Subgroups — C₂×C₂₉×#2238 #20328554

Order 7424.

604 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₉×#469	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₉×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×#2238	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₉×#365	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₉×#361	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₉×#349	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₂₉	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₉	—	2	1	no	—
9	C₂²×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×C₂₉×#121	—	2	1	no	—
11	C₂×C₂₉×M₃₂	C₂²	1	2	yes	—
12	C₂₉×#469	—	2	2	no	—
13	C₂₉×#469	C₂²	1	4	yes	—
14	C₂₉×#430	—	2	2	no	—
15	C₂₉×#430	C₂²	1	4	yes	complemented
16	C₂×C₂₉×#108	—	2	1	no	—
17	C₂×C₂₉×#108	C₂²	1	2	yes	—
18	C₂×C₂₉×#106	C₂²	1	1	yes	characteristic
19	C₂×C₂₉×#102	—	2	1	no	—
20	C₂₉×#365	C₂²	1	8	yes	complemented
21	C₂₉×#361	C₂²	1	8	yes	complemented
22	C₂₉×#349	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₂₉	—	2	1	no	—
24	C₂²×D₄×C₂₉	—	4	1	no	—
25	C₂³×C₄×C₂₉	D₄	1	1	yes	characteristic
26	C₂×D₈×C₂₉	—	2	1	no	—
27	C₂×D₈×C₂₉	—	2	2	no	—
28	C₂×D₈×C₂₉	—	2	2	no	—
29	C₂×D₈×C₂₉	C₂³	1	2	yes	—
30	C₂×M₁₆×C₂₉	—	2	2	no	—
31	C₂×M₁₆×C₂₉	—	2	2	no	—
32	C₂×M₁₆×C₂₉	—	2	4	no	—
33	C₂×M₁₆×C₂₉	—	2	2	no	—
34	C₂×M₁₆×C₂₉	—	4	2	no	—
35	C₂×M₁₆×C₂₉	—	4	1	no	—
36	C₂×M₁₆×C₂₉	D₄	1	2	yes	—
37	C₂×M₁₆×C₂₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₉	—	4	1	no	—
39	C₂²×C₈×C₂₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₉	C₂³	1	1	yes	characteristic
41	C₂₉×#121	—	2	4	no	—
42	C₂₉×#121	—	4	2	no	—
43	C₂×C₄⋊C₄×C₂₉	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₂₉	—	4	1	no	—
45	C₂₉×M₃₂	—	2	2	no	—
46	C₂₉×M₃₂	C₂³	1	4	yes	—
47	C₂×C₁₆×C₂₉	—	2	2	no	—
48	C₂₉×#108	—	2	4	no	—
49	C₂₉×#108	—	2	2	no	—
50	C₂₉×#108	C₂³	1	4	yes	—
51	C₂₉×#106	—	2	1	no	—
52	C₂₉×#106	C₂³	1	2	yes	—
53	C₂₉×#102	—	2	4	no	—
54	C₂⁴×C₂₉	—	4	1	no	—
55	C₂⁴×C₂₉	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₂₉	—	2	2	no	—
57	C₂×D₄×C₂₉	—	4	4	no	—
58	C₂×D₄×C₂₉	—	4	1	no	—
59	C₂×D₄×C₂₉	—	4	1	no	—
60	C₂×D₄×C₂₉	—	4	1	no	—
61	C₂×D₄×C₂₉	—	4	1	no	—
62	C₂×D₄×C₂₉	—	4	1	no	—
63	C₂×D₄×C₂₉	—	8	4	no	—
64	C₂²×C₄×C₂₉	—	2	1	no	—
65	C₂²×C₄×C₂₉	—	2	1	no	—
66	C₂²×C₄×C₂₉	—	2	1	no	—
67	C₂²×C₄×C₂₉	—	2	1	no	—
68	C₂²×C₄×C₂₉	—	4	1	no	—
69	C₂²×C₄×C₂₉	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₂₉	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₂₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
73	D₈×C₂₉	—	2	4	no	—
74	D₈×C₂₉	—	4	2	no	—
75	M₁₆×C₂₉	—	2	4	no	—
76	M₁₆×C₂₉	—	2	4	no	—
77	M₁₆×C₂₉	—	2	4	no	—
78	M₁₆×C₂₉	—	4	4	no	—
79	M₁₆×C₂₉	—	4	2	no	—
80	C₂×C₈×C₂₉	—	2	1	no	—
81	C₂×C₈×C₂₉	—	2	1	no	—
82	C₂×C₈×C₂₉	—	4	4	no	—
83	C₂×C₈×C₂₉	—	4	2	no	—
84	C₂×C₈×C₂₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₂₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₂₉	#41	1	2	yes	—
87	C₂×C₈×C₂₉	#41	1	2	yes	—
88	C₂×C₈×C₂₉	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₂₉	—	2	2	no	—
90	C₄⋊C₄×C₂₉	—	4	1	no	—
91	C₂²⋊C₄×C₂₉	—	4	4	no	—
92	C₁₆×C₂₉	—	2	4	no	—
93	C₂×#2238	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₂₉	—	2	1	no	—
95	C₂³×C₂₉	—	2	1	no	—
96	C₂³×C₂₉	—	4	2	no	—
97	C₂³×C₂₉	—	4	2	no	—
98	C₂³×C₂₉	—	8	2	no	—
99	C₂³×C₂₉	—	8	2	no	—
100	C₂³×C₂₉	—	8	1	no	—
101	C₂³×C₂₉	—	8	1	no	—
102	C₂³×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂³×C₂₉	C₂×SD₁₆	1	2	yes	—
104	D₄×C₂₉	—	4	2	no	—
105	D₄×C₂₉	—	4	2	no	—
106	D₄×C₂₉	—	8	8	no	—
107	D₄×C₂₉	—	8	1	no	—
108	D₄×C₂₉	—	8	1	no	—
109	C₂×C₄×C₂₉	—	2	1	no	—
110	C₂×C₄×C₂₉	—	2	1	no	—
111	C₂×C₄×C₂₉	—	2	2	no	—
112	C₂×C₄×C₂₉	—	2	2	no	—
113	C₂×C₄×C₂₉	—	2	2	no	—
114	C₂×C₄×C₂₉	—	2	2	no	—
115	C₂×C₄×C₂₉	—	4	2	no	—
116	C₂×C₄×C₂₉	—	8	2	no	—
117	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₂₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₂₉	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₂₉	—	2	2	no	—
125	C₈×C₂₉	—	4	4	no	—
126	C₈×C₂₉	#123	1	2	yes	—
127	C₈×C₂₉	C₂×#41	1	2	yes	—
128	C₂×#469	C₂×C₂₉	1	1	yes	characteristic
129	C₂×#430	C₂×C₂₉	1	1	yes	characteristic, complemented
130	#2238	C₂×C₂₉	1	8	yes	direct product factor
131	C₂×#365	C₂×C₂₉	1	2	yes	complemented
132	C₂×#361	C₂×C₂₉	1	2	yes	complemented
133	C₂×#349	C₂×C₂₉	1	1	yes	characteristic, complemented
134	C₂²×C₂₉	—	2	2	no	—
135	C₂²×C₂₉	—	2	2	no	—
136	C₂²×C₂₉	—	4	2	no	—
137	C₂²×C₂₉	—	4	2	no	—
138	C₂²×C₂₉	—	8	1	no	—
139	C₂²×C₂₉	—	8	2	no	—
140	C₂²×C₂₉	—	8	2	no	—
141	C₂²×C₂₉	—	8	2	no	—
142	C₂²×C₂₉	—	16	2	no	—
143	C₂²×C₂₉	—	16	2	no	—
144	C₂²×C₂₉	#481	1	1	yes	characteristic
145	C₂²×C₂₉	#481	1	1	yes	characteristic
146	C₂²×C₂₉	C₂²×SD₁₆	1	1	yes	characteristic
147	C₄×C₂₉	—	2	2	no	—
148	C₄×C₂₉	—	8	2	no	—
149	C₄×C₂₉	#481	1	1	yes	characteristic
150	C₄×C₂₉	#481	1	1	yes	characteristic
151	C₄×C₂₉	C₂×#123	1	1	yes	characteristic
152	C₄×C₂₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×D₈	C₂²×C₂₉	1	1	yes	characteristic
154	C₂²×M₁₆	—	2	1	no	—
155	C₂²×M₁₆	C₂²×C₂₉	1	1	yes	characteristic
156	C₂×#121	—	2	1	no	—
157	C₂×M₃₂	C₂²×C₂₉	1	2	yes	—
158	#469	—	2	2	no	—
159	#469	C₂²×C₂₉	1	4	yes	—
160	#430	—	2	2	no	—
161	#430	C₂²×C₂₉	1	4	yes	complemented
162	C₂×#108	—	2	1	no	—
163	C₂×#108	C₂²×C₂₉	1	2	yes	—
164	C₂×#106	C₂²×C₂₉	1	1	yes	characteristic
165	C₂×#102	—	2	1	no	—
166	#365	C₂²×C₂₉	1	8	yes	complemented
167	#361	C₂²×C₂₉	1	8	yes	complemented
168	#349	C₂²×C₂₉	1	4	yes	complemented
169	C₂×C₂₉	—	2	1	no	—
170	C₂×C₂₉	—	2	1	no	—
171	C₂×C₂₉	—	4	2	no	—
172	C₂×C₂₉	—	16	2	no	—
173	C₂×C₂₉	#2238	1	2	yes	direct product factor
174	C₂×C₂₉	C₂×#481	1	1	yes	characteristic
175	C₂²×D₄	—	2	1	no	—
176	C₂²×D₄	—	4	1	no	—
177	C₂³×C₄	D₄×C₂₉	1	1	yes	characteristic
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	C₂³×C₂₉	1	2	yes	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	4	2	no	—
187	C₂×M₁₆	—	4	1	no	—
188	C₂×M₁₆	D₄×C₂₉	1	2	yes	—
189	C₂×M₁₆	C₂³×C₂₉	1	2	yes	—
190	C₂²×C₈	—	4	1	no	—
191	C₂²×C₈	D₄×C₂₉	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₂₉	1	1	yes	characteristic
193	#121	—	2	4	no	—
194	#121	—	4	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	4	1	no	—
197	M₃₂	—	2	2	no	—
198	M₃₂	C₂³×C₂₉	1	4	yes	—
199	C₂×C₁₆	—	2	2	no	—
200	#108	—	2	4	no	—
201	#108	—	2	2	no	—
202	#108	C₂³×C₂₉	1	4	yes	—
203	#106	—	2	1	no	—
204	#106	C₂³×C₂₉	1	2	yes	—
205	#102	—	2	4	no	—
206	C₂₉	C₂×#2238	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	4	1	no	—
208	C₂⁴	SD₁₆×C₂₉	1	1	yes	characteristic
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	4	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	8	4	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	SD₁₆×C₂₉	1	1	yes	characteristic
223	C₂²×C₄	SD₁₆×C₂₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₂₉	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₂₉	1	1	yes	characteristic
226	D₈	—	2	4	no	—
227	D₈	—	4	2	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	4	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×D₄×C₂₉	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₂₉	1	1	yes	characteristic
239	C₂×C₈	#41×C₂₉	1	2	yes	—
240	C₂×C₈	#41×C₂₉	1	2	yes	—
241	C₂×C₈	C₂⁴×C₂₉	1	1	yes	characteristic
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	1	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₁₆	—	2	4	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	—	8	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	1	no	—
254	C₂³	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
255	C₂³	C₂×SD₁₆×C₂₉	1	2	yes	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	8	8	no	—
259	D₄	—	8	1	no	—
260	D₄	—	8	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
270	C₂×C₄	C₂₉×#123	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₂₉	1	2	yes	—
272	C₂×C₄	C₂×SD₁₆×C₂₉	1	1	yes	characteristic
273	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂×C₄	C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₄×C₂₉	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	4	4	no	—
278	C₈	C₂₉×#123	1	2	yes	—
279	C₈	C₂×#41×C₂₉	1	2	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	8	1	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	16	2	no	—
289	C₂²	—	16	2	no	—
290	C₂²	C₂₉×#481	1	1	yes	characteristic
291	C₂²	C₂₉×#481	1	1	yes	characteristic
292	C₂²	C₂²×SD₁₆×C₂₉	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₂₉×#481	1	1	yes	characteristic
296	C₄	C₂₉×#481	1	1	yes	characteristic
297	C₄	C₂×C₂₉×#123	1	1	yes	characteristic
298	C₄	C₂×C₂₉×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	4	2	no	—
302	C₂	—	16	2	no	—
303	C₂	C₂₉×#2238	1	2	yes	direct product factor
304	C₂	C₂×C₂₉×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₉	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₉
prime radical	C₂×C₂₉×#469	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₉ ◁ C₂²×C₄×C₂₉ ◁ C₂²×M₁₆×C₂₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₉×#430	C₂	semidirect
3. C₂₉×#2238	C₂	direct (N × H)
4. C₂×C₂₉×#365	C₂	semidirect
5. C₂×C₂₉×#361	C₂	semidirect
6. C₂×C₂₉×#349	C₂	semidirect
15. C₂₉×#430	C₂²	semidirect
20. C₂₉×#365	C₂²	semidirect
21. C₂₉×#361	C₂²	semidirect
22. C₂₉×#349	C₂²	semidirect
93. C₂×#2238	C₂₉	direct (N × H)
129. C₂×#430	C₂×C₂₉	semidirect
130. #2238	C₂×C₂₉	direct (N × H)
131. C₂×#365	C₂×C₂₉	semidirect
132. C₂×#361	C₂×C₂₉	semidirect
133. C₂×#349	C₂×C₂₉	semidirect
161. #430	C₂²×C₂₉	semidirect
166. #365	C₂²×C₂₉	semidirect
167. #361	C₂²×C₂₉	semidirect
168. #349	C₂²×C₂₉	semidirect
173. C₂×C₂₉	#2238	direct (N × H)
206. C₂₉	C₂×#2238	direct (N × H)
303. C₂	C₂₉×#2238	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×C₂₉×#469	2
C₂×C₂₉×#430	2
C₂₉×#2238	2
C₂×C₂₉×#365	2
C₂×C₂₉×#361	2
C₂×C₂₉×#349	2
C₂×#2238	29

Subgroups — C2×C29×#2238 #20328554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₉×#2238 #20328554