Subgroups — C₂×#21423993 #28586744

Order 15104.

1318 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273661	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14273659	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊Dic₅₉	C₂	1	2	yes	maximal
4	#21423993	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16837165	C₂	1	2	yes	maximal
6	C₂²×#97×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₅₉	—	2	2	no	—
10	C₂×#14273661	C₄	1	8	yes	complemented
11	C₂×#14273661	C₂²	1	4	yes	—
12	C₂×#14273659	C₄	1	8	yes	complemented
13	C₂×#14273659	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₅₉	—	2	4	no	—
15	C₄×C₄⋊Dic₅₉	C₂²	1	8	yes	—
16	C₂×C₅₉⋊#95	—	2	2	no	—
17	#16837165	C₂²	1	8	yes	—
18	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×#97×C₅₉	C₄	1	8	yes	complemented
22	C₂×#97×C₅₉	C₂²	1	4	yes	—
23	C₂³×Dic₅₉	—	2	2	no	—
24	C₂³×Dic₅₉	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	—	2	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	2	no	—
27	C₂×C₄⋊Dic₅₉	—	2	4	no	—
28	C₂×C₄⋊Dic₅₉	—	2	4	no	—
29	C₂×C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊Dic₅₉	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
33	C₂×C₄×Dic₅₉	—	2	8	no	—
34	C₂×C₄×Dic₅₉	—	4	8	no	—
35	#14273661	C₂×C₄	1	16	yes	complemented
36	#14273659	C₂×C₄	1	16	yes	complemented
37	C₅₉⋊#95	—	2	8	no	—
38	C₂³×C₄×C₅₉	D₄	1	2	yes	—
39	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₉	—	2	2	no	—
41	C₂²×C₈×C₅₉	—	2	2	no	—
42	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₅₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₅₉	D₄	1	2	yes	—
50	C₂×C₄²×C₅₉	Q₈	1	4	yes	—
51	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
52	#97×C₅₉	C₂×C₄	1	16	yes	complemented
53	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	—	2	4	no	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₉	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅₉	#41	1	2	yes	—
64	C₂²×C₄×C₅₉	#41	1	2	yes	—
65	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₅₉	—	2	8	no	—
67	C₂²×Dic₅₉	—	2	8	no	—
68	C₂²×Dic₅₉	—	2	4	no	—
69	C₂²×Dic₅₉	—	2	4	no	—
70	C₂²×Dic₅₉	—	4	4	no	—
71	C₂²×Dic₅₉	—	4	4	no	—
72	C₂²×Dic₅₉	—	4	4	no	—
73	C₂²×Dic₅₉	—	4	4	no	—
74	C₂×C₈×C₅₉	—	2	8	no	—
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	C₄²	1	16	yes	complemented
78	C₂×C₈×C₅₉	C₂²×C₄	1	4	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	C₄²×C₅₉	C₄⋊C₄	1	8	yes	—
83	C₄²×C₅₉	C₂²×C₄	1	4	yes	—
84	C₄²×C₅₉	C₂×Q₈	1	4	yes	—
85	C₄⋊Dic₅₉	—	2	8	no	—
86	C₄⋊Dic₅₉	—	2	8	no	—
87	C₄⋊Dic₅₉	—	2	4	no	—
88	C₄⋊Dic₅₉	—	2	4	no	—
89	C₄⋊Dic₅₉	C₂²×C₄	1	8	yes	—
90	C₄⋊Dic₅₉	C₂²×C₄	1	8	yes	—
91	C₄×Dic₅₉	—	4	16	no	—
92	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅₉	C₄×D₄	1	2	yes	—
94	C₂³×C₅₉	C₄×D₄	1	2	yes	—
95	C₂³×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
96	C₂³×C₅₉	#137	1	4	yes	—
97	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₉	—	2	4	no	—
99	C₂×C₄×C₅₉	—	2	4	no	—
100	C₂×C₄×C₅₉	—	2	16	no	—
101	C₂×C₄×C₅₉	C₂×C₄²	1	4	yes	—
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	8	yes	—
103	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₉	#117	1	4	yes	—
105	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₅₉	C₄×Q₈	1	4	yes	—
108	C₂×C₄×C₅₉	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₅₉	—	2	8	no	—
116	C₂×Dic₅₉	—	2	8	no	—
117	C₂×Dic₅₉	—	4	4	no	—
118	C₂×Dic₅₉	—	4	4	no	—
119	C₂×Dic₅₉	—	4	4	no	—
120	C₂×Dic₅₉	—	4	4	no	—
121	C₂×Dic₅₉	—	4	16	no	—
122	C₂×Dic₅₉	—	4	16	no	—
123	C₈×C₅₉	—	2	4	no	—
124	C₈×C₅₉	C₂×C₄²	1	8	yes	complemented
125	C₂×#1776	—	59	1	no	maximal, Sylow
126	C₂²×C₅₉	C₄×C₄⋊C₄	1	4	yes	—
127	C₂²×C₅₉	#428	1	8	yes	—
128	C₂²×C₅₉	#440	1	8	yes	—
129	C₂²×C₅₉	#441	1	4	yes	—
130	C₂²×C₅₉	#442	1	4	yes	—
131	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₂²×C₅₉	C₂×#137	1	2	yes	—
136	C₄×C₅₉	—	2	8	no	—
137	C₄×C₅₉	C₄×C₄⋊C₄	1	4	yes	—
138	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₄×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
142	Dic₅₉	—	4	8	no	—
143	Dic₅₉	—	4	8	no	—
144	C₂²×#107	—	59	1	no	—
145	C₂²×#106	—	59	1	no	—
146	C₂²×#97	D₅₉	1	1	yes	characteristic
147	C₂×C₄×C₄⋊C₄	—	59	2	no	—
148	#1776	—	59	8	no	—
149	C₂×#340	—	59	2	no	—
150	C₂×C₅₉	#1776	1	8	yes	complemented
151	C₂×C₅₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₅₉	C₂×#428	1	2	yes	—
153	C₂×C₅₉	C₂×#440	1	2	yes	—
154	C₂×C₅₉	C₂×#441	1	1	yes	characteristic
155	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
156	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
158	C₂²×C₄⋊C₄	—	59	2	no	—
159	C₂²×C₄⋊C₄	—	59	2	no	—
160	C₂²×C₄²	—	118	2	no	—
161	C₂²×C₄²	C₂×D₅₉	1	1	yes	characteristic
162	C₂×#107	—	59	8	no	—
163	C₂×#107	—	59	4	no	—
164	C₂×#106	—	59	8	no	—
165	C₂×#106	—	59	4	no	—
166	C₂×#97	Dic₅₉	1	8	yes	complemented
167	C₂×#97	C₂×D₅₉	1	4	yes	—
168	C₄×C₄⋊C₄	—	59	8	no	—
169	C₄×C₄⋊C₄	—	118	4	no	—
170	C₂×#95	—	118	2	no	—
171	#340	—	59	8	no	—
172	C₅₉	C₂×#1776	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	118	2	no	—
174	C₂³×C₄	—	118	2	no	—
175	C₂³×C₄	D₂₃₆	1	2	yes	—
176	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₄×D₅₉	1	4	yes	—
180	C₂²×C₈	C₄×D₅₉	1	4	yes	—
181	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
182	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
183	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
184	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
185	C₂×C₄⋊C₄	—	59	8	no	—
186	C₂×C₄⋊C₄	—	59	8	no	—
187	C₂×C₄⋊C₄	—	59	4	no	—
188	C₂×C₄⋊C₄	—	59	4	no	—
189	C₂×C₄⋊C₄	—	118	2	no	—
190	C₂×C₄⋊C₄	—	118	2	no	—
191	C₂×C₄⋊C₄	—	118	4	no	—
192	C₂×C₄⋊C₄	—	118	4	no	—
193	C₂×C₄²	—	118	8	no	—
194	C₂×C₄²	—	236	8	no	—
195	C₂×C₄²	Dic₁₁₈	1	4	yes	—
196	C₂×C₄²	D₂₃₆	1	2	yes	—
197	C₂×C₄²	C₂×Dic₅₉	1	4	yes	—
198	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
199	#107	—	59	16	no	—
200	#106	—	59	16	no	—
201	#97	C₂×Dic₅₉	1	16	yes	complemented
202	#95	—	118	8	no	—
203	C₂⁴	C₂×D₂₃₆	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	118	8	no	—
207	C₂²×C₄	—	118	8	no	—
208	C₂²×C₄	—	118	4	no	—
209	C₂²×C₄	—	118	4	no	—
210	C₂²×C₄	—	236	4	no	—
211	C₂²×C₄	—	236	4	no	—
212	C₂²×C₄	—	236	4	no	—
213	C₂²×C₄	—	236	4	no	—
214	C₂²×C₄	C₄⋊Dic₅₉	1	8	yes	—
215	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
218	C₂²×C₄	C₂×D₂₃₆	1	4	yes	—
219	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
220	C₂²×C₄	#11745685	1	2	yes	—
221	C₂²×C₄	#11745685	1	2	yes	—
222	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
223	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	C₄×Dic₅₉	1	16	yes	complemented
228	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
229	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
230	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
231	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
232	C₄⋊C₄	—	59	8	no	—
233	C₄⋊C₄	—	59	8	no	—
234	C₄⋊C₄	—	118	8	no	—
235	C₄⋊C₄	—	118	8	no	—
236	C₄⋊C₄	—	118	4	no	—
237	C₄⋊C₄	—	118	4	no	—
238	C₄²	—	236	16	no	—
239	C₄²	C₄⋊Dic₅₉	1	8	yes	—
240	C₄²	C₂×Dic₁₁₈	1	4	yes	—
241	C₄²	C₂²×Dic₅₉	1	4	yes	—
242	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
243	C₂³	C₄×D₂₃₆	1	2	yes	—
244	C₂³	C₄×D₂₃₆	1	2	yes	—
245	C₂³	#14273755	1	4	yes	—
246	C₂³	#14273760	1	4	yes	—
247	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	16	no	—
251	C₂×C₄	—	118	8	no	—
252	C₂×C₄	—	118	8	no	—
253	C₂×C₄	—	236	4	no	—
254	C₂×C₄	—	236	4	no	—
255	C₂×C₄	—	236	4	no	—
256	C₂×C₄	—	236	4	no	—
257	C₂×C₄	—	236	16	no	—
258	C₂×C₄	—	236	16	no	—
259	C₂×C₄	C₂×C₄×Dic₅₉	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₅₉	1	8	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
262	C₂×C₄	#14273686	1	4	yes	—
263	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
264	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
265	C₂×C₄	C₄×Dic₁₁₈	1	4	yes	—
266	C₂×C₄	C₄×Dic₁₁₈	1	4	yes	—
267	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
269	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
270	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
271	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
272	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	C₂×C₄×Dic₅₉	1	8	yes	complemented
275	C₂²	C₄×C₄⋊Dic₅₉	1	4	yes	—
276	C₂²	#16837444	1	8	yes	—
277	C₂²	#16837504	1	8	yes	—
278	C₂²	#16837510	1	4	yes	—
279	C₂²	#16837515	1	4	yes	—
280	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
281	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
282	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
283	C₂²	C₂×#14273755	1	2	yes	—
284	C₂²	C₂×#14273760	1	2	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	236	8	no	—
287	C₄	—	236	8	no	—
288	C₄	C₄×C₄⋊Dic₅₉	1	4	yes	—
289	C₄	C₂²×C₄×Dic₅₉	1	1	yes	characteristic
290	C₄	C₂×#14273686	1	1	yes	characteristic
291	C₄	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
292	C₄	C₂×C₄×Dic₁₁₈	1	1	yes	characteristic
293	C₂	#21423993	1	8	yes	direct product factor
294	C₂	C₂×C₄×C₄⋊Dic₅₉	1	1	yes	characteristic
295	C₂	C₂×#16837444	1	2	yes	—
296	C₂	C₂×#16837504	1	2	yes	—
297	C₂	C₂×#16837510	1	1	yes	characteristic
298	C₂	C₂×#16837515	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₂₃₆
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#97×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#97	D₅₉
hypocenter	C₅₉	C₂×#1776
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #21423993	C₂	direct (N × H)
10. C₂×#14273661	C₄	semidirect
12. C₂×#14273659	C₄	semidirect
21. C₂×#97×C₅₉	C₄	semidirect
35. #14273661	C₂×C₄	semidirect
36. #14273659	C₂×C₄	semidirect
52. #97×C₅₉	C₂×C₄	semidirect
77. C₂×C₈×C₅₉	C₄²	semidirect
124. C₈×C₅₉	C₂×C₄²	semidirect
150. C₂×C₅₉	#1776	semidirect
166. C₂×#97	Dic₅₉	semidirect
172. C₅₉	C₂×#1776	semidirect
201. #97	C₂×Dic₅₉	semidirect
227. C₂×C₈	C₄×Dic₅₉	semidirect
274. C₈	C₂×C₄×Dic₅₉	semidirect
293. C₂	#21423993	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×#14273661	2
C₂²×#14273659	2
C₂×C₄×C₄⋊Dic₅₉	2
#21423993	2
C₂×#16837165	2
C₂²×#97×C₅₉	2
C₂×#1776	59

Subgroups — C2×#21423993 #28586744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21423993 #28586744