Subgroups — #28507192

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837786	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837214	C₂	1	1	yes	maximal, characteristic
5	#21425815	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425811	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
9	C₂²×#11745660	—	2	1	no	—
10	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
11	C₂×#14273715	—	2	1	no	—
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	#16837786	—	2	2	no	—
15	#16837786	C₄	1	4	yes	complemented
16	#16837223	—	2	2	no	—
17	#16837217	—	2	2	no	—
18	#16837214	—	2	2	no	—
19	#16837205	—	2	2	no	—
20	#16837202	—	2	2	no	—
21	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#427	C₂²	1	1	yes	characteristic
23	C₅₉×#405	—	2	1	no	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#339	—	2	1	no	—
26	C₂²×Dic₁₁₈	—	2	1	no	—
27	C₂²×D₂₃₆	—	2	1	no	—
28	C₂×Dic₂₃₆	—	2	1	no	—
29	C₂×Dic₂₃₆	—	2	1	no	—
30	C₂×Dic₂₃₆	—	2	1	no	—
31	C₂×Dic₂₃₆	—	2	1	no	—
32	C₂×Dic₂₃₆	—	2	1	no	—
33	C₂×Dic₂₃₆	C₂×C₄	1	2	yes	—
34	C₂×#11745660	—	2	4	no	—
35	C₂×#11745660	—	4	1	no	—
36	C₂×#11745660	—	4	1	no	—
37	C₂×#11745660	—	4	2	no	—
38	C₂×D₄₇₂	—	2	1	no	—
39	C₂×D₄₇₂	—	2	1	no	—
40	C₂×D₄₇₂	—	2	1	no	—
41	C₂×D₄₇₂	—	2	1	no	—
42	C₂×D₄₇₂	—	2	1	no	—
43	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
44	#14273715	—	2	4	no	—
45	#14273715	—	4	2	no	—
46	C₂×(C₂×D₅₉)⋊C₄	—	4	2	no	—
47	#14273667	—	2	1	no	—
48	#14273667	—	2	1	no	—
49	#14273667	—	2	1	no	—
50	#14273667	—	2	1	no	—
51	C₂×C₅₉⋊C₁₆	—	2	1	no	—
52	C₂×C₅₉⋊C₁₆	—	2	1	no	—
53	C₂×M₁₆×C₅₉	—	2	1	no	—
54	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
56	#117×C₅₉	—	2	1	no	—
57	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
58	#108×C₅₉	—	2	1	no	—
59	#108×C₅₉	D₄	1	2	yes	—
60	#107×C₅₉	D₄	1	2	yes	—
61	#106×C₅₉	—	2	1	no	—
62	#97×C₅₉	—	4	1	no	—
63	C₄×C₈×C₅₉	—	4	1	no	—
64	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₉	D₄	1	2	yes	—
66	C₂³×D₅₉	—	4	1	no	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₅₉	—	4	1	no	—
70	C₂×D₂₃₆	—	2	2	no	—
71	C₂×D₂₃₆	—	4	1	no	—
72	C₂×D₂₃₆	—	4	1	no	—
73	C₂×D₂₃₆	—	4	1	no	—
74	C₂×D₂₃₆	—	4	1	no	—
75	C₂×D₂₃₆	—	4	1	no	—
76	C₂×Dic₁₁₈	—	2	2	no	—
77	C₂×Dic₁₁₈	—	4	1	no	—
78	C₂×Dic₁₁₈	—	4	1	no	—
79	C₂×Dic₁₁₈	—	4	1	no	—
80	C₂×Dic₁₁₈	—	4	1	no	—
81	C₂×Dic₁₁₈	—	4	1	no	—
82	M₁₆×C₅₉	—	2	2	no	—
83	M₁₆×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
87	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
88	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
89	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
94	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
95	C₄⋊C₄×C₅₉	—	2	2	no	—
96	C₂²⋊C₄×C₅₉	—	4	1	no	—
97	C₄²×C₅₉	—	2	1	no	—
98	C₄²×C₅₉	—	4	1	no	—
99	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
100	(C₂×D₅₉)⋊C₄	—	4	8	no	—
101	Dic₂₃₆	—	2	2	no	—
102	Dic₂₃₆	—	2	2	no	—
103	Dic₂₃₆	—	4	1	no	—
104	Dic₂₃₆	—	4	1	no	—
105	D₄₇₂	—	2	2	no	—
106	D₄₇₂	—	2	2	no	—
107	D₄₇₂	—	4	1	no	—
108	D₄₇₂	—	4	1	no	—
109	#11745660	—	4	4	no	—
110	#11745660	—	4	4	no	—
111	C₅₉⋊C₁₆	—	4	1	no	—
112	C₅₉⋊C₁₆	—	4	1	no	—
113	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂²×D₅₉	—	4	2	no	—
115	C₂²×D₅₉	—	8	4	no	—
116	C₂²×D₅₉	—	8	1	no	—
117	C₂²×D₅₉	—	8	1	no	—
118	C₂×C₄×C₅₉	—	2	2	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	8	1	no	—
129	C₂×Dic₅₉	—	8	1	no	—
130	D₂₃₆	—	4	2	no	—
131	D₂₃₆	—	4	2	no	—
132	D₂₃₆	—	8	1	no	—
133	D₂₃₆	—	8	1	no	—
134	Dic₁₁₈	—	4	2	no	—
135	Dic₁₁₈	—	4	2	no	—
136	Dic₁₁₈	—	8	1	no	—
137	Dic₁₁₈	—	8	1	no	—
138	C₈×C₅₉	—	4	2	no	—
139	C₈×C₅₉	#102	1	2	yes	—
140	C₈×C₅₉	#102	1	2	yes	—
141	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
143	C₈×C₅₉	#124	1	1	yes	characteristic
144	C₈×C₅₉	#127	1	1	yes	characteristic
145	#13260	—	59	1	no	maximal, Sylow
146	C₂²×C₅₉	—	2	2	no	—
147	C₂²×C₅₉	#418	1	1	yes	characteristic
148	C₂²×C₅₉	#459	1	1	yes	characteristic
149	C₂²×C₅₉	#460	1	1	yes	characteristic
150	C₂×D₅₉	—	8	4	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₂×D₅₉	—	16	2	no	—
153	C₂×D₅₉	—	16	2	no	—
154	C₄×C₅₉	—	4	2	no	—
155	C₄×C₅₉	—	8	2	no	—
156	C₄×C₅₉	#390	1	1	yes	characteristic
157	C₄×C₅₉	C₂×#102	1	1	yes	characteristic
158	C₄×C₅₉	#459	1	1	yes	characteristic
159	C₄×C₅₉	#460	1	1	yes	characteristic
160	Dic₅₉	—	8	2	no	—
161	C₂×#495	—	59	1	no	—
162	C₂×#364	—	59	1	no	—
163	C₂×#362	—	59	1	no	—
164	C₂×#361	—	59	1	no	—
165	#2141	—	59	1	no	—
166	#2140	—	59	1	no	—
167	#1772	D₅₉	1	1	yes	characteristic, complemented
168	C₂×C₅₉	—	2	1	no	—
169	C₂×C₅₉	—	2	1	no	—
170	C₂×C₅₉	#1982	1	1	yes	characteristic
171	C₂×C₅₉	#2212	1	2	yes	—
172	D₅₉	—	16	2	no	—
173	C₂²×Q₁₆	—	59	1	no	—
174	C₂²×SD₁₆	—	118	1	no	—
175	C₂²×D₈	—	59	1	no	—
176	C₂×#124	—	118	1	no	—
177	C₂×M₃₂	—	59	1	no	—
178	C₂×M₃₂	—	59	1	no	—
179	#495	—	59	4	no	—
180	#495	—	118	2	no	—
181	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
182	#427	C₂×D₅₉	1	1	yes	characteristic
183	#405	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
185	#364	—	118	2	no	—
186	#362	—	118	2	no	—
187	#361	—	118	2	no	—
188	#358	—	118	2	no	—
189	#357	—	118	2	no	—
190	#339	—	2	1	no	—
191	C₅₉	#13260	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	118	1	no	—
193	C₂²×D₄	—	118	1	no	—
194	C₂×Q₁₆	—	59	2	no	—
195	C₂×Q₁₆	—	118	1	no	—
196	C₂×Q₁₆	—	118	1	no	—
197	C₂×Q₁₆	—	118	1	no	—
198	C₂×Q₁₆	—	118	1	no	—
199	C₂×Q₁₆	—	118	1	no	—
200	C₂×SD₁₆	—	118	4	no	—
201	C₂×SD₁₆	—	236	1	no	—
202	C₂×SD₁₆	—	236	1	no	—
203	C₂×SD₁₆	—	236	2	no	—
204	C₂×D₈	—	59	2	no	—
205	C₂×D₈	—	118	1	no	—
206	C₂×D₈	—	118	1	no	—
207	C₂×D₈	—	118	1	no	—
208	C₂×D₈	—	118	1	no	—
209	C₂×D₈	—	118	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
213	#124	—	118	4	no	—
214	#124	—	236	2	no	—
215	#117	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	236	2	no	—
217	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
218	M₃₂	—	118	1	no	—
219	M₃₂	—	118	1	no	—
220	M₃₂	—	118	1	no	—
221	M₃₂	—	118	1	no	—
222	C₂×C₁₆	—	118	1	no	—
223	C₂×C₁₆	—	118	1	no	—
224	#108	—	2	1	no	—
225	#108	D₂₃₆	1	2	yes	—
226	#107	D₂₃₆	1	2	yes	—
227	#106	—	2	1	no	—
228	#97	—	4	1	no	—
229	C₄×C₈	—	4	1	no	—
230	C₄×C₈	C₄×D₅₉	1	2	yes	—
231	C₄×C₈	D₂₃₆	1	2	yes	—
232	C₂⁴	—	236	1	no	—
233	C₂×Q₈	—	118	2	no	—
234	C₂×Q₈	—	236	1	no	—
235	C₂×Q₈	—	236	1	no	—
236	C₂×Q₈	—	236	1	no	—
237	C₂×Q₈	—	236	1	no	—
238	C₂×Q₈	—	236	1	no	—
239	C₂×D₄	—	118	2	no	—
240	C₂×D₄	—	236	1	no	—
241	C₂×D₄	—	236	1	no	—
242	C₂×D₄	—	236	1	no	—
243	C₂×D₄	—	236	1	no	—
244	C₂×D₄	—	236	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	236	1	no	—
247	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
248	Q₁₆	—	118	2	no	—
249	Q₁₆	—	118	2	no	—
250	Q₁₆	—	236	1	no	—
251	Q₁₆	—	236	1	no	—
252	SD₁₆	—	236	4	no	—
253	SD₁₆	—	236	4	no	—
254	D₈	—	118	2	no	—
255	D₈	—	118	2	no	—
256	D₈	—	236	1	no	—
257	D₈	—	236	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
262	C₂×C₈	#11745669	1	1	yes	characteristic
263	C₂×C₈	#11745669	1	1	yes	characteristic
264	C₂×C₈	#11745671	1	2	yes	—
265	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
266	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
267	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
268	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
269	C₂×C₈	#11745685	1	1	yes	characteristic
270	C₂×C₈	#11745685	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	236	8	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
277	C₁₆	—	236	1	no	—
278	C₁₆	—	236	1	no	—
279	C₂³	—	472	4	no	—
280	C₂³	—	472	1	no	—
281	C₂³	—	472	1	no	—
282	C₂³	—	236	2	no	—
283	C₂³	#14273752	1	1	yes	characteristic
284	Q₈	—	472	1	no	—
285	Q₈	—	472	1	no	—
286	Q₈	—	236	2	no	—
287	Q₈	—	236	2	no	—
288	D₄	—	472	1	no	—
289	D₄	—	472	1	no	—
290	D₄	—	236	2	no	—
291	D₄	—	236	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	472	1	no	—
296	C₂×C₄	—	472	1	no	—
297	C₂×C₄	—	236	2	no	—
298	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#14273701	1	1	yes	characteristic
300	C₂×C₄	#14273701	1	1	yes	characteristic
301	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
302	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
303	C₂×C₄	#14273752	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	#14273647	1	2	yes	—
306	C₈	#14273647	1	2	yes	—
307	C₈	C₄×D₂₃₆	1	1	yes	characteristic
308	C₈	C₄×D₂₃₆	1	1	yes	characteristic
309	C₈	#14273715	1	1	yes	characteristic
310	C₈	#14273724	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	944	2	no	—
313	C₂²	—	944	2	no	—
314	C₂²	—	472	4	no	—
315	C₂²	—	472	2	no	—
316	C₂²	#16837406	1	1	yes	characteristic
317	C₂²	#16837597	1	1	yes	characteristic
318	C₂²	#16837602	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	472	2	no	—
322	C₄	#16837304	1	1	yes	characteristic
323	C₄	C₂×#14273647	1	1	yes	characteristic
324	C₄	#16837597	1	1	yes	characteristic
325	C₄	#16837602	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	944	2	no	—
329	C₂	#21425023	1	1	yes	characteristic
330	C₂	#21426165	1	2	yes	—

	subgroup	G/H
center	C₂²	#16837597
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1772	C₂
socle	C₂²×C₅₉	#459
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1772	D₅₉
hypocenter	C₅₉	#13260
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
2. C₂×#16837223	C₂	semidirect
3. C₂×#16837217	C₂	semidirect
5. #21425815	C₂	semidirect
7. C₅₉×#1772	C₂	semidirect
15. #16837786	C₄	semidirect
167. #1772	D₅₉	semidirect
191. C₅₉	#13260	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837786	2
C₂×#16837223	2
C₂×#16837217	2
C₂×#16837214	2
#21425815	2
#21425811	2
C₅₉×#1772	2
#13260	59