Subgroups — C₂₉×#21344 #20316244

Order 7424.

666 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₉×#2295	C₂	1	1	yes	maximal, characteristic
2	C₂₉×#1985	C₂	1	2	yes	maximal
3	C₂₉×#1940	C₂	1	2	yes	maximal
4	C₂₉×#1916	C₂	1	4	yes	maximal
5	C₂₉×#1768	C₂	1	2	yes	maximal
6	C₄×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#1746	C₂	1	2	yes	maximal
8	C₄×C₂₉×#97	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₉×#97	C₂²	1	2	yes	—
13	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
14	C₂×C₂₉×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₉	C₂²	1	2	yes	—
17	C₂₉×#391	C₄	1	2	yes	—
18	C₂₉×#389	C₄	1	2	yes	—
19	C₂₉×#387	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₉	C₂²	1	1	yes	characteristic
22	C₂₉×#376	C₂²	1	2	yes	—
23	C₄³×C₂₉	C₂²	1	1	yes	characteristic
24	C₂₉×#336	C₂²	1	2	yes	—
25	C₂₉×#336	C₂²	1	4	yes	—
26	C₂₉×#336	C₂²	1	4	yes	—
27	C₂₉×#336	C₂²	1	4	yes	—
28	C₂²×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
30	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₂₉	C₂³	1	2	yes	—
32	C₄×Q₈×C₂₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
40	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₂₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₉	C₂³	1	1	yes	characteristic
44	C₂₉×C₄⋊C₈	—	2	8	no	—
45	C₂₉×C₄⋊C₈	—	2	4	no	—
46	C₂₉×#97	—	2	4	no	—
47	C₂₉×#97	—	2	2	no	—
48	C₂₉×#97	Q₈	1	4	yes	complemented
49	C₄×C₈×C₂₉	—	2	4	no	—
50	C₄×C₈×C₂₉	—	2	4	no	—
51	C₂₉×#95	C₂×C₄	1	4	yes	—
52	C₂₉×#95	C₂×C₄	1	4	yes	—
53	C₂₉×#95	C₂×C₄	1	2	yes	—
54	C₂₉×#95	C₂×C₄	1	4	yes	—
55	C₂×Q₈×C₂₉	—	2	2	no	—
56	C₂×Q₈×C₂₉	—	4	2	no	—
57	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₂₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₂₉	—	2	2	no	—
66	C₂×C₈×C₂₉	—	2	2	no	—
67	C₂×C₈×C₂₉	—	2	2	no	—
68	C₂×C₈×C₂₉	—	2	2	no	—
69	C₂×C₈×C₂₉	—	2	4	no	—
70	C₂×C₈×C₂₉	—	2	4	no	—
71	C₂×C₈×C₂₉	—	2	4	no	—
72	C₂×C₈×C₂₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₂₉	#41	1	4	yes	—
74	C₄⋊C₄×C₂₉	—	2	2	no	—
75	C₄⋊C₄×C₂₉	—	2	4	no	—
76	C₄⋊C₄×C₂₉	—	2	8	no	—
77	C₄⋊C₄×C₂₉	—	4	2	no	—
78	C₄²×C₂₉	—	2	4	no	—
79	C₄²×C₂₉	—	2	4	no	—
80	C₄²×C₂₉	—	2	2	no	—
81	C₄²×C₂₉	—	2	4	no	—
82	C₄²×C₂₉	—	4	2	no	—
83	C₄²×C₂₉	C₂×Q₈	1	2	yes	—
84	C₄²×C₂₉	#41	1	4	yes	—
85	C₄²×C₂₉	#41	1	2	yes	—
86	#21344	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₉	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₂₉	—	4	4	no	—
89	C₂×C₄×C₂₉	—	2	2	no	—
90	C₂×C₄×C₂₉	—	2	2	no	—
91	C₂×C₄×C₂₉	—	2	2	no	—
92	C₂×C₄×C₂₉	—	2	2	no	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	—	2	2	no	—
95	C₂×C₄×C₂₉	—	2	2	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	2	4	no	—
98	C₂×C₄×C₂₉	—	2	4	no	—
99	C₂×C₄×C₂₉	—	2	4	no	—
100	C₂×C₄×C₂₉	—	2	4	no	—
101	C₂×C₄×C₂₉	—	2	4	no	—
102	C₂×C₄×C₂₉	#117	1	2	yes	—
103	C₂×C₄×C₂₉	#117	1	2	yes	—
104	C₂×C₄×C₂₉	C₄×Q₈	1	4	yes	—
105	C₂×C₄×C₂₉	#131	1	4	yes	—
106	C₂×C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
109	C₂×C₄×C₂₉	C₂×#41	1	2	yes	—
110	C₈×C₂₉	—	2	4	no	—
111	C₈×C₂₉	—	2	4	no	—
112	C₈×C₂₉	—	4	4	no	—
113	#2295	C₂×C₂₉	1	1	yes	characteristic
114	#1985	C₂×C₂₉	1	2	yes	—
115	#1940	C₂×C₂₉	1	2	yes	—
116	#1916	C₂×C₂₉	1	4	yes	—
117	#1768	C₂×C₂₉	1	2	yes	—
118	C₄×C₄⋊C₈	C₂×C₂₉	1	1	yes	characteristic
119	#1746	C₂×C₂₉	1	2	yes	—
120	C₄×#97	C₂×C₂₉	1	1	yes	characteristic
121	C₂²×C₂₉	C₂×#117	1	1	yes	characteristic
122	C₂²×C₂₉	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂²×C₂₉	C₄×#41	1	1	yes	characteristic
124	C₂²×C₂₉	C₂×#131	1	2	yes	—
125	C₂²×C₂₉	#568	1	2	yes	—
126	C₄×C₂₉	—	2	2	no	—
127	C₄×C₂₉	—	2	2	no	—
128	C₄×C₂₉	—	2	4	no	—
129	C₄×C₂₉	—	2	4	no	—
130	C₄×C₂₉	—	4	2	no	—
131	C₄×C₂₉	—	4	2	no	—
132	C₄×C₂₉	—	4	4	no	—
133	C₄×C₂₉	#529	1	2	yes	—
134	C₄×C₂₉	#533	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₂₉	1	1	yes	characteristic
136	C₂×C₄⋊C₈	C₂²×C₂₉	1	4	yes	—
137	C₂×C₄⋊C₈	C₂²×C₂₉	1	2	yes	—
138	C₂×#97	C₂²×C₂₉	1	2	yes	—
139	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
140	C₂×#97	C₂²×C₂₉	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₂₉	1	2	yes	—
143	#391	C₄×C₂₉	1	2	yes	—
144	#389	C₄×C₂₉	1	2	yes	—
145	#387	C₄×C₂₉	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₂₉	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₂₉	1	1	yes	characteristic
148	#376	C₂²×C₂₉	1	2	yes	—
149	C₄³	C₂²×C₂₉	1	1	yes	characteristic
150	#336	C₂²×C₂₉	1	2	yes	—
151	#336	C₂²×C₂₉	1	4	yes	—
152	#336	C₂²×C₂₉	1	4	yes	—
153	#336	C₂²×C₂₉	1	4	yes	—
154	C₂×C₂₉	#2295	1	1	yes	characteristic
155	C₂×C₂₉	#2921	1	1	yes	characteristic
156	C₂×C₂₉	#2922	1	1	yes	characteristic
157	C₂×C₂₉	#2962	1	2	yes	—
158	C₂×C₂₉	#2968	1	2	yes	—
159	C₂²×Q₈	C₂×C₄×C₂₉	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
161	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
162	C₂²×C₈	C₂³×C₂₉	1	2	yes	—
163	C₄×Q₈	—	2	4	no	—
164	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	2	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	4	yes	—
167	C₂×C₄⋊C₄	C₂×C₄×C₂₉	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
169	C₂×C₄²	C₂×C₄×C₂₉	1	2	yes	—
170	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
171	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
172	C₂×C₄²	C₂³×C₂₉	1	2	yes	—
173	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₂₉	1	1	yes	characteristic
175	C₄⋊C₈	—	2	8	no	—
176	C₄⋊C₈	—	2	4	no	—
177	#97	—	2	4	no	—
178	#97	—	2	2	no	—
179	#97	Q₈×C₂₉	1	4	yes	complemented
180	C₄×C₈	—	2	4	no	—
181	C₄×C₈	—	2	4	no	—
182	#95	C₂×C₄×C₂₉	1	4	yes	—
183	#95	C₂×C₄×C₂₉	1	4	yes	—
184	#95	C₂×C₄×C₂₉	1	2	yes	—
185	#95	C₂×C₄×C₂₉	1	4	yes	—
186	C₂₉	#21344	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	C₂²×C₄×C₂₉	1	4	yes	—
190	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
191	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
192	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₄×C₂₉	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₂₉	1	2	yes	—
196	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	C₂×Q₈×C₂₉	1	4	yes	—
205	C₂×C₈	#41×C₂₉	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	8	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	4	no	—
213	C₄²	—	2	4	no	—
214	C₄²	—	4	2	no	—
215	C₄²	C₂×Q₈×C₂₉	1	2	yes	—
216	C₄²	#41×C₂₉	1	4	yes	—
217	C₄²	#41×C₂₉	1	2	yes	—
218	C₂³	C₂³×C₄×C₂₉	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	C₂₉×#117	1	2	yes	—
234	C₂×C₄	C₂₉×#117	1	2	yes	—
235	C₂×C₄	C₄×Q₈×C₂₉	1	4	yes	—
236	C₂×C₄	C₂₉×#131	1	4	yes	—
237	C₂×C₄	C₂²×Q₈×C₂₉	1	1	yes	characteristic
238	C₂×C₄	C₂×#41×C₂₉	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
240	C₂×C₄	C₂×#41×C₂₉	1	2	yes	—
241	C₈	—	2	4	no	—
242	C₈	—	2	4	no	—
243	C₈	—	4	4	no	—
244	C₂²	C₂×C₂₉×#117	1	1	yes	characteristic
245	C₂²	C₂×C₄×Q₈×C₂₉	1	1	yes	characteristic
246	C₂²	C₄×#41×C₂₉	1	1	yes	characteristic
247	C₂²	C₂×C₂₉×#131	1	2	yes	—
248	C₂²	C₂₉×#568	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	C₂₉×#529	1	2	yes	—
257	C₄	C₂₉×#533	1	2	yes	—
258	C₂	C₂₉×#2295	1	1	yes	characteristic
259	C₂	C₂₉×#2921	1	1	yes	characteristic
260	C₂	C₂₉×#2922	1	1	yes	characteristic
261	C₂	C₂₉×#2962	1	2	yes	—
262	C₂	C₂₉×#2968	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
48. C₂₉×#97	Q₈	semidirect
86. #21344	C₂₉	direct (N × H)
179. #97	Q₈×C₂₉	semidirect
186. C₂₉	#21344	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×#2295	2
C₂₉×#1985	2
C₂₉×#1940	2
C₂₉×#1916	2
C₂₉×#1768	2
C₄×C₂₉×C₄⋊C₈	2
C₂₉×#1746	2
C₄×C₂₉×#97	2
#21344	29

Subgroups — C29×#21344 #20316244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₉×#21344 #20316244