Subgroups — #28503918

Order 15104.

442 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837209	C₂	1	2	yes	maximal
2	C₂×#16837178	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic, complemented
4	#21422191	C₂	1	2	yes	maximal, complemented
5	C₄×#135×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14273663	C₂²	1	2	yes	—
8	C₄×#11745664	—	2	1	no	—
9	#16837209	C₄	1	8	yes	complemented
10	#16837178	—	2	2	no	—
11	#16837167	—	2	1	no	—
12	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
13	C₄×#41×C₅₉	—	2	1	no	—
14	C₄×Q₁₆×C₅₉	C₄	1	2	yes	—
15	C₄×SD₁₆×C₅₉	—	2	2	no	—
16	C₄×D₈×C₅₉	C₄	1	2	yes	—
17	C₅₉×#427	C₂²	1	1	yes	characteristic
18	C₅₉×#416	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#11745664	—	2	2	no	—
21	C₂×C₅₉⋊C₁₆	—	2	2	no	—
22	C₂×C₅₉⋊C₁₆	—	2	2	no	—
23	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
24	#14273663	—	2	2	no	—
25	#14273663	C₂×C₄	1	4	yes	complemented
26	#14273638	—	4	1	no	—
27	C₄×C₅₉⋊C₈	—	4	1	no	—
28	C₂×#41×C₅₉	—	2	1	no	—
29	#135×C₅₉	—	2	2	no	—
30	#135×C₅₉	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
32	C₂×SD₁₆×C₅₉	—	2	1	no	—
33	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₅₉	—	2	2	no	—
37	C₄×D₄×C₅₉	—	2	2	no	—
38	C₄×D₄×C₅₉	—	4	2	no	—
39	#117×C₅₉	—	2	1	no	—
40	#117×C₅₉	—	4	1	no	—
41	C₂×C₄²×C₅₉	—	4	1	no	—
42	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
43	#107×C₅₉	C₂×C₄	1	2	yes	—
44	#106×C₅₉	—	2	1	no	—
45	#103×C₅₉	—	2	2	no	—
46	#102×C₅₉	—	2	2	no	—
47	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₉	D₄	1	2	yes	—
49	#41×C₅₉	—	2	4	no	—
50	#41×C₅₉	—	4	2	no	—
51	C₂×Q₈×C₅₉	—	2	1	no	—
52	C₂×D₄×C₅₉	—	2	1	no	—
53	C₂×D₄×C₅₉	—	4	1	no	—
54	C₂²×C₄×C₅₉	—	4	1	no	—
55	C₂²×C₄×C₅₉	—	4	2	no	—
56	C₂²×C₄×C₅₉	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	Q₁₆×C₅₉	—	2	1	no	—
59	Q₁₆×C₅₉	C₄²	1	2	yes	—
60	SD₁₆×C₅₉	—	2	4	no	—
61	D₈×C₅₉	—	2	1	no	—
62	D₈×C₅₉	C₄²	1	2	yes	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₉	#41	1	2	yes	—
69	C₄⋊C₄×C₅₉	—	2	2	no	—
70	C₄⋊C₄×C₅₉	—	4	2	no	—
71	C₂²⋊C₄×C₅₉	—	4	1	no	—
72	C₂²⋊C₄×C₅₉	—	4	2	no	—
73	C₄²×C₅₉	—	4	2	no	—
74	C₄²×C₅₉	—	4	1	no	—
75	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₅₉	D₈	1	1	yes	characteristic
77	C₄²×C₅₉	D₈	1	1	yes	characteristic
78	#11745664	—	2	4	no	—
79	#11745664	—	4	2	no	—
80	C₂×C₅₉⋊C₈	—	4	2	no	—
81	C₅₉⋊C₁₆	—	2	4	no	—
82	C₂³×C₅₉	—	4	1	no	—
83	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
84	Q₈×C₅₉	—	2	2	no	—
85	Q₈×C₅₉	—	4	1	no	—
86	D₄×C₅₉	—	2	2	no	—
87	D₄×C₅₉	—	4	4	no	—
88	D₄×C₅₉	—	4	1	no	—
89	C₂×C₄×C₅₉	—	4	2	no	—
90	C₂×C₄×C₅₉	—	4	1	no	—
91	C₂×C₄×C₅₉	—	4	4	no	—
92	C₂×C₄×C₅₉	—	4	2	no	—
93	C₂×C₄×C₅₉	—	8	4	no	—
94	C₂×C₄×C₅₉	#102	1	2	yes	—
95	C₂×C₄×C₅₉	#102	1	2	yes	—
96	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₉	#135	1	2	yes	—
100	C₈×C₅₉	—	2	2	no	—
101	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₅₉	#117	1	1	yes	characteristic
103	C₈×C₅₉	C₄×D₄	1	2	yes	—
104	C₅₉⋊C₈	—	4	4	no	—
105	#12640	—	59	1	no	maximal, Sylow
106	C₂²×C₅₉	—	2	2	no	—
107	C₂²×C₅₉	—	4	2	no	—
108	C₂²×C₅₉	—	8	2	no	—
109	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
110	C₂²×C₅₉	C₄×SD₁₆	1	2	yes	—
111	C₄×C₅₉	—	2	2	no	—
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	—	4	2	no	—
114	C₄×C₅₉	—	8	2	no	—
115	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₅₉	#416	1	1	yes	characteristic
117	C₄×C₅₉	C₄×D₈	1	2	yes	—
118	C₄×#135	D₅₉	1	1	yes	characteristic
119	C₂×#359	—	59	2	no	—
120	C₂×#346	—	59	1	no	—
121	C₄×#108	—	59	1	no	—
122	#1382	—	59	2	no	—
123	C₂×C₅₉	—	2	2	no	—
124	C₂×C₅₉	—	8	2	no	—
125	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
126	C₂×C₅₉	#2143	1	2	yes	—
127	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
128	C₄×#41	—	2	1	no	—
129	C₂²×C₁₆	—	59	2	no	—
130	C₄×Q₁₆	Dic₅₉	1	2	yes	—
131	C₄×SD₁₆	—	2	2	no	—
132	C₄×D₈	Dic₅₉	1	2	yes	—
133	C₂×#108	—	59	2	no	—
134	#427	C₂×D₅₉	1	1	yes	characteristic
135	#416	—	2	1	no	—
136	C₄×M₁₆	—	118	1	no	—
137	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
138	#359	—	59	8	no	—
139	#346	—	118	2	no	—
140	#341	—	118	1	no	—
141	C₅₉	#12640	1	1	yes	characteristic, complemented, Sylow
142	C₂×#41	—	2	1	no	—
143	#135	—	2	2	no	—
144	#135	C₄×D₅₉	1	4	yes	—
145	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
146	C₂×SD₁₆	—	2	1	no	—
147	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
148	C₂×M₁₆	—	118	2	no	—
149	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	#117	—	2	1	no	—
155	#117	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
158	C₂×C₁₆	—	59	4	no	—
159	C₂×C₁₆	—	118	2	no	—
160	C₂×C₁₆	—	118	2	no	—
161	#108	—	59	4	no	—
162	#108	—	118	2	no	—
163	#107	C₂×Dic₅₉	1	2	yes	—
164	#106	—	2	1	no	—
165	#103	—	2	2	no	—
166	#102	—	2	2	no	—
167	#97	—	236	1	no	—
168	C₄×C₈	—	236	1	no	—
169	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
171	#41	—	2	4	no	—
172	#41	—	4	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	#11745670	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₄×Dic₅₉	1	2	yes	—
182	SD₁₆	—	2	4	no	—
183	D₈	—	2	1	no	—
184	D₈	C₄×Dic₅₉	1	2	yes	—
185	M₁₆	—	118	4	no	—
186	M₁₆	—	236	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	236	2	no	—
189	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
190	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
191	C₂×C₈	#11745685	1	2	yes	—
192	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
193	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	1	no	—
200	C₄²	#11745669	1	1	yes	characteristic
201	C₄²	#11745669	1	1	yes	characteristic
202	C₄²	C₂²⋊Dic₅₉	1	2	yes	—
203	C₁₆	—	118	4	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂×#11745670	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#14273648	1	2	yes	—
217	C₂×C₄	#14273648	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
219	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
220	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
221	C₂×C₄	#14273747	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	236	4	no	—
224	C₈	C₂×C₄×Dic₅₉	1	1	yes	characteristic
225	C₈	#14273686	1	1	yes	characteristic
226	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×#14273648	1	1	yes	characteristic
231	C₂²	C₄×#11745670	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
237	C₄	#16837398	1	1	yes	characteristic
238	C₄	C₄×#11745669	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×#14273648	1	1	yes	characteristic
242	C₂	#21425824	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#135×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#102
prime radical	C₂×#135×C₅₉	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#135	D₅₉
hypocenter	C₅₉	#12640
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#16837178	C₂	semidirect
3. C₄×#14273663	C₂	semidirect
4. #21422191	C₂	semidirect
9. #16837209	C₄	semidirect
23. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
25. #14273663	C₂×C₄	semidirect
141. C₅₉	#12640	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837209	2
C₂×#16837178	2
C₄×#14273663	2
#21422191	2
C₄×#135×C₅₉	2
#12640	59