Subgroups — #16828513

Order 3712.

500 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	C₄×D₄×D₂₉	C₂	1	1	yes	maximal, characteristic
2	C₈×D₁₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#6597	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271228	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₂₉	C₂	1	2	yes	maximal, complemented
6	#14271226	C₂	1	2	yes	maximal, complemented
7	#14271215	C₂	1	1	yes	maximal, characteristic, complemented
8	#14271214	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11745168	C₂	1	2	yes	maximal, complemented
10	C₂²⋊C₈×D₂₉	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₂₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₉	C₂²	1	2	yes	complemented
15	C₄×D₄×C₂₉	C₂²	1	1	yes	characteristic
16	D₄×Dic₂₉	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₉	C₂²	1	2	yes	—
18	C₂²×#6597	C₂²	1	2	yes	complemented
19	C₂×C₈×D₂₉	C₂²	1	2	yes	complemented
20	C₂×C₈×D₂₉	C₂²	1	2	yes	complemented
21	#11745255	C₄	1	1	yes	characteristic
22	C₄⋊C₄×D₂₉	C₄	1	1	yes	characteristic
23	#11745243	C₄	1	2	yes	—
24	C₂²⋊C₄×D₂₉	C₄	1	2	yes	—
25	C₄×D₁₁₆	C₂²	1	1	yes	characteristic
26	C₄²×D₂₉	C₂²	1	1	yes	characteristic
27	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₂₉×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²⋊#6597	C₂²	1	2	yes	complemented
31	#11745168	C₂²	1	1	yes	characteristic
32	#11745168	C₂²	1	2	yes	—
33	#11745168	C₂²	1	1	yes	characteristic
34	#11745168	C₂²	1	2	yes	complemented
35	#11745168	C₂²	1	2	yes	complemented
36	#11745162	C₂²	1	1	yes	characteristic, complemented
37	#11745162	C₂²	1	1	yes	characteristic, complemented
38	C₈×Dic₂₉	C₂²	1	2	yes	complemented
39	C₄⋊#6597	C₂²	1	1	yes	characteristic, complemented
40	C₄×#6597	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₂₉	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₉	C₂³	1	2	yes	—
44	C₂×C₂²⋊D₂₉	C₂×C₄	1	2	yes	—
45	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
46	D₄×D₂₉	C₈	1	8	yes	complemented
47	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₉	—	4	2	no	—
49	C₂×C₄×D₂₉	—	4	2	no	—
50	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₂₉	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₂₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₂₉	C₂³	1	2	yes	—
55	C₂×C₄×D₂₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₂₉	—	4	2	no	—
57	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₉	C₂³	1	2	yes	—
59	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
62	C₄²×C₂₉	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₂₉	C₂×C₄	1	2	yes	—
64	#90781	C₂×C₄	1	2	yes	—
65	#90781	C₂×C₄	1	2	yes	—
66	C₄⋊Dic₂₉	C₂×C₄	1	1	yes	characteristic
67	#90779	C₂×C₄	1	2	yes	—
68	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₂₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₉	C₂³	1	1	yes	characteristic
71	C₂×#6597	—	4	2	no	—
72	C₂×#6597	C₂³	1	1	yes	characteristic
73	C₂×#6597	C₂³	1	2	yes	—
74	C₂×#6597	C₂³	1	1	yes	characteristic
75	C₈×D₂₉	—	4	2	no	—
76	C₈×D₂₉	—	4	2	no	—
77	C₂³×C₂₉	C₂²×C₄	1	2	yes	—
78	C₂²×D₂₉	—	4	2	no	—
79	C₂²×D₂₉	—	4	2	no	—
80	C₂²×D₂₉	C₂×C₈	1	8	yes	complemented
81	C₂²×D₂₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₂₉	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₂₉	C₂×C₈	1	4	yes	complemented
84	C₂×C₄×C₂₉	—	2	2	no	—
85	C₂×C₄×C₂₉	—	2	2	no	—
86	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₂₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₂₉	C₂×C₈	1	8	yes	complemented
91	C₂×Dic₂₉	—	4	2	no	—
92	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₂₉	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₉	C₂²×C₄	1	1	yes	characteristic
96	D₁₁₆	C₂×C₈	1	4	yes	complemented
97	C₄×D₂₉	—	2	2	no	—
98	C₄×D₂₉	—	2	2	no	—
99	C₄×D₂₉	—	2	1	no	—
100	C₄×D₂₉	—	2	1	no	—
101	C₄×D₂₉	—	4	2	no	—
102	C₄×D₂₉	—	4	2	no	—
103	C₄×D₂₉	—	4	1	no	—
104	C₄×D₂₉	C₂×C₈	1	4	yes	complemented
105	C₈×C₂₉	—	2	1	no	—
106	C₈×C₂₉	—	2	2	no	—
107	C₈×C₂₉	—	2	1	no	—
108	#6597	—	2	1	no	—
109	#6597	—	2	2	no	—
110	#6597	—	2	1	no	—
111	#2972	—	29	1	no	maximal, Sylow
112	C₂²×C₂₉	—	2	2	no	—
113	C₂²×C₂₉	—	2	2	no	—
114	C₂²×C₂₉	C₂²×C₈	1	4	yes	—
115	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₂₉	—	2	1	no	—
117	C₂×D₂₉	—	2	2	no	—
118	C₂×D₂₉	—	2	1	no	—
119	C₂×D₂₉	—	2	2	no	—
120	C₂×D₂₉	—	4	8	no	—
121	C₂×D₂₉	C₂²×C₈	1	2	yes	—
122	C₂×D₂₉	C₂²×C₈	1	4	yes	—
123	C₄×C₂₉	—	2	2	no	—
124	C₄×C₂₉	—	2	1	no	—
125	C₄×C₂₉	C₂²×C₈	1	2	yes	—
126	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
128	Dic₂₉	—	2	1	no	—
129	Dic₂₉	—	2	2	no	—
130	Dic₂₉	—	2	1	no	—
131	Dic₂₉	C₂²×C₈	1	2	yes	—
132	C₂×C₄×D₄	—	29	1	no	—
133	C₈×D₄	—	29	2	no	—
134	C₈×D₄	—	29	1	no	—
135	C₈×D₄	—	29	1	no	—
136	C₈×D₄	—	29	2	no	—
137	C₈×D₄	—	29	1	no	—
138	C₈×D₄	D₂₉	1	1	yes	characteristic, complemented
139	#431	—	29	1	no	—
140	#431	—	29	1	no	—
141	C₂×C₂²⋊C₈	—	29	2	no	—
142	C₂×C₂²⋊C₈	—	29	2	no	—
143	C₂×C₂₉	—	2	4	no	—
144	C₂×C₂₉	#518	1	1	yes	characteristic
145	C₂×C₂₉	C₂³×C₈	1	1	yes	characteristic
146	C₂×C₂₉	#568	1	1	yes	characteristic
147	D₂₉	—	2	2	no	—
148	D₂₉	—	2	4	no	—
149	C₂²×D₄	—	29	1	no	—
150	C₂³×C₄	—	29	2	no	—
151	C₂²×C₈	—	29	2	no	—
152	C₂²×C₈	—	29	2	no	—
153	C₂²×C₈	—	29	2	no	—
154	C₂²×C₈	C₂×D₂₉	1	2	yes	complemented
155	C₄×D₄	—	29	1	no	—
156	C₄×D₄	—	29	2	no	—
157	C₄×D₄	—	29	1	no	—
158	C₄×D₄	—	29	2	no	—
159	C₄×D₄	—	29	1	no	—
160	C₄×D₄	C₂×D₂₉	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	29	1	no	—
162	C₂×C₂²⋊C₄	—	29	2	no	—
163	C₂×C₄²	—	29	1	no	—
164	C₄⋊C₈	—	29	1	no	—
165	C₄⋊C₈	—	29	1	no	—
166	C₄⋊C₈	—	29	1	no	—
167	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
168	C₂²⋊C₈	—	29	2	no	—
169	C₂²⋊C₈	—	29	2	no	—
170	C₂²⋊C₈	—	29	1	no	—
171	C₂²⋊C₈	—	29	1	no	—
172	C₂²⋊C₈	—	29	2	no	—
173	C₂²⋊C₈	—	29	2	no	—
174	C₂²⋊C₈	C₂×D₂₉	1	2	yes	complemented
175	C₄×C₈	—	29	2	no	—
176	C₄×C₈	—	29	1	no	—
177	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic, complemented
178	C₂₉	#2972	1	1	yes	characteristic, complemented, Sylow
179	C₂⁴	—	29	2	no	—
180	C₂×D₄	—	29	2	no	—
181	C₂×D₄	—	29	8	no	—
182	C₂×D₄	—	29	1	no	—
183	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
184	C₂²×C₄	—	29	1	no	—
185	C₂²×C₄	—	29	2	no	—
186	C₂²×C₄	—	29	1	no	—
187	C₂²×C₄	—	29	2	no	—
188	C₂²×C₄	—	29	1	no	—
189	C₂²×C₄	—	29	2	no	—
190	C₂²×C₄	—	29	2	no	—
191	C₂²×C₄	—	116	2	no	—
192	C₂²×C₄	—	116	2	no	—
193	C₂²×C₄	C₂²×D₂₉	1	2	yes	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	29	1	no	—
196	C₂×C₈	—	29	2	no	—
197	C₂×C₈	—	29	1	no	—
198	C₂×C₈	—	116	2	no	—
199	C₂×C₈	—	116	2	no	—
200	C₂×C₈	—	116	2	no	—
201	C₂×C₈	C₂²×D₂₉	1	2	yes	—
202	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
204	C₄⋊C₄	—	29	2	no	—
205	C₄⋊C₄	—	29	1	no	—
206	C₄⋊C₄	C₄×D₂₉	1	1	yes	characteristic
207	C₂²⋊C₄	—	29	2	no	—
208	C₂²⋊C₄	—	29	2	no	—
209	C₂²⋊C₄	—	29	2	no	—
210	C₂²⋊C₄	C₄×D₂₉	1	2	yes	—
211	C₄²	—	29	1	no	—
212	C₄²	—	29	1	no	—
213	C₄²	—	29	1	no	—
214	C₄²	C₂²×D₂₉	1	1	yes	characteristic
215	C₂³	—	29	2	no	—
216	C₂³	—	29	1	no	—
217	C₂³	—	29	8	no	—
218	C₂³	—	116	2	no	—
219	C₂³	—	116	2	no	—
220	C₂³	C₂×C₄×D₂₉	1	2	yes	—
221	D₄	—	29	8	no	—
222	D₄	—	29	4	no	—
223	D₄	C₈×D₂₉	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	29	1	no	—
227	C₂×C₄	—	29	4	no	—
228	C₂×C₄	—	29	1	no	—
229	C₂×C₄	—	29	1	no	—
230	C₂×C₄	—	29	2	no	—
231	C₂×C₄	—	58	2	no	—
232	C₂×C₄	—	58	1	no	—
233	C₂×C₄	—	58	2	no	—
234	C₂×C₄	—	58	1	no	—
235	C₂×C₄	—	116	2	no	—
236	C₂×C₄	—	116	2	no	—
237	C₂×C₄	—	116	1	no	—
238	C₂×C₄	—	116	2	no	—
239	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₂₉	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	58	1	no	—
247	C₈	—	58	2	no	—
248	C₈	—	58	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	29	2	no	—
252	C₂²	—	29	4	no	—
253	C₂²	—	58	1	no	—
254	C₂²	—	58	2	no	—
255	C₂²	—	58	1	no	—
256	C₂²	—	58	2	no	—
257	C₂²	—	116	8	no	—
258	C₂²	C₂×C₈×D₂₉	1	4	yes	—
259	C₂²	C₂²×C₄×D₂₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	29	2	no	—
263	C₄	—	58	1	no	—
264	C₄	—	58	2	no	—
265	C₄	—	58	1	no	—
266	C₄	C₂×C₈×D₂₉	1	2	yes	—
267	C₄	#11745359	1	1	yes	characteristic
268	C₄	#11745364	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	58	2	no	—
271	C₂	—	58	4	no	—
272	C₂	#14271635	1	1	yes	characteristic
273	C₂	C₂²×C₈×D₂₉	1	1	yes	characteristic
274	C₂	#14271928	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator	C₂×C₂₉	C₂³×C₈
Fitting	C₈×D₄×C₂₉	C₂
socle	C₂²×C₂₉	C₂³×C₄
prime radical	C₂×D₄×D₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₈×D₄	D₂₉
hypocenter	C₂₉	#2972
LCS	G ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×D₁₁₆	C₂	semidirect
3. D₄×#6597	C₂	semidirect
4. #14271228	C₂	semidirect
5. C₈×C₂²⋊D₂₉	C₂	semidirect
6. #14271226	C₂	semidirect
7. #14271215	C₂	semidirect
8. #14271214	C₂	semidirect
9. C₂×#11745168	C₂	semidirect
10. C₂²⋊C₈×D₂₉	C₂	semidirect
11. C₈×D₄×C₂₉	C₂	semidirect
14. C₂²×C₈×C₂₉	C₂²	semidirect
18. C₂²×#6597	C₂²	semidirect
19. C₂×C₈×D₂₉	C₂²	semidirect
20. C₂×C₈×D₂₉	C₂²	semidirect
27. C₂₉×C₄⋊C₈	C₂²	semidirect
28. C₂₉×C₂²⋊C₈	C₂²	semidirect
29. C₄×C₈×C₂₉	C₂²	semidirect
30. C₂²⋊#6597	C₂²	semidirect
34. #11745168	C₂²	semidirect
35. #11745168	C₂²	semidirect
36. #11745162	C₂²	semidirect
37. #11745162	C₂²	semidirect
38. C₈×Dic₂₉	C₂²	semidirect
39. C₄⋊#6597	C₂²	semidirect
40. C₄×#6597	C₂²	semidirect
46. D₄×D₂₉	C₈	semidirect
80. C₂²×D₂₉	C₂×C₈	semidirect
83. D₄×C₂₉	C₂×C₈	semidirect
90. C₂²⋊D₂₉	C₂×C₈	semidirect
96. D₁₁₆	C₂×C₈	semidirect
104. C₄×D₂₉	C₂×C₈	semidirect
138. C₈×D₄	D₂₉	semidirect
154. C₂²×C₈	C₂×D₂₉	semidirect
167. C₄⋊C₈	C₂×D₂₉	semidirect
174. C₂²⋊C₈	C₂×D₂₉	semidirect
177. C₄×C₈	C₂×D₂₉	semidirect
178. C₂₉	#2972	semidirect
223. D₄	C₈×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₂₉	2
C₈×D₁₁₆	2
D₄×#6597	2
#14271228	2
C₈×C₂²⋊D₂₉	2
#14271226	2
#14271215	2
#14271214	2
C₂×#11745168	2
C₂²⋊C₈×D₂₉	2
C₈×D₄×C₂₉	2
#2972	29