Subgroups — C₂₉×#2451 #16816165

Order 3712.

484 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂₉×#398	C₂	1	2	yes	maximal
4	C₂₉×#395	C₂	1	2	yes	maximal
5	C₂₉×#391	C₂	1	2	yes	maximal
6	C₂₉×#391	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#391	C₂	1	1	yes	maximal, characteristic
8	C₂₉×#389	C₂	1	1	yes	maximal, characteristic
9	C₂₉×#389	C₂	1	1	yes	maximal, characteristic
10	C₂₉×#387	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₉	C₂²	1	2	yes	—
13	C₂²×Q₈×C₂₉	C₂²	1	1	yes	characteristic
14	C₂₉×C₄⋊Q₈	—	2	4	no	—
15	C₄×Q₈×C₂₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
29	C₂₉×#95	C₂²	1	1	yes	characteristic
30	C₂₉×#95	C₂²	1	1	yes	characteristic
31	C₂₉×#95	C₂²	1	2	yes	—
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	C₂₉×#95	C₂²	1	2	yes	—
37	C₂×Q₈×C₂₉	—	2	2	no	—
38	C₂×Q₈×C₂₉	—	2	4	no	—
39	C₂×Q₈×C₂₉	—	2	2	no	—
40	C₂×Q₈×C₂₉	—	2	4	no	—
41	C₂×Q₈×C₂₉	—	4	2	no	—
42	C₂×Q₈×C₂₉	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₉	—	2	4	no	—
55	C₄⋊C₄×C₂₉	—	2	4	no	—
56	C₄⋊C₄×C₂₉	—	2	2	no	—
57	C₄⋊C₄×C₂₉	—	2	2	no	—
58	C₄⋊C₄×C₂₉	—	2	2	no	—
59	C₄⋊C₄×C₂₉	—	4	2	no	—
60	C₄⋊C₄×C₂₉	—	4	2	no	—
61	C₄⋊C₄×C₂₉	D₄	1	4	yes	—
62	C₄⋊C₄×C₂₉	Q₈	1	4	yes	complemented
63	C₄²×C₂₉	—	2	4	no	—
64	C₄²×C₂₉	—	2	2	no	—
65	C₄²×C₂₉	—	2	2	no	—
66	C₄²×C₂₉	—	2	2	no	—
67	C₂³×C₂₉	C₂⁴	1	1	yes	characteristic
68	Q₈×C₂₉	—	2	8	no	—
69	Q₈×C₂₉	—	4	8	no	—
70	C₂×C₄×C₂₉	—	2	1	no	—
71	C₂×C₄×C₂₉	—	2	1	no	—
72	C₂×C₄×C₂₉	—	2	2	no	—
73	C₂×C₄×C₂₉	—	2	1	no	—
74	C₂×C₄×C₂₉	—	2	1	no	—
75	C₂×C₄×C₂₉	—	2	1	no	—
76	C₂×C₄×C₂₉	—	2	2	no	—
77	C₂×C₄×C₂₉	—	2	2	no	—
78	C₂×C₄×C₂₉	—	2	2	no	—
79	C₂×C₄×C₂₉	—	2	1	no	—
80	C₂×C₄×C₂₉	—	2	1	no	—
81	C₂×C₄×C₂₉	—	2	1	no	—
82	C₂×C₄×C₂₉	—	2	1	no	—
83	C₂×C₄×C₂₉	—	2	2	no	—
84	C₂×C₄×C₂₉	—	2	1	no	—
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	—	2	1	no	—
87	C₂×C₄×C₂₉	—	2	2	no	—
88	C₂×C₄×C₂₉	—	2	1	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	1	no	—
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	—	2	1	no	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
95	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₉	C₂×Q₈	1	4	yes	—
100	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₂₉	#41	1	2	yes	—
102	C₂×C₄×C₂₉	#41	1	2	yes	—
103	#2451	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×C₂₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
108	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
111	C₄×C₂₉	—	2	4	no	—
112	C₄×C₂₉	—	2	2	no	—
113	C₄×C₂₉	—	2	4	no	—
114	C₄×C₂₉	—	2	2	no	—
115	C₄×C₂₉	—	2	2	no	—
116	C₄×C₂₉	—	2	2	no	—
117	C₄×C₂₉	—	4	1	no	—
118	C₄×C₂₉	—	4	2	no	—
119	C₄×C₂₉	—	4	1	no	—
120	C₄×C₂₉	—	4	2	no	—
121	C₄×C₂₉	C₂²⋊Q₈	1	4	yes	—
122	C₂×C₄⋊Q₈	C₂×C₂₉	1	1	yes	characteristic
123	C₂×C₄×Q₈	C₂×C₂₉	1	1	yes	characteristic
124	#398	C₂×C₂₉	1	2	yes	—
125	#395	C₂×C₂₉	1	2	yes	—
126	#391	C₂×C₂₉	1	2	yes	—
127	#391	C₂×C₂₉	1	1	yes	characteristic
128	#391	C₂×C₂₉	1	1	yes	characteristic
129	#389	C₂×C₂₉	1	1	yes	characteristic
130	#389	C₂×C₂₉	1	1	yes	characteristic
131	#387	C₂×C₂₉	1	2	yes	—
132	C₄×C₄⋊C₄	C₂×C₂₉	1	1	yes	characteristic
133	C₂×C₂₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₂₉	#533	1	1	yes	characteristic
135	C₂×C₂₉	#542	1	1	yes	characteristic
136	C₂×C₂₉	#546	1	1	yes	characteristic
137	C₂×C₂₉	D₄×Q₈	1	1	yes	characteristic
138	C₂×C₂₉	#557	1	1	yes	characteristic
139	C₂×C₂₉	Q₈²	1	1	yes	characteristic
140	C₂²×Q₈	C₂²×C₂₉	1	2	yes	—
141	C₂²×Q₈	C₂²×C₂₉	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	4	no	—
143	C₄×Q₈	—	2	4	no	—
144	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
149	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
151	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
153	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₂₉	1	2	yes	—
155	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
156	C₂×C₄²	C₂²×C₂₉	1	1	yes	characteristic
157	#95	C₂²×C₂₉	1	1	yes	characteristic
158	#95	C₂²×C₂₉	1	1	yes	characteristic
159	#95	C₂²×C₂₉	1	2	yes	—
160	#95	C₂²×C₂₉	1	2	yes	—
161	#95	C₂²×C₂₉	1	2	yes	—
162	#95	C₂²×C₂₉	1	2	yes	—
163	#95	C₂²×C₂₉	1	2	yes	—
164	#95	C₂²×C₂₉	1	2	yes	—
165	C₂₉	#2451	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	4	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	Q₈×C₂₉	1	4	yes	complemented
172	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₉	1	2	yes	—
179	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₂₉	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	D₄×C₂₉	1	4	yes	—
191	C₄⋊C₄	Q₈×C₂₉	1	4	yes	complemented
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₂³	C₂⁴×C₂₉	1	1	yes	characteristic
197	Q₈	—	2	8	no	—
198	Q₈	—	4	8	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₂×D₄×C₂₉	1	4	yes	—
224	C₂×C₄	C₂×D₄×C₂₉	1	2	yes	—
225	C₂×C₄	C₂×Q₈×C₂₉	1	2	yes	—
226	C₂×C₄	C₂×Q₈×C₂₉	1	2	yes	—
227	C₂×C₄	C₂×Q₈×C₂₉	1	2	yes	—
228	C₂×C₄	C₂×Q₈×C₂₉	1	4	yes	—
229	C₂×C₄	C₂×Q₈×C₂₉	1	2	yes	—
230	C₂×C₄	#41×C₂₉	1	2	yes	—
231	C₂×C₄	#41×C₂₉	1	2	yes	—
232	C₂²	C₂²×D₄×C₂₉	1	1	yes	characteristic
233	C₂²	C₂²×Q₈×C₂₉	1	1	yes	characteristic
234	C₂²	C₂²×Q₈×C₂₉	1	1	yes	characteristic
235	C₂²	C₂×#41×C₂₉	1	1	yes	characteristic
236	C₂²	C₂×#41×C₂₉	1	1	yes	characteristic
237	C₂²	C₂₉×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂²	C₂₉×2₋¹⁺⁴	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	4	1	no	—
246	C₄	—	4	2	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	C₂₉×C₂²⋊Q₈	1	4	yes	—
250	C₂	C₂×C₂₉×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂	C₂₉×#533	1	1	yes	characteristic
252	C₂	C₂₉×#542	1	1	yes	characteristic
253	C₂	C₂₉×#546	1	1	yes	characteristic
254	C₂	D₄×Q₈×C₂₉	1	1	yes	characteristic
255	C₂	C₂₉×#557	1	1	yes	characteristic
256	C₂	Q₈²×C₂₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
42. C₂×Q₈×C₂₉	Q₈	semidirect
62. C₄⋊C₄×C₂₉	Q₈	semidirect
103. #2451	C₂₉	direct (N × H)
165. C₂₉	#2451	direct (N × H)
171. C₂×Q₈	Q₈×C₂₉	semidirect
191. C₄⋊C₄	Q₈×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×C₄⋊Q₈	2
C₂×C₄×Q₈×C₂₉	2
C₂₉×#398	2
C₂₉×#395	2
C₂₉×#391	2
C₂₉×#389	2
C₂₉×#387	2
C₄×C₄⋊C₄×C₂₉	2
#2451	29

Subgroups — C29×#2451 #16816165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#2451 #16816165