Subgroups — #28499987

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434610	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837155	C₂	1	1	yes	maximal, characteristic
3	#21422153	C₂	1	2	yes	maximal, complemented
4	#21422150	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
6	#16837959	—	2	1	no	—
7	#16837807	C₂²	1	1	yes	characteristic
8	#16837807	C₂²	1	1	yes	characteristic
9	#16837804	—	2	1	no	—
10	#16837804	C₄	1	2	yes	—
11	#16837800	—	2	1	no	—
12	#16837440	—	2	1	no	—
13	#16837440	C₄	1	2	yes	—
14	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
15	#16837155	—	2	1	no	—
16	#16837155	C₂²	1	2	yes	—
17	C₂×M₃₂×C₅₉	—	2	1	no	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×C₄⋊C₁₆	—	2	1	no	—
20	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
21	C₂²×C₅₉⋊C₈	—	4	1	no	—
22	#14273727	—	2	1	no	—
23	#14273727	—	2	1	no	—
24	#14273719	—	2	2	no	—
25	#14273706	—	4	1	no	—
26	#14273706	—	4	1	no	—
27	C₄×Dic₁₁₈	—	4	1	no	—
28	C₄×Dic₁₁₈	—	4	1	no	—
29	#14273686	—	2	2	no	—
30	#14273661	—	2	1	no	—
31	#14273661	—	2	1	no	—
32	#14273661	C₂×C₄	1	2	yes	—
33	#14273661	C₂×C₄	1	2	yes	—
34	#14273659	—	2	2	no	—
35	#14273659	—	2	2	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₅₉⋊C₈)	—	4	1	no	—
38	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
41	M₃₂×C₅₉	—	4	2	no	—
42	C₂×C₁₆×C₅₉	—	2	2	no	—
43	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
50	C₂×Dic₁₁₈	—	4	1	no	—
51	C₂×Dic₁₁₈	—	4	1	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₉	D₈	1	1	yes	characteristic
61	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
62	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²⋊Dic₅₉	—	4	2	no	—
64	C₄⋊Dic₅₉	—	2	2	no	—
65	C₄⋊Dic₅₉	—	2	2	no	—
66	C₄⋊Dic₅₉	—	4	1	no	—
67	C₄⋊Dic₅₉	—	4	1	no	—
68	Dic₅₉⋊C₄	—	4	2	no	—
69	Dic₅₉⋊C₄	—	4	2	no	—
70	C₄×Dic₅₉	—	4	2	no	—
71	C₂×C₅₉⋊C₈	—	4	2	no	—
72	C₂×C₅₉⋊C₈	—	8	2	no	—
73	C₁₆×C₅₉	—	4	2	no	—
74	C₂³×C₅₉	#102	1	1	yes	characteristic
75	C₂×C₄×C₅₉	—	2	1	no	—
76	C₂×C₄×C₅₉	—	2	1	no	—
77	C₂×C₄×C₅₉	—	4	1	no	—
78	C₂×C₄×C₅₉	#101	1	2	yes	—
79	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#103	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₅₉	—	4	2	no	—
87	C₂×Dic₅₉	—	4	2	no	—
88	Dic₁₁₈	—	8	2	no	—
89	Dic₁₁₈	—	8	1	no	—
90	Dic₁₁₈	—	8	1	no	—
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	—	2	1	no	—
93	C₈×C₅₉	#103	1	2	yes	—
94	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
96	C₅₉⋊C₈	—	8	2	no	—
97	#11942	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	#332	1	2	yes	—
101	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
102	C₄×C₅₉	—	2	1	no	—
103	C₄×C₅₉	—	2	1	no	—
104	C₄×C₅₉	—	2	1	no	—
105	C₄×C₅₉	#332	1	2	yes	—
106	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
107	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
108	C₄×C₅₉	#509	1	1	yes	characteristic
109	C₄×C₅₉	#510	1	1	yes	characteristic
110	Dic₅₉	—	8	2	no	—
111	Dic₅₉	—	8	2	no	—
112	#3160	—	59	1	no	—
113	#2150	D₅₉	1	1	yes	characteristic
114	C₂×#335	—	59	1	no	—
115	#1365	—	59	2	no	—
116	#1364	—	59	2	no	—
117	C₂×C₅₉	—	2	2	no	—
118	C₂×C₅₉	C₂×#332	1	1	yes	characteristic
119	C₂×C₅₉	#2139	1	1	yes	characteristic
120	C₂×C₅₉	#2148	1	1	yes	characteristic
121	#533	—	118	1	no	—
122	C₂×M₃₂	—	2	1	no	—
123	#500	—	59	1	no	—
124	#500	—	59	1	no	—
125	#499	—	59	2	no	—
126	#499	—	118	1	no	—
127	#498	—	118	1	no	—
128	#427	—	59	2	no	—
129	#427	—	118	1	no	—
130	C₂×C₄⋊C₈	—	118	1	no	—
131	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
134	#335	—	59	2	no	—
135	#335	—	118	1	no	—
136	C₅₉	#11942	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	236	1	no	—
138	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
139	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
140	C₄⋊Q₈	—	118	1	no	—
141	C₄⋊Q₈	—	118	1	no	—
142	#125	—	118	2	no	—
143	C₂²⋊Q₈	—	236	1	no	—
144	C₂²⋊Q₈	—	236	1	no	—
145	C₄×Q₈	—	236	1	no	—
146	C₄×Q₈	—	236	1	no	—
147	#117	—	118	2	no	—
148	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
149	M₃₂	—	4	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	#107	—	59	2	no	—
152	#107	—	59	2	no	—
153	#107	—	118	1	no	—
154	#107	—	118	1	no	—
155	#106	—	118	2	no	—
156	#106	—	118	2	no	—
157	C₄⋊C₈	—	118	2	no	—
158	C₄⋊C₈	—	236	1	no	—
159	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
160	C₄×C₈	D₂₃₆	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
163	C₂×Q₈	—	236	1	no	—
164	C₂×Q₈	—	236	1	no	—
165	C₂²×C₄	#11745660	1	1	yes	characteristic
166	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
167	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	472	2	no	—
170	C₂×C₈	—	236	2	no	—
171	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
172	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	#11745670	1	2	yes	—
174	C₂×C₈	#11745672	1	2	yes	—
175	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
177	C₄⋊C₄	—	118	2	no	—
178	C₄⋊C₄	—	118	2	no	—
179	C₄⋊C₄	—	236	2	no	—
180	C₄⋊C₄	—	236	2	no	—
181	C₄⋊C₄	—	236	1	no	—
182	C₄⋊C₄	—	236	1	no	—
183	C₂²⋊C₄	—	236	2	no	—
184	C₄²	—	236	2	no	—
185	C₄²	#11745660	1	1	yes	characteristic
186	C₄²	D₄₇₂	1	1	yes	characteristic
187	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
188	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₂³	#14273649	1	1	yes	characteristic
191	Q₈	—	472	2	no	—
192	Q₈	—	472	1	no	—
193	Q₈	—	472	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	236	2	no	—
198	C₂×C₄	—	236	2	no	—
199	C₂×C₄	#14273645	1	2	yes	—
200	C₂×C₄	#14273649	1	1	yes	characteristic
201	C₂×C₄	#14273649	1	1	yes	characteristic
202	C₂×C₄	#14273649	1	1	yes	characteristic
203	C₂×C₄	#14273650	1	2	yes	—
204	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
206	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	472	2	no	—
210	C₈	#14273650	1	2	yes	—
211	C₈	C₂×#11745670	1	1	yes	characteristic
212	C₈	C₂×#11745672	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16837147	1	2	yes	—
216	C₂²	C₂×#14273649	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	472	2	no	—
221	C₄	—	472	2	no	—
222	C₄	#16837147	1	2	yes	—
223	C₄	C₂×#14273645	1	1	yes	characteristic
224	C₄	C₂×#14273650	1	1	yes	characteristic
225	C₄	#16837840	1	1	yes	characteristic
226	C₄	#16837845	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#16837147	1	1	yes	characteristic
229	C₂	#21425809	1	1	yes	characteristic
230	C₂	#21425846	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273649
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₉
Fitting	C₅₉×#2150	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂³×C₅₉	#102
cosocle	C₄×C₈×C₅₉	C₂³
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#11942
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #21422153	C₂	semidirect
4. #21422150	C₂	semidirect
136. C₅₉	#11942	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
C₂×#16837155	2
#21422153	2
#21422150	2
C₅₉×#2150	2
#11942	59