Subgroups — C₄×#16837207 #28503899

Order 15104.

470 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837207	C₂	1	2	yes	maximal
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
4	#21422188	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₁₈	—	2	1	no	—
8	C₄×Dic₂₃₆	—	2	2	no	—
9	C₄×Dic₂₃₆	C₄	1	4	yes	complemented
10	C₂×#14273661	C₂²	1	1	yes	characteristic
11	C₂×#14273661	C₂²	1	2	yes	—
12	C₂×#14273652	—	2	1	no	—
13	C₄×C₄⋊Dic₅₉	—	2	1	no	—
14	#16837207	C₄	1	8	yes	direct product factor
15	#16837165	—	2	1	no	—
16	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₅₉	—	2	2	no	—
19	C₂²×Dic₁₁₈	—	2	1	no	—
20	C₂×Dic₂₃₆	—	2	1	no	—
21	C₂×Dic₂₃₆	—	2	2	no	—
22	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₁₈	—	2	4	no	—
25	C₄×Dic₁₁₈	—	4	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	1	no	—
27	C₂×C₄⋊Dic₅₉	—	2	2	no	—
28	C₂×Dic₅₉⋊C₄	—	4	1	no	—
29	C₂×C₄×Dic₅₉	—	4	1	no	—
30	C₂×C₄×Dic₅₉	—	4	1	no	—
31	#14273661	—	2	1	no	—
32	#14273661	—	2	2	no	—
33	#14273661	C₂×C₄	1	4	yes	—
34	#14273661	C₂×C₄	1	2	yes	—
35	#14273652	—	2	4	no	—
36	C₅₉⋊#95	—	4	1	no	—
37	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₉	—	2	2	no	—
41	C₂×C₁₆×C₅₉	—	2	2	no	—
42	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₉	D₄	1	2	yes	—
45	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅₉	—	4	1	no	—
49	C₂²×Dic₅₉	—	4	2	no	—
50	C₂²×Dic₅₉	—	4	1	no	—
51	C₂×Dic₁₁₈	—	2	2	no	—
52	C₂×Dic₁₁₈	—	2	4	no	—
53	C₂×Dic₁₁₈	—	4	1	no	—
54	C₂×Dic₁₁₈	—	4	2	no	—
55	C₂×C₈×C₅₉	—	2	2	no	—
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₉	#41	1	2	yes	—
61	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₉	SD₁₆	1	2	yes	—
63	C₄⋊Dic₅₉	—	2	2	no	—
64	C₄⋊Dic₅₉	—	2	4	no	—
65	C₄⋊Dic₅₉	—	4	1	no	—
66	C₄⋊Dic₅₉	—	4	2	no	—
67	Dic₅₉⋊C₄	—	4	4	no	—
68	C₄×Dic₅₉	—	4	4	no	—
69	C₄×Dic₅₉	—	8	2	no	—
70	Dic₂₃₆	—	2	2	no	—
71	Dic₂₃₆	—	2	4	no	—
72	Dic₂₃₆	C₄²	1	4	yes	complemented
73	C₁₆×C₅₉	—	2	4	no	—
74	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	#102	1	2	yes	—
77	C₂×C₄×C₅₉	#102	1	2	yes	—
78	C₂×C₄×C₅₉	SD₃₂	1	2	yes	—
79	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₉	#135	1	2	yes	—
84	C₂×Dic₅₉	—	4	2	no	—
85	C₂×Dic₅₉	—	4	4	no	—
86	C₂×Dic₅₉	—	4	2	no	—
87	C₂×Dic₅₉	—	4	4	no	—
88	C₂×Dic₅₉	—	8	2	no	—
89	C₂×Dic₅₉	—	8	2	no	—
90	C₂×Dic₅₉	—	8	2	no	—
91	Dic₁₁₈	—	2	4	no	—
92	Dic₁₁₈	—	4	2	no	—
93	Dic₁₁₈	—	4	4	no	—
94	C₈×C₅₉	—	2	2	no	—
95	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
96	C₈×C₅₉	#117	1	1	yes	characteristic
97	C₈×C₅₉	C₄×D₄	1	2	yes	—
98	C₄×#358	—	59	1	no	maximal, Sylow
99	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
100	C₂²×C₅₉	C₄×D₈	1	2	yes	—
101	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
102	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
103	C₂²×C₅₉	#508	1	2	yes	—
104	C₄×C₅₉	—	2	2	no	—
105	C₄×C₅₉	#358	1	4	yes	complemented
106	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₅₉	#416	1	1	yes	characteristic
108	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
109	Dic₅₉	—	4	4	no	—
110	Dic₅₉	—	8	2	no	—
111	Dic₅₉	—	8	4	no	—
112	C₂×C₄×Q₁₆	—	59	1	no	—
113	C₂×#358	—	59	2	no	—
114	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic
115	C₄×#107	—	59	1	no	—
116	#1380	—	59	2	no	—
117	C₂×C₅₉	C₄×#102	1	1	yes	characteristic
118	C₂×C₅₉	C₂×#358	1	1	yes	characteristic
119	C₂×C₅₉	#2138	1	1	yes	characteristic
120	C₂×C₅₉	C₄×SD₃₂	1	2	yes	—
121	C₂×C₅₉	C₄×Q₃₂	1	2	yes	—
122	C₂²×Q₁₆	—	59	1	no	—
123	C₂×C₄×Q₈	—	118	1	no	—
124	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
125	C₄×Q₁₆	—	59	4	no	—
126	C₄×Q₁₆	—	118	2	no	—
127	C₂×#107	—	59	2	no	—
128	C₂×#107	—	59	1	no	—
129	C₂×#103	—	118	1	no	—
130	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
131	C₄×C₄⋊C₄	—	118	1	no	—
132	#358	—	59	8	no	—
133	C₄×C₁₆	—	2	2	no	—
134	#340	—	118	1	no	—
135	C₅₉	C₄×#358	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	118	1	no	—
137	C₂×Q₁₆	—	59	4	no	—
138	C₂×Q₁₆	—	59	2	no	—
139	C₂×Q₁₆	—	118	2	no	—
140	C₂×Q₁₆	—	118	1	no	—
141	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
143	C₄×Q₈	—	118	4	no	—
144	C₄×Q₈	—	236	2	no	—
145	C₂×C₄⋊C₄	—	118	1	no	—
146	C₂×C₄⋊C₄	—	118	2	no	—
147	C₂×C₄⋊C₄	—	236	1	no	—
148	C₂×C₄²	—	236	1	no	—
149	C₂×C₄²	—	236	1	no	—
150	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	C₄×D₅₉	1	4	yes	—
154	#107	—	59	4	no	—
155	#107	—	59	2	no	—
156	#107	—	118	2	no	—
157	#107	—	118	1	no	—
158	#103	—	118	4	no	—
159	C₄×C₈	C₄×D₅₉	1	2	yes	—
160	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
161	#95	—	236	1	no	—
162	C₂×Q₈	—	118	2	no	—
163	C₂×Q₈	—	118	4	no	—
164	C₂×Q₈	—	236	1	no	—
165	C₂×Q₈	—	236	2	no	—
166	C₂²×C₄	—	236	2	no	—
167	C₂²×C₄	—	236	1	no	—
168	C₂²×C₄	—	236	1	no	—
169	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
170	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
172	Q₁₆	—	59	4	no	—
173	Q₁₆	—	118	2	no	—
174	Q₁₆	—	118	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
178	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
179	C₂×C₈	#11745685	1	2	yes	—
180	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
181	C₄⋊C₄	—	118	2	no	—
182	C₄⋊C₄	—	118	4	no	—
183	C₄⋊C₄	—	236	4	no	—
184	C₄⋊C₄	—	236	1	no	—
185	C₄⋊C₄	—	236	2	no	—
186	C₄²	—	472	2	no	—
187	C₄²	—	236	4	no	—
188	C₄²	#11745660	1	2	yes	—
189	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
190	C₁₆	—	2	4	no	—
191	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
192	Q₈	—	118	4	no	—
193	Q₈	—	236	2	no	—
194	Q₈	—	236	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	472	2	no	—
197	C₂×C₄	—	472	2	no	—
198	C₂×C₄	—	472	2	no	—
199	C₂×C₄	—	236	2	no	—
200	C₂×C₄	—	236	4	no	—
201	C₂×C₄	—	236	2	no	—
202	C₂×C₄	—	236	4	no	—
203	C₂×C₄	#14273649	1	2	yes	—
204	C₂×C₄	#14273649	1	2	yes	—
205	C₂×C₄	#14273672	1	2	yes	—
206	C₂×C₄	Dic₄₇₂	1	2	yes	—
207	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
208	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
209	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
210	C₂×C₄	#14273749	1	2	yes	—
211	C₈	—	2	2	no	—
212	C₈	C₄²×D₅₉	1	1	yes	characteristic
213	C₈	#14273684	1	1	yes	characteristic
214	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
215	C₂²	C₂×#14273649	1	1	yes	characteristic
216	C₂²	C₄×D₄₇₂	1	2	yes	—
217	C₂²	C₂×#14273672	1	1	yes	characteristic
218	C₂²	C₂×Dic₄₇₂	1	1	yes	characteristic
219	C₂²	#16837834	1	2	yes	—
220	C₄	—	2	2	no	—
221	C₄	—	472	4	no	—
222	C₄	—	472	2	no	—
223	C₄	—	236	4	no	—
224	C₄	#16837207	1	4	yes	direct product factor
225	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
226	C₄	#16837399	1	1	yes	characteristic
227	C₄	C₄×#11745660	1	2	yes	—
228	C₂	C₄×#14273649	1	1	yes	characteristic
229	C₂	C₂×#16837207	1	1	yes	characteristic
230	C₂	#21425804	1	1	yes	characteristic
231	C₂	C₄×#14273672	1	2	yes	—
232	C₂	C₄×Dic₄₇₂	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄₇₂
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	C₄×#358
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₃₆	C₄	semidirect
14. #16837207	C₄	direct (N × H)
72. Dic₂₃₆	C₄²	semidirect
105. C₄×C₅₉	#358	semidirect
135. C₅₉	C₄×#358	semidirect
224. C₄	#16837207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃₆	2
C₂×#16837207	2
C₄×#14273661	2
#21422188	2
C₂×C₄×C₁₆×C₅₉	2
C₄×#358	59

Subgroups — C4×#16837207 #28503899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837207 #28503899