Subgroups — #28746047

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436444	C₂	1	2	yes	maximal, complemented
2	#21426371	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426277	C₂	1	4	yes	maximal, complemented
4	#21426234	C₂	1	2	yes	maximal, complemented
5	#21426182	C₂	1	2	yes	maximal, complemented
6	C₄×D₉₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16837228	C₂	1	2	yes	maximal, complemented
8	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#16838056	—	2	2	no	—
10	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
11	#16837807	C₂²	1	2	yes	—
12	#16837716	—	2	4	no	—
13	#16837683	—	2	2	no	—
14	#16837638	C₂²	1	4	yes	—
15	C₄×D₄₇₂	C₂²	1	2	yes	—
16	C₂×#14273661	C₂²	1	4	yes	—
17	#16837228	C₂²	1	2	yes	—
18	#16837228	C₂²	1	1	yes	characteristic
19	#16837228	C₂²	1	4	yes	—
20	#16837228	C₂²	1	2	yes	—
21	#16837204	C₂²	1	4	yes	—
22	#16837204	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
24	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₅₉×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#11745687	—	4	2	no	—
29	C₂×D₄₇₂	C₂³	1	2	yes	—
30	#14273727	—	2	2	no	—
31	#14273711	—	4	4	no	—
32	#14273707	—	4	2	no	—
33	#14273701	—	2	4	no	—
34	C₄×D₂₃₆	—	2	2	no	—
35	C₂×C₄⋊Dic₅₉	—	2	4	no	—
36	D₉₄₄	—	2	2	no	—
37	#14273661	—	2	4	no	—
38	#14273661	—	2	2	no	—
39	#14273661	C₂³	1	4	yes	—
40	#14273661	C₂³	1	2	yes	—
41	#14273649	—	2	4	no	—
42	#14273649	—	2	2	no	—
43	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
44	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₅₉	—	2	4	no	—
46	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
48	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₅₉	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
54	C₂×C₂²⋊D₅₉	—	4	4	no	—
55	C₂²×Dic₅₉	—	4	4	no	—
56	#11745687	—	8	8	no	—
57	C₂×D₂₃₆	—	2	2	no	—
58	C₂×C₄×D₅₉	—	4	2	no	—
59	C₂×Dic₁₁₈	—	4	2	no	—
60	C₂×C₈×C₅₉	—	2	2	no	—
61	C₂×C₈×C₅₉	—	2	2	no	—
62	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
66	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
67	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
68	(C₂×D₅₉)⋊C₄	—	4	2	no	—
69	(C₂×D₅₉)⋊C₄	—	4	4	no	—
70	C₄⋊Dic₅₉	—	2	2	no	—
71	C₄⋊Dic₅₉	—	2	4	no	—
72	C₄⋊Dic₅₉	—	4	4	no	—
73	C₄⋊Dic₅₉	—	4	2	no	—
74	D₄₇₂	—	2	2	no	—
75	D₄₇₂	—	2	2	no	—
76	C₁₆×C₅₉	—	2	2	no	—
77	C₁₆×C₅₉	—	2	1	no	—
78	C₁₆×C₅₉	#41	1	2	yes	complemented
79	C₂³×C₅₉	C₂×D₈	1	2	yes	—
80	C₂²×D₅₉	—	4	2	no	—
81	D₄×C₅₉	D₁₆	1	4	yes	complemented
82	C₂×C₄×C₅₉	—	2	2	no	—
83	C₂×C₄×C₅₉	—	2	2	no	—
84	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
88	C₂²⋊D₅₉	—	8	8	no	—
89	C₂×Dic₅₉	—	4	4	no	—
90	C₂×Dic₅₉	—	4	2	no	—
91	C₂×Dic₅₉	—	8	8	no	—
92	D₂₃₆	—	4	2	no	—
93	D₂₃₆	—	4	2	no	—
94	C₄×D₅₉	—	8	4	no	—
95	Dic₁₁₈	—	8	4	no	—
96	C₈×C₅₉	—	2	2	no	—
97	C₈×C₅₉	—	2	1	no	—
98	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
99	C₈×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
100	#33901	—	59	1	no	maximal, Sylow
101	C₂²×C₅₉	—	2	2	no	—
102	C₂²×C₅₉	—	2	2	no	—
103	C₂²×C₅₉	C₂×D₁₆	1	4	yes	—
104	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
105	C₂×D₅₉	—	8	2	no	—
106	C₂×D₅₉	—	8	2	no	—
107	C₂×D₅₉	—	8	2	no	—
108	C₄×C₅₉	—	2	2	no	—
109	C₄×C₅₉	—	2	1	no	—
110	C₄×C₅₉	C₂×D₁₆	1	2	yes	—
111	C₄×C₅₉	#547	1	1	yes	characteristic
112	C₄×C₅₉	#578	1	1	yes	characteristic
113	Dic₅₉	—	8	4	no	—
114	Dic₅₉	—	8	2	no	—
115	#3334	—	59	2	no	—
116	#2252	—	59	1	no	—
117	#2232	—	59	4	no	—
118	#2224	—	59	2	no	—
119	#2215	—	59	2	no	—
120	C₄×D₁₆	—	59	1	no	—
121	D₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
122	C₂×#366	—	59	2	no	—
123	C₂×C₅₉	—	2	4	no	—
124	C₂×C₅₉	#3334	1	1	yes	characteristic
125	C₂×C₅₉	C₂²×D₁₆	1	1	yes	characteristic
126	C₂×C₅₉	#3417	1	1	yes	characteristic
127	D₅₉	—	16	2	no	—
128	#547	—	118	2	no	—
129	C₂×D₁₆	—	59	1	no	—
130	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
131	#500	—	59	2	no	—
132	#480	—	118	4	no	—
133	#474	—	118	2	no	—
134	#466	—	59	4	no	—
135	C₄×D₈	—	59	2	no	—
136	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
137	C₂×#107	—	59	4	no	—
138	#366	—	59	1	no	—
139	#366	—	59	2	no	—
140	#366	—	59	4	no	—
141	#366	—	59	2	no	—
142	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
143	#357	—	59	2	no	—
144	#357	—	59	4	no	—
145	C₂²⋊C₁₆	C₂×D₅₉	1	2	yes	—
146	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
147	C₅₉	#33901	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	236	2	no	—
149	C₂×D₈	—	59	2	no	—
150	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
151	C₄⋊Q₈	—	118	2	no	—
152	#123	—	236	4	no	—
153	C₂²⋊Q₈	—	236	2	no	—
154	#121	—	118	4	no	—
155	C₄×D₄	—	118	2	no	—
156	C₄×D₄	D₂₃₆	1	1	yes	characteristic
157	C₂×C₄⋊C₄	—	118	4	no	—
158	D₁₆	—	118	2	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
161	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
162	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
163	#107	—	59	2	no	—
164	#107	—	59	4	no	—
165	#107	—	118	4	no	—
166	#107	—	118	2	no	—
167	C₄⋊C₈	D₂₃₆	1	1	yes	characteristic
168	#102	—	118	2	no	—
169	#102	—	118	4	no	—
170	C₂²⋊C₈	D₂₃₆	1	2	yes	—
171	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
172	#41	—	472	8	no	—
173	C₂×Q₈	—	236	2	no	—
174	C₂×D₄	—	118	2	no	—
175	C₂×D₄	—	236	4	no	—
176	C₂×D₄	D₄₇₂	1	1	yes	characteristic
177	C₂²×C₄	—	236	4	no	—
178	C₂²×C₄	—	236	2	no	—
179	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
180	D₈	—	118	2	no	—
181	D₈	—	118	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
185	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
186	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
187	C₄⋊C₄	—	118	2	no	—
188	C₄⋊C₄	—	118	4	no	—
189	C₄⋊C₄	—	236	4	no	—
190	C₄⋊C₄	—	236	2	no	—
191	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
192	C₂²⋊C₄	—	236	2	no	—
193	C₂²⋊C₄	—	236	4	no	—
194	C₂²⋊C₄	D₄₇₂	1	2	yes	—
195	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#11745687	1	2	yes	complemented
199	C₂³	—	236	2	no	—
200	C₂³	C₂×D₄₇₂	1	2	yes	—
201	Q₈	—	472	4	no	—
202	D₄	—	472	8	no	—
203	D₄	—	236	2	no	—
204	D₄	—	236	2	no	—
205	D₄	D₉₄₄	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	472	8	no	—
209	C₂×C₄	—	472	4	no	—
210	C₂×C₄	—	236	2	no	—
211	C₂×C₄	—	236	4	no	—
212	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
214	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	C₂×#11745687	1	1	yes	characteristic
219	C₈	#14273774	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	472	2	no	—
223	C₂²	—	472	2	no	—
224	C₂²	—	472	2	no	—
225	C₂²	C₂×D₉₄₄	1	4	yes	—
226	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	472	2	no	—
230	C₄	—	472	4	no	—
231	C₄	C₂×D₉₄₄	1	2	yes	—
232	C₄	#16838056	1	1	yes	characteristic
233	C₄	#16838219	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	944	2	no	—
236	C₂	#21436444	1	1	yes	characteristic
237	C₂	C₂²×D₉₄₄	1	1	yes	characteristic
238	C₂	#21437249	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	#33901
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #21436444	C₂	semidirect
2. #21426371	C₂	semidirect
3. #21426277	C₂	semidirect
4. #21426234	C₂	semidirect
5. #21426182	C₂	semidirect
6. C₄×D₉₄₄	C₂	semidirect
7. C₂×#16837228	C₂	semidirect
8. D₄×C₁₆×C₅₉	C₂	semidirect
78. C₁₆×C₅₉	#41	semidirect
81. D₄×C₅₉	D₁₆	semidirect
121. D₄×C₁₆	D₅₉	semidirect
147. C₅₉	#33901	semidirect
198. C₁₆	#11745687	semidirect
205. D₄	D₉₄₄	semidirect

Maximal subgroups

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

subgroup	index
#21436444	2
#21426371	2
#21426277	2
#21426234	2
#21426182	2
C₄×D₉₄₄	2
C₂×#16837228	2
D₄×C₁₆×C₅₉	2
#33901	59