Subgroups — #16828398

Order 3712.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14271961	C₂	1	1	yes	maximal, characteristic
2	#14271930	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271929	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271282	C₂	1	4	yes	maximal, complemented
5	#14271277	C₂	1	4	yes	maximal, complemented
6	#14271211	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11745153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#90777	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₉×#104	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×#90805	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
14	#11745313	C₄	1	2	yes	complemented
15	#11745312	C₂²	1	4	yes	—
16	#11745311	C₄	1	2	yes	complemented
17	#11745310	—	2	2	no	—
18	#11745310	C₂²	1	4	yes	complemented
19	#11745304	C₄	1	2	yes	complemented
20	#11745294	C₄	1	2	yes	complemented
21	C₂×#90777	—	2	2	no	—
22	C₂×#90777	C₂²	1	1	yes	characteristic
23	C₂×#90777	C₂²	1	1	yes	characteristic
24	C₂×#90777	C₂²	1	1	yes	characteristic
25	#11745269	C₂²	1	4	yes	complemented
26	M₁₆×D₂₉	—	2	2	no	—
27	#11745263	—	2	2	no	—
28	C₂₉×#104	C₂²	1	2	yes	complemented
29	C₂₉×#104	C₂²	1	2	yes	complemented
30	#11745170	C₂²	1	4	yes	complemented
31	#11745153	C₂²	1	2	yes	complemented
32	#11745153	C₂²	1	2	yes	complemented
33	#11745151	C₂²	1	2	yes	complemented
34	C₄×#6597	C₂²	1	2	yes	complemented
35	#41×C₂₉	—	2	1	no	—
36	#41×C₂₉	—	2	1	no	—
37	#41×C₂₉	C₂×C₄	1	2	yes	—
38	#41×C₂₉	C₂³	1	2	yes	—
39	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₂₉	—	2	1	no	—
41	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₉	—	2	1	no	—
43	C₂²×C₄×C₂₉	—	2	1	no	—
44	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₂₉	—	2	1	no	—
46	#90805	—	2	1	no	—
47	#90805	—	2	1	no	—
48	#90805	C₂×C₄	1	2	yes	—
49	#90805	C₂³	1	2	yes	—
50	C₂×D₁₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₉	—	2	1	no	—
52	C₂×Dic₅₈	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₂₉	—	2	1	no	—
54	M₁₆×C₂₉	C₂³	1	2	yes	—
55	C₂×C₈×C₂₉	—	2	1	no	—
56	C₄²×C₂₉	—	2	1	no	—
57	C₄²×C₂₉	C₂³	1	1	yes	characteristic
58	C₄²×C₂₉	C₂³	1	1	yes	characteristic
59	#90785	C₂×C₄	1	4	yes	complemented
60	#90784	C₂×C₄	1	4	yes	complemented
61	#90783	C₂×C₄	1	4	yes	complemented
62	#90782	C₂×C₄	1	4	yes	complemented
63	#90777	—	2	1	no	—
64	#90777	—	2	4	no	—
65	#90777	C₂×C₄	1	2	yes	—
66	#90777	C₂×C₄	1	2	yes	—
67	#90777	D₄	1	2	yes	—
68	#90777	D₄	1	2	yes	—
69	#90777	C₂³	1	2	yes	—
70	C₂×#6597	—	2	1	no	—
71	C₂×#6597	—	2	4	no	—
72	C₂×#6597	C₂³	1	2	yes	—
73	C₂×#6597	C₂³	1	2	yes	—
74	#90772	—	2	4	no	—
75	C₈×D₂₉	—	2	4	no	—
76	C₂³×C₂₉	—	2	1	no	—
77	C₂³×C₂₉	#41	1	1	yes	characteristic
78	C₂²×D₂₉	—	2	1	no	—
79	Q₈×C₂₉	—	2	1	no	—
80	Q₈×C₂₉	C₂²×C₄	1	2	yes	—
81	D₄×C₂₉	—	2	2	no	—
82	D₄×C₂₉	—	2	2	no	—
83	D₄×C₂₉	—	2	1	no	—
84	D₄×C₂₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₉	—	2	1	no	—
86	C₂×C₄×C₂₉	—	2	1	no	—
87	C₂×C₄×C₂₉	—	2	2	no	—
88	C₂×C₄×C₂₉	—	2	2	no	—
89	C₂×C₄×C₂₉	—	2	1	no	—
90	C₂×C₄×C₂₉	—	4	2	no	—
91	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
96	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₉	—	2	2	no	—
98	C₂²⋊D₂₉	—	2	2	no	—
99	C₂×Dic₂₉	—	2	1	no	—
100	D₁₁₆	—	2	1	no	—
101	D₁₁₆	C₂²×C₄	1	2	yes	—
102	C₄×D₂₉	—	2	2	no	—
103	C₄×D₂₉	—	2	2	no	—
104	Dic₅₈	—	2	1	no	—
105	Dic₅₈	C₂²×C₄	1	2	yes	—
106	C₈×C₂₉	—	2	2	no	—
107	#6597	—	2	2	no	—
108	#6597	C₂²×C₄	1	4	yes	—
109	#6597	C₂×D₄	1	4	yes	—
110	#2956	—	29	1	no	maximal, Sylow
111	C₂²×C₂₉	—	2	2	no	—
112	C₂²×C₂₉	—	4	1	no	—
113	C₂²×C₂₉	—	8	1	no	—
114	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₂₉	C₂×#41	1	1	yes	characteristic
117	C₂×D₂₉	—	2	2	no	—
118	C₂×D₂₉	—	8	1	no	—
119	C₄×C₂₉	—	2	2	no	—
120	C₄×C₂₉	—	2	2	no	—
121	C₄×C₂₉	—	4	1	no	—
122	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
123	C₄×C₂₉	C₄×D₄	1	1	yes	characteristic
124	C₄×C₂₉	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₂₉	C₂²×D₄	1	1	yes	characteristic
126	Dic₂₉	—	2	2	no	—
127	#575	—	29	1	no	—
128	#568	—	29	1	no	—
129	#568	—	29	1	no	—
130	#444	—	29	4	no	—
131	#443	—	29	4	no	—
132	#430	—	29	1	no	—
133	C₂×#104	—	29	1	no	—
134	C₂×#104	D₂₉	1	1	yes	characteristic, complemented
135	C₄×M₁₆	—	29	1	no	—
136	C₂×C₂₉	—	2	1	no	—
137	C₂×C₂₉	—	2	1	no	—
138	C₂×C₂₉	—	2	1	no	—
139	C₂×C₂₉	—	4	2	no	—
140	C₂×C₂₉	C₂×C₄×D₄	1	1	yes	characteristic
141	D₂₉	—	4	2	no	—
142	C₂×#41	—	29	1	no	—
143	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
144	#137	—	29	2	no	—
145	#137	—	29	2	no	—
146	AGL(1,ℤ/8)	—	29	2	no	—
147	AGL(1,ℤ/8)	—	29	2	no	—
148	#135	—	29	4	no	—
149	#131	—	29	4	no	—
150	#131	—	29	4	no	—
151	#131	—	58	2	no	—
152	#131	—	58	2	no	—
153	C₂×M₁₆	—	29	1	no	—
154	C₂×M₁₆	—	29	1	no	—
155	C₂×M₁₆	—	29	1	no	—
156	C₂×M₁₆	—	58	2	no	—
157	C₂×M₁₆	—	58	2	no	—
158	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
159	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
160	#108	—	29	4	no	—
161	#104	—	29	2	no	—
162	#104	—	29	2	no	—
163	#104	C₂×D₂₉	1	2	yes	complemented
164	#104	C₂×D₂₉	1	2	yes	complemented
165	#97	—	29	2	no	—
166	C₄×C₈	—	29	2	no	—
167	C₂₉	#2956	1	1	yes	characteristic, complemented, Sylow
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	29	2	no	—
171	#41	—	29	2	no	—
172	#41	—	58	1	no	—
173	#41	—	58	1	no	—
174	#41	C₄×D₂₉	1	2	yes	—
175	#41	C₂²×D₂₉	1	2	yes	—
176	C₂×Q₈	—	29	1	no	—
177	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	29	1	no	—
180	C₂×D₄	—	58	1	no	—
181	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	58	1	no	—
185	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
186	Q₁₆	—	29	4	no	—
187	SD₁₆	—	29	4	no	—
188	SD₁₆	—	29	4	no	—
189	D₈	—	29	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	29	2	no	—
192	M₁₆	—	29	2	no	—
193	M₁₆	—	29	2	no	—
194	M₁₆	—	29	2	no	—
195	M₁₆	—	29	2	no	—
196	M₁₆	—	58	4	no	—
197	M₁₆	—	58	4	no	—
198	M₁₆	—	58	1	no	—
199	M₁₆	C₂²×D₂₉	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	29	2	no	—
202	C₂×C₈	—	29	2	no	—
203	C₂×C₈	—	58	4	no	—
204	C₂×C₈	—	58	4	no	—
205	C₂×C₈	—	58	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×D₂₉	1	1	yes	characteristic
208	C₄²	C₂²×D₂₉	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	58	1	no	—
211	C₂³	#90807	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	29	2	no	—
214	Q₈	—	58	1	no	—
215	Q₈	C₂×C₄×D₂₉	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	29	2	no	—
220	D₄	—	58	2	no	—
221	D₄	—	58	2	no	—
222	D₄	—	58	1	no	—
223	D₄	C₂×C₄×D₂₉	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	58	1	no	—
231	C₂×C₄	—	58	2	no	—
232	C₂×C₄	—	58	2	no	—
233	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
235	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
236	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
237	C₂×C₄	#90807	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	29	4	no	—
241	C₈	—	29	4	no	—
242	C₈	—	58	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	58	2	no	—
247	C₂²	—	232	1	no	—
248	C₂²	#11745243	1	1	yes	characteristic
249	C₂²	#11745243	1	1	yes	characteristic
250	C₂²	C₂×#90807	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	58	2	no	—
255	C₄	#11745243	1	1	yes	characteristic
256	C₄	#11745243	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₂₉	1	1	yes	characteristic
258	C₄	C₂×D₄×D₂₉	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	116	2	no	—
264	C₂	C₂×#11745243	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₉
commutator	C₄×C₂₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂×C₂₉×#104	C₂
socle	C₂×C₂₉	C₂×C₄×D₄
prime radical	#14271961	C₂
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₂×#104	D₂₉
hypocenter	C₂₉	#2956
LCS	G ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #14271930	C₂	semidirect
3. #14271929	C₂	semidirect
4. #14271282	C₂	semidirect
5. #14271277	C₂	semidirect
6. #14271211	C₂	semidirect
7. C₂×#11745153	C₂	semidirect
8. C₄×#90777	C₂	semidirect
9. C₂×C₂₉×#104	C₂	semidirect
14. #11745313	C₄	semidirect
16. #11745311	C₄	semidirect
18. #11745310	C₂²	semidirect
19. #11745304	C₄	semidirect
20. #11745294	C₄	semidirect
25. #11745269	C₂²	semidirect
28. C₂₉×#104	C₂²	semidirect
29. C₂₉×#104	C₂²	semidirect
30. #11745170	C₂²	semidirect
31. #11745153	C₂²	semidirect
32. #11745153	C₂²	semidirect
33. #11745151	C₂²	semidirect
34. C₄×#6597	C₂²	semidirect
59. #90785	C₂×C₄	semidirect
60. #90784	C₂×C₄	semidirect
61. #90783	C₂×C₄	semidirect
62. #90782	C₂×C₄	semidirect
134. C₂×#104	D₂₉	semidirect
163. #104	C₂×D₂₉	semidirect
164. #104	C₂×D₂₉	semidirect
167. C₂₉	#2956	semidirect

Maximal subgroups

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

subgroup	index
#14271961	2
#14271930	2
#14271929	2
#14271282	2
#14271277	2
#14271211	2
C₂×#11745153	2
C₄×#90777	2
C₂×C₂₉×#104	2
#2956	29