Subgroups — C₂×#21424877 #28592230

Order 15104.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837444	C₂	1	2	yes	maximal
2	C₂²×#14273659	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
4	#21424877	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16837170	C₂	1	2	yes	maximal
6	C₂²×#108×C₅₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11745664	C₂²	1	1	yes	characteristic
8	C₂×#14273686	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₅₉	C₂²	1	2	yes	—
10	#16837444	—	2	4	no	—
11	#16837444	C₂²	1	8	yes	—
12	C₂×#14273661	—	2	2	no	—
13	C₂×#14273659	—	2	2	no	—
14	C₂×#14273659	—	2	1	no	—
15	C₂×#14273659	—	2	1	no	—
16	C₂×#14273659	—	2	2	no	—
17	C₂×#14273659	C₂²	1	4	yes	—
18	#16837357	—	2	2	no	—
19	#16837357	C₂²	1	4	yes	—
20	C₂×#14273637	—	2	1	no	—
21	C₂×C₈×Dic₅₉	—	2	1	no	—
22	#16837170	C₂²	1	8	yes	—
23	C₂²×M₁₆×C₅₉	C₂²	1	2	yes	—
24	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#108×C₅₉	C₄	1	8	yes	complemented
26	C₂×#108×C₅₉	C₂²	1	4	yes	—
27	C₂³×Dic₅₉	—	2	2	no	—
28	C₂×#11745664	—	2	1	no	—
29	C₂×#11745664	—	2	2	no	—
30	C₂×#11745664	Q₈	1	4	yes	—
31	C₂×#11745664	C₂³	1	2	yes	—
32	C₂²×C₅₉⋊C₈	—	2	1	no	—
33	#14273686	—	2	1	no	—
34	#14273686	—	2	1	no	—
35	#14273686	D₄	1	2	yes	—
36	#14273686	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₅₉	—	2	2	no	—
38	C₂×C₄⋊Dic₅₉	—	2	2	no	—
39	C₂×C₄⋊Dic₅₉	—	2	2	no	—
40	C₂×C₄⋊Dic₅₉	—	2	2	no	—
41	C₂×C₄⋊Dic₅₉	—	2	2	no	—
42	C₂×C₄⋊Dic₅₉	D₄	1	2	yes	—
43	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
44	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
45	C₂×C₄×Dic₅₉	—	2	1	no	—
46	#14273661	—	2	8	no	—
47	#14273659	—	2	4	no	—
48	#14273659	—	2	4	no	—
49	#14273659	—	2	8	no	—
50	#14273637	—	2	4	no	—
51	C₈×Dic₅₉	—	2	4	no	—
52	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	—	2	2	no	—
54	C₂×M₁₆×C₅₉	—	2	4	no	—
55	C₂×M₁₆×C₅₉	C₂×C₄	1	8	yes	—
56	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
57	C₂²×C₈×C₅₉	—	2	2	no	—
58	C₂²×C₈×C₅₉	C₂×C₄	1	4	yes	—
59	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	D₄	1	2	yes	—
62	C₂²×C₈×C₅₉	Q₈	1	4	yes	—
63	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
64	#108×C₅₉	C₂×C₄	1	16	yes	complemented
65	C₂⁴×C₅₉	#41	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₉	#41	1	2	yes	—
74	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₅₉	—	2	2	no	—
77	C₂²×Dic₅₉	—	2	4	no	—
78	C₂²×Dic₅₉	—	4	2	no	—
79	C₂²×Dic₅₉	—	4	2	no	—
80	C₂²×Dic₅₉	—	8	4	no	—
81	M₁₆×C₅₉	—	2	8	no	—
82	M₁₆×C₅₉	—	2	4	no	—
83	M₁₆×C₅₉	C₂²×C₄	1	8	yes	—
84	C₂×C₈×C₅₉	—	2	8	no	—
85	C₂×C₈×C₅₉	—	2	4	no	—
86	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
87	C₂×C₈×C₅₉	C₄⋊C₄	1	4	yes	—
88	C₂×C₈×C₅₉	C₄⋊C₄	1	8	yes	—
89	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
93	C₂²⋊Dic₅₉	—	2	2	no	—
94	C₂²⋊Dic₅₉	—	2	2	no	—
95	C₄⋊Dic₅₉	—	2	2	no	—
96	C₄⋊Dic₅₉	—	2	4	no	—
97	C₄⋊Dic₅₉	—	2	4	no	—
98	C₄⋊Dic₅₉	—	4	2	no	—
99	C₄⋊Dic₅₉	—	4	2	no	—
100	C₄⋊Dic₅₉	C₂×D₄	1	4	yes	—
101	C₄×Dic₅₉	—	2	2	no	—
102	C₄×Dic₅₉	—	2	2	no	—
103	#11745664	—	2	4	no	—
104	#11745664	—	2	2	no	—
105	#11745664	C₂×Q₈	1	4	yes	—
106	C₂×C₅₉⋊C₈	—	2	4	no	—
107	C₂×C₅₉⋊C₈	—	2	2	no	—
108	C₂³×C₅₉	—	2	4	no	—
109	C₂³×C₅₉	C₄×Q₈	1	2	yes	—
110	C₂³×C₅₉	#121	1	1	yes	characteristic
111	C₂³×C₅₉	#121	1	1	yes	characteristic
112	C₂³×C₅₉	#125	1	2	yes	—
113	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₅₉	#125	1	2	yes	—
124	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₅₉	—	2	4	no	—
133	C₂×Dic₅₉	—	4	4	no	—
134	C₂×Dic₅₉	—	4	4	no	—
135	C₂×Dic₅₉	—	8	4	no	—
136	C₂×Dic₅₉	—	8	4	no	—
137	C₂×Dic₅₉	—	8	4	no	—
138	C₈×C₅₉	—	2	8	no	—
139	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
140	C₈×C₅₉	C₂×C₄⋊C₄	1	4	yes	—
141	C₅₉⋊C₈	—	2	4	no	—
142	C₂×#1951	—	59	1	no	maximal, Sylow
143	C₂²×C₅₉	—	2	4	no	—
144	C₂²×C₅₉	—	2	4	no	—
145	C₂²×C₅₉	—	2	2	no	—
146	C₂²×C₅₉	—	2	2	no	—
147	C₂²×C₅₉	#389	1	2	yes	—
148	C₂²×C₅₉	#389	1	2	yes	—
149	C₂²×C₅₉	#471	1	4	yes	—
150	C₂²×C₅₉	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
152	C₂²×C₅₉	C₂×#125	1	1	yes	characteristic
153	C₄×C₅₉	#389	1	2	yes	—
154	C₄×C₅₉	#389	1	2	yes	—
155	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	Dic₅₉	—	4	4	no	—
160	Dic₅₉	—	8	4	no	—
161	C₂²×#108	D₅₉	1	1	yes	characteristic
162	C₂×#428	—	59	2	no	—
163	C₂²×#106	—	59	1	no	—
164	C₂×#405	—	59	1	no	—
165	#1951	—	59	8	no	—
166	C₂×#343	—	59	2	no	—
167	C₂×C₅₉	—	2	1	no	—
168	C₂×C₅₉	—	2	1	no	—
169	C₂×C₅₉	—	2	2	no	—
170	C₂×C₅₉	#1951	1	4	yes	complemented
171	C₂×C₅₉	C₂×#389	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#471	1	2	yes	—
173	C₂²×M₁₆	—	59	1	no	—
174	C₂²×M₁₆	C₂×D₅₉	1	2	yes	—
175	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
176	C₂×#117	—	59	1	no	—
177	C₂²×C₄⋊C₄	—	59	2	no	—
178	C₂×#108	Dic₅₉	1	8	yes	complemented
179	C₂×#108	C₂×D₅₉	1	4	yes	—
180	#428	—	59	8	no	—
181	#428	—	118	4	no	—
182	C₂×#107	—	118	2	no	—
183	C₂×#106	—	59	4	no	—
184	C₂×#106	—	118	2	no	—
185	C₂×#106	—	118	1	no	—
186	C₂×#106	—	118	1	no	—
187	C₂×#106	—	118	2	no	—
188	#405	—	59	4	no	—
189	#405	—	118	2	no	—
190	C₂×#97	—	118	1	no	—
191	C₂×C₄×C₈	—	118	1	no	—
192	#343	—	59	8	no	—
193	C₅₉	C₂×#1951	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₄	—	118	2	no	—
195	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	59	4	no	—
199	C₂×M₁₆	—	59	2	no	—
200	C₂×M₁₆	—	118	1	no	—
201	C₂×M₁₆	—	118	2	no	—
202	C₂×M₁₆	C₂×Dic₅₉	1	8	yes	—
203	C₂×M₁₆	C₂²×D₅₉	1	4	yes	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	118	1	no	—
206	C₂²×C₈	Dic₁₁₈	1	4	yes	—
207	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
208	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
209	C₂²×C₈	D₂₃₆	1	2	yes	—
210	C₂²×C₈	C₂×Dic₅₉	1	4	yes	—
211	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
212	#117	—	59	2	no	—
213	#117	—	59	2	no	—
214	#117	—	118	1	no	—
215	#117	—	118	1	no	—
216	C₂×C₄⋊C₄	—	59	2	no	—
217	C₂×C₄⋊C₄	—	59	4	no	—
218	C₂×C₄⋊C₄	—	118	2	no	—
219	C₂×C₄⋊C₄	—	118	2	no	—
220	C₂×C₄⋊C₄	—	118	2	no	—
221	C₂×C₄⋊C₄	—	118	2	no	—
222	C₂×C₄⋊C₄	—	118	2	no	—
223	C₂×C₂²⋊C₄	—	118	1	no	—
224	C₂×C₄²	—	118	1	no	—
225	#108	C₂×Dic₅₉	1	16	yes	complemented
226	#107	—	118	8	no	—
227	#106	—	118	8	no	—
228	#106	—	118	4	no	—
229	#106	—	118	4	no	—
230	#97	—	118	4	no	—
231	C₄×C₈	—	118	4	no	—
232	C₂⁴	#11745689	1	1	yes	characteristic
233	C₂²×C₄	—	118	2	no	—
234	C₂²×C₄	—	118	4	no	—
235	C₂²×C₄	—	472	4	no	—
236	C₂²×C₄	—	236	2	no	—
237	C₂²×C₄	—	236	2	no	—
238	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
239	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
243	C₂²×C₄	#11745687	1	2	yes	—
244	C₂²×C₄	Q₈×D₅₉	1	2	yes	—
245	C₂²×C₄	#11745689	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
247	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	59	4	no	—
251	M₁₆	—	118	4	no	—
252	M₁₆	—	118	2	no	—
253	M₁₆	C₂²×Dic₅₉	1	8	yes	—
254	C₂×C₈	—	2	8	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	118	4	no	—
257	C₂×C₈	—	118	2	no	—
258	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
259	C₂×C₈	C₄⋊Dic₅₉	1	4	yes	—
260	C₂×C₈	C₄⋊Dic₅₉	1	8	yes	—
261	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
262	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
263	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
264	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
265	C₄⋊C₄	—	59	4	no	—
266	C₄⋊C₄	—	118	2	no	—
267	C₄⋊C₄	—	118	4	no	—
268	C₄⋊C₄	—	118	4	no	—
269	C₄⋊C₄	—	236	2	no	—
270	C₄⋊C₄	—	236	2	no	—
271	C₂²⋊C₄	—	118	2	no	—
272	C₂²⋊C₄	—	118	2	no	—
273	C₄²	—	118	2	no	—
274	C₄²	—	118	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	Q₈×Dic₅₉	1	2	yes	—
277	C₂³	#14273702	1	1	yes	characteristic
278	C₂³	#14273702	1	1	yes	characteristic
279	C₂³	#14273718	1	2	yes	—
280	C₂³	C₂×#11745689	1	1	yes	characteristic
281	C₂×C₄	—	118	4	no	—
282	C₂×C₄	—	472	4	no	—
283	C₂×C₄	—	472	4	no	—
284	C₂×C₄	—	472	4	no	—
285	C₂×C₄	—	236	4	no	—
286	C₂×C₄	—	236	4	no	—
287	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
289	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
290	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
291	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
292	C₂×C₄	#14273702	1	1	yes	characteristic
293	C₂×C₄	#14273702	1	1	yes	characteristic
294	C₂×C₄	#14273707	1	2	yes	—
295	C₂×C₄	#14273707	1	2	yes	—
296	C₂×C₄	#14273718	1	2	yes	—
297	C₂×C₄	#14273726	1	2	yes	—
298	C₂×C₄	#14273726	1	2	yes	—
299	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
304	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
305	C₈	—	2	8	no	—
306	C₈	—	118	4	no	—
307	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
308	C₈	C₂×C₄⋊Dic₅₉	1	4	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	#16837301	1	2	yes	—
314	C₂²	#16837301	1	2	yes	—
315	C₂²	#16837666	1	4	yes	—
316	C₂²	C₂×Q₈×Dic₅₉	1	1	yes	characteristic
317	C₂²	C₂×#14273702	1	1	yes	characteristic
318	C₂²	C₂×#14273718	1	1	yes	characteristic
319	C₄	—	472	4	no	—
320	C₄	—	236	4	no	—
321	C₄	#16837301	1	2	yes	—
322	C₄	#16837301	1	2	yes	—
323	C₄	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
324	C₄	C₂×D₄×Dic₅₉	1	1	yes	characteristic
325	C₄	C₂×#14273707	1	1	yes	characteristic
326	C₄	C₂×#14273726	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	#21424877	1	4	yes	direct product factor
331	C₂	C₂×#16837301	1	1	yes	characteristic
332	C₂	C₂×#16837666	1	2	yes	—

	subgroup	G/H
center	C₂³	#14273702
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂²×#108×C₅₉	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂⁴×C₅₉	#41
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₂²×#108	D₅₉
hypocenter	C₅₉	C₂×#1951
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #21424877	C₂	direct (N × H)
25. C₂×#108×C₅₉	C₄	semidirect
64. #108×C₅₉	C₂×C₄	semidirect
170. C₂×C₅₉	#1951	semidirect
178. C₂×#108	Dic₅₉	semidirect
193. C₅₉	C₂×#1951	semidirect
225. #108	C₂×Dic₅₉	semidirect
330. C₂	#21424877	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×#16837444	2
C₂²×#14273659	2
C₂×#16837357	2
#21424877	2
C₂×#16837170	2
C₂²×#108×C₅₉	2
C₂×#1951	59

Subgroups — C2×#21424877 #28592230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21424877 #28592230