Subgroups — #16828392

Order 3712.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271960	C₂	1	1	yes	maximal, characteristic
2	#131×D₂₉	C₂	1	2	yes	maximal, complemented
3	#14271286	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271283	C₂	1	2	yes	maximal, complemented
5	#14271280	C₂	1	1	yes	maximal, characteristic, complemented
6	#14271279	C₂	1	1	yes	maximal, characteristic, complemented
7	#14271277	C₂	1	2	yes	maximal, complemented
8	#14271210	C₂	1	1	yes	maximal, characteristic, complemented
9	#14271178	C₂	1	2	yes	maximal, complemented
10	#14271120	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₉×#444	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₉	C₂²	1	2	yes	—
13	C₂₉×#135	C₂²	1	1	yes	characteristic
14	C₂₉×#131	C₂²	1	2	yes	complemented
15	#11745312	C₂²	1	2	yes	—
16	#11745310	C₂²	1	2	yes	complemented
17	#11745280	C₄	1	2	yes	complemented
18	#11745277	C₄	1	2	yes	complemented
19	#11745276	C₄	1	2	yes	complemented
20	#11745273	C₄	1	2	yes	complemented
21	#11745269	—	2	2	no	—
22	#11745269	C₂²	1	2	yes	complemented
23	M₁₆×D₂₉	—	2	2	no	—
24	M₁₆×D₂₉	C₂²	1	2	yes	—
25	#11745266	C₂²	1	1	yes	characteristic
26	#11745263	—	2	2	no	—
27	#11745263	C₂²	1	2	yes	complemented
28	C₂×#90772	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₂₉	—	2	2	no	—
30	C₂×C₈×D₂₉	C₂²	1	1	yes	characteristic
31	#11745234	C₂²	1	1	yes	characteristic
32	C₂₉×#108	C₂²	1	1	yes	characteristic, complemented
33	C₂₉×#104	C₂²	1	2	yes	complemented
34	C₂₉×#97	C₂²	1	1	yes	characteristic, complemented
35	#11745185	C₂²	1	2	yes	complemented
36	#11745173	C₂²	1	2	yes	complemented
37	#11745170	C₂²	1	2	yes	complemented
38	#11745167	C₂²	1	1	yes	characteristic, complemented
39	#11745163	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₂₉	C₂²	1	1	yes	characteristic, complemented
41	#11745153	C₂²	1	2	yes	complemented
42	C₄×#6597	C₂²	1	1	yes	characteristic, complemented
43	#41×C₂₉	C₂³	1	2	yes	—
44	#90809	C₂×C₄	1	2	yes	—
45	Q₈×D₂₉	C₂×C₄	1	2	yes	—
46	#90807	—	2	2	no	—
47	#90807	C₂×C₄	1	2	yes	—
48	D₄×D₂₉	—	2	2	no	—
49	D₄×D₂₉	C₂×C₄	1	2	yes	—
50	#90805	—	2	2	no	—
51	#90805	C₂³	1	2	yes	—
52	C₂×C₄×D₂₉	—	2	2	no	—
53	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
54	Q₁₆×C₂₉	C₂×C₄	1	1	yes	characteristic
55	SD₁₆×C₂₉	C₂×C₄	1	2	yes	—
56	D₈×C₂₉	C₂×C₄	1	1	yes	characteristic
57	M₁₆×C₂₉	—	2	2	no	—
58	M₁₆×C₂₉	C₂³	1	2	yes	—
59	C₂×C₈×C₂₉	—	2	2	no	—
60	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
62	C₄²×C₂₉	C₂³	1	1	yes	characteristic
63	#90785	C₂×C₄	1	2	yes	—
64	#90784	C₂×C₄	1	2	yes	—
65	#90783	C₂×C₄	1	2	yes	—
66	#90782	C₂×C₄	1	2	yes	—
67	#90781	—	2	1	no	—
68	#90779	—	2	1	no	—
69	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
70	#90777	—	2	2	no	—
71	#90777	C₂³	1	2	yes	—
72	C₂×#6597	—	2	2	no	—
73	C₂×#6597	C₂³	1	1	yes	characteristic
74	C₂×#6597	C₂³	1	1	yes	characteristic
75	Dic₁₁₆	C₂×C₄	1	1	yes	characteristic
76	D₂₃₂	C₂×C₄	1	1	yes	characteristic
77	#90773	C₂×C₄	1	2	yes	—
78	#90772	—	2	2	no	—
79	#90772	—	2	2	no	—
80	#90772	—	2	2	no	—
81	#90772	C₂×C₄	1	2	yes	—
82	#90772	C₂×C₄	1	2	yes	—
83	C₈×D₂₉	—	2	2	no	—
84	C₈×D₂₉	—	2	2	no	—
85	C₈×D₂₉	—	2	2	no	—
86	C₈×D₂₉	D₄	1	2	yes	—
87	C₈×D₂₉	D₄	1	2	yes	—
88	C₂²×D₂₉	—	2	2	no	—
89	C₂²×D₂₉	#41	1	1	yes	characteristic
90	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
91	D₄×C₂₉	—	2	2	no	—
92	D₄×C₂₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₉	—	2	2	no	—
94	C₂×C₄×C₂₉	—	2	1	no	—
95	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₂₉	—	2	2	no	—
97	C₂²⋊D₂₉	—	2	2	no	—
98	C₂²⋊D₂₉	—	2	2	no	—
99	C₂×Dic₂₉	—	2	2	no	—
100	C₂×Dic₂₉	—	2	1	no	—
101	C₂×Dic₂₉	#41	1	1	yes	characteristic
102	D₁₁₆	—	2	2	no	—
103	D₁₁₆	C₂²×C₄	1	2	yes	—
104	C₄×D₂₉	—	2	2	no	—
105	C₄×D₂₉	—	2	2	no	—
106	C₄×D₂₉	—	2	2	no	—
107	C₄×D₂₉	C₂²×C₄	1	2	yes	—
108	C₄×D₂₉	C₂×D₄	1	2	yes	—
109	Dic₅₈	—	2	2	no	—
110	Dic₅₈	C₂²×C₄	1	2	yes	—
111	C₈×C₂₉	—	2	2	no	—
112	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
113	C₈×C₂₉	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₂₉	C₂×D₄	1	2	yes	—
115	#6597	—	2	2	no	—
116	#6597	C₂²×C₄	1	2	yes	—
117	#6597	C₂×D₄	1	1	yes	characteristic
118	#6597	C₂×D₄	1	1	yes	characteristic
119	#2955	—	29	1	no	maximal, Sylow
120	C₂²×C₂₉	—	2	2	no	—
121	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
122	C₂×D₂₉	—	2	2	no	—
123	C₂×D₂₉	—	4	1	no	—
124	C₂×D₂₉	—	4	4	no	—
125	C₂×D₂₉	C₄×D₄	1	2	yes	—
126	C₄×C₂₉	—	2	2	no	—
127	C₄×C₂₉	—	4	1	no	—
128	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
130	Dic₂₉	—	2	2	no	—
131	Dic₂₉	—	4	1	no	—
132	Dic₂₉	C₄×D₄	1	2	yes	—
133	#575	—	29	1	no	—
134	C₂×#131	—	29	2	no	—
135	#444	—	29	2	no	—
136	#444	—	29	1	no	—
137	#444	D₂₉	1	1	yes	characteristic, complemented
138	#443	—	29	1	no	—
139	#443	—	29	1	no	—
140	#443	—	29	2	no	—
141	#430	—	29	1	no	—
142	#421	—	29	2	no	—
143	#405	—	29	1	no	—
144	C₂×C₂₉	—	2	1	no	—
145	C₂×C₂₉	—	4	2	no	—
146	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
147	D₂₉	—	2	2	no	—
148	D₂₉	—	4	2	no	—
149	C₂×#41	—	29	2	no	—
150	#137	—	29	2	no	—
151	#137	—	29	2	no	—
152	AGL(1,ℤ/8)	—	29	2	no	—
153	AGL(1,ℤ/8)	—	29	2	no	—
154	#135	—	29	2	no	—
155	#135	—	29	1	no	—
156	#135	C₂×D₂₉	1	1	yes	characteristic
157	#131	—	29	2	no	—
158	#131	—	29	2	no	—
159	#131	—	29	2	no	—
160	#131	—	58	2	no	—
161	#131	—	58	2	no	—
162	#131	C₂×D₂₉	1	2	yes	complemented
163	C₂×M₁₆	—	29	1	no	—
164	C₂×M₁₆	—	29	2	no	—
165	C₂×M₁₆	—	58	2	no	—
166	C₂²×C₈	—	29	1	no	—
167	C₂²×C₈	—	58	2	no	—
168	#117	—	29	1	no	—
169	#108	—	29	2	no	—
170	#108	—	29	1	no	—
171	#108	C₂×D₂₉	1	1	yes	characteristic, complemented
172	#104	—	29	2	no	—
173	#104	—	29	2	no	—
174	#104	—	29	2	no	—
175	#104	C₂×D₂₉	1	2	yes	complemented
176	#97	—	29	1	no	—
177	#97	C₂×D₂₉	1	1	yes	characteristic, complemented
178	C₄×C₈	—	29	1	no	—
179	C₄×C₈	—	29	1	no	—
180	C₂₉	#2955	1	1	yes	characteristic, complemented, Sylow
181	#41	—	29	2	no	—
182	#41	—	29	2	no	—
183	#41	—	29	2	no	—
184	#41	—	58	2	no	—
185	#41	—	58	2	no	—
186	#41	C₂²×D₂₉	1	2	yes	—
187	C₂×Q₈	—	29	2	no	—
188	C₂×D₄	—	29	2	no	—
189	C₂×D₄	—	58	2	no	—
190	C₂²×C₄	—	29	1	no	—
191	C₂²×C₄	—	58	2	no	—
192	Q₁₆	—	29	2	no	—
193	Q₁₆	—	29	1	no	—
194	Q₁₆	C₄×D₂₉	1	1	yes	characteristic
195	SD₁₆	—	29	2	no	—
196	SD₁₆	—	29	2	no	—
197	SD₁₆	—	29	2	no	—
198	SD₁₆	C₄×D₂₉	1	2	yes	—
199	D₈	—	29	2	no	—
200	D₈	—	29	1	no	—
201	D₈	C₄×D₂₉	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	29	2	no	—
204	M₁₆	—	29	2	no	—
205	M₁₆	—	29	2	no	—
206	M₁₆	—	58	2	no	—
207	M₁₆	—	58	2	no	—
208	M₁₆	—	58	2	no	—
209	M₁₆	—	58	2	no	—
210	M₁₆	C₂²×D₂₉	1	2	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	29	1	no	—
213	C₂×C₈	—	29	1	no	—
214	C₂×C₈	—	29	2	no	—
215	C₂×C₈	—	29	2	no	—
216	C₂×C₈	—	58	2	no	—
217	C₂×C₈	—	58	2	no	—
218	C₂×C₈	—	58	2	no	—
219	C₂×C₈	—	58	2	no	—
220	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
222	C₄⋊C₄	—	58	1	no	—
223	C₂²⋊C₄	—	58	1	no	—
224	C₄²	—	29	1	no	—
225	C₄²	C₂²×D₂₉	1	1	yes	characteristic
226	C₂³	—	29	1	no	—
227	C₂³	—	58	2	no	—
228	Q₈	—	29	2	no	—
229	Q₈	—	58	2	no	—
230	Q₈	C₂×C₄×D₂₉	1	2	yes	—
231	D₄	—	2	2	no	—
232	D₄	—	29	2	no	—
233	D₄	—	58	2	no	—
234	D₄	—	58	2	no	—
235	D₄	—	58	2	no	—
236	D₄	—	58	2	no	—
237	D₄	C₂×C₄×D₂₉	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	29	1	no	—
241	C₂×C₄	—	29	2	no	—
242	C₂×C₄	—	29	2	no	—
243	C₂×C₄	—	58	2	no	—
244	C₂×C₄	—	58	2	no	—
245	C₂×C₄	—	58	2	no	—
246	C₂×C₄	—	58	2	no	—
247	C₂×C₄	—	58	1	no	—
248	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	29	2	no	—
251	C₈	—	29	1	no	—
252	C₈	—	29	1	no	—
253	C₈	—	58	2	no	—
254	C₈	C₂×C₄×D₂₉	1	1	yes	characteristic
255	C₈	C₂×C₄×D₂₉	1	1	yes	characteristic
256	C₈	D₄×D₂₉	1	2	yes	—
257	C₂²	—	2	2	no	—
258	C₂²	—	29	2	no	—
259	C₂²	—	58	2	no	—
260	C₂²	—	116	1	no	—
261	C₂²	—	116	4	no	—
262	C₂²	#41×D₂₉	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	29	2	no	—
266	C₄	—	58	2	no	—
267	C₄	—	116	1	no	—
268	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
269	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	58	2	no	—
273	C₂	—	116	2	no	—
274	C₂	C₄×D₄×D₂₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#444	C₂
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	#14271960	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	#444	D₂₉
hypocenter	C₂₉	#2955
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. #131×D₂₉	C₂	semidirect
3. #14271286	C₂	semidirect
4. #14271283	C₂	semidirect
5. #14271280	C₂	semidirect
6. #14271279	C₂	semidirect
7. #14271277	C₂	semidirect
8. #14271210	C₂	semidirect
9. #14271178	C₂	semidirect
10. #14271120	C₂	semidirect
11. C₂₉×#444	C₂	semidirect
14. C₂₉×#131	C₂²	semidirect
16. #11745310	C₂²	semidirect
17. #11745280	C₄	semidirect
18. #11745277	C₄	semidirect
19. #11745276	C₄	semidirect
20. #11745273	C₄	semidirect
22. #11745269	C₂²	semidirect
27. #11745263	C₂²	semidirect
32. C₂₉×#108	C₂²	semidirect
33. C₂₉×#104	C₂²	semidirect
34. C₂₉×#97	C₂²	semidirect
35. #11745185	C₂²	semidirect
36. #11745173	C₂²	semidirect
37. #11745170	C₂²	semidirect
38. #11745167	C₂²	semidirect
39. #11745163	C₂²	semidirect
40. C₈×Dic₂₉	C₂²	semidirect
41. #11745153	C₂²	semidirect
42. C₄×#6597	C₂²	semidirect
137. #444	D₂₉	semidirect
162. #131	C₂×D₂₉	semidirect
171. #108	C₂×D₂₉	semidirect
175. #104	C₂×D₂₉	semidirect
177. #97	C₂×D₂₉	semidirect
180. C₂₉	#2955	semidirect

Maximal subgroups

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

subgroup	index
#14271960	2
#131×D₂₉	2
#14271286	2
#14271283	2
#14271280	2
#14271279	2
#14271277	2
#14271210	2
#14271178	2
#14271120	2
C₂₉×#444	2
#2955	29