Subgroups — C₂²×#16837806 #29458286

Order 15104.

2398 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273726	C₂	1	2	yes	maximal
2	C₂×#16837806	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14273660	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₁₈	—	2	2	no	—
8	C₂³×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14273726	—	2	12	no	—
10	C₂×#14273726	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
12	C₂²×Dic₅₉⋊C₄	—	2	2	no	—
13	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
14	#16837806	C₂²	1	64	yes	direct product factor
15	C₂×#14273661	C₂²	1	12	yes	—
16	C₂×#14273660	C₂²	1	24	yes	—
17	C₂×C₈×Dic₅₉	C₂²	1	12	yes	—
18	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
20	C₂×#107×C₅₉	C₂²	1	12	yes	—
21	C₂²×Dic₁₁₈	—	2	24	no	—
22	C₂²×Dic₁₁₈	—	4	16	no	—
23	C₂³×Dic₅₉	—	2	2	no	—
24	C₂³×Dic₅₉	D₄	1	1	yes	characteristic
25	C₂³×Dic₅₉	D₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	Q₈	1	8	yes	complemented
27	C₂²×C₅₉⋊C₈	C₂³	1	6	yes	—
28	#14273726	—	2	48	no	—
29	#14273726	C₂³	1	32	yes	—
30	C₂×C₄⋊Dic₅₉	—	2	6	no	—
31	C₂×C₄⋊Dic₅₉	C₂³	1	12	yes	—
32	C₂×Dic₅₉⋊C₄	—	2	24	no	—
33	C₂×C₄×Dic₅₉	D₄	1	6	yes	—
34	C₂×C₄×Dic₅₉	C₂³	1	6	yes	—
35	#14273661	C₂³	1	16	yes	—
36	#14273660	C₂³	1	32	yes	—
37	C₈×Dic₅₉	C₂³	1	16	yes	—
38	C₂³×C₄×C₅₉	—	2	2	no	—
39	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	Q₈	1	8	yes	complemented
41	C₂²×C₈×C₅₉	C₂³	1	6	yes	—
42	C₂×C₄⋊C₄×C₅₉	—	2	6	no	—
43	C₂×C₄⋊C₄×C₅₉	C₂³	1	12	yes	—
44	#107×C₅₉	C₂³	1	16	yes	—
45	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	—	2	12	no	—
47	C₂²×C₄×C₅₉	—	4	8	no	—
48	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₅₉	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₅₉	C₂⁴	1	3	yes	—
52	C₂²×Dic₅₉	—	2	12	no	—
53	C₂²×Dic₅₉	—	4	8	no	—
54	C₂²×Dic₅₉	D₈	1	8	yes	—
55	C₂²×Dic₅₉	Q₁₆	1	8	yes	—
56	C₂²×Dic₅₉	C₂×D₄	1	6	yes	—
57	C₂²×Dic₅₉	C₂×D₄	1	6	yes	—
58	C₂×Dic₁₁₈	—	2	32	no	—
59	C₂×Dic₁₁₈	—	4	96	no	—
60	C₂×C₈×C₅₉	C₂×Q₈	1	24	yes	complemented
61	C₂×C₈×C₅₉	C₂⁴	1	4	yes	—
62	C₄⋊C₄×C₅₉	—	2	12	no	—
63	C₄⋊C₄×C₅₉	C₂⁴	1	8	yes	—
64	C₄⋊Dic₅₉	—	2	12	no	—
65	C₄⋊Dic₅₉	C₂⁴	1	8	yes	—
66	Dic₅₉⋊C₄	—	2	32	no	—
67	C₄×Dic₅₉	C₂×D₄	1	12	yes	—
68	C₄×Dic₅₉	C₂⁴	1	4	yes	—
69	C₂×C₅₉⋊C₈	C₂×Q₈	1	24	yes	complemented
70	C₂×C₅₉⋊C₈	C₂⁴	1	4	yes	—
71	C₂³×C₅₉	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₅₉	C₂×D₈	1	4	yes	—
73	C₂³×C₅₉	C₂×Q₁₆	1	4	yes	—
74	C₂³×C₅₉	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₅₉	—	2	8	no	—
76	C₂×C₄×C₅₉	—	4	24	no	—
77	C₂×C₄×C₅₉	C₄⋊Q₈	1	12	yes	—
78	C₂×C₄×C₅₉	C₂²×D₄	1	3	yes	—
79	C₂×C₄×C₅₉	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₅₉	C₂²×Q₈	1	6	yes	—
81	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
82	C₂×Dic₅₉	—	2	8	no	—
83	C₂×Dic₅₉	—	4	24	no	—
84	C₂×Dic₅₉	C₂×D₈	1	24	yes	—
85	C₂×Dic₅₉	C₂×Q₁₆	1	24	yes	—
86	C₂×Dic₅₉	C₂²×D₄	1	4	yes	—
87	C₂×Dic₅₉	C₂²×D₄	1	4	yes	—
88	Dic₁₁₈	—	4	64	no	—
89	C₈×C₅₉	C₂²×Q₈	1	8	yes	complemented
90	C₅₉⋊C₈	C₂²×Q₈	1	8	yes	complemented
91	C₂²×#500	—	59	1	no	maximal, Sylow
92	C₂²×C₅₉	#500	1	16	yes	complemented
93	C₂²×C₅₉	C₂×C₄⋊Q₈	1	6	yes	—
94	C₂²×C₅₉	C₂²×D₈	1	6	yes	—
95	C₂²×C₅₉	C₂²×Q₁₆	1	6	yes	—
96	C₂²×C₅₉	C₂³×D₄	1	1	yes	characteristic
97	C₄×C₅₉	—	4	8	no	—
98	C₄×C₅₉	C₂×C₄⋊Q₈	1	6	yes	—
99	C₄×C₅₉	C₂³×Q₈	1	1	yes	characteristic
100	C₄×C₅₉	C₂³×Q₈	1	1	yes	characteristic
101	Dic₅₉	—	4	8	no	—
102	Dic₅₉	C₂²×D₈	1	8	yes	—
103	Dic₅₉	C₂²×Q₁₆	1	8	yes	—
104	C₂²×C₄⋊Q₈	—	59	2	no	—
105	C₂×#500	—	59	24	no	—
106	C₂²×#107	—	59	1	no	—
107	C₂²×#107	—	59	2	no	—
108	C₂²×#107	D₅₉	1	1	yes	characteristic
109	C₂²×C₄×C₈	—	59	1	no	—
110	C₂×C₅₉	C₂×#500	1	12	yes	complemented
111	C₂×C₅₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₅₉	C₂³×D₈	1	1	yes	characteristic
113	C₂×C₅₉	C₂³×Q₁₆	1	1	yes	characteristic
114	C₂³×Q₈	—	118	2	no	—
115	C₂³×C₈	—	59	1	no	—
116	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	—	59	24	no	—
118	C₂×C₄⋊Q₈	—	118	12	no	—
119	C₂²×C₄⋊C₄	—	59	2	no	—
120	C₂²×C₄⋊C₄	—	118	2	no	—
121	C₂²×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
122	C₂²×C₄²	—	59	1	no	—
123	#500	—	59	64	no	—
124	C₂×#107	—	59	12	no	—
125	C₂×#107	—	59	24	no	—
126	C₂×#107	C₂×D₅₉	1	12	yes	—
127	C₂×C₄×C₈	—	59	12	no	—
128	C₅₉	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	118	24	no	—
130	C₂²×Q₈	—	236	16	no	—
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	—	59	1	no	—
133	C₂³×C₄	—	59	1	no	—
134	C₂³×C₄	—	118	2	no	—
135	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
136	C₂²×C₈	—	59	8	no	—
137	C₂²×C₈	—	59	6	no	—
138	C₂²×C₈	Dic₁₁₈	1	8	yes	complemented
139	C₂²×C₈	C₂²×D₅₉	1	6	yes	—
140	C₄⋊Q₈	—	59	32	no	—
141	C₄⋊Q₈	—	118	48	no	—
142	C₂×C₄⋊C₄	—	2	6	no	—
143	C₂×C₄⋊C₄	—	59	12	no	—
144	C₂×C₄⋊C₄	—	118	24	no	—
145	C₂×C₄⋊C₄	—	118	6	no	—
146	C₂×C₄⋊C₄	C₂²×D₅₉	1	12	yes	—
147	C₂×C₄²	—	59	6	no	—
148	C₂×C₄²	—	59	6	no	—
149	#107	—	59	16	no	—
150	#107	—	59	32	no	—
151	#107	C₂²×D₅₉	1	16	yes	—
152	C₄×C₈	—	59	16	no	—
153	C₂⁴	D₄×D₅₉	1	1	yes	characteristic
154	C₂×Q₈	—	118	32	no	—
155	C₂×Q₈	—	236	96	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	4	8	no	—
158	C₂²×C₄	—	59	8	no	—
159	C₂²×C₄	—	59	8	no	—
160	C₂²×C₄	—	59	6	no	—
161	C₂²×C₄	—	59	6	no	—
162	C₂²×C₄	—	118	12	no	—
163	C₂²×C₄	—	236	8	no	—
164	C₂²×C₄	C₂×Dic₁₁₈	1	4	yes	—
165	C₂²×C₄	D₄×D₅₉	1	3	yes	—
166	C₂²×C₄	Q₈×D₅₉	1	4	yes	—
167	C₂²×C₄	C₂³×D₅₉	1	3	yes	—
168	C₂×C₈	—	59	24	no	—
169	C₂×C₈	—	59	4	no	—
170	C₂×C₈	C₂×Dic₁₁₈	1	24	yes	complemented
171	C₂×C₈	C₂³×D₅₉	1	4	yes	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	59	8	no	—
174	C₄⋊C₄	—	118	32	no	—
175	C₄⋊C₄	—	118	12	no	—
176	C₄⋊C₄	C₂³×D₅₉	1	8	yes	—
177	C₄²	—	59	12	no	—
178	C₄²	—	59	4	no	—
179	C₂³	#14273726	1	4	yes	—
180	C₂³	D₈×D₅₉	1	4	yes	—
181	C₂³	Q₁₆×D₅₉	1	4	yes	—
182	C₂³	C₂×D₄×D₅₉	1	3	yes	—
183	Q₈	—	236	64	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	59	24	no	—
187	C₂×C₄	—	59	24	no	—
188	C₂×C₄	—	59	4	no	—
189	C₂×C₄	—	59	4	no	—
190	C₂×C₄	—	118	8	no	—
191	C₂×C₄	—	236	24	no	—
192	C₂×C₄	#14273726	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₅₉	1	3	yes	—
194	C₂×C₄	C₂×Q₈×D₅₉	1	6	yes	—
195	C₂×C₄	C₂²×Dic₁₁₈	1	6	yes	—
196	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
197	C₈	—	59	8	no	—
198	C₈	C₂²×Dic₁₁₈	1	8	yes	complemented
199	C₂²	#16837806	1	16	yes	direct product factor
200	C₂²	C₂×#14273726	1	6	yes	—
201	C₂²	C₂×D₈×D₅₉	1	6	yes	—
202	C₂²	C₂×Q₁₆×D₅₉	1	6	yes	—
203	C₂²	C₂²×D₄×D₅₉	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	59	8	no	—
206	C₄	—	59	8	no	—
207	C₄	—	236	8	no	—
208	C₄	C₂×#14273726	1	6	yes	—
209	C₄	C₂²×Q₈×D₅₉	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₁₈	1	1	yes	characteristic
211	C₂	C₂×#16837806	1	12	yes	direct product factor
212	C₂	C₂²×#14273726	1	1	yes	characteristic
213	C₂	C₂²×D₈×D₅₉	1	1	yes	characteristic
214	C₂	C₂²×Q₁₆×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅₉
commutator, cosocle	C₂×C₄×C₅₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₉
Fitting	C₂²×#107×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
hypercenter	C₂²×#107	D₅₉
hypocenter	C₅₉	C₂²×#500
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#16837806	C₂	direct (N × H)
14. #16837806	C₂²	direct (N × H)
26. C₂²×C₅₉⋊C₈	Q₈	semidirect
40. C₂²×C₈×C₅₉	Q₈	semidirect
60. C₂×C₈×C₅₉	C₂×Q₈	semidirect
69. C₂×C₅₉⋊C₈	C₂×Q₈	semidirect
89. C₈×C₅₉	C₂²×Q₈	semidirect
90. C₅₉⋊C₈	C₂²×Q₈	semidirect
92. C₂²×C₅₉	#500	semidirect
110. C₂×C₅₉	C₂×#500	semidirect
128. C₅₉	C₂²×#500	semidirect
138. C₂²×C₈	Dic₁₁₈	semidirect
170. C₂×C₈	C₂×Dic₁₁₈	semidirect
198. C₈	C₂²×Dic₁₁₈	semidirect
199. C₂²	#16837806	direct (N × H)
211. C₂	C₂×#16837806	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14273726	2
C₂×#16837806	2
C₂²×#14273661	2
C₂²×#14273660	2
C₂²×C₈×Dic₅₉	2
C₂²×#107×C₅₉	2
C₂²×#500	59

Subgroups — C22×#16837806 #29458286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16837806 #29458286