Subgroups — #28560417

Order 15104.

618 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	#21433183	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21423006	C₂	1	2	yes	maximal, complemented
4	#21422948	C₂	1	4	yes	maximal, complemented
5	#21422916	C₂	1	2	yes	maximal, complemented
6	#21422859	C₂	1	1	yes	maximal, characteristic
7	#21422544	C₂	1	2	yes	maximal
8	C₅₉×#1451	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₃₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
11	#16837532	C₂²	1	4	yes	—
12	#16837531	—	2	4	no	—
13	C₈×Dic₁₁₈	C₂²	1	4	yes	—
14	C₄×Dic₂₃₆	C₄	1	2	yes	—
15	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
16	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
17	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₃₆	C₂²	1	2	yes	—
19	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
20	#16837459	C₂²	1	2	yes	—
21	#16837456	—	2	2	no	—
22	#16837452	—	2	2	no	—
23	C₂×#14273661	C₄	1	1	yes	characteristic
24	C₂×#14273652	C₄	1	2	yes	—
25	#16837155	C₂²	1	2	yes	—
26	#16837153	C₂²	1	2	yes	—
27	#16837134	C₂²	1	4	yes	—
28	#16837134	C₂²	1	4	yes	—
29	C₅₉×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₅₉×#322	C₂²	1	2	yes	—
33	C₅₉×#322	C₂²	1	2	yes	—
34	C₂²×Dic₁₁₈	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₃₆	—	2	4	no	—
36	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
37	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₁₈	—	2	2	no	—
40	C₄×Dic₁₁₈	—	2	2	no	—
41	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₁₈	C₂³	1	2	yes	—
44	C₄×Dic₁₁₈	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
46	C₂×Dic₅₉⋊C₄	—	2	2	no	—
47	C₂×C₄×Dic₅₉	—	2	2	no	—
48	#14273661	C₂×C₄	1	2	yes	—
49	#14273661	C₂×C₄	1	1	yes	characteristic
50	#14273661	C₂×C₄	1	1	yes	characteristic
51	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
52	Dic₅₉⋊C₈	—	2	4	no	—
53	Dic₅₉⋊C₈	—	2	4	no	—
54	#14273652	C₂×C₄	1	2	yes	—
55	#14273652	C₂×C₄	1	2	yes	—
56	#14273652	C₂×C₄	1	2	yes	—
57	#14273652	C₂×C₄	1	2	yes	—
58	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
59	#14273638	—	2	2	no	—
60	#14273637	—	2	4	no	—
61	C₈×Dic₅₉	—	2	4	no	—
62	C₂×M₁₆×C₅₉	—	2	1	no	—
63	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
67	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
68	#97×C₅₉	—	2	1	no	—
69	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₉	C₂³	1	2	yes	—
72	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	2	yes	—
75	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
78	C₂²×Dic₅₉	—	2	2	no	—
79	C₂²×Dic₅₉	—	2	2	no	—
80	C₂×Dic₁₁₈	—	2	4	no	—
81	C₂×Dic₁₁₈	—	2	2	no	—
82	C₂×Dic₁₁₈	—	4	2	no	—
83	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
84	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
85	M₁₆×C₅₉	—	4	2	no	—
86	C₂×C₈×C₅₉	—	2	2	no	—
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	—	4	1	no	—
89	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
94	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₉	#41	1	1	yes	characteristic
97	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊Dic₅₉	—	2	2	no	—
99	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
101	Dic₅₉⋊C₄	—	2	2	no	—
102	Dic₅₉⋊C₄	—	2	2	no	—
103	Dic₅₉⋊C₄	—	2	2	no	—
104	Dic₅₉⋊C₄	—	2	2	no	—
105	C₄×Dic₅₉	—	2	2	no	—
106	C₄×Dic₅₉	—	2	2	no	—
107	C₄×Dic₅₉	—	2	2	no	—
108	C₄×Dic₅₉	—	2	2	no	—
109	C₂×C₅₉⋊C₈	—	2	4	no	—
110	Dic₂₃₆	—	2	4	no	—
111	Dic₂₃₆	—	2	2	no	—
112	Dic₂₃₆	M₁₆	1	4	yes	complemented
113	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	2	2	no	—
127	C₂×Dic₅₉	—	2	2	no	—
128	C₂×Dic₅₉	—	4	4	no	—
129	C₂×Dic₅₉	—	4	4	no	—
130	Dic₁₁₈	—	2	2	no	—
131	Dic₁₁₈	—	2	2	no	—
132	Dic₁₁₈	—	2	2	no	—
133	Dic₁₁₈	—	4	4	no	—
134	Dic₁₁₈	C₂×M₁₆	1	4	yes	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	4	no	—
139	C₈×C₅₉	—	4	2	no	—
140	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
141	C₅₉⋊C₈	—	4	4	no	—
142	#18062	—	59	1	no	maximal, Sylow
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	#444	1	2	yes	—
146	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
152	C₄×C₅₉	#568	1	1	yes	characteristic
153	C₄×C₅₉	#578	1	2	yes	—
154	Dic₅₉	—	2	4	no	—
155	Dic₅₉	—	4	2	no	—
156	Dic₅₉	—	4	2	no	—
157	Dic₅₉	—	4	2	no	—
158	#2984	—	59	2	no	—
159	C₂×C₄×Q₁₆	—	59	1	no	—
160	#1593	—	59	2	no	—
161	#1584	—	59	4	no	—
162	#1579	—	59	2	no	—
163	#1568	—	59	1	no	—
164	#1491	—	59	2	no	—
165	#1451	D₅₉	1	1	yes	characteristic
166	C₂×C₅₉	—	2	2	no	—
167	C₂×C₅₉	D₄×M₁₆	1	1	yes	characteristic
168	C₂×C₅₉	#2950	1	1	yes	characteristic
169	C₂×C₅₉	C₂×#444	1	1	yes	characteristic
170	C₂²×Q₁₆	—	59	1	no	—
171	C₂×C₄×Q₈	—	59	2	no	—
172	#446	—	59	4	no	—
173	#446	—	118	4	no	—
174	C₈×Q₈	—	59	4	no	—
175	C₄×Q₁₆	—	59	1	no	—
176	C₄×Q₁₆	—	59	2	no	—
177	C₄×Q₁₆	—	59	1	no	—
178	C₄×Q₁₆	—	59	1	no	—
179	C₄×Q₁₆	—	59	2	no	—
180	C₄×Q₁₆	—	59	1	no	—
181	#432	—	59	2	no	—
182	#432	—	118	2	no	—
183	#431	—	118	2	no	—
184	#431	C₂×D₅₉	1	1	yes	characteristic
185	C₂×#107	—	59	1	no	—
186	C₂×#103	—	59	2	no	—
187	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
189	#335	—	59	2	no	—
190	#334	—	59	2	no	—
191	Q₈⋊C₈	—	59	4	no	—
192	Q₈⋊C₈	—	59	4	no	—
193	#322	C₂×D₅₉	1	2	yes	—
194	#322	C₂×D₅₉	1	2	yes	—
195	C₅₉	#18062	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	59	2	no	—
197	C₂×Q₁₆	—	59	1	no	—
198	C₂×Q₁₆	—	59	2	no	—
199	C₂×Q₁₆	—	59	1	no	—
200	C₂×Q₁₆	—	118	4	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
204	C₄×Q₈	—	59	2	no	—
205	C₄×Q₈	—	59	2	no	—
206	C₄×Q₈	—	59	2	no	—
207	C₄×Q₈	—	59	2	no	—
208	C₄×Q₈	—	118	2	no	—
209	C₄×Q₈	—	118	2	no	—
210	C₂×C₄⋊C₄	—	59	2	no	—
211	C₂×C₄⋊C₄	—	118	2	no	—
212	C₂×C₄²	—	118	2	no	—
213	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
214	#107	—	59	2	no	—
215	#107	—	59	1	no	—
216	#107	—	59	1	no	—
217	C₄⋊C₈	—	59	4	no	—
218	C₄⋊C₈	—	118	4	no	—
219	C₄⋊C₈	—	118	4	no	—
220	C₄⋊C₈	D₂₃₆	1	2	yes	—
221	#103	—	59	2	no	—
222	#103	—	59	2	no	—
223	#103	—	59	2	no	—
224	#103	—	59	2	no	—
225	C₂²⋊C₈	—	118	2	no	—
226	C₂²⋊C₈	D₂₃₆	1	2	yes	—
227	#97	—	2	1	no	—
228	#97	—	118	4	no	—
229	#97	—	118	2	no	—
230	C₄×C₈	—	118	4	no	—
231	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
232	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₅₉	1	2	yes	—
234	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₉	1	2	yes	—
237	C₂×Q₈	—	59	2	no	—
238	C₂×Q₈	—	59	2	no	—
239	C₂×Q₈	—	118	4	no	—
240	C₂×Q₈	—	118	2	no	—
241	C₂×Q₈	—	236	2	no	—
242	C₂²×C₄	—	118	2	no	—
243	C₂²×C₄	—	118	2	no	—
244	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
246	C₂²×C₄	#11745685	1	1	yes	characteristic
247	Q₁₆	—	59	4	no	—
248	Q₁₆	—	118	4	no	—
249	Q₁₆	—	118	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	118	4	no	—
255	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
260	C₄⋊C₄	—	59	2	no	—
261	C₄⋊C₄	—	59	2	no	—
262	C₄⋊C₄	—	118	2	no	—
263	C₄⋊C₄	—	118	2	no	—
264	C₄⋊C₄	—	118	2	no	—
265	C₄⋊C₄	—	118	2	no	—
266	C₄⋊C₄	—	118	2	no	—
267	C₄²	—	118	2	no	—
268	C₄²	—	118	2	no	—
269	C₄²	—	118	2	no	—
270	C₄²	—	118	2	no	—
271	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
272	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
273	C₄²	#11745685	1	1	yes	characteristic
274	C₄²	C₂³×D₅₉	1	1	yes	characteristic
275	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
276	Q₈	—	59	4	no	—
277	Q₈	—	118	2	no	—
278	Q₈	—	118	2	no	—
279	Q₈	—	118	2	no	—
280	Q₈	—	236	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	118	2	no	—
286	C₂×C₄	—	118	2	no	—
287	C₂×C₄	—	118	2	no	—
288	C₂×C₄	—	118	2	no	—
289	C₂×C₄	—	236	4	no	—
290	C₂×C₄	—	236	4	no	—
291	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
292	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
293	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
296	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	4	no	—
301	C₈	—	4	2	no	—
302	C₈	—	236	4	no	—
303	C₈	M₁₆×D₅₉	1	2	yes	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	#16837525	1	2	yes	—
307	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	118	4	no	—
313	C₄	—	236	2	no	—
314	C₄	—	236	2	no	—
315	C₄	—	236	2	no	—
316	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
317	C₄	#16838168	1	1	yes	characteristic
318	C₄	#16838219	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	M₁₆×D₂₃₆	1	1	yes	characteristic
321	C₂	#21432965	1	1	yes	characteristic
322	C₂	C₂×#16837525	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1451	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1451	D₅₉
hypocenter	C₅₉	#18062
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #21423006	C₂	semidirect
4. #21422948	C₂	semidirect
5. #21422916	C₂	semidirect
112. Dic₂₃₆	M₁₆	semidirect
195. C₅₉	#18062	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21433183	2
C₂×C₄×Dic₂₃₆	2
#21423006	2
#21422948	2
#21422916	2
#21422859	2
#21422544	2
C₅₉×#1451	2
#18062	59