Subgroups — #28506813

Order 15104.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837224	C₂	1	2	yes	maximal
2	#21425852	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
4	#21422194	C₂	1	2	yes	maximal
5	C₅₉×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	2	yes	—
7	C₂×#14273663	C₂²	1	1	yes	characteristic
8	C₂×#14273663	C₂²	1	1	yes	characteristic
9	C₄×#11745664	—	2	1	no	—
10	#16837224	—	2	4	no	—
11	C₄⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	complemented
12	#16837167	—	2	1	no	—
13	C₂×#106×C₅₉	—	2	1	no	—
14	C₂×#106×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#384	—	2	1	no	—
17	C₂×#11745664	—	2	1	no	—
18	C₂×#11745664	—	2	1	no	—
19	#14273667	—	2	2	no	—
20	#14273667	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
22	#14273663	—	2	1	no	—
23	#14273663	—	2	1	no	—
24	#14273663	D₄	1	2	yes	—
25	#14273663	Q₈	1	2	yes	—
26	#14273638	—	4	1	no	—
27	C₄×C₅₉⋊C₈	—	4	1	no	—
28	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₉	—	4	2	no	—
32	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
33	#106×C₅₉	—	2	2	no	—
34	#106×C₅₉	—	2	2	no	—
35	#106×C₅₉	—	4	2	no	—
36	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₉	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₉	—	4	2	no	—
39	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
40	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₅₉	—	2	1	no	—
42	C₂×C₈×C₅₉	—	2	1	no	—
43	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
49	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
50	C₄⋊C₄×C₅₉	—	4	2	no	—
51	C₄⋊C₄×C₅₉	—	4	2	no	—
52	C₄⋊C₄×C₅₉	—	8	4	no	—
53	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
55	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
56	#11745664	—	2	2	no	—
57	#11745664	—	2	2	no	—
58	#11745664	—	4	1	no	—
59	#11745664	—	4	1	no	—
60	C₂×C₅₉⋊C₈	—	4	1	no	—
61	C₂×C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₁₆	—	2	4	no	—
63	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₉	—	2	4	no	—
65	C₂×C₄×C₅₉	—	8	4	no	—
66	C₂×C₄×C₅₉	—	8	2	no	—
67	C₂×C₄×C₅₉	#106	1	2	yes	—
68	C₂×C₄×C₅₉	#106	1	2	yes	—
69	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
71	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
75	C₈×C₅₉	—	2	2	no	—
76	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
77	C₈×C₅₉	C₄×Q₈	1	1	yes	characteristic
78	C₈×C₅₉	#125	1	1	yes	characteristic
79	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
80	C₅₉⋊C₈	—	4	2	no	—
81	C₅₉⋊C₈	—	4	2	no	—
82	#13183	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	2	no	—
84	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
85	C₂²×C₅₉	#460	1	1	yes	characteristic
86	C₂²×C₅₉	#475	1	1	yes	characteristic
87	C₄×C₅₉	—	2	2	no	—
88	C₄×C₅₉	—	16	2	no	—
89	C₄×C₅₉	#365	1	4	yes	complemented
90	C₄×C₅₉	#389	1	1	yes	characteristic
91	C₄×C₅₉	#428	1	1	yes	characteristic
92	C₄×C₅₉	#461	1	1	yes	characteristic
93	C₄×C₅₉	#476	1	1	yes	characteristic
94	C₂×#365	—	59	2	no	—
95	#2150	—	59	1	no	—
96	#1844	D₅₉	1	1	yes	characteristic
97	C₄×#108	—	59	1	no	—
98	#1383	—	59	2	no	—
99	C₂×C₅₉	—	2	1	no	—
100	C₂×C₅₉	—	2	1	no	—
101	C₂×C₅₉	#1948	1	1	yes	characteristic
102	C₂×C₅₉	C₂×#365	1	2	yes	—
103	C₂×M₃₂	—	59	2	no	—
104	C₂×#108	—	59	1	no	—
105	C₂×#108	—	59	1	no	—
106	C₂×#106	—	2	1	no	—
107	C₂×#106	C₂×D₅₉	1	2	yes	—
108	C₄×M₁₆	—	118	1	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	#384	—	2	1	no	—
111	#365	—	118	4	no	—
112	C₄⋊C₁₆	—	59	4	no	—
113	#341	—	118	1	no	—
114	C₅₉	#13183	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	118	1	no	—
116	C₂×M₁₆	—	118	1	no	—
117	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	2	no	—
121	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
122	M₃₂	—	118	2	no	—
123	M₃₂	—	118	2	no	—
124	C₂×C₁₆	—	59	4	no	—
125	#108	—	59	2	no	—
126	#108	—	59	2	no	—
127	#108	—	118	1	no	—
128	#108	—	118	1	no	—
129	#106	—	2	2	no	—
130	#106	—	2	2	no	—
131	#106	—	4	2	no	—
132	#97	—	236	1	no	—
133	C₄×C₈	—	236	1	no	—
134	C₄×C₈	Dic₁₁₈	1	2	yes	—
135	C₄×C₈	C₄×D₅₉	1	2	yes	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	#11745670	1	2	yes	—
138	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
139	M₁₆	—	118	2	no	—
140	M₁₆	—	118	2	no	—
141	M₁₆	—	236	1	no	—
142	M₁₆	—	236	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	236	1	no	—
146	C₂×C₈	—	236	1	no	—
147	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
148	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
150	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
151	C₂×C₈	#11745687	1	1	yes	characteristic
152	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
153	C₂×C₈	#11745689	1	1	yes	characteristic
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	8	4	no	—
157	C₄²	Dic₅₉⋊C₄	1	2	yes	—
158	C₄²	#11745671	1	1	yes	characteristic
159	C₄²	#11745671	1	1	yes	characteristic
160	C₁₆	—	118	4	no	—
161	C₂³	C₂×#11745670	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	8	4	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	#14273658	1	2	yes	—
166	C₂×C₄	#14273658	1	2	yes	—
167	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#14273698	1	1	yes	characteristic
169	C₂×C₄	#14273703	1	1	yes	characteristic
170	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
171	C₂×C₄	#14273753	1	1	yes	characteristic
172	C₂×C₄	#14273758	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	236	2	no	—
175	C₈	—	236	2	no	—
176	C₈	#14273689	1	1	yes	characteristic
177	C₈	#14273692	1	1	yes	characteristic
178	C₈	#14273718	1	1	yes	characteristic
179	C₈	#14273726	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14273658	1	1	yes	characteristic
182	C₂²	#16837599	1	1	yes	characteristic
183	C₂²	#16837684	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	16	2	no	—
186	C₄	#16837224	1	4	yes	complemented
187	C₄	#16837298	1	1	yes	characteristic
188	C₄	#16837442	1	1	yes	characteristic
189	C₄	#16837604	1	1	yes	characteristic
190	C₄	#16837689	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#21424857	1	1	yes	characteristic
194	C₂	C₂×#16837224	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14273658
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1844	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1844	D₅₉
hypocenter	C₅₉	#13183
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1844

Complements

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

N	H ≅ G/N	type
11. C₄⋊(C₅₉⋊C₁₆)	C₄	semidirect
89. C₄×C₅₉	#365	semidirect
114. C₅₉	#13183	semidirect
186. C₄	#16837224	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837224	2
#21425852	2
C₄×#14273663	2
#21422194	2
C₅₉×#1844	2
#13183	59