Subgroups — C₂×#21425958 #28741976

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14273670	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	#21425958	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16837227	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837206	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837202	C₂	1	1	yes	maximal, characteristic
7	C₂×#16837178	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₃₆	—	2	1	no	—
11	C₂×#14273670	C₄	1	8	yes	complemented
12	C₂×#14273670	C₂²	1	4	yes	—
13	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₄×D₄₇₂	—	2	2	no	—
16	C₄×D₄₇₂	C₂²	1	4	yes	—
17	C₂×#14273661	C₂²	1	1	yes	characteristic
18	C₂×#14273649	—	2	1	no	—
19	#16837227	C₂²	1	4	yes	complemented
20	#16837206	C₂²	1	4	yes	complemented
21	#16837202	C₂²	1	4	yes	—
22	#16837178	C₂²	1	4	yes	complemented
23	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
25	C₄×Q₁₆×C₅₉	—	2	2	no	—
26	C₄×Q₁₆×C₅₉	C₂²	1	4	yes	complemented
27	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₅₉	—	2	1	no	—
29	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₂²×D₂₃₆	—	2	1	no	—
31	C₂²×C₄×D₅₉	—	4	1	no	—
32	C₂×D₄₇₂	—	2	1	no	—
33	C₂×D₄₇₂	—	2	2	no	—
34	C₂×D₄₇₂	C₂×C₄	1	4	yes	—
35	C₂×D₄₇₂	C₂³	1	2	yes	—
36	C₄×D₂₃₆	—	2	4	no	—
37	C₄×D₂₃₆	—	4	2	no	—
38	C₂×C₄⋊Dic₅₉	—	2	1	no	—
39	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
40	#14273670	C₂×C₄	1	16	yes	complemented
41	C₂×C₅₉⋊C₁₆	—	2	2	no	—
42	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
44	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
45	#14273661	—	2	1	no	—
46	#14273661	C₂³	1	2	yes	—
47	#14273649	—	2	4	no	—
48	C₂²×Q₈×C₅₉	—	2	1	no	—
49	C₂×Q₁₆×C₅₉	—	2	1	no	—
50	C₂×Q₁₆×C₅₉	—	2	2	no	—
51	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
52	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	—	2	4	no	—
56	C₄×Q₈×C₅₉	—	4	2	no	—
57	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
59	C₂×C₄²×C₅₉	—	4	1	no	—
60	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
61	#107×C₅₉	—	2	1	no	—
62	#107×C₅₉	C₂³	1	2	yes	—
63	#103×C₅₉	—	2	4	no	—
64	C₄×C₈×C₅₉	D₄	1	2	yes	—
65	C₄×C₈×C₅₉	C₂³	1	2	yes	—
66	C₂³×D₅₉	—	4	1	no	—
67	C₂×Q₈×C₅₉	—	2	4	no	—
68	C₂×Q₈×C₅₉	—	2	2	no	—
69	C₂×Q₈×C₅₉	—	4	1	no	—
70	C₂×Q₈×C₅₉	—	4	2	no	—
71	C₂²×C₄×C₅₉	—	4	1	no	—
72	C₂²×C₄×C₅₉	—	4	1	no	—
73	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₅₉	—	4	1	no	—
77	C₂×D₂₃₆	—	2	4	no	—
78	C₂×D₂₃₆	—	2	2	no	—
79	C₂×D₂₃₆	—	4	1	no	—
80	C₂×D₂₃₆	—	4	2	no	—
81	C₂×C₄×D₅₉	—	4	4	no	—
82	C₂×C₄×D₅₉	—	8	4	no	—
83	Q₁₆×C₅₉	—	2	4	no	—
84	Q₁₆×C₅₉	—	2	2	no	—
85	Q₁₆×C₅₉	C₂²×C₄	1	4	yes	complemented
86	C₂×C₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	#41	1	2	yes	—
91	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₉	—	2	2	no	—
93	C₄⋊C₄×C₅₉	—	4	1	no	—
94	C₄⋊C₄×C₅₉	—	4	4	no	—
95	C₄²×C₅₉	—	4	4	no	—
96	C₄²×C₅₉	D₈	1	2	yes	—
97	C₄²×C₅₉	C₂×D₄	1	2	yes	—
98	(C₂×D₅₉)⋊C₄	—	4	4	no	—
99	C₄⋊Dic₅₉	—	2	2	no	—
100	C₄⋊Dic₅₉	—	4	1	no	—
101	D₄₇₂	—	2	4	no	—
102	D₄₇₂	—	2	2	no	—
103	D₄₇₂	C₂²×C₄	1	4	yes	—
104	C₅₉⋊C₁₆	—	2	2	no	—
105	C₅₉⋊C₁₆	C₂²×C₄	1	4	yes	complemented
106	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₅₉	—	4	2	no	—
108	C₂²×D₅₉	—	4	4	no	—
109	C₂²×D₅₉	—	8	2	no	—
110	C₂²×D₅₉	—	8	2	no	—
111	Q₈×C₅₉	—	2	4	no	—
112	Q₈×C₅₉	—	4	2	no	—
113	Q₈×C₅₉	—	4	4	no	—
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	4	4	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	8	2	no	—
120	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	#135	1	2	yes	—
125	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	8	2	no	—
128	D₂₃₆	—	2	4	no	—
129	D₂₃₆	—	4	2	no	—
130	D₂₃₆	—	4	4	no	—
131	C₄×D₅₉	—	8	8	no	—
132	C₈×C₅₉	—	2	2	no	—
133	C₈×C₅₉	C₄×D₄	1	2	yes	—
134	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
135	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
136	C₂×#2175	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	C₄×D₈	1	2	yes	—
138	C₂²×C₅₉	#509	1	2	yes	—
139	C₂²×C₅₉	#510	1	2	yes	—
140	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₅₉	—	4	4	no	—
142	C₂×D₅₉	—	8	2	no	—
143	C₂×D₅₉	—	8	2	no	—
144	C₂×D₅₉	—	8	8	no	—
145	C₄×C₅₉	—	2	2	no	—
146	C₄×C₅₉	—	2	2	no	—
147	C₄×C₅₉	—	4	4	no	—
148	C₄×C₅₉	—	8	2	no	—
149	C₄×C₅₉	C₄×D₈	1	2	yes	—
150	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
152	Dic₅₉	—	8	2	no	—
153	C₂²×SD₃₂	—	59	1	no	—
154	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
155	C₂×C₄×D₈	—	59	1	no	—
156	#2175	—	59	8	no	—
157	C₂×#366	—	59	1	no	—
158	C₂×#358	—	59	1	no	—
159	C₂×#357	—	59	1	no	—
160	C₂×#346	—	59	1	no	—
161	C₂×C₅₉	#2175	1	4	yes	complemented
162	C₂×C₅₉	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
164	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
165	D₅₉	—	8	4	no	—
166	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
167	C₂²×D₈	—	59	1	no	—
168	C₂×C₄×Q₈	—	2	1	no	—
169	C₂×C₄×D₄	—	118	1	no	—
170	C₂×SD₃₂	—	59	8	no	—
171	C₂×SD₃₂	—	59	4	no	—
172	C₂²×C₁₆	—	59	1	no	—
173	C₂²×C₁₆	—	59	1	no	—
174	C₄×Q₁₆	—	2	2	no	—
175	C₄×Q₁₆	C₂×D₅₉	1	4	yes	complemented
176	C₄×D₈	—	59	4	no	—
177	C₄×D₈	—	118	2	no	—
178	C₂×#107	—	59	1	no	—
179	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
180	C₂×#103	—	2	1	no	—
181	C₂×#102	—	118	1	no	—
182	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
183	#366	—	59	4	no	—
184	#358	—	59	4	no	—
185	#357	—	59	4	no	—
186	#346	—	59	4	no	—
187	C₅₉	C₂×#2175	1	1	yes	characteristic, complemented, Sylow
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	118	1	no	—
190	C₂³×C₄	—	236	1	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	1	no	—
193	C₂×Q₁₆	C₄×D₅₉	1	4	yes	complemented
194	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
195	C₂×D₈	—	59	4	no	—
196	C₂×D₈	—	59	2	no	—
197	C₂×D₈	—	118	2	no	—
198	C₂×D₈	—	118	1	no	—
199	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
201	C₄×Q₈	—	2	4	no	—
202	C₄×Q₈	—	4	2	no	—
203	C₄×D₄	—	118	4	no	—
204	C₄×D₄	—	236	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	4	1	no	—
207	C₂×C₄⋊C₄	—	118	1	no	—
208	C₂×C₂²⋊C₄	—	236	1	no	—
209	C₂×C₄²	—	4	1	no	—
210	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
211	SD₃₂	—	59	16	no	—
212	C₂×C₁₆	—	59	4	no	—
213	C₂×C₁₆	—	59	2	no	—
214	C₂×C₁₆	—	59	2	no	—
215	C₂×C₁₆	—	118	2	no	—
216	#107	—	2	1	no	—
217	#107	—	59	2	no	—
218	#107	—	118	1	no	—
219	#107	C₂²×D₅₉	1	2	yes	—
220	#103	—	2	4	no	—
221	#102	—	118	4	no	—
222	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
223	C₄×C₈	C₂²×D₅₉	1	2	yes	—
224	C₂⁴	—	236	1	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	2	no	—
229	C₂×D₄	—	118	4	no	—
230	C₂×D₄	—	118	2	no	—
231	C₂×D₄	—	236	1	no	—
232	C₂×D₄	—	236	2	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	472	4	no	—
236	C₂²×C₄	—	236	1	no	—
237	C₂²×C₄	—	236	4	no	—
238	C₂²×C₄	#11745669	1	1	yes	characteristic
239	C₂²×C₄	#11745669	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
241	Q₁₆	—	2	4	no	—
242	Q₁₆	—	2	2	no	—
243	Q₁₆	C₂×C₄×D₅₉	1	4	yes	complemented
244	D₈	—	59	4	no	—
245	D₈	—	118	4	no	—
246	D₈	—	118	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
249	C₂×C₈	#11745685	1	2	yes	—
250	C₂×C₈	C₂×C₂²⋊D₅₉	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	118	2	no	—
257	C₄⋊C₄	—	236	1	no	—
258	C₂²⋊C₄	—	236	4	no	—
259	C₄²	—	4	4	no	—
260	C₄²	#11745669	1	2	yes	—
261	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
262	C₁₆	—	59	4	no	—
263	C₁₆	—	118	2	no	—
264	C₂³	—	472	2	no	—
265	C₂³	—	472	2	no	—
266	C₂³	—	236	2	no	—
267	C₂³	—	236	4	no	—
268	C₂³	C₂×#11745669	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	4	2	no	—
271	Q₈	—	4	4	no	—
272	D₄	—	118	4	no	—
273	D₄	—	236	2	no	—
274	D₄	—	236	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	472	8	no	—
282	C₂×C₄	—	472	2	no	—
283	C₂×C₄	—	236	2	no	—
284	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
285	C₂×C₄	C₂×#11745669	1	2	yes	—
286	C₂×C₄	C₂×#11745669	1	2	yes	—
287	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
288	C₂×C₄	#14273747	1	2	yes	—
289	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	C₄×C₂²⋊D₅₉	1	2	yes	—
292	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
293	C₈	C₂×#11745685	1	1	yes	characteristic
294	C₂²	—	472	2	no	—
295	C₂²	—	472	8	no	—
296	C₂²	—	472	2	no	—
297	C₂²	—	236	4	no	—
298	C₂²	C₄×#11745669	1	2	yes	—
299	C₂²	#16837837	1	2	yes	—
300	C₂²	#16837843	1	2	yes	—
301	C₂²	C₂²×#11745669	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	4	no	—
305	C₄	—	8	2	no	—
306	C₄	—	472	2	no	—
307	C₄	C₄×#11745669	1	2	yes	—
308	C₄	C₂×C₄×C₂²⋊D₅₉	1	1	yes	characteristic
309	C₄	C₂×#14273747	1	1	yes	characteristic
310	C₂	—	472	4	no	—
311	C₂	#21425958	1	4	yes	direct product factor
312	C₂	C₂×C₄×#11745669	1	1	yes	characteristic
313	C₂	C₂×#16837837	1	1	yes	characteristic
314	C₂	C₂×#16837843	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11745669
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	C₂×#2175
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #21425958	C₂	direct (N × H)
4. C₂×#16837227	C₂	semidirect
5. C₂×#16837206	C₂	semidirect
7. C₂×#16837178	C₂	semidirect
8. C₂×C₄×Q₁₆×C₅₉	C₂	semidirect
11. C₂×#14273670	C₄	semidirect
19. #16837227	C₂²	semidirect
20. #16837206	C₂²	semidirect
22. #16837178	C₂²	semidirect
26. C₄×Q₁₆×C₅₉	C₂²	semidirect
40. #14273670	C₂×C₄	semidirect
42. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
51. C₂×Q₁₆×C₅₉	C₂×C₄	semidirect
85. Q₁₆×C₅₉	C₂²×C₄	semidirect
105. C₅₉⋊C₁₆	C₂²×C₄	semidirect
154. C₂×C₄×Q₁₆	D₅₉	semidirect
161. C₂×C₅₉	#2175	semidirect
175. C₄×Q₁₆	C₂×D₅₉	semidirect
187. C₅₉	C₂×#2175	semidirect
193. C₂×Q₁₆	C₄×D₅₉	semidirect
243. Q₁₆	C₂×C₄×D₅₉	semidirect
311. C₂	#21425958	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×#14273670	2
C₂×C₄×D₄₇₂	2
#21425958	2
C₂×#16837227	2
C₂×#16837206	2
C₂×#16837202	2
C₂×#16837178	2
C₂×C₄×Q₁₆×C₅₉	2
C₂×#2175	59

Subgroups — C2×#21425958 #28741976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21425958 #28741976