Subgroups — C₄×C₂₉×#361 #20307544

Order 7424.

498 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#361	C₂	1	2	yes	maximal
3	C₄×C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₉×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₂₉	—	2	1	no	—
8	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
9	C₄×D₈×C₂₉	—	2	1	no	—
10	C₄×D₈×C₂₉	—	2	1	no	—
11	C₄×D₈×C₂₉	C₄	1	2	yes	—
12	C₄×D₈×C₂₉	C₄	1	2	yes	—
13	C₂×C₂₉×#108	C₂²	1	2	yes	—
14	C₂×C₂₉×#107	C₂²	1	1	yes	characteristic
15	C₂×C₂₉×#102	—	2	1	no	—
16	C₄×M₁₆×C₂₉	—	2	1	no	—
17	C₂×C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
18	C₂₉×#361	C₄	1	8	yes	direct product factor
19	C₂₉×#346	—	2	1	no	—
20	C₄×C₁₆×C₂₉	—	2	1	no	—
21	C₂₉×#341	—	2	1	no	—
22	C₂²×D₄×C₂₉	—	2	1	no	—
23	C₂³×C₄×C₂₉	—	4	1	no	—
24	C₂×D₈×C₂₉	—	2	1	no	—
25	C₂×D₈×C₂₉	—	2	2	no	—
26	C₂×D₈×C₂₉	C₂×C₄	1	2	yes	—
27	C₂×D₈×C₂₉	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₂₉	—	2	2	no	—
29	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₂₉	—	2	2	no	—
32	C₄×D₄×C₂₉	—	2	2	no	—
33	C₄×D₄×C₂₉	—	4	1	no	—
34	C₄×D₄×C₂₉	—	4	1	no	—
35	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₂₉	—	4	1	no	—
37	C₂×C₄²×C₂₉	D₄	1	1	yes	characteristic
38	C₂₉×M₃₂	—	2	2	no	—
39	C₂₉×M₃₂	C₂×C₄	1	4	yes	complemented
40	C₂×C₁₆×C₂₉	—	2	2	no	—
41	C₂₉×#108	—	2	2	no	—
42	C₂₉×#108	C₂×C₄	1	4	yes	complemented
43	C₂₉×#107	—	2	1	no	—
44	C₂₉×#107	C₂×C₄	1	2	yes	—
45	C₂₉×#102	—	2	2	no	—
46	C₂₉×#102	—	2	2	no	—
47	C₂₉×#97	—	4	1	no	—
48	C₄×C₈×C₂₉	—	4	1	no	—
49	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₂₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₂₉	D₄	1	1	yes	characteristic
53	C₂⁴×C₂₉	—	4	1	no	—
54	C₂×D₄×C₂₉	—	2	2	no	—
55	C₂×D₄×C₂₉	—	2	4	no	—
56	C₂×D₄×C₂₉	—	4	1	no	—
57	C₂×D₄×C₂₉	—	4	2	no	—
58	C₂²×C₄×C₂₉	—	4	1	no	—
59	C₂²×C₄×C₂₉	—	4	2	no	—
60	C₂²×C₄×C₂₉	—	4	2	no	—
61	C₂²×C₄×C₂₉	—	8	4	no	—
62	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
65	D₈×C₂₉	—	2	2	no	—
66	D₈×C₂₉	—	2	4	no	—
67	D₈×C₂₉	C₄²	1	4	yes	—
68	M₁₆×C₂₉	—	2	4	no	—
69	M₁₆×C₂₉	—	4	2	no	—
70	C₂×C₈×C₂₉	—	2	2	no	—
71	C₂×C₈×C₂₉	—	4	2	no	—
72	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₂₉	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₂₉	#41	1	2	yes	—
78	C₄⋊C₄×C₂₉	—	2	2	no	—
79	C₄⋊C₄×C₂₉	—	4	1	no	—
80	C₂²⋊C₄×C₂₉	—	4	2	no	—
81	C₂²⋊C₄×C₂₉	—	4	2	no	—
82	C₄²×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₂₉	D₈	1	1	yes	characteristic
85	C₄²×C₂₉	D₈	1	1	yes	characteristic
86	C₁₆×C₂₉	—	2	4	no	—
87	C₄×#361	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₉	—	4	2	no	—
89	C₂³×C₂₉	—	4	4	no	—
90	C₂³×C₂₉	—	8	2	no	—
91	C₂³×C₂₉	—	8	2	no	—
92	C₂³×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
93	D₄×C₂₉	—	2	4	no	—
94	D₄×C₂₉	—	4	2	no	—
95	D₄×C₂₉	—	4	4	no	—
96	C₂×C₄×C₂₉	—	2	2	no	—
97	C₂×C₄×C₂₉	—	4	1	no	—
98	C₂×C₄×C₂₉	—	4	2	no	—
99	C₂×C₄×C₂₉	—	8	4	no	—
100	C₂×C₄×C₂₉	—	8	2	no	—
101	C₂×C₄×C₂₉	—	8	4	no	—
102	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
103	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
104	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
105	C₂×C₄×C₂₉	#102	1	1	yes	characteristic
106	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₉	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₂₉	#135	1	2	yes	—
110	C₈×C₂₉	—	2	1	no	—
111	C₈×C₂₉	—	2	1	no	—
112	C₈×C₂₉	—	4	4	no	—
113	C₈×C₂₉	C₂×C₄²	1	1	yes	characteristic
114	C₈×C₂₉	#117	1	1	yes	characteristic
115	C₈×C₂₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×D₈	C₂×C₂₉	1	1	yes	characteristic
117	C₂×#361	C₂×C₂₉	1	2	yes	—
118	C₄×M₃₂	C₂×C₂₉	1	1	yes	characteristic, complemented
119	C₄×#108	C₂×C₂₉	1	1	yes	characteristic, complemented
120	#1387	C₂×C₂₉	1	2	yes	complemented
121	C₂²×C₂₉	—	2	1	no	—
122	C₂²×C₂₉	—	2	1	no	—
123	C₂²×C₂₉	—	4	4	no	—
124	C₂²×C₂₉	—	8	8	no	—
125	C₂²×C₂₉	—	8	2	no	—
126	C₂²×C₂₉	—	8	2	no	—
127	C₂²×C₂₉	C₂×#102	1	1	yes	characteristic
128	C₂²×C₂₉	C₄×SD₁₆	1	2	yes	—
129	C₄×C₂₉	—	2	1	no	—
130	C₄×C₂₉	—	2	1	no	—
131	C₄×C₂₉	—	2	1	no	—
132	C₄×C₂₉	—	8	2	no	—
133	C₄×C₂₉	#361	1	2	yes	direct product factor
134	C₄×C₂₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₂₉	#416	1	1	yes	characteristic
136	C₄×C₂₉	C₄×D₈	1	2	yes	—
137	C₂²×D₈	C₂²×C₂₉	1	1	yes	characteristic
138	C₂×C₄×D₄	—	2	1	no	—
139	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
140	C₄×D₈	—	2	1	no	—
141	C₄×D₈	—	2	1	no	—
142	C₄×D₈	C₄×C₂₉	1	2	yes	—
143	C₄×D₈	C₄×C₂₉	1	2	yes	—
144	C₂×#108	C₂²×C₂₉	1	2	yes	—
145	C₂×#107	C₂²×C₂₉	1	1	yes	characteristic
146	C₂×#102	—	2	1	no	—
147	C₄×M₁₆	—	2	1	no	—
148	C₂×C₄×C₈	C₂²×C₂₉	1	1	yes	characteristic
149	#361	C₄×C₂₉	1	8	yes	direct product factor
150	#346	—	2	1	no	—
151	C₄×C₁₆	—	2	1	no	—
152	#341	—	2	1	no	—
153	C₂×C₂₉	—	2	2	no	—
154	C₂×C₂₉	—	8	4	no	—
155	C₂×C₂₉	C₄×#102	1	1	yes	characteristic
156	C₂×C₂₉	C₂×#361	1	1	yes	characteristic
157	C₂×C₂₉	#2148	1	1	yes	characteristic
158	C₂²×D₄	—	2	1	no	—
159	C₂³×C₄	—	4	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	2	no	—
162	C₂×D₈	C₂×C₄×C₂₉	1	4	yes	—
163	C₂×D₈	C₂×C₄×C₂₉	1	2	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₂²⋊C₄	—	4	1	no	—
173	C₂×C₄²	D₄×C₂₉	1	1	yes	characteristic
174	M₃₂	—	2	2	no	—
175	M₃₂	C₂×C₄×C₂₉	1	4	yes	complemented
176	C₂×C₁₆	—	2	2	no	—
177	#108	—	2	2	no	—
178	#108	C₂×C₄×C₂₉	1	4	yes	complemented
179	#107	—	2	1	no	—
180	#107	C₂×C₄×C₂₉	1	2	yes	—
181	#102	—	2	2	no	—
182	#102	—	2	2	no	—
183	#97	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₂₉	1	1	yes	characteristic
187	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
188	C₄×C₈	D₄×C₂₉	1	1	yes	characteristic
189	C₂₉	C₄×#361	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	—	4	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	8	4	no	—
199	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
200	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
202	D₈	—	2	2	no	—
203	D₈	—	2	4	no	—
204	D₈	C₄²×C₂₉	1	4	yes	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₂₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₂₉	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
214	C₂×C₈	#41×C₂₉	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	—	4	1	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
220	C₄²	C₂²⋊C₄×C₂₉	1	1	yes	characteristic
221	C₄²	D₈×C₂₉	1	1	yes	characteristic
222	C₄²	D₈×C₂₉	1	1	yes	characteristic
223	C₁₆	—	2	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
239	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
240	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
241	C₂×C₄	C₂₉×#102	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₂₉	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₂₉	1	1	yes	characteristic
245	C₂×C₄	C₂₉×#135	1	2	yes	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	4	4	no	—
249	C₈	C₂×C₄²×C₂₉	1	1	yes	characteristic
250	C₈	C₂₉×#117	1	1	yes	characteristic
251	C₈	C₄×D₄×C₂₉	1	2	yes	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	8	8	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×C₂₉×#102	1	1	yes	characteristic
259	C₂²	C₄×SD₁₆×C₂₉	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	8	2	no	—
264	C₄	C₂₉×#361	1	2	yes	direct product factor
265	C₄	C₄×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
266	C₄	C₂₉×#416	1	1	yes	characteristic
267	C₄	C₄×D₈×C₂₉	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₄×C₂₉×#102	1	1	yes	characteristic
271	C₂	C₂×C₂₉×#361	1	1	yes	characteristic
272	C₂	C₂₉×#2148	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₉	#102
commutator	C₈	C₂×C₄²×C₂₉
Frattini, cosocle	C₂²×C₈	C₂³×C₂₉
socle	C₂²×C₂₉	C₂×#102
prime radical	C₂²×D₈×C₂₉	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₉ ◁ C₂×C₄²×C₂₉ ◁ C₂×C₄×C₈×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₉×M₃₂	C₂	semidirect
4. C₄×C₂₉×#108	C₂	semidirect
5. C₂₉×#1387	C₂	semidirect
18. C₂₉×#361	C₄	direct (N × H)
39. C₂₉×M₃₂	C₂×C₄	semidirect
42. C₂₉×#108	C₂×C₄	semidirect
87. C₄×#361	C₂₉	direct (N × H)
118. C₄×M₃₂	C₂×C₂₉	semidirect
119. C₄×#108	C₂×C₂₉	semidirect
120. #1387	C₂×C₂₉	semidirect
133. C₄×C₂₉	#361	direct (N × H)
149. #361	C₄×C₂₉	direct (N × H)
175. M₃₂	C₂×C₄×C₂₉	semidirect
178. #108	C₂×C₄×C₂₉	semidirect
189. C₂₉	C₄×#361	direct (N × H)
264. C₄	C₂₉×#361	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₂₉	2
C₂×C₂₉×#361	2
C₄×C₂₉×M₃₂	2
C₄×C₂₉×#108	2
C₂₉×#1387	2
C₄×#361	29

Subgroups — C4×C29×#361 #20307544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂₉×#361 #20307544