Subgroups — #28744198

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434635	C₂	1	2	yes	maximal
2	#21426382	C₂	1	4	yes	maximal, complemented
3	#21426375	C₂	1	2	yes	maximal, complemented
4	#21426371	C₂	1	2	yes	maximal, complemented
5	#21425884	C₂	1	2	yes	maximal
6	#21425880	C₂	1	1	yes	maximal, characteristic
7	C₂×#16837228	C₂	1	1	yes	maximal, characteristic
8	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
9	#16838030	—	2	2	no	—
10	#16837811	—	2	4	no	—
11	#16837807	C₂²	1	4	yes	—
12	#16837804	C₂²	1	4	yes	—
13	#16837444	—	2	2	no	—
14	#16837440	C₂²	1	2	yes	—
15	C₂×#14273661	C₂²	1	2	yes	—
16	#16837230	C₂²	1	4	yes	—
17	#16837230	C₂²	1	2	yes	—
18	#16837228	C₂²	1	1	yes	characteristic
19	#16837228	C₂²	1	2	yes	—
20	#16837228	C₂²	1	4	yes	—
21	#16837228	C₂²	1	1	yes	characteristic
22	#16837228	C₂²	1	2	yes	—
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#405	C₂²	1	1	yes	characteristic
26	C₅₉×#346	C₂²	1	2	yes	—
27	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
28	#14273727	—	2	4	no	—
29	#14273726	—	4	2	no	—
30	#14273719	—	2	4	no	—
31	#14273718	—	4	2	no	—
32	#14273704	—	4	4	no	—
33	#14273686	—	2	2	no	—
34	C₂×C₄⋊Dic₅₉	—	2	2	no	—
35	#14273661	—	2	2	no	—
36	#14273661	—	2	4	no	—
37	#14273661	C₂³	1	4	yes	—
38	#14273661	C₂³	1	2	yes	—
39	#14273661	C₂³	1	2	yes	—
40	#14273659	—	2	2	no	—
41	#14273659	—	2	4	no	—
42	C₂×M₁₆×C₅₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
44	#117×C₅₉	D₄	1	1	yes	characteristic
45	M₃₂×C₅₉	Q₈	1	4	yes	complemented
46	C₂×C₁₆×C₅₉	Q₈	1	4	yes	complemented
47	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
49	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
50	#97×C₅₉	D₄	1	2	yes	—
51	C₄×C₈×C₅₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₉	—	4	2	no	—
54	C₂×Dic₁₁₈	—	4	4	no	—
55	M₁₆×C₅₉	Q₁₆	1	4	yes	—
56	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	D₈	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₉	D₈	1	2	yes	—
64	C₄²×C₅₉	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₅₉	—	4	2	no	—
66	C₄⋊Dic₅₉	—	2	2	no	—
67	C₄⋊Dic₅₉	—	2	4	no	—
68	C₄⋊Dic₅₉	—	2	2	no	—
69	C₄⋊Dic₅₉	—	4	2	no	—
70	C₄⋊Dic₅₉	—	4	4	no	—
71	Dic₅₉⋊C₄	—	4	4	no	—
72	Dic₅₉⋊C₄	—	4	4	no	—
73	C₄×Dic₅₉	—	4	2	no	—
74	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
75	C₁₆×C₅₉	C₂×Q₈	1	2	yes	—
76	C₁₆×C₅₉	C₂×Q₈	1	4	yes	—
77	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
79	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₅₉	—	4	2	no	—
85	C₂×Dic₅₉	—	4	2	no	—
86	C₂×Dic₅₉	—	4	4	no	—
87	C₂×Dic₅₉	—	16	2	no	—
88	Dic₁₁₈	—	16	4	no	—
89	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
90	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
91	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
92	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
93	#33729	—	59	1	no	maximal, Sylow
94	C₂²×C₅₉	—	2	1	no	—
95	C₂²×C₅₉	—	2	1	no	—
96	C₂²×C₅₉	#500	1	2	yes	—
97	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₉	—	2	2	no	—
99	C₄×C₅₉	—	2	2	no	—
100	C₄×C₅₉	#500	1	2	yes	—
101	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₅₉	—	8	2	no	—
104	Dic₅₉	—	8	4	no	—
105	Dic₅₉	—	8	2	no	—
106	#3164	—	59	2	no	—
107	#2255	—	59	4	no	—
108	#2253	—	59	2	no	—
109	#2252	—	59	2	no	—
110	#2159	—	59	2	no	—
111	#2158	—	59	1	no	—
112	C₂×#366	—	59	1	no	—
113	#2108	D₅₉	1	1	yes	characteristic
114	C₂×C₅₉	—	2	2	no	—
115	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
116	C₂×C₅₉	#3415	1	1	yes	characteristic
117	C₂×C₅₉	#3417	1	1	yes	characteristic
118	#544	—	118	2	no	—
119	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
120	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
121	#501	—	118	4	no	—
122	#500	—	59	4	no	—
123	#499	—	59	4	no	—
124	#428	—	118	2	no	—
125	#427	—	59	2	no	—
126	C₂×#107	—	59	2	no	—
127	#405	C₂×D₅₉	1	1	yes	characteristic
128	#367	—	59	2	no	—
129	#367	—	59	4	no	—
130	#366	—	59	2	no	—
131	#366	—	59	2	no	—
132	#366	—	59	1	no	—
133	#366	—	59	1	no	—
134	#366	—	59	4	no	—
135	#346	C₂×D₅₉	1	2	yes	—
136	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
137	C₅₉	#33729	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	D₂₃₆	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
140	C₄⋊Q₈	—	118	4	no	—
141	C₄⋊Q₈	—	236	2	no	—
142	#125	—	118	4	no	—
143	#125	—	236	2	no	—
144	C₂²⋊Q₈	—	236	4	no	—
145	#117	—	118	2	no	—
146	#117	D₂₃₆	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	118	2	no	—
148	M₃₂	Dic₁₁₈	1	4	yes	complemented
149	C₂×C₁₆	Dic₁₁₈	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
152	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
153	#107	—	59	4	no	—
154	#107	—	59	2	no	—
155	#107	—	59	2	no	—
156	#107	—	118	2	no	—
157	#107	—	118	4	no	—
158	#106	—	118	2	no	—
159	#106	—	118	4	no	—
160	#97	D₂₃₆	1	2	yes	—
161	C₄×C₈	C₂²×D₅₉	1	2	yes	—
162	C₂×Q₈	—	236	4	no	—
163	C₂²×C₄	—	236	2	no	—
164	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
165	M₁₆	Dic₂₃₆	1	4	yes	—
166	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₁₈	1	2	yes	—
168	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
170	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
171	C₄⋊C₄	—	118	2	no	—
172	C₄⋊C₄	—	118	4	no	—
173	C₄⋊C₄	—	118	2	no	—
174	C₄⋊C₄	—	236	2	no	—
175	C₄⋊C₄	—	236	4	no	—
176	C₄⋊C₄	—	236	4	no	—
177	C₄⋊C₄	—	236	4	no	—
178	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
179	C₄⋊C₄	D₄₇₂	1	1	yes	characteristic
180	C₂²⋊C₄	—	236	2	no	—
181	C₂²⋊C₄	D₄₇₂	1	2	yes	—
182	C₄²	—	236	2	no	—
183	C₄²	C₂×D₂₃₆	1	2	yes	—
184	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
185	C₁₆	C₂×Dic₁₁₈	1	2	yes	—
186	C₁₆	C₂×Dic₁₁₈	1	4	yes	—
187	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
188	Q₈	—	944	4	no	—
189	C₂×C₄	—	944	2	no	—
190	C₂×C₄	—	236	2	no	—
191	C₂×C₄	—	236	2	no	—
192	C₂×C₄	—	236	4	no	—
193	C₂×C₄	#14273727	1	2	yes	—
194	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
195	C₂×C₄	C₂×D₄₇₂	1	2	yes	—
196	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
198	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
199	C₈	#14273727	1	2	yes	—
200	C₈	C₂×Dic₂₃₆	1	4	yes	—
201	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₁₈	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#16837807	1	2	yes	—
206	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	472	4	no	—
210	C₄	—	472	2	no	—
211	C₄	—	472	2	no	—
212	C₄	#16837807	1	2	yes	—
213	C₄	C₂×#14273727	1	1	yes	characteristic
214	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#16837807	1	1	yes	characteristic
217	C₂	#21437237	1	1	yes	characteristic
218	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	C₅₉×#2108	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#33729
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #21426382	C₂	semidirect
3. #21426375	C₂	semidirect
4. #21426371	C₂	semidirect
45. M₃₂×C₅₉	Q₈	semidirect
46. C₂×C₁₆×C₅₉	Q₈	semidirect
137. C₅₉	#33729	semidirect
148. M₃₂	Dic₁₁₈	semidirect
149. C₂×C₁₆	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
#21434635	2
#21426382	2
#21426375	2
#21426371	2
#21425884	2
#21425880	2
C₂×#16837228	2
C₅₉×#2108	2
#33729	59