Subgroups — C₂×#21422885 #28557774

Order 15104.

714 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837455	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837426	C₂	1	1	yes	maximal, characteristic
3	#21422885	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837155	C₂	1	2	yes	maximal, complemented
5	C₂×#16837153	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₉×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	—	2	1	no	—
8	#16837455	—	2	2	no	—
9	#16837455	C₂²	1	4	yes	—
10	#16837426	—	2	2	no	—
11	#16837426	C₂²	1	4	yes	—
12	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
17	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
18	#16837155	C₂²	1	8	yes	complemented
19	#16837153	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₅₉×#431	C₄	1	4	yes	—
22	C₅₉×#431	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₅₉	C₄	1	2	yes	—
24	C₂×C₂²⋊C₈×C₅₉	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
26	C₂×#11745664	—	2	4	no	—
27	C₂×#11745664	—	4	4	no	—
28	C₂²×C₅₉⋊C₈	—	2	2	no	—
29	C₂²×C₅₉⋊C₈	—	2	1	no	—
30	C₂²×C₅₉⋊C₈	—	2	1	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
32	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
34	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
35	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
36	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
37	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
38	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
39	C₄×C₅₉⋊C₈	—	2	4	no	—
40	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₉	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₉	Q₈	1	1	yes	characteristic
43	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	D₄	1	2	yes	—
48	C₂×C₄²×C₅₉	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₅₉	Q₈	1	2	yes	—
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
52	C₄⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₅₉	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₅₉	C₂³	1	4	yes	—
56	C₂⁴×C₅₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	—	2	4	no	—
58	C₂²×C₄×C₅₉	—	2	4	no	—
59	C₂²×C₄×C₅₉	—	2	2	no	—
60	C₂²×C₄×C₅₉	—	2	2	no	—
61	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₅₉	—	2	8	no	—
74	C₂×C₈×C₅₉	C₂²×C₄	1	8	yes	—
75	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₉	C₂×D₄	1	2	yes	—
81	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
83	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
85	#11745664	—	4	8	no	—
86	C₂×C₅₉⋊C₈	—	2	2	no	—
87	C₂×C₅₉⋊C₈	—	2	2	no	—
88	C₂×C₅₉⋊C₈	—	2	4	no	—
89	C₂×C₅₉⋊C₈	—	4	2	no	—
90	C₂×C₅₉⋊C₈	—	4	4	no	—
91	C₂×C₅₉⋊C₈	—	4	2	no	—
92	C₂³×C₅₉	—	2	2	no	—
93	C₂³×C₅₉	—	2	2	no	—
94	C₂³×C₅₉	—	2	2	no	—
95	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	—	2	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	2	4	no	—
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	2	4	no	—
104	C₂×C₄×C₅₉	—	2	4	no	—
105	C₂×C₄×C₅₉	—	2	4	no	—
106	C₂×C₄×C₅₉	—	2	4	no	—
107	C₂×C₄×C₅₉	#108	1	4	yes	—
108	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₉	#131	1	4	yes	—
115	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₅₉	#137	1	2	yes	—
117	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₅₉	—	2	8	no	—
121	C₅₉⋊C₈	—	4	2	no	—
122	C₅₉⋊C₈	—	4	4	no	—
123	C₅₉⋊C₈	—	4	2	no	—
124	C₂×#1574	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	2	2	no	—
126	C₂²×C₅₉	—	2	2	no	—
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	4	4	no	—
129	C₂²×C₅₉	#424	1	2	yes	—
130	C₂²×C₅₉	#428	1	2	yes	—
131	C₂²×C₅₉	C₂×#108	1	2	yes	—
132	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₅₉	—	2	2	no	—
134	C₄×C₅₉	—	2	2	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	#424	1	2	yes	—
138	C₄×C₅₉	#428	1	2	yes	—
139	C₄×C₅₉	C₂×#108	1	4	yes	—
140	C₄×C₅₉	C₂×#131	1	2	yes	—
141	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
143	C₂×#431	—	59	1	no	—
144	C₂×#431	D₅₉	1	1	yes	characteristic
145	C₂×#423	—	59	1	no	—
146	#1574	—	59	8	no	—
147	C₂×#335	—	59	2	no	—
148	C₂×#334	—	59	2	no	—
149	C₂×C₅₉	—	4	2	no	—
150	C₂×C₅₉	#1574	1	4	yes	complemented
151	C₂×C₅₉	C₂×#424	1	1	yes	characteristic
152	C₂×C₅₉	C₂×#428	1	1	yes	characteristic
153	C₂×C₅₉	C₂²×#108	1	1	yes	characteristic
154	C₂²×M₁₆	—	118	1	no	—
155	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
156	#431	—	59	4	no	—
157	#431	—	118	2	no	—
158	#431	Dic₅₉	1	4	yes	—
159	#431	C₂×D₅₉	1	4	yes	—
160	#423	—	59	4	no	—
161	#423	—	118	2	no	—
162	C₂×C₄⋊C₈	—	59	1	no	—
163	C₂×C₄⋊C₈	—	59	2	no	—
164	C₂×C₄⋊C₈	—	59	1	no	—
165	C₂×C₄⋊C₈	—	118	1	no	—
166	C₂×C₄⋊C₈	Dic₅₉	1	2	yes	—
167	C₂×C₂²⋊C₈	—	118	1	no	—
168	C₂×C₂²⋊C₈	Dic₅₉	1	2	yes	—
169	C₂×C₄×C₈	—	118	1	no	—
170	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
171	#335	—	59	8	no	—
172	#334	—	59	8	no	—
173	C₅₉	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	Dic₁₁₈	1	1	yes	characteristic
175	C₂³×C₄	D₂₃₆	1	1	yes	characteristic
176	C₂³×C₄	C₂×Dic₅₉	1	1	yes	characteristic
177	C₂×M₁₆	—	118	4	no	—
178	C₂×M₁₆	—	236	4	no	—
179	C₂²×C₈	—	118	2	no	—
180	C₂²×C₈	—	118	1	no	—
181	C₂²×C₈	—	118	1	no	—
182	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
183	C₂×C₄²	Dic₁₁₈	1	1	yes	characteristic
184	C₂×C₄²	Dic₁₁₈	1	2	yes	—
185	C₂×C₄²	D₂₃₆	1	2	yes	—
186	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
187	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
189	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
190	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
191	C₄⋊C₈	—	59	2	no	—
192	C₄⋊C₈	—	59	4	no	—
193	C₄⋊C₈	—	59	2	no	—
194	C₄⋊C₈	—	118	2	no	—
195	C₄⋊C₈	—	118	1	no	—
196	C₄⋊C₈	—	118	1	no	—
197	C₄⋊C₈	—	118	4	no	—
198	C₄⋊C₈	C₂×Dic₅₉	1	8	yes	—
199	C₂²⋊C₈	—	118	4	no	—
200	C₂²⋊C₈	C₂×Dic₅₉	1	8	yes	—
201	C₄×C₈	—	118	4	no	—
202	C₄×C₈	C₂×Dic₅₉	1	4	yes	—
203	C₄×C₈	C₂²×D₅₉	1	4	yes	—
204	C₂⁴	C₄⋊Dic₅₉	1	1	yes	characteristic
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
211	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₅₉	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
217	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
218	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
221	M₁₆	—	236	8	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	—	118	2	no	—
224	C₂×C₈	—	118	2	no	—
225	C₂×C₈	—	118	4	no	—
226	C₂×C₈	—	236	2	no	—
227	C₂×C₈	—	236	4	no	—
228	C₂×C₈	—	236	2	no	—
229	C₂×C₈	C₂²×Dic₅₉	1	8	yes	—
230	C₄²	C₄⋊Dic₅₉	1	2	yes	—
231	C₄²	C₄⋊Dic₅₉	1	2	yes	—
232	C₄²	C₄⋊Dic₅₉	1	2	yes	—
233	C₄²	C₂×Dic₁₁₈	1	2	yes	—
234	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
235	C₄²	C₂×D₂₃₆	1	2	yes	—
236	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
237	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
238	C₄²	C₂²×Dic₅₉	1	2	yes	—
239	C₄²	C₂³×D₅₉	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
244	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	#14273663	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
262	C₂×C₄	#14273734	1	4	yes	—
263	C₂×C₄	#14273755	1	2	yes	—
264	C₂×C₄	#14273760	1	2	yes	—
265	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
267	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	236	2	no	—
270	C₈	—	236	4	no	—
271	C₈	—	236	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	4	no	—
276	C₂²	#16837429	1	2	yes	—
277	C₂²	#16837444	1	2	yes	—
278	C₂²	C₂×#14273663	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	#16837429	1	2	yes	—
285	C₄	#16837444	1	2	yes	—
286	C₄	C₂×#14273663	1	4	yes	—
287	C₄	C₂×#14273734	1	2	yes	—
288	C₄	C₂×#14273755	1	1	yes	characteristic
289	C₄	C₂×#14273760	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#21422885	1	4	yes	direct product factor
292	C₂	C₂×#16837429	1	1	yes	characteristic
293	C₂	C₂×#16837444	1	1	yes	characteristic
294	C₂	C₂²×#14273663	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#431	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₄⋊C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₂×#431	D₅₉
hypocenter	C₅₉	C₂×#1574
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #21422885	C₂	direct (N × H)
4. C₂×#16837155	C₂	semidirect
5. C₂×#16837153	C₂	semidirect
18. #16837155	C₂²	semidirect
19. #16837153	C₂²	semidirect
150. C₂×C₅₉	#1574	semidirect
173. C₅₉	C₂×#1574	semidirect
291. C₂	#21422885	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×#16837455	2
C₂×#16837426	2
#21422885	2
C₂×#16837155	2
C₂×#16837153	2
C₂×C₅₉×#431	2
C₂×#1574	59

Subgroups — C2×#21422885 #28557774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21422885 #28557774