Subgroups — C₂×#21424004 #28586792

Order 15104.

1046 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273663	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×Dic₅₉	C₂	1	2	yes	maximal
3	C₂²×#14273637	C₂	1	1	yes	maximal, characteristic
4	#21424004	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16837166	C₂	1	2	yes	maximal
6	C₂²×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	C₂²	1	2	yes	—
8	C₂³×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
10	C₂×#14273663	C₄	1	8	yes	complemented
11	C₂×#14273663	C₂²	1	4	yes	—
12	M₁₆×Dic₅₉	—	2	4	no	—
13	M₁₆×Dic₅₉	C₂²	1	8	yes	—
14	C₂×#14273637	—	2	2	no	—
15	C₂×#14273637	C₄	1	4	yes	—
16	C₂×#14273637	C₄	1	4	yes	—
17	C₂×#14273637	C₂²	1	4	yes	—
18	C₂×C₈×Dic₅₉	—	2	2	no	—
19	#16837166	C₂²	1	8	yes	—
20	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
21	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#108×C₅₉	C₄	1	8	yes	complemented
23	C₂×#108×C₅₉	C₂²	1	4	yes	—
24	C₂³×Dic₅₉	Q₈	1	2	yes	—
25	C₂×#11745664	—	2	2	no	—
26	C₂×#11745664	—	2	4	no	—
27	C₂×#11745664	C₂×C₄	1	8	yes	—
28	C₂×#11745664	C₂³	1	4	yes	—
29	C₂²×C₅₉⋊C₈	—	2	2	no	—
30	C₂²×C₅₉⋊C₈	—	2	2	no	—
31	C₂²×C₅₉⋊C₈	—	2	2	no	—
32	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅₉⋊C₈	C₂³	1	2	yes	—
35	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
38	C₂×C₄×Dic₅₉	D₄	1	2	yes	—
39	C₂×C₄×Dic₅₉	Q₈	1	2	yes	—
40	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
41	#14273663	C₂×C₄	1	16	yes	complemented
42	#14273637	—	2	8	no	—
43	#14273637	C₂×C₄	1	16	yes	—
44	C₈×Dic₅₉	—	2	8	no	—
45	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₅₉	—	2	2	no	—
47	C₂×M₁₆×C₅₉	—	2	4	no	—
48	C₂×M₁₆×C₅₉	C₂×C₄	1	8	yes	—
49	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
50	C₂²×C₈×C₅₉	—	2	2	no	—
51	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
55	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
56	#108×C₅₉	C₂×C₄	1	16	yes	complemented
57	C₂⁴×C₅₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	#41	1	2	yes	—
66	C₂²×C₄×C₅₉	#41	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	4	4	no	—
69	C₂²×Dic₅₉	C₄⋊C₄	1	4	yes	—
70	C₂²×Dic₅₉	C₄⋊C₄	1	4	yes	—
71	C₂²×Dic₅₉	C₂×Q₈	1	4	yes	—
72	M₁₆×C₅₉	—	2	8	no	—
73	M₁₆×C₅₉	—	2	4	no	—
74	M₁₆×C₅₉	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₅₉	—	2	8	no	—
76	C₂×C₈×C₅₉	—	2	4	no	—
77	C₂×C₈×C₅₉	C₄²	1	8	yes	—
78	C₂×C₈×C₅₉	C₄²	1	8	yes	—
79	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
82	C₄×Dic₅₉	C₄⋊C₄	1	4	yes	—
83	C₄×Dic₅₉	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₅₉	C₂²×C₄	1	4	yes	—
85	C₄×Dic₅₉	C₂×D₄	1	4	yes	—
86	#11745664	—	2	8	no	—
87	#11745664	—	2	4	no	—
88	#11745664	C₂²×C₄	1	8	yes	—
89	C₂×C₅₉⋊C₈	—	2	4	no	—
90	C₂×C₅₉⋊C₈	—	2	4	no	—
91	C₂×C₅₉⋊C₈	—	2	8	no	—
92	C₂×C₅₉⋊C₈	—	2	4	no	—
93	C₂×C₅₉⋊C₈	—	2	2	no	—
94	C₂×C₅₉⋊C₈	—	2	2	no	—
95	C₂×C₅₉⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂³×C₅₉	—	2	4	no	—
97	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₅₉	C₄×Q₈	1	2	yes	—
100	C₂³×C₅₉	C₄×Q₈	1	2	yes	—
101	C₂³×C₅₉	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#117	1	2	yes	—
107	C₂×C₄×C₅₉	#117	1	2	yes	—
108	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
120	C₂×Dic₅₉	—	2	8	no	—
121	C₂×Dic₅₉	—	4	4	no	—
122	C₂×Dic₅₉	—	4	4	no	—
123	C₂×Dic₅₉	C₂×C₄⋊C₄	1	8	yes	—
124	C₈×C₅₉	—	2	8	no	—
125	C₈×C₅₉	C₂×C₄²	1	8	yes	—
126	C₅₉⋊C₈	—	2	4	no	—
127	C₅₉⋊C₈	—	2	8	no	—
128	C₂×#1778	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	—	2	2	no	—
130	C₂²×C₅₉	—	2	2	no	—
131	C₂²×C₅₉	—	2	4	no	—
132	C₂²×C₅₉	—	2	4	no	—
133	C₂²×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
134	C₂²×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
135	C₂²×C₅₉	#430	1	4	yes	—
136	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
140	C₄×C₅₉	C₄×C₄⋊C₄	1	2	yes	—
141	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
143	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
145	Dic₅₉	—	2	8	no	—
146	C₂²×#108	—	59	1	no	—
147	C₂²×#108	D₅₉	1	1	yes	characteristic
148	C₂×C₄×M₁₆	—	59	2	no	—
149	C₂²×#97	—	59	1	no	—
150	#1778	—	59	8	no	—
151	C₂×#341	—	59	2	no	—
152	C₂×C₅₉	—	2	1	no	—
153	C₂×C₅₉	—	2	1	no	—
154	C₂×C₅₉	—	2	2	no	—
155	C₂×C₅₉	#1778	1	4	yes	complemented
156	C₂×C₅₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₅₉	C₂×#430	1	2	yes	—
158	C₂²×M₁₆	—	59	2	no	—
159	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
160	C₂³×C₈	—	59	1	no	—
161	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
162	C₂²×C₄²	—	59	1	no	—
163	C₂×#108	—	59	8	no	—
164	C₂×#108	—	59	4	no	—
165	C₂×#108	Dic₅₉	1	8	yes	complemented
166	C₂×#108	C₂×D₅₉	1	4	yes	—
167	C₄×M₁₆	—	59	8	no	—
168	C₄×M₁₆	—	118	4	no	—
169	C₂×#97	—	59	4	no	—
170	C₂×#97	—	59	4	no	—
171	C₂×#97	—	59	4	no	—
172	C₂×#97	—	118	2	no	—
173	C₂×C₄×C₈	—	118	2	no	—
174	#341	—	59	8	no	—
175	C₅₉	C₂×#1778	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	59	2	no	—
177	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	59	8	no	—
181	C₂×M₁₆	—	59	4	no	—
182	C₂×M₁₆	—	118	2	no	—
183	C₂×M₁₆	—	118	4	no	—
184	C₂×M₁₆	C₂×Dic₅₉	1	8	yes	—
185	C₂×M₁₆	C₂²×D₅₉	1	4	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	59	2	no	—
188	C₂²×C₈	—	59	2	no	—
189	C₂²×C₈	—	59	2	no	—
190	C₂²×C₈	—	118	2	no	—
191	C₂²×C₈	—	118	2	no	—
192	C₂²×C₈	—	118	2	no	—
193	C₂²×C₈	C₄×D₅₉	1	2	yes	—
194	C₂²×C₈	C₄×D₅₉	1	2	yes	—
195	C₂²×C₈	C₄×D₅₉	1	4	yes	—
196	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
197	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
198	C₂×C₄²	—	59	2	no	—
199	C₂×C₄²	—	59	2	no	—
200	C₂×C₄²	—	59	2	no	—
201	C₂×C₄²	—	59	2	no	—
202	C₂×C₄²	—	59	2	no	—
203	C₂×C₄²	—	59	2	no	—
204	#108	—	59	16	no	—
205	#108	C₂×Dic₅₉	1	16	yes	complemented
206	#97	—	59	16	no	—
207	#97	—	118	8	no	—
208	C₄×C₈	—	118	8	no	—
209	C₂⁴	Q₈×D₅₉	1	1	yes	characteristic
210	C₂²×C₄	—	59	4	no	—
211	C₂²×C₄	—	59	4	no	—
212	C₂²×C₄	—	59	4	no	—
213	C₂²×C₄	—	236	4	no	—
214	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
217	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
219	C₂²×C₄	#11745687	1	2	yes	—
220	C₂²×C₄	Q₈×D₅₉	1	1	yes	characteristic
221	C₂²×C₄	#11745689	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
223	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
224	M₁₆	—	2	8	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	59	8	no	—
227	M₁₆	—	118	8	no	—
228	M₁₆	—	118	4	no	—
229	M₁₆	C₂²×Dic₅₉	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	59	4	no	—
233	C₂×C₈	—	118	4	no	—
234	C₂×C₈	—	118	4	no	—
235	C₂×C₈	—	118	8	no	—
236	C₂×C₈	—	118	4	no	—
237	C₂×C₈	—	118	2	no	—
238	C₂×C₈	—	118	2	no	—
239	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
240	C₂×C₈	C₄×Dic₅₉	1	8	yes	—
241	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
242	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
243	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
244	C₄²	—	59	4	no	—
245	C₄²	—	59	4	no	—
246	C₄²	—	59	4	no	—
247	C₄²	—	59	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
250	C₂³	C₄⋊C₄×D₅₉	1	1	yes	characteristic
251	C₂³	#14273692	1	2	yes	—
252	C₂³	Q₈×Dic₅₉	1	2	yes	—
253	C₂³	C₂×Q₈×D₅₉	1	1	yes	characteristic
254	C₂×C₄	—	59	8	no	—
255	C₂×C₄	—	118	8	no	—
256	C₂×C₄	—	236	4	no	—
257	C₂×C₄	—	236	4	no	—
258	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₅₉	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
261	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
262	C₂×C₄	#14273685	1	2	yes	—
263	C₂×C₄	#14273685	1	2	yes	—
264	C₂×C₄	#14273689	1	2	yes	—
265	C₂×C₄	#14273689	1	2	yes	—
266	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
267	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
268	C₂×C₄	#14273692	1	2	yes	—
269	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
274	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
275	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
276	C₈	—	2	8	no	—
277	C₈	—	118	4	no	—
278	C₈	—	118	8	no	—
279	C₈	C₂×C₄×Dic₅₉	1	8	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	C₄⋊C₄×Dic₅₉	1	2	yes	—
285	C₂²	C₄⋊C₄×Dic₅₉	1	2	yes	—
286	C₂²	#16837449	1	4	yes	—
287	C₂²	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
288	C₂²	C₂×#14273692	1	1	yes	characteristic
289	C₂²	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
290	C₄	—	118	8	no	—
291	C₄	C₄⋊C₄×Dic₅₉	1	2	yes	—
292	C₄	C₄⋊C₄×Dic₅₉	1	2	yes	—
293	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
294	C₄	C₂×#14273685	1	1	yes	characteristic
295	C₄	C₂×#14273689	1	1	yes	characteristic
296	C₄	C₂×D₄×Dic₅₉	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	#21424004	1	4	yes	direct product factor
301	C₂	C₂×C₄⋊C₄×Dic₅₉	1	1	yes	characteristic
302	C₂	C₂×#16837449	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#108×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₉	C₂×Q₈
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#108	D₅₉
hypocenter	C₅₉	C₂×#1778
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #21424004	C₂	direct (N × H)
10. C₂×#14273663	C₄	semidirect
22. C₂×#108×C₅₉	C₄	semidirect
41. #14273663	C₂×C₄	semidirect
56. #108×C₅₉	C₂×C₄	semidirect
155. C₂×C₅₉	#1778	semidirect
165. C₂×#108	Dic₅₉	semidirect
175. C₅₉	C₂×#1778	semidirect
205. #108	C₂×Dic₅₉	semidirect
300. C₂	#21424004	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×#14273663	2
C₂×M₁₆×Dic₅₉	2
C₂²×#14273637	2
#21424004	2
C₂×#16837166	2
C₂²×#108×C₅₉	2
C₂×#1778	59

Subgroups — C2×#21424004 #28586792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21424004 #28586792