Subgroups — #28499948

Order 15104.

398 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837782	C₂	1	1	yes	maximal, characteristic
2	#21422844	C₂	1	1	yes	maximal, characteristic
3	#21422146	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₆	—	2	1	no	—
6	C₂²×Dic₂₃₆	C₄	1	2	yes	—
7	C₂×#14273727	—	2	1	no	—
8	#16837782	—	2	1	no	—
9	#16837782	—	2	1	no	—
10	#16837782	C₄	1	2	yes	—
11	#16837782	C₂²	1	2	yes	—
12	#16837455	—	2	1	no	—
13	#16837155	C₂²	1	2	yes	—
14	#16837153	—	2	1	no	—
15	C₂×M₃₂×C₅₉	—	2	1	no	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×C₄⋊C₁₆	—	2	1	no	—
18	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₂²×Dic₁₁₈	—	2	2	no	—
20	C₂×Dic₂₃₆	—	2	2	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	—	2	2	no	—
23	C₂×Dic₂₃₆	—	4	4	no	—
24	C₂×Dic₂₃₆	—	4	2	no	—
25	C₂×Dic₂₃₆	—	4	2	no	—
26	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
27	#14273727	—	2	2	no	—
28	#14273727	—	2	2	no	—
29	#14273727	—	4	1	no	—
30	#14273727	—	4	1	no	—
31	C₂×C₄⋊Dic₅₉	—	4	1	no	—
32	C₂×C₄⋊Dic₅₉	—	4	1	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₅₉⋊C₈)	—	4	1	no	—
35	C₄×C₅₉⋊C₈	—	4	1	no	—
36	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
39	M₃₂×C₅₉	—	4	2	no	—
40	C₂×C₁₆×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	4	2	no	—
49	C₂×Dic₁₁₈	—	2	4	no	—
50	C₂×Dic₁₁₈	—	4	4	no	—
51	C₂×Dic₁₁₈	—	4	4	no	—
52	C₂×Dic₁₁₈	—	4	2	no	—
53	C₂×C₈×C₅₉	—	2	2	no	—
54	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	D₈	1	2	yes	—
57	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₉	D₈	1	1	yes	characteristic
62	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
63	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Dic₅₉	—	4	2	no	—
65	C₄⋊Dic₅₉	—	4	2	no	—
66	C₄⋊Dic₅₉	—	4	2	no	—
67	C₄⋊Dic₅₉	—	4	2	no	—
68	C₂×C₅₉⋊C₈	—	4	2	no	—
69	Dic₂₃₆	—	4	4	no	—
70	Dic₂₃₆	—	4	4	no	—
71	Dic₂₃₆	—	4	4	no	—
72	Dic₂₃₆	—	4	2	no	—
73	Dic₂₃₆	—	4	2	no	—
74	C₁₆×C₅₉	—	4	2	no	—
75	C₂³×C₅₉	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	—	2	2	no	—
78	C₂×C₄×C₅₉	#100	1	2	yes	—
79	C₂×C₄×C₅₉	#102	1	2	yes	—
80	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₉	#102	1	1	yes	characteristic
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	4	no	—
87	C₂×Dic₅₉	—	8	4	no	—
88	Dic₁₁₈	—	4	4	no	—
89	Dic₁₁₈	—	4	4	no	—
90	Dic₁₁₈	—	8	4	no	—
91	C₈×C₅₉	—	2	1	no	—
92	C₈×C₅₉	—	2	1	no	—
93	C₈×C₅₉	#102	1	2	yes	—
94	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
95	C₈×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₅₉⋊C₈	—	8	2	no	—
97	#11932	—	59	1	no	maximal, Sylow
98	C₂²×C₅₉	—	2	1	no	—
99	C₂²×C₅₉	—	2	1	no	—
100	C₂²×C₅₉	#331	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	#331	1	2	yes	—
105	C₄×C₅₉	#362	1	2	yes	—
106	C₄×C₅₉	C₂×#100	1	1	yes	characteristic
107	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
108	C₄×C₅₉	#510	1	2	yes	—
109	Dic₅₉	—	8	4	no	—
110	C₂×#494	—	59	1	no	—
111	#2150	D₅₉	1	1	yes	characteristic
112	#1564	—	59	1	no	—
113	#1362	—	59	4	no	—
114	C₂×C₅₉	—	2	2	no	—
115	C₂×C₅₉	C₂×#331	1	1	yes	characteristic
116	C₂×C₅₉	#2141	1	1	yes	characteristic
117	C₂×C₅₉	C₂×#362	1	1	yes	characteristic
118	C₂²×Q₁₆	—	59	2	no	—
119	C₂²×Q₁₆	—	118	1	no	—
120	C₂×C₄⋊Q₈	—	118	1	no	—
121	C₂×M₃₂	—	2	1	no	—
122	#494	—	59	2	no	—
123	#494	—	59	2	no	—
124	#494	—	118	1	no	—
125	#494	—	118	1	no	—
126	#431	—	118	1	no	—
127	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
128	C₄⋊C₁₆	—	2	1	no	—
129	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
130	#335	—	59	2	no	—
131	#334	—	118	1	no	—
132	C₅₉	#11932	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	118	2	no	—
134	C₂×Q₁₆	—	59	4	no	—
135	C₂×Q₁₆	—	118	2	no	—
136	C₂×Q₁₆	—	118	2	no	—
137	C₂×Q₁₆	—	118	2	no	—
138	C₂×Q₁₆	—	236	4	no	—
139	C₂×Q₁₆	—	236	2	no	—
140	C₂×Q₁₆	—	236	2	no	—
141	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
143	C₄⋊Q₈	—	118	2	no	—
144	C₄⋊Q₈	—	118	2	no	—
145	C₄⋊Q₈	—	236	1	no	—
146	C₄⋊Q₈	—	236	1	no	—
147	C₂×C₄⋊C₄	—	236	1	no	—
148	C₂×C₄⋊C₄	—	236	1	no	—
149	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
150	M₃₂	—	4	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	118	2	no	—
153	C₂²⋊C₈	—	236	1	no	—
154	C₄×C₈	—	236	1	no	—
155	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
156	C₄×C₈	D₂₃₆	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
159	C₂×Q₈	—	118	4	no	—
160	C₂×Q₈	—	236	4	no	—
161	C₂×Q₈	—	236	4	no	—
162	C₂×Q₈	—	236	2	no	—
163	C₂²×C₄	—	236	2	no	—
164	C₂²×C₄	#11745660	1	1	yes	characteristic
165	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
166	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
167	Q₁₆	—	236	4	no	—
168	Q₁₆	—	236	4	no	—
169	Q₁₆	—	236	4	no	—
170	Q₁₆	—	236	2	no	—
171	Q₁₆	—	236	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	236	2	no	—
174	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	#11745669	1	2	yes	—
177	C₂×C₈	#11745671	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	C₄⋊C₄	—	236	2	no	—
181	C₄⋊C₄	—	236	2	no	—
182	C₄⋊C₄	—	236	2	no	—
183	C₄⋊C₄	—	236	2	no	—
184	C₄²	#11745660	1	1	yes	characteristic
185	C₄²	D₄₇₂	1	1	yes	characteristic
186	C₄²	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
187	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₂³	#14273649	1	1	yes	characteristic
190	Q₈	—	472	4	no	—
191	Q₈	—	236	4	no	—
192	Q₈	—	236	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	472	4	no	—
196	C₂×C₄	—	236	4	no	—
197	C₂×C₄	#14273643	1	2	yes	—
198	C₂×C₄	#14273647	1	2	yes	—
199	C₂×C₄	#14273649	1	1	yes	characteristic
200	C₂×C₄	#14273649	1	1	yes	characteristic
201	C₂×C₄	#14273649	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
204	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	472	2	no	—
208	C₈	#14273647	1	2	yes	—
209	C₈	C₂×#11745669	1	1	yes	characteristic
210	C₈	C₂×#11745671	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16837145	1	2	yes	—
214	C₂²	C₂×#14273649	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	472	4	no	—
218	C₄	#16837145	1	2	yes	—
219	C₄	#16837219	1	2	yes	—
220	C₄	C₂×#14273643	1	1	yes	characteristic
221	C₄	C₂×#14273647	1	1	yes	characteristic
222	C₄	#16837845	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₂×#16837145	1	1	yes	characteristic
225	C₂	#21425817	1	1	yes	characteristic
226	C₂	C₂×#16837219	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	#11932
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. #21422146	C₂	semidirect
132. C₅₉	#11932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837782	2
#21422844	2
#21422146	2
C₅₉×#2150	2
#11932	59