2859	C₂×C₂₉	1	1	yes	characteristic
108	C₄×C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic
109	#2026	C₂×C₂₉	1	4	yes	—
110	#1960	C₂×C₂₉	1	2	yes	—
111	#1836	C₂×C₂₉	1	2	yes	—
112	#1803	C₂×C₂₉	1	4	yes	—
113	#1767	C₂×C₂₉	1	1	yes	characteristic
114	C₂²×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₂₉	C₂×#124	1	1	yes	characteristic
116	C₂²×C₂₉	#535	1	1	yes	characteristic
117	C₂²×C₂₉	C₂×#137	1	2	yes	—
118	C₂²×C₂₉	#575	1	2	yes	—
119	C₄×C₂₉	—	2	4	no	—
120	C₄×C₂₉	—	2	2	no	—
121	C₄×C₂₉	—	4	4	no	—
122	C₄×C₂₉	—	4	2	no	—
123	C₄×C₂₉	—	8	2	no	—
124	C₄×C₂₉	—	8	4	no	—
125	C₄×C₂₉	#491	1	4	yes	—
126	C₄×C₂₉	#554	1	2	yes	—
127	C₄×C₂₉	#555	1	2	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₂₉	1	1	yes	characteristic
129	C₂×#125	C₂²×C₂₉	1	2	yes	—
130	C₂×C₄×Q₈	C₂²×C₂₉	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
132	C₂×#103	C₂²×C₂₉	1	8	yes	—
133	C₂×#97	C₂²×C₂₉	1	2	yes	—
134	#391	C₂²×C₂₉	1	2	yes	—
135	#389	—	2	2	no	—
136	#389	—	2	2	no	—
137	#389	—	2	2	no	—
138	#384	C₂²×C₂₉	1	1	yes	characteristic
139	#384	C₂²×C₂₉	1	2	yes	—
140	#382	C₂²×C₂₉	1	2	yes	—
141	C₄×C₄⋊C₄	—	2	2	no	—
142	C₄³	C₂²×C₂₉	1	1	yes	characteristic
143	#340	C₂²×C₂₉	1	8	yes	—
144	#336	C₂²×C₂₉	1	2	yes	—
145	C₂×C₂₉	#2563	1	1	yes	characteristic
146	C₂×C₂₉	#3078	1	2	yes	—
147	C₂×C₂₉	C₂×#491	1	1	yes	characteristic
148	C₂×C₂₉	#3142	1	1	yes	characteristic
149	C₂×C₂₉	#3216	1	2	yes	—
150	C₂²×Q₈	C₂³×C₂₉	1	2	yes	—
151	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	Q₈×C₂₉	1	4	yes	—
154	#125	—	4	4	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂³×C₂₉	1	2	yes	—
163	C₂×C₄⋊C₄	C₂³×C₂₉	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	D₄×C₂₉	1	2	yes	—
166	C₂×C₄²	D₄×C₂₉	1	2	yes	—
167	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	#103	—	4	8	no	—
172	#97	—	2	4	no	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₂₉	#29150	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	C₂×Q₈×C₂₉	1	4	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
187	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	4	4	no	—
203	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
204	C₄²	#41×C₂₉	1	2	yes	—
205	C₄²	#41×C₂₉	1	4	yes	—
206	C₂³	C₂²×D₄×C₂₉	1	1	yes	characteristic
207	Q₈	—	2	8	no	—
208	Q₈	—	4	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₂₉×C₂²⋊Q₈	1	4	yes	—
224	C₂×C₄	C₂₉×#124	1	2	yes	—
225	C₂×C₄	C₂₉×#124	1	2	yes	—
226	C₂×C₄	C₂₉×#137	1	4	yes	—
227	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
228	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
229	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
230	C₂×C₄	C₂₉×H(5,𝔽₂)	1	2	yes	—
231	C₈	—	4	4	no	—
232	C₂²	C₂×C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂²	C₂×C₂₉×#124	1	1	yes	characteristic
234	C₂²	C₂₉×#535	1	1	yes	characteristic
235	C₂²	C₂×C₂₉×#137	1	2	yes	—
236	C₂²	C₂₉×#575	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	2	no	—
243	C₄	C₂₉×#491	1	4	yes	—
244	C₄	C₂₉×#554	1	2	yes	—
245	C₄	C₂₉×#555	1	2	yes	—
246	C₂	C₂₉×#2563	1	1	yes	characteristic
247	C₂	C₂₉×#3078	1	2	yes	—
248	C₂	C₂×C₂₉×#491	1	1	yes	characteristic
249	C₂	C₂₉×#3142	1	1	yes	characteristic
250	C₂	C₂₉×#3216	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 ◁ C₄³×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
80. #29150	C₂₉	direct (N × H)
175. C₂₉	#29150	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×#2859	2
C₄×C₂₉×C₄⋊Q₈	2
C₂₉×#2026	2
C₂₉×#1960	2
C₂₉×#1836	2
C₂₉×#1803	2
C₂₉×#1767	2
#29150	29

Subgroups — C29×#29150 #20324050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#29150 #20324050