Subgroups — C₂×#21427277 #28751324

Order 15104.

2326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21427277	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16837309	C₂	1	4	yes	maximal
3	C₂×#16837288	C₂	1	8	yes	maximal
4	C₂×#16837270	C₂	1	2	yes	maximal
5	C₂²×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₅₉	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₅₉	C₂²	1	4	yes	—
8	#16837309	C₂²	1	32	yes	—
9	#16837288	C₂²	1	64	yes	—
10	#16837270	C₂²	1	16	yes	—
11	C₂×C₅₉⋊#95	C₂²	1	8	yes	—
12	C₂×C₅₉⋊#95	C₂²	1	4	yes	—
13	C₂×C₅₉⋊#95	C₂²	1	8	yes	—
14	C₂³×Q₈×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₅₉	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₅₉	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₅₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
19	C₂³×Dic₅₉	C₂³	1	8	yes	—
20	C₂×C₄⋊Dic₅₉	—	4	8	no	—
21	C₂×C₄⋊Dic₅₉	C₂³	1	16	yes	—
22	C₂×C₄×Dic₅₉	—	4	8	no	—
23	C₂×C₄×Dic₅₉	C₂³	1	16	yes	—
24	C₅₉⋊#95	C₂³	1	32	yes	—
25	C₅₉⋊#95	C₂³	1	16	yes	—
26	C₅₉⋊#95	C₂³	1	32	yes	—
27	C₂²×Q₈×C₅₉	—	4	4	no	—
28	C₂²×Q₈×C₅₉	—	4	4	no	—
29	C₂²×Q₈×C₅₉	C₂×C₄	1	16	yes	—
30	C₂²×Q₈×C₅₉	C₂³	1	8	yes	—
31	C₂³×C₄×C₅₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅₉	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₅₉	C₂×C₄	1	64	yes	complemented
35	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	32	yes	—
36	C₂×C₄⋊C₄×C₅₉	C₂³	1	16	yes	—
37	C₂×C₄²×C₅₉	C₂×C₄	1	8	yes	—
38	C₂×C₄²×C₅₉	C₂³	1	4	yes	—
39	C₂⁴×C₅₉	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₅₉	—	4	32	no	—
41	C₂×Q₈×C₅₉	—	4	8	no	—
42	C₂×Q₈×C₅₉	—	4	8	no	—
43	C₂×Q₈×C₅₉	C₂²×C₄	1	32	yes	—
44	C₂²×C₄×C₅₉	—	2	4	no	—
45	C₂²×C₄×C₅₉	—	2	8	no	—
46	C₂²×C₄×C₅₉	—	2	8	no	—
47	C₂²×C₄×C₅₉	—	2	8	no	—
48	C₂²×C₄×C₅₉	C₂²×C₄	1	8	yes	—
49	C₂²×C₄×C₅₉	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₅₉	C₂²×C₄	1	16	yes	—
51	C₂²×C₄×C₅₉	C₂⁴	1	4	yes	—
52	C₂²×C₄×C₅₉	C₂⁴	1	2	yes	—
53	C₂²×C₄×C₅₉	C₂⁴	1	8	yes	—
54	C₂²×Dic₅₉	—	2	16	no	—
55	C₂²×Dic₅₉	—	2	32	no	—
56	C₂²×Dic₅₉	C₂⁴	1	16	yes	—
57	C₄⋊C₄×C₅₉	C₂²×C₄	1	64	yes	—
58	C₄²×C₅₉	C₂²×C₄	1	16	yes	—
59	C₄⋊Dic₅₉	—	4	16	no	—
60	C₄×Dic₅₉	—	4	16	no	—
61	C₂³×C₅₉	C₂³×C₄	1	2	yes	—
62	C₂³×C₅₉	H(5,𝔽₂)	1	4	yes	—
63	C₂³×C₅₉	2₋¹⁺⁴	1	8	yes	—
64	C₂³×C₅₉	C₂⁵	1	1	yes	characteristic
65	Q₈×C₅₉	—	4	32	no	—
66	C₂×C₄×C₅₉	—	2	32	no	—
67	C₂×C₄×C₅₉	—	2	8	no	—
68	C₂×C₄×C₅₉	—	2	16	no	—
69	C₂×C₄×C₅₉	—	2	8	no	—
70	C₂×C₄×C₅₉	—	2	8	no	—
71	C₂×C₄×C₅₉	—	2	4	no	—
72	C₂×C₄×C₅₉	—	2	8	no	—
73	C₂×C₄×C₅₉	C₂³×C₄	1	8	yes	—
74	C₂×C₄×C₅₉	C₂³×C₄	1	4	yes	—
75	C₂×C₄×C₅₉	C₂³×C₄	1	16	yes	—
76	C₂×Dic₅₉	—	2	16	no	—
77	C₂×Dic₅₉	—	2	32	no	—
78	C₂×Dic₅₉	—	4	32	no	—
79	C₂×#2381	—	59	1	no	maximal, Sylow
80	C₂²×C₅₉	#518	1	4	yes	—
81	C₂²×C₅₉	#519	1	8	yes	—
82	C₂²×C₅₉	#563	1	16	yes	—
83	C₂²×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
84	C₂²×C₅₉	C₂×H(5,𝔽₂)	1	2	yes	—
85	C₂²×C₅₉	C₂×2₋¹⁺⁴	1	4	yes	—
86	C₄×C₅₉	—	2	16	no	—
87	C₄×C₅₉	—	2	4	no	—
88	C₄×C₅₉	—	2	8	no	—
89	Dic₅₉	—	4	16	no	—
90	C₂²×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
91	#2381	—	59	16	no	—
92	C₂×#391	—	59	4	no	—
93	C₂×#387	—	59	8	no	—
94	C₂×#383	—	59	2	no	—
95	C₂×C₅₉	#2381	1	8	yes	complemented
96	C₂×C₅₉	C₂×#518	1	1	yes	characteristic
97	C₂×C₅₉	C₂×#519	1	2	yes	—
98	C₂×C₅₉	C₂×#563	1	4	yes	—
99	C₂³×Q₈	C₂×D₅₉	1	2	yes	—
100	C₂×C₄⋊Q₈	Dic₅₉	1	16	yes	complemented
101	C₂×C₄⋊Q₈	C₂×D₅₉	1	8	yes	—
102	C₂²×C₄⋊C₄	—	59	4	no	—
103	C₂²×C₄⋊C₄	C₂×D₅₉	1	4	yes	—
104	C₂²×C₄²	—	59	4	no	—
105	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
106	#391	—	59	32	no	—
107	#387	—	59	64	no	—
108	#383	—	59	16	no	—
109	C₂×#95	—	59	8	no	—
110	C₂×#95	—	59	4	no	—
111	C₂×#95	—	59	8	no	—
112	C₅₉	C₂×#2381	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	4	4	no	—
114	C₂²×Q₈	—	4	4	no	—
115	C₂²×Q₈	C₂×Dic₅₉	1	16	yes	—
116	C₂²×Q₈	C₂²×D₅₉	1	8	yes	—
117	C₂³×C₄	—	59	8	no	—
118	C₂³×C₄	C₂²×D₅₉	1	4	yes	—
119	C₂³×C₄	C₂²×D₅₉	1	2	yes	—
120	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
121	C₄⋊Q₈	C₂×Dic₅₉	1	64	yes	complemented
122	C₂×C₄⋊C₄	—	59	16	no	—
123	C₂×C₄⋊C₄	—	236	8	no	—
124	C₂×C₄⋊C₄	C₂×Dic₅₉	1	32	yes	—
125	C₂×C₄⋊C₄	C₂²×D₅₉	1	16	yes	—
126	C₂×C₄²	—	59	16	no	—
127	C₂×C₄²	—	236	8	no	—
128	C₂×C₄²	C₂×Dic₅₉	1	8	yes	—
129	C₂×C₄²	C₂²×D₅₉	1	4	yes	—
130	#95	—	59	32	no	—
131	#95	—	59	16	no	—
132	#95	—	59	32	no	—
133	C₂⁴	C₂³×D₅₉	1	1	yes	characteristic
134	C₂×Q₈	—	4	32	no	—
135	C₂×Q₈	—	4	8	no	—
136	C₂×Q₈	—	4	8	no	—
137	C₂×Q₈	C₂²×Dic₅₉	1	32	yes	—
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	59	16	no	—
143	C₂²×C₄	—	118	16	no	—
144	C₂²×C₄	—	118	32	no	—
145	C₂²×C₄	C₂²×Dic₅₉	1	8	yes	—
146	C₂²×C₄	C₂²×Dic₅₉	1	4	yes	—
147	C₂²×C₄	C₂²×Dic₅₉	1	16	yes	—
148	C₂²×C₄	C₂³×D₅₉	1	4	yes	—
149	C₂²×C₄	C₂³×D₅₉	1	2	yes	—
150	C₂²×C₄	C₂³×D₅₉	1	8	yes	—
151	C₄⋊C₄	—	236	16	no	—
152	C₄⋊C₄	C₂²×Dic₅₉	1	64	yes	—
153	C₄²	—	236	16	no	—
154	C₄²	C₂²×Dic₅₉	1	16	yes	—
155	C₂³	C₂³×Dic₅₉	1	2	yes	—
156	C₂³	#14273772	1	4	yes	—
157	C₂³	#14273775	1	8	yes	—
158	C₂³	C₂⁴×D₅₉	1	1	yes	characteristic
159	Q₈	—	4	32	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	32	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	118	16	no	—
168	C₂×C₄	—	118	32	no	—
169	C₂×C₄	—	236	32	no	—
170	C₂×C₄	C₂³×Dic₅₉	1	8	yes	—
171	C₂×C₄	C₂³×Dic₅₉	1	4	yes	—
172	C₂×C₄	C₂³×Dic₅₉	1	16	yes	—
173	C₂²	#16837877	1	4	yes	—
174	C₂²	#16837880	1	8	yes	—
175	C₂²	#16838151	1	16	yes	—
176	C₂²	C₂⁴×Dic₅₉	1	1	yes	characteristic
177	C₂²	C₂×#14273772	1	2	yes	—
178	C₂²	C₂×#14273775	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	16	no	—
181	C₄	—	2	8	no	—
182	C₄	—	236	16	no	—
183	C₂	#21427277	1	8	yes	direct product factor
184	C₂	C₂×#16837877	1	1	yes	characteristic
185	C₂	C₂×#16837880	1	2	yes	—
186	C₂	C₂×#16838151	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₉
commutator	C₂²×C₅₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₉
Fitting	C₂²×C₄⋊Q₈×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂⁴
cosocle	C₂³×C₅₉	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	C₂×#2381
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #21427277	C₂	direct (N × H)
15. C₂×C₄⋊Q₈×C₅₉	C₄	semidirect
34. C₄⋊Q₈×C₅₉	C₂×C₄	semidirect
95. C₂×C₅₉	#2381	semidirect
100. C₂×C₄⋊Q₈	Dic₅₉	semidirect
112. C₅₉	C₂×#2381	semidirect
121. C₄⋊Q₈	C₂×Dic₅₉	semidirect
183. C₂	#21427277	direct (N × H)

Maximal subgroups

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

subgroup	index
#21427277	2
C₂×#16837309	2
C₂×#16837288	2
C₂×#16837270	2
C₂²×C₄⋊Q₈×C₅₉	2
C₂×#2381	59

Subgroups — C2×#21427277 #28751324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21427277 #28751324