Subgroups — C₂²×C₂₉×#341 #20315249

Order 7424.

2238 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₂₉	C₂	1	2	yes	maximal
2	C₂⁴×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₉×#341	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₂₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₂₉	—	2	8	no	—
6	C₂²×M₁₆×C₂₉	—	2	6	no	—
7	C₂²×M₁₆×C₂₉	C₄	1	16	yes	—
8	C₂²×M₁₆×C₂₉	C₂²	1	12	yes	—
9	C₂³×C₈×C₂₉	—	2	8	no	—
10	C₂³×C₈×C₂₉	—	2	2	no	—
11	C₂³×C₈×C₂₉	C₄	1	2	yes	—
12	C₂³×C₈×C₂₉	C₄	1	6	yes	—
13	C₂³×C₈×C₂₉	C₂²	1	6	yes	—
14	C₂₉×#341	C₂²	1	16	yes	direct product factor
15	C₂⁵×C₂₉	Q₈	1	1	yes	characteristic
16	C₂³×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₂₉	C₂×C₄	1	3	yes	—
18	C₂³×C₄×C₂₉	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₂₉	D₄	1	8	yes	—
20	C₂³×C₄×C₂₉	D₄	1	3	yes	—
21	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₉	Q₈	1	3	yes	—
23	C₂³×C₄×C₂₉	C₂³	1	3	yes	—
24	C₂×M₁₆×C₂₉	—	2	48	no	—
25	C₂×M₁₆×C₂₉	—	2	24	no	—
26	C₂×M₁₆×C₂₉	—	2	12	no	—
27	C₂×M₁₆×C₂₉	C₂×C₄	1	48	yes	—
28	C₂×M₁₆×C₂₉	C₂³	1	8	yes	—
29	C₂²×C₈×C₂₉	—	2	24	no	—
30	C₂²×C₈×C₂₉	—	2	8	no	—
31	C₂²×C₈×C₂₉	—	2	16	no	—
32	C₂²×C₈×C₂₉	—	2	24	no	—
33	C₂²×C₈×C₂₉	—	2	12	no	—
34	C₂²×C₈×C₂₉	C₂×C₄	1	12	yes	—
35	C₂²×C₈×C₂₉	C₂×C₄	1	12	yes	—
36	C₂²×C₈×C₂₉	C₂³	1	4	yes	—
37	C₂⁴×C₂₉	—	2	8	no	—
38	C₂⁴×C₂₉	C₄⋊C₄	1	8	yes	—
39	C₂⁴×C₂₉	C₄⋊C₄	1	3	yes	—
40	C₂⁴×C₂₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂⁴×C₂₉	C₂×Q₈	1	3	yes	—
42	C₂²×C₄×C₂₉	C₄²	1	12	yes	—
43	C₂²×C₄×C₂₉	C₄²	1	4	yes	—
44	C₂²×C₄×C₂₉	C₂²⋊C₄	1	12	yes	—
45	C₂²×C₄×C₂₉	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₂₉	C₂²⋊C₄	1	8	yes	—
47	C₂²×C₄×C₂₉	C₂²⋊C₄	1	24	yes	—
48	C₂²×C₄×C₂₉	C₄⋊C₄	1	6	yes	—
49	C₂²×C₄×C₂₉	C₄⋊C₄	1	24	yes	—
50	C₂²×C₄×C₂₉	C₄⋊C₄	1	3	yes	—
51	C₂²×C₄×C₂₉	C₄⋊C₄	1	3	yes	—
52	C₂²×C₄×C₂₉	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₂₉	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₂₉	C₂²×C₄	1	12	yes	—
55	C₂²×C₄×C₂₉	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₂₉	C₂×D₄	1	12	yes	—
57	C₂²×C₄×C₂₉	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₂₉	C₂×Q₈	1	3	yes	—
59	C₂²×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
60	M₁₆×C₂₉	—	2	48	no	—
61	M₁₆×C₂₉	—	2	8	no	—
62	M₁₆×C₂₉	C₂²×C₄	1	16	yes	—
63	C₂×C₈×C₂₉	—	2	24	no	—
64	C₂×C₈×C₂₉	—	2	24	no	—
65	C₂×C₈×C₂₉	—	2	48	no	—
66	C₂×C₈×C₂₉	—	2	8	no	—
67	C₂×C₈×C₂₉	—	2	8	no	—
68	C₂×C₈×C₂₉	C₂²×C₄	1	8	yes	—
69	C₂²×#341	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₂₉	—	2	32	no	—
71	C₂³×C₂₉	—	2	12	no	—
72	C₂³×C₂₉	—	2	12	no	—
73	C₂³×C₂₉	#95	1	12	yes	—
74	C₂³×C₂₉	#95	1	4	yes	—
75	C₂³×C₂₉	#108	1	8	yes	—
76	C₂³×C₂₉	C₂×C₄⋊C₄	1	3	yes	—
77	C₂³×C₂₉	C₂×C₄⋊C₄	1	12	yes	—
78	C₂³×C₂₉	C₂×C₄⋊C₄	1	3	yes	—
79	C₂³×C₂₉	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₂₉	#95	1	24	yes	—
81	C₂×C₄×C₂₉	#95	1	12	yes	—
82	C₂×C₄×C₂₉	#95	1	12	yes	—
83	C₂×C₄×C₂₉	C₂×C₄²	1	6	yes	—
84	C₂×C₄×C₂₉	C₂×C₄²	1	6	yes	—
85	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	6	yes	—
86	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	6	yes	—
87	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	12	yes	—
88	C₂×C₄×C₂₉	C₂×C₂²⋊C₄	1	12	yes	—
89	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	12	yes	—
90	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₂₉	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	C₂²×D₄	1	2	yes	—
96	C₈×C₂₉	—	2	8	no	—
97	C₈×C₂₉	—	2	16	no	—
98	C₂³×M₁₆	C₂×C₂₉	1	2	yes	—
99	C₂⁴×C₈	C₂×C₂₉	1	1	yes	characteristic
100	C₂×#341	C₂×C₂₉	1	12	yes	direct product factor
101	C₂²×C₂₉	—	2	6	no	—
102	C₂²×C₂₉	—	2	2	no	—
103	C₂²×C₂₉	—	2	24	no	—
104	C₂²×C₂₉	—	2	24	no	—
105	C₂²×C₂₉	#341	1	16	yes	direct product factor
106	C₂²×C₂₉	C₂×#95	1	6	yes	—
107	C₂²×C₂₉	C₂×#95	1	6	yes	—
108	C₂²×C₂₉	C₂×#108	1	12	yes	—
109	C₂²×C₂₉	C₂²×C₄⋊C₄	1	2	yes	—
110	C₂²×C₂₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₂₉	C₂×#95	1	6	yes	—
112	C₄×C₂₉	C₂×#95	1	6	yes	—
113	C₄×C₂₉	C₂²×C₄²	1	1	yes	characteristic
114	C₄×C₂₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₂₉	C₂²×C₂²⋊C₄	1	2	yes	—
116	C₂⁴×C₄	C₂²×C₂₉	1	1	yes	characteristic
117	C₂²×M₁₆	—	2	8	no	—
118	C₂²×M₁₆	—	2	6	no	—
119	C₂²×M₁₆	C₄×C₂₉	1	16	yes	—
120	C₂²×M₁₆	C₂²×C₂₉	1	12	yes	—
121	C₂³×C₈	—	2	8	no	—
122	C₂³×C₈	—	2	2	no	—
123	C₂³×C₈	C₄×C₂₉	1	6	yes	—
124	C₂³×C₈	C₄×C₂₉	1	2	yes	—
125	C₂³×C₈	C₂²×C₂₉	1	6	yes	—
126	#341	C₂²×C₂₉	1	16	yes	direct product factor
127	C₂×C₂₉	—	2	2	no	—
128	C₂×C₂₉	—	2	6	no	—
129	C₂×C₂₉	C₂×#341	1	12	yes	direct product factor
130	C₂×C₂₉	C₂²×#95	1	1	yes	characteristic
131	C₂×C₂₉	C₂²×#108	1	2	yes	—
132	C₂⁵	Q₈×C₂₉	1	1	yes	characteristic
133	C₂³×C₄	C₂×C₄×C₂₉	1	3	yes	—
134	C₂³×C₄	C₂×C₄×C₂₉	1	1	yes	characteristic
135	C₂³×C₄	C₂×C₄×C₂₉	1	8	yes	—
136	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
137	C₂³×C₄	D₄×C₂₉	1	8	yes	—
138	C₂³×C₄	D₄×C₂₉	1	3	yes	—
139	C₂³×C₄	Q₈×C₂₉	1	3	yes	—
140	C₂³×C₄	C₂³×C₂₉	1	3	yes	—
141	C₂×M₁₆	—	2	48	no	—
142	C₂×M₁₆	—	2	24	no	—
143	C₂×M₁₆	—	2	12	no	—
144	C₂×M₁₆	C₂×C₄×C₂₉	1	48	yes	—
145	C₂×M₁₆	C₂³×C₂₉	1	8	yes	—
146	C₂²×C₈	—	2	24	no	—
147	C₂²×C₈	—	2	8	no	—
148	C₂²×C₈	—	2	16	no	—
149	C₂²×C₈	—	2	24	no	—
150	C₂²×C₈	—	2	12	no	—
151	C₂²×C₈	C₂×C₄×C₂₉	1	12	yes	—
152	C₂²×C₈	C₂×C₄×C₂₉	1	12	yes	—
153	C₂²×C₈	C₂³×C₂₉	1	4	yes	—
154	C₂₉	C₂²×#341	1	1	yes	characteristic, direct product factor, Sylow
155	C₂⁴	—	2	8	no	—
156	C₂⁴	C₄⋊C₄×C₂₉	1	3	yes	—
157	C₂⁴	C₄⋊C₄×C₂₉	1	8	yes	—
158	C₂⁴	C₄⋊C₄×C₂₉	1	1	yes	characteristic
159	C₂⁴	C₂×Q₈×C₂₉	1	3	yes	—
160	C₂²×C₄	C₄²×C₂₉	1	12	yes	—
161	C₂²×C₄	C₄²×C₂₉	1	4	yes	—
162	C₂²×C₄	C₂²⋊C₄×C₂₉	1	12	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₂₉	1	4	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₂₉	1	8	yes	—
165	C₂²×C₄	C₂²⋊C₄×C₂₉	1	24	yes	—
166	C₂²×C₄	C₄⋊C₄×C₂₉	1	6	yes	—
167	C₂²×C₄	C₄⋊C₄×C₂₉	1	24	yes	—
168	C₂²×C₄	C₄⋊C₄×C₂₉	1	3	yes	—
169	C₂²×C₄	C₄⋊C₄×C₂₉	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₂₉	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₂₉	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₂₉	1	12	yes	—
173	C₂²×C₄	C₂×D₄×C₂₉	1	3	yes	—
174	C₂²×C₄	C₂×D₄×C₂₉	1	12	yes	—
175	C₂²×C₄	C₂×D₄×C₂₉	1	3	yes	—
176	C₂²×C₄	C₂×Q₈×C₂₉	1	3	yes	—
177	C₂²×C₄	C₂⁴×C₂₉	1	1	yes	characteristic
178	M₁₆	—	2	48	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	C₂²×C₄×C₂₉	1	16	yes	—
181	C₂×C₈	—	2	24	no	—
182	C₂×C₈	—	2	24	no	—
183	C₂×C₈	—	2	48	no	—
184	C₂×C₈	—	2	8	no	—
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	C₂²×C₄×C₂₉	1	8	yes	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	12	no	—
189	C₂³	—	2	32	no	—
190	C₂³	C₂₉×#95	1	12	yes	—
191	C₂³	C₂₉×#95	1	4	yes	—
192	C₂³	C₂₉×#108	1	8	yes	—
193	C₂³	C₂×C₄⋊C₄×C₂₉	1	12	yes	—
194	C₂³	C₂×C₄⋊C₄×C₂₉	1	3	yes	—
195	C₂³	C₂×C₄⋊C₄×C₂₉	1	3	yes	—
196	C₂³	C₂²×Q₈×C₂₉	1	1	yes	characteristic
197	C₂×C₄	C₂₉×#95	1	24	yes	—
198	C₂×C₄	C₂₉×#95	1	12	yes	—
199	C₂×C₄	C₂₉×#95	1	12	yes	—
200	C₂×C₄	C₂×C₄²×C₂₉	1	6	yes	—
201	C₂×C₄	C₂×C₄²×C₂₉	1	6	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	6	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	6	yes	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	12	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₂₉	1	12	yes	—
206	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₂₉	1	3	yes	—
209	C₂×C₄	C₂³×C₄×C₂₉	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₄×C₂₉	1	2	yes	—
211	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₂₉	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₈	—	2	16	no	—
215	C₂²	—	2	24	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	6	no	—
218	C₂²	—	2	2	no	—
219	C₂²	C₂₉×#341	1	16	yes	direct product factor
220	C₂²	C₂×C₂₉×#95	1	6	yes	—
221	C₂²	C₂×C₂₉×#95	1	6	yes	—
222	C₂²	C₂×C₂₉×#108	1	12	yes	—
223	C₂²	C₂²×C₄⋊C₄×C₂₉	1	2	yes	—
224	C₂²	C₂²×C₄⋊C₄×C₂₉	1	1	yes	characteristic
225	C₄	C₂×C₂₉×#95	1	6	yes	—
226	C₄	C₂×C₂₉×#95	1	6	yes	—
227	C₄	C₂²×C₄²×C₂₉	1	1	yes	characteristic
228	C₄	C₂²×C₂²⋊C₄×C₂₉	1	1	yes	characteristic
229	C₄	C₂²×C₂²⋊C₄×C₂₉	1	2	yes	—
230	C₂	—	2	2	no	—
231	C₂	—	2	6	no	—
232	C₂	C₂×C₂₉×#341	1	12	yes	direct product factor
233	C₂	C₂²×C₂₉×#95	1	1	yes	characteristic
234	C₂	C₂²×C₂₉×#108	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₂₉	D₄
commutator	C₄	C₂²×C₄²×C₂₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₉
socle	C₂⁴×C₂₉	C₄⋊C₄
prime radical	C₂⁵×C₂₉	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₂₉ ◁ C₂⁴×C₄×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₉×#341	C₂	direct (N × H)
14. C₂₉×#341	C₂²	direct (N × H)
69. C₂²×#341	C₂₉	direct (N × H)
100. C₂×#341	C₂×C₂₉	direct (N × H)
105. C₂²×C₂₉	#341	direct (N × H)
126. #341	C₂²×C₂₉	direct (N × H)
129. C₂×C₂₉	C₂×#341	direct (N × H)
154. C₂₉	C₂²×#341	direct (N × H)
219. C₂²	C₂₉×#341	direct (N × H)
232. C₂	C₂×C₂₉×#341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₂₉	2
C₂⁴×C₈×C₂₉	2
C₂×C₂₉×#341	2
C₂²×#341	29

Subgroups — C22×C29×#341 #20315249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₉×#341 #20315249