Subgroups — C₄×#14273726 #21426666

Order 7552.

588 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273726	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₁₈	C₂	1	2	yes	maximal
3	#16837306	C₂	1	2	yes	maximal
4	#16837301	C₂	1	1	yes	maximal, characteristic
5	#16837299	C₂	1	2	yes	maximal
6	#16837298	C₂	1	1	yes	maximal, characteristic
7	#16837271	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₅₉⋊C₄	C₂	1	2	yes	maximal
9	C₄×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic
10	C₄²×Dic₅₉	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₁₈	C₂²	1	2	yes	—
13	#14273726	C₄	1	8	yes	direct product factor
14	C₄×Dic₁₁₈	—	2	8	no	—
15	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₉⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₅₉⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₅₉⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₅₉⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₅₉	C₂²	1	2	yes	—
26	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	C₅₉⋊#95	C₂²	1	2	yes	—
30	C₅₉⋊#95	C₂²	1	2	yes	—
31	C₅₉⋊#95	C₂²	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₅₉	C₂²	1	2	yes	—
35	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
36	#95×C₅₉	C₂²	1	2	yes	—
37	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₅₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₂²×Dic₅₉	C₂³	1	2	yes	—
43	C₂²×Dic₅₉	C₂³	1	2	yes	—
44	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
45	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
46	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
47	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₁₈	—	2	4	no	—
49	C₂×Dic₁₁₈	—	2	4	no	—
50	C₂×Dic₁₁₈	C₂×C₄	1	8	yes	—
51	C₄⋊C₄×C₅₉	—	2	2	no	—
52	C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
53	C₄²×C₅₉	—	2	4	no	—
54	C₄²×C₅₉	Q₈	1	4	yes	complemented
55	C₄⋊Dic₅₉	—	2	2	no	—
56	C₄⋊Dic₅₉	—	2	4	no	—
57	C₄⋊Dic₅₉	C₂×C₄	1	4	yes	—
58	Dic₅₉⋊C₄	—	2	4	no	—
59	Dic₅₉⋊C₄	—	2	4	no	—
60	Dic₅₉⋊C₄	—	2	4	no	—
61	Dic₅₉⋊C₄	C₂×C₄	1	8	yes	—
62	C₄×Dic₅₉	—	2	4	no	—
63	C₄×Dic₅₉	—	2	2	no	—
64	C₄×Dic₅₉	—	2	2	no	—
65	C₄×Dic₅₉	—	2	2	no	—
66	C₄×Dic₅₉	C₂×C₄	1	4	yes	—
67	C₄×Dic₅₉	D₄	1	4	yes	—
68	C₄×Dic₅₉	Q₈	1	4	yes	complemented
69	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	1	no	—
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	2	2	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	2	2	no	—
76	C₂×C₄×C₅₉	—	2	2	no	—
77	C₂×C₄×C₅₉	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₅₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₅₉	#41	1	2	yes	—
85	C₂×Dic₅₉	—	2	2	no	—
86	C₂×Dic₅₉	—	2	1	no	—
87	C₂×Dic₅₉	—	2	1	no	—
88	C₂×Dic₅₉	—	2	1	no	—
89	C₂×Dic₅₉	—	2	1	no	—
90	C₂×Dic₅₉	—	2	2	no	—
91	C₂×Dic₅₉	—	2	2	no	—
92	C₂×Dic₅₉	—	2	2	no	—
93	C₂×Dic₅₉	—	2	2	no	—
94	C₂×Dic₅₉	C₂²×C₄	1	4	yes	—
95	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
96	C₂×Dic₅₉	C₂²×C₄	1	2	yes	—
97	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
98	C₂×Dic₅₉	C₂×D₄	1	2	yes	—
99	C₂×Dic₅₉	C₂×Q₈	1	2	yes	—
100	C₂×Dic₅₉	C₂×Q₈	1	2	yes	—
101	C₂×Dic₅₉	#41	1	2	yes	—
102	C₂×Dic₅₉	#41	1	2	yes	—
103	Dic₁₁₈	—	2	16	no	—
104	C₂²×C₅₉	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
107	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
109	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
110	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
111	C₄×C₅₉	—	2	4	no	—
112	C₄×C₅₉	—	2	2	no	—
113	C₄×C₅₉	—	4	2	no	—
114	C₄×C₅₉	C₄×Q₈	1	4	yes	complemented
115	C₄×C₅₉	C₄⋊Q₈	1	4	yes	complemented
116	Dic₅₉	—	2	4	no	—
117	Dic₅₉	—	2	2	no	—
118	Dic₅₉	—	2	2	no	—
119	Dic₅₉	—	4	2	no	—
120	Dic₅₉	C₄×D₄	1	4	yes	—
121	Dic₅₉	C₄×Q₈	1	4	yes	complemented
122	C₄×C₄⋊Q₈	—	59	1	no	maximal, Sylow
123	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₂×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
126	C₂×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₅₉	#532	1	1	yes	characteristic
128	C₂×C₅₉	#533	1	1	yes	characteristic
129	C₂×C₅₉	#533	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	—	59	1	no	—
131	C₂×C₄×Q₈	—	59	2	no	—
132	#391	—	59	2	no	—
133	#389	—	59	1	no	—
134	#389	—	59	1	no	—
135	#389	—	59	2	no	—
136	#384	—	59	1	no	—
137	C₄×C₄⋊C₄	—	59	1	no	—
138	C₄×C₄⋊C₄	—	59	2	no	—
139	C₄×C₄⋊C₄	D₅₉	1	1	yes	characteristic
140	C₄³	—	59	1	no	—
141	C₅₉	C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	59	2	no	—
143	C₄⋊Q₈	—	59	8	no	—
144	C₄×Q₈	—	118	8	no	—
145	C₂×C₄⋊C₄	—	59	2	no	—
146	C₂×C₄⋊C₄	—	59	1	no	—
147	C₂×C₄⋊C₄	—	59	2	no	—
148	C₂×C₄⋊C₄	—	59	2	no	—
149	C₂×C₄⋊C₄	—	59	2	no	—
150	C₂×C₄⋊C₄	—	59	2	no	—
151	C₂×C₄⋊C₄	—	59	1	no	—
152	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
154	C₂×C₄²	—	59	1	no	—
155	C₂×C₄²	—	59	1	no	—
156	C₂×C₄²	—	59	1	no	—
157	C₂×C₄²	—	59	1	no	—
158	C₂×C₄²	—	59	1	no	—
159	C₂×C₄²	—	59	1	no	—
160	C₂×C₄²	—	59	2	no	—
161	C₂×C₄²	C₂×D₅₉	1	2	yes	—
162	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
163	#95	—	59	2	no	—
164	#95	—	59	2	no	—
165	#95	—	59	2	no	—
166	#95	C₂×D₅₉	1	2	yes	—
167	C₂×Q₈	—	59	8	no	—
168	C₂×Q₈	—	118	4	no	—
169	C₂×Q₈	—	118	4	no	—
170	C₂²×C₄	—	59	1	no	—
171	C₂²×C₄	—	59	1	no	—
172	C₂²×C₄	—	59	1	no	—
173	C₂²×C₄	—	59	2	no	—
174	C₂²×C₄	—	59	1	no	—
175	C₂²×C₄	—	59	2	no	—
176	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
177	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₅₉	1	2	yes	—
179	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	59	8	no	—
183	C₄⋊C₄	—	59	4	no	—
184	C₄⋊C₄	—	118	4	no	—
185	C₄⋊C₄	—	118	4	no	—
186	C₄⋊C₄	—	118	4	no	—
187	C₄⋊C₄	—	118	4	no	—
188	C₄⋊C₄	—	118	2	no	—
189	C₄⋊C₄	C₄×D₅₉	1	4	yes	—
190	C₄²	—	2	4	no	—
191	C₄²	—	59	4	no	—
192	C₄²	—	59	4	no	—
193	C₄²	—	59	4	no	—
194	C₄²	—	118	2	no	—
195	C₄²	—	118	2	no	—
196	C₄²	—	118	2	no	—
197	C₄²	—	118	4	no	—
198	C₄²	Dic₁₁₈	1	4	yes	complemented
199	C₂³	C₂³×D₅₉	1	1	yes	characteristic
200	Q₈	—	118	16	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	59	2	no	—
209	C₂×C₄	—	59	2	no	—
210	C₂×C₄	—	59	2	no	—
211	C₂×C₄	—	59	2	no	—
212	C₂×C₄	—	59	2	no	—
213	C₂×C₄	—	59	2	no	—
214	C₂×C₄	—	59	4	no	—
215	C₂×C₄	—	59	2	no	—
216	C₂×C₄	—	59	2	no	—
217	C₂×C₄	—	118	2	no	—
218	C₂×C₄	—	118	1	no	—
219	C₂×C₄	—	118	1	no	—
220	C₂×C₄	—	118	2	no	—
221	C₂×C₄	—	118	1	no	—
222	C₂×C₄	—	118	1	no	—
223	C₂×C₄	—	118	2	no	—
224	C₂×C₄	—	118	2	no	—
225	C₂×C₄	—	118	2	no	—
226	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
227	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
228	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
229	C₂×C₄	C₂×C₄×D₅₉	1	4	yes	—
230	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
231	C₂×C₄	#11745685	1	2	yes	—
232	C₂×C₄	D₄×D₅₉	1	2	yes	—
233	C₂×C₄	Q₈×D₅₉	1	2	yes	—
234	C₂²	C₂²×C₄×D₅₉	1	1	yes	characteristic
235	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
236	C₂²	C₂×Q₈×D₅₉	1	1	yes	characteristic
237	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
238	C₂²	#41×D₅₉	1	1	yes	characteristic
239	C₂²	#41×D₅₉	1	1	yes	characteristic
240	C₂²	C₂×#11745685	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	59	4	no	—
245	C₄	—	59	4	no	—
246	C₄	—	118	2	no	—
247	C₄	—	118	2	no	—
248	C₄	—	118	4	no	—
249	C₄	—	236	2	no	—
250	C₄	C₄×Dic₁₁₈	1	4	yes	complemented
251	C₄	#14273726	1	4	yes	direct product factor
252	C₂	C₄×D₄×D₅₉	1	1	yes	characteristic
253	C₂	C₄×Q₈×D₅₉	1	1	yes	characteristic
254	C₂	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
255	C₂	C₂×#14273726	1	1	yes	characteristic
256	C₂	#16837949	1	1	yes	characteristic
257	C₂	#16837955	1	1	yes	characteristic
258	C₂	#16837956	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂³×C₄
Frattini	C₂³	C₂³×D₅₉
Fitting	C₄×C₄⋊C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₅₉
hypocenter	C₅₉	C₄×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
13. #14273726	C₄	direct (N × H)
54. C₄²×C₅₉	Q₈	semidirect
68. C₄×Dic₅₉	Q₈	semidirect
114. C₄×C₅₉	C₄×Q₈	semidirect
115. C₄×C₅₉	C₄⋊Q₈	semidirect
121. Dic₅₉	C₄×Q₈	semidirect
141. C₅₉	C₄×C₄⋊Q₈	semidirect
198. C₄²	Dic₁₁₈	semidirect
250. C₄	C₄×Dic₁₁₈	semidirect
251. C₄	#14273726	direct (N × H)

Subgroups — C4×#14273726 #21426666

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14273726 #21426666