Subgroups — C₂×#21437197 #29495156

Order 15104.

2074 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#21437197	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16837820	C₂	1	3	yes	maximal, complemented
4	C₂×#16837819	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837818	C₂	1	3	yes	maximal, complemented
6	C₂²×#14273666	C₂	1	3	yes	maximal, complemented
7	C₂×M₃₂×D₅₉	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₉×#504	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×D₅₉	C₄	1	1	yes	characteristic
10	C₂×#14273735	C₄	1	3	yes	—
11	C₂×#14273734	C₄	1	1	yes	characteristic
12	C₂×#14273733	C₂²	1	3	yes	—
13	#131×D₅₉	C₄	1	8	yes	—
14	#131×D₅₉	C₂²	1	8	yes	—
15	C₂×M₁₆×D₅₉	C₄	1	3	yes	—
16	C₂²×C₈×D₅₉	C₂²	1	3	yes	—
17	#16837820	C₂²	1	24	yes	complemented
18	#16837819	C₂²	1	8	yes	complemented
19	#16837818	C₂²	1	24	yes	complemented
20	C₂×#14273667	C₂²	1	3	yes	complemented
21	C₂×#14273666	C₂²	1	3	yes	—
22	C₂×#14273666	C₂²	1	1	yes	characteristic
23	C₂×#14273666	C₂²	1	3	yes	complemented
24	C₂×#14273666	C₂²	1	3	yes	complemented
25	C₂×#14273666	C₂²	1	24	yes	complemented
26	M₃₂×D₅₉	C₂²	1	24	yes	complemented
27	C₂²×C₅₉⋊C₁₆	C₂²	1	3	yes	complemented
28	C₂×C₁₆×D₅₉	C₂²	1	6	yes	complemented
29	C₂×#131×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×#504	C₂²	1	8	yes	complemented
31	C₂×M₃₂×C₅₉	C₂²	1	3	yes	complemented
32	C₂²×C₁₆×C₅₉	C₂²	1	3	yes	complemented
33	C₂×#11745689	C₈	1	1	yes	characteristic
34	C₂×#11745685	C₂×C₄	1	3	yes	—
35	C₂×#11745687	C₈	1	3	yes	—
36	#41×D₅₉	C₈	1	8	yes	—
37	#41×D₅₉	C₂×C₄	1	8	yes	—
38	C₂×Q₈×D₅₉	C₈	1	1	yes	characteristic
39	C₂×D₄×D₅₉	C₈	1	3	yes	—
40	C₂²×C₄×D₅₉	C₂×C₄	1	3	yes	—
41	#14273735	C₂×C₄	1	24	yes	—
42	#14273734	C₂×C₄	1	8	yes	—
43	#14273733	C₂×C₄	1	12	yes	—
44	#14273733	C₂³	1	12	yes	—
45	C₂×#11745664	C₂×C₄	1	3	yes	—
46	C₂×#11745659	C₂×C₄	1	6	yes	—
47	M₁₆×D₅₉	C₂×C₄	1	24	yes	—
48	C₂²×C₅₉⋊C₈	C₂×C₄	1	3	yes	—
49	C₂×C₈×D₅₉	C₂×C₄	1	3	yes	—
50	C₂×C₈×D₅₉	C₂×C₄	1	12	yes	—
51	C₂×C₈×D₅₉	C₂×C₄	1	3	yes	—
52	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₅₉	C₂³	1	12	yes	—
54	C₂×C₈×D₅₉	C₂³	1	3	yes	—
55	#14273667	C₂³	1	12	yes	complemented
56	#14273666	C₂³	1	12	yes	—
57	#14273666	C₂³	1	4	yes	—
58	#14273666	C₂³	1	12	yes	complemented
59	#14273666	C₂³	1	12	yes	complemented
60	C₂×C₅₉⋊C₁₆	C₂³	1	3	yes	—
61	C₂×C₅₉⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₅₉⋊C₁₆	C₂³	1	12	yes	complemented
63	C₁₆×D₅₉	C₂³	1	24	yes	complemented
64	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
65	#131×C₅₉	C₂×C₄	1	4	yes	—
66	#131×C₅₉	C₂³	1	4	yes	—
67	C₂×M₁₆×C₅₉	C₂×C₄	1	3	yes	—
68	C₂²×C₈×C₅₉	C₂³	1	3	yes	—
69	M₃₂×C₅₉	C₂³	1	12	yes	complemented
70	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₉	C₂³	1	3	yes	—
72	C₂×C₁₆×C₅₉	C₂³	1	12	yes	complemented
73	C₂³×D₅₉	C₂×C₈	1	3	yes	—
74	#41×C₅₉	C₂×C₈	1	4	yes	—
75	#41×C₅₉	C₂²×C₄	1	4	yes	—
76	C₂×Q₈×C₅₉	C₂×C₈	1	1	yes	characteristic
77	C₂×D₄×C₅₉	C₂×C₈	1	3	yes	—
78	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
79	C₂×C₂²⋊D₅₉	C₂×C₈	1	6	yes	—
80	C₂²×Dic₅₉	C₂×C₈	1	3	yes	—
81	#11745689	C₂×C₈	1	8	yes	—
82	Q₈×D₅₉	C₂×C₈	1	8	yes	—
83	#11745687	C₂×C₈	1	24	yes	—
84	D₄×D₅₉	C₂×C₈	1	24	yes	—
85	#11745685	C₂×C₈	1	12	yes	—
86	#11745685	C₂²×C₄	1	12	yes	—
87	C₂×D₂₃₆	C₂×C₈	1	3	yes	—
88	C₂×C₄×D₅₉	C₂×C₈	1	3	yes	—
89	C₂×C₄×D₅₉	C₂×C₈	1	12	yes	—
90	C₂×C₄×D₅₉	C₂×C₈	1	3	yes	—
91	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₅₉	C₂²×C₄	1	12	yes	—
93	C₂×C₄×D₅₉	C₂²×C₄	1	3	yes	—
94	C₂×Dic₁₁₈	C₂×C₈	1	3	yes	—
95	M₁₆×C₅₉	C₂²×C₄	1	12	yes	—
96	C₂×C₈×C₅₉	C₂²×C₄	1	3	yes	—
97	C₂×C₈×C₅₉	C₂²×C₄	1	6	yes	—
98	C₂×C₈×C₅₉	C₂⁴	1	6	yes	—
99	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
100	#11745664	C₂²×C₄	1	12	yes	—
101	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂²×C₄	1	3	yes	—
103	C₂×C₅₉⋊C₈	C₂²×C₄	1	12	yes	—
104	#11745659	C₂²×C₄	1	24	yes	—
105	C₈×D₅₉	C₂²×C₄	1	2	yes	—
106	C₈×D₅₉	C₂²×C₄	1	12	yes	—
107	C₈×D₅₉	C₂²×C₄	1	6	yes	—
108	C₈×D₅₉	C₂²×C₄	1	12	yes	—
109	C₈×D₅₉	C₂⁴	1	6	yes	—
110	C₈×D₅₉	C₂⁴	1	2	yes	—
111	C₁₆×C₅₉	C₂⁴	1	2	yes	—
112	C₁₆×C₅₉	C₂⁴	1	6	yes	—
113	C₅₉⋊C₁₆	C₂⁴	1	6	yes	—
114	C₅₉⋊C₁₆	C₂⁴	1	2	yes	—
115	C₂³×C₅₉	C₂²×C₈	1	3	yes	—
116	C₂²×D₅₉	—	4	6	no	—
117	C₂²×D₅₉	C₂²×C₈	1	1	yes	characteristic
118	C₂²×D₅₉	C₂²×C₈	1	3	yes	—
119	C₂²×D₅₉	C₂²×C₈	1	12	yes	—
120	Q₈×C₅₉	C₂²×C₈	1	4	yes	—
121	D₄×C₅₉	C₂²×C₈	1	12	yes	—
122	C₂×C₄×C₅₉	C₂²×C₈	1	3	yes	—
123	C₂×C₄×C₅₉	C₂²×C₈	1	6	yes	—
124	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
126	C₂²⋊D₅₉	C₂²×C₈	1	24	yes	—
127	C₂×Dic₅₉	C₂²×C₈	1	3	yes	—
128	C₂×Dic₅₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂²×C₈	1	12	yes	—
130	D₂₃₆	C₂²×C₈	1	12	yes	—
131	C₄×D₅₉	C₂²×C₈	1	2	yes	—
132	C₄×D₅₉	C₂²×C₈	1	6	yes	—
133	C₄×D₅₉	C₂²×C₈	1	12	yes	—
134	C₄×D₅₉	C₂²×C₈	1	12	yes	—
135	C₄×D₅₉	C₂³×C₄	1	2	yes	—
136	C₄×D₅₉	C₂³×C₄	1	6	yes	—
137	Dic₁₁₈	C₂²×C₈	1	12	yes	—
138	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
139	C₈×C₅₉	C₂³×C₄	1	6	yes	—
140	C₈×C₅₉	C₂⁵	1	1	yes	characteristic
141	C₅₉⋊C₈	C₂³×C₄	1	2	yes	—
142	C₅₉⋊C₈	C₂³×C₄	1	6	yes	—
143	C₂×#3407	—	59	1	no	maximal, Sylow
144	C₂²×C₅₉	—	2	6	no	—
145	C₂²×C₅₉	C₂³×C₈	1	6	yes	—
146	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₅₉	—	2	2	no	—
148	C₂×D₅₉	—	2	6	no	—
149	C₂×D₅₉	—	4	12	no	—
150	C₂×D₅₉	C₂³×C₈	1	2	yes	—
151	C₂×D₅₉	C₂³×C₈	1	6	yes	—
152	C₄×C₅₉	C₂³×C₈	1	6	yes	—
153	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
155	Dic₅₉	C₂³×C₈	1	6	yes	—
156	Dic₅₉	C₂³×C₈	1	2	yes	—
157	C₂²×#131	—	59	1	no	—
158	#3407	—	59	16	no	—
159	C₂×#504	—	59	3	no	—
160	C₂×#504	—	59	3	no	—
161	C₂×#504	—	59	1	no	—
162	C₂×#504	D₅₉	1	1	yes	characteristic, complemented
163	C₂²×M₃₂	—	59	3	no	—
164	C₂²×M₃₂	—	59	3	no	—
165	C₂×C₅₉	—	2	6	no	—
166	C₂×C₅₉	#3407	1	2	yes	complemented
167	C₂×C₅₉	C₂⁴×C₈	1	1	yes	characteristic
168	D₅₉	—	2	2	no	—
169	D₅₉	—	2	6	no	—
170	C₂²×#41	—	59	1	no	—
171	C₂×#131	—	59	1	no	—
172	C₂×#131	—	59	8	no	—
173	C₂×#131	—	59	8	no	—
174	C₂×#131	—	59	3	no	—
175	C₂×#131	—	59	3	no	—
176	C₂×#131	C₂×D₅₉	1	1	yes	characteristic
177	C₂²×M₁₆	—	59	3	no	—
178	C₂³×C₈	—	59	3	no	—
179	#504	—	59	8	no	—
180	#504	—	59	24	no	—
181	#504	—	59	24	no	—
182	#504	C₂×D₅₉	1	8	yes	complemented
183	C₂×M₃₂	—	59	3	no	—
184	C₂×M₃₂	—	59	24	no	—
185	C₂×M₃₂	—	59	3	no	—
186	C₂×M₃₂	—	59	1	no	—
187	C₂×M₃₂	—	59	3	no	—
188	C₂×M₃₂	—	59	3	no	—
189	C₂×M₃₂	—	59	24	no	—
190	C₂×M₃₂	C₂×D₅₉	1	3	yes	complemented
191	C₂²×C₁₆	—	59	3	no	—
192	C₂²×C₁₆	—	59	6	no	—
193	C₂²×C₁₆	C₂×D₅₉	1	3	yes	complemented
194	C₅₉	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	59	8	no	—
196	C₂×#41	—	59	8	no	—
197	C₂×#41	—	59	1	no	—
198	C₂×#41	—	59	3	no	—
199	C₂×#41	—	59	3	no	—
200	C₂×#41	C₄×D₅₉	1	1	yes	characteristic
201	C₂²×Q₈	—	59	1	no	—
202	C₂²×D₄	—	59	3	no	—
203	C₂³×C₄	—	59	3	no	—
204	#131	—	59	8	no	—
205	#131	—	59	24	no	—
206	#131	—	59	12	no	—
207	#131	—	59	12	no	—
208	#131	C₄×D₅₉	1	4	yes	—
209	#131	C₂²×D₅₉	1	4	yes	—
210	C₂×M₁₆	—	59	3	no	—
211	C₂×M₁₆	—	59	24	no	—
212	C₂×M₁₆	—	59	6	no	—
213	C₂×M₁₆	C₄×D₅₉	1	3	yes	—
214	C₂²×C₈	—	59	3	no	—
215	C₂²×C₈	—	59	3	no	—
216	C₂²×C₈	—	59	3	no	—
217	C₂²×C₈	—	59	12	no	—
218	C₂²×C₈	—	59	12	no	—
219	C₂²×C₈	—	59	1	no	—
220	C₂²×C₈	—	59	3	no	—
221	C₂²×C₈	C₂²×D₅₉	1	3	yes	—
222	M₃₂	—	59	12	no	—
223	M₃₂	—	59	12	no	—
224	M₃₂	—	59	4	no	—
225	M₃₂	—	59	12	no	—
226	M₃₂	—	59	12	no	—
227	M₃₂	C₂²×D₅₉	1	12	yes	complemented
228	C₂×C₁₆	—	59	12	no	—
229	C₂×C₁₆	—	59	3	no	—
230	C₂×C₁₆	—	59	1	no	—
231	C₂×C₁₆	—	59	24	no	—
232	C₂×C₁₆	C₂²×D₅₉	1	3	yes	—
233	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₅₉	1	12	yes	complemented
235	C₂⁴	—	59	3	no	—
236	#41	—	59	8	no	—
237	#41	—	59	24	no	—
238	#41	—	59	12	no	—
239	#41	—	59	12	no	—
240	#41	C₈×D₅₉	1	4	yes	—
241	#41	C₂×C₄×D₅₉	1	4	yes	—
242	C₂×Q₈	—	59	8	no	—
243	C₂×Q₈	—	59	3	no	—
244	C₂×Q₈	C₈×D₅₉	1	1	yes	characteristic
245	C₂×D₄	—	59	6	no	—
246	C₂×D₄	—	59	24	no	—
247	C₂×D₄	—	59	3	no	—
248	C₂×D₄	C₈×D₅₉	1	3	yes	—
249	C₂²×C₄	—	59	3	no	—
250	C₂²×C₄	—	59	3	no	—
251	C₂²×C₄	—	59	3	no	—
252	C₂²×C₄	—	59	12	no	—
253	C₂²×C₄	—	59	1	no	—
254	C₂²×C₄	—	59	3	no	—
255	C₂²×C₄	—	59	12	no	—
256	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
257	M₁₆	—	59	12	no	—
258	M₁₆	—	59	24	no	—
259	M₁₆	C₂×C₄×D₅₉	1	12	yes	—
260	C₂×C₈	—	59	1	no	—
261	C₂×C₈	—	59	3	no	—
262	C₂×C₈	—	59	12	no	—
263	C₂×C₈	—	59	2	no	—
264	C₂×C₈	—	59	12	no	—
265	C₂×C₈	—	59	12	no	—
266	C₂×C₈	—	59	6	no	—
267	C₂×C₈	—	59	6	no	—
268	C₂×C₈	—	59	2	no	—
269	C₂×C₈	C₂×C₄×D₅₉	1	3	yes	—
270	C₂×C₈	C₂×C₄×D₅₉	1	6	yes	—
271	C₂×C₈	C₂³×D₅₉	1	6	yes	—
272	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
273	C₁₆	—	59	2	no	—
274	C₁₆	—	59	6	no	—
275	C₁₆	C₂³×D₅₉	1	6	yes	—
276	C₁₆	C₂³×D₅₉	1	2	yes	—
277	C₂³	—	59	1	no	—
278	C₂³	—	59	3	no	—
279	C₂³	—	59	12	no	—
280	C₂³	—	236	6	no	—
281	C₂³	C₂×C₈×D₅₉	1	3	yes	—
282	Q₈	—	59	12	no	—
283	Q₈	C₂×C₈×D₅₉	1	4	yes	—
284	D₄	—	59	24	no	—
285	D₄	—	59	12	no	—
286	D₄	C₂×C₈×D₅₉	1	12	yes	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	59	3	no	—
289	C₂×C₄	—	59	12	no	—
290	C₂×C₄	—	59	2	no	—
291	C₂×C₄	—	59	12	no	—
292	C₂×C₄	—	59	12	no	—
293	C₂×C₄	—	59	6	no	—
294	C₂×C₄	—	59	2	no	—
295	C₂×C₄	—	59	6	no	—
296	C₂×C₄	C₂×C₈×D₅₉	1	3	yes	—
297	C₂×C₄	C₂×C₈×D₅₉	1	6	yes	—
298	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
300	C₈	—	59	2	no	—
301	C₈	—	59	6	no	—
302	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
303	C₈	C₂²×C₄×D₅₉	1	6	yes	—
304	C₈	C₂⁴×D₅₉	1	1	yes	characteristic
305	C₂²	—	2	6	no	—
306	C₂²	—	59	2	no	—
307	C₂²	—	59	6	no	—
308	C₂²	—	118	2	no	—
309	C₂²	—	118	6	no	—
310	C₂²	—	236	12	no	—
311	C₂²	C₂²×C₈×D₅₉	1	6	yes	—
312	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
313	C₄	—	59	2	no	—
314	C₄	—	59	6	no	—
315	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
316	C₄	C₂²×C₈×D₅₉	1	6	yes	—
317	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
318	C₂	—	2	6	no	—
319	C₂	—	118	2	no	—
320	C₂	—	118	6	no	—
321	C₂	#21437197	1	2	yes	direct product factor
322	C₂	C₂³×C₈×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂³×D₅₉
commutator	C₂×C₅₉	C₂⁴×C₈
Frattini	C₈	C₂⁴×D₅₉
Fitting	C₂×C₅₉×#504	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂×#41×D₅₉	C₄
cosocle	C₈×C₅₉	C₂⁵
hypercenter	C₂×#504	D₅₉
hypocenter	C₅₉	C₂×#3407
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #21437197	C₂	direct (N × H)
3. C₂×#16837820	C₂	semidirect
4. C₂×#16837819	C₂	semidirect
5. C₂×#16837818	C₂	semidirect
6. C₂²×#14273666	C₂	semidirect
7. C₂×M₃₂×D₅₉	C₂	semidirect
8. C₂×C₅₉×#504	C₂	semidirect
17. #16837820	C₂²	semidirect
18. #16837819	C₂²	semidirect
19. #16837818	C₂²	semidirect
20. C₂×#14273667	C₂²	semidirect
23. C₂×#14273666	C₂²	semidirect
24. C₂×#14273666	C₂²	semidirect
25. C₂×#14273666	C₂²	semidirect
26. M₃₂×D₅₉	C₂²	semidirect
27. C₂²×C₅₉⋊C₁₆	C₂²	semidirect
28. C₂×C₁₆×D₅₉	C₂²	semidirect
30. C₅₉×#504	C₂²	semidirect
31. C₂×M₃₂×C₅₉	C₂²	semidirect
32. C₂²×C₁₆×C₅₉	C₂²	semidirect
55. #14273667	C₂³	semidirect
58. #14273666	C₂³	semidirect
59. #14273666	C₂³	semidirect
62. C₂×C₅₉⋊C₁₆	C₂³	semidirect
63. C₁₆×D₅₉	C₂³	semidirect
69. M₃₂×C₅₉	C₂³	semidirect
72. C₂×C₁₆×C₅₉	C₂³	semidirect
162. C₂×#504	D₅₉	semidirect
166. C₂×C₅₉	#3407	semidirect
182. #504	C₂×D₅₉	semidirect
190. C₂×M₃₂	C₂×D₅₉	semidirect
193. C₂²×C₁₆	C₂×D₅₉	semidirect
194. C₅₉	C₂×#3407	semidirect
227. M₃₂	C₂²×D₅₉	semidirect
234. C₂×C₁₆	C₂²×D₅₉	semidirect
321. C₂	#21437197	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×#131×D₅₉	2
#21437197	2
C₂×#16837820	2
C₂×#16837819	2
C₂×#16837818	2
C₂²×#14273666	2
C₂×M₃₂×D₅₉	2
C₂×C₅₉×#504	2
C₂×#3407	59

Subgroups — C2×#21437197 #29495156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21437197 #29495156