Subgroups — #28608441

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic
2	#21427051	C₂	1	2	yes	maximal
3	#21424957	C₂	1	2	yes	maximal
4	#21424738	C₂	1	4	yes	maximal
5	#21424256	C₂	1	1	yes	maximal, characteristic
6	#21424102	C₂	1	2	yes	maximal
7	C₄×#14273661	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₁₈	C₂²	1	2	yes	—
11	C₄×Dic₂₃₆	C₄	1	8	yes	complemented
12	C₂×#14273661	C₂²	1	2	yes	—
13	C₂×#14273661	C₂²	1	2	yes	—
14	C₂×#14273661	C₂²	1	1	yes	characteristic
15	C₂×#14273652	C₂²	1	4	yes	—
16	C₂×#14273652	C₂²	1	2	yes	—
17	#16837308	C₂²	1	4	yes	—
18	#16837289	—	2	4	no	—
19	#16837272	C₂²	1	2	yes	—
20	#16837267	—	2	2	no	—
21	C₄⋊C₄×Dic₅₉	—	2	2	no	—
22	C₄×C₄⋊Dic₅₉	C₂²	1	4	yes	—
23	#16837165	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#336	C₂²	1	2	yes	—
29	C₂²×Dic₁₁₈	C₂³	1	2	yes	—
30	C₂×Dic₂₃₆	—	2	2	no	—
31	C₂×Dic₂₃₆	—	2	2	no	—
32	C₂×Dic₂₃₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₁₈	—	2	4	no	—
34	C₄×Dic₁₁₈	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊Dic₅₉	—	2	2	no	—
36	C₂×C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₄⋊Dic₅₉	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₅₉	C₂³	1	2	yes	—
39	C₂×Dic₅₉⋊C₄	—	2	2	no	—
40	C₂×C₄×Dic₅₉	—	2	2	no	—
41	C₂×C₄×Dic₅₉	—	2	4	no	—
42	#14273661	—	2	4	no	—
43	#14273661	—	4	2	no	—
44	#14273661	C₂×C₄	1	4	yes	—
45	#14273652	—	4	4	no	—
46	#14273652	C₂×C₄	1	8	yes	—
47	C₅₉⋊#95	—	2	4	no	—
48	C₅₉⋊#95	—	2	4	no	—
49	C₅₉⋊#95	—	2	4	no	—
50	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₉	—	2	2	no	—
58	C₄×C₈×C₅₉	—	2	4	no	—
59	C₄×C₈×C₅₉	C₂×C₄	1	4	yes	—
60	#95×C₅₉	D₄	1	2	yes	—
61	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₅₉	—	2	2	no	—
67	C₂²×Dic₅₉	—	2	2	no	—
68	C₂²×Dic₅₉	—	2	4	no	—
69	C₂×Dic₁₁₈	—	2	2	no	—
70	C₂×Dic₁₁₈	—	2	2	no	—
71	C₂×Dic₁₁₈	—	2	2	no	—
72	C₂×Dic₁₁₈	—	2	2	no	—
73	C₂×Dic₁₁₈	—	2	2	no	—
74	C₂×Dic₁₁₈	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	1	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₉	#41	1	2	yes	—
83	C₂×C₈×C₅₉	#41	1	2	yes	—
84	C₄⋊C₄×C₅₉	—	2	2	no	—
85	C₄⋊C₄×C₅₉	Q₁₆	1	4	yes	complemented
86	C₄²×C₅₉	—	2	2	no	—
87	C₄²×C₅₉	—	2	2	no	—
88	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₉	#41	1	2	yes	—
90	C₄⋊Dic₅₉	—	2	2	no	—
91	C₄⋊Dic₅₉	—	2	4	no	—
92	C₄⋊Dic₅₉	—	2	4	no	—
93	C₄⋊Dic₅₉	—	4	4	no	—
94	C₄⋊Dic₅₉	—	8	2	no	—
95	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
96	Dic₅₉⋊C₄	—	2	8	no	—
97	C₄×Dic₅₉	—	2	8	no	—
98	C₄×Dic₅₉	—	8	4	no	—
99	Dic₂₃₆	—	2	8	no	—
100	C₂³×C₅₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	—	2	2	no	—
102	C₂×C₄×C₅₉	—	2	2	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	2	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	2	2	no	—
108	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
111	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₅₉	#135	1	2	yes	—
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×Dic₅₉	—	2	4	no	—
120	C₂×Dic₅₉	—	2	4	no	—
121	C₂×Dic₅₉	—	4	2	no	—
122	C₂×Dic₅₉	—	4	2	no	—
123	C₂×Dic₅₉	—	4	2	no	—
124	C₂×Dic₅₉	—	4	2	no	—
125	C₂×Dic₅₉	—	4	4	no	—
126	C₂×Dic₅₉	—	4	4	no	—
127	C₂×Dic₅₉	—	4	4	no	—
128	Dic₁₁₈	—	2	4	no	—
129	Dic₁₁₈	—	2	4	no	—
130	Dic₁₁₈	—	4	2	no	—
131	Dic₁₁₈	—	4	2	no	—
132	C₈×C₅₉	—	2	2	no	—
133	C₈×C₅₉	—	4	2	no	—
134	C₈×C₅₉	#117	1	4	yes	—
135	#22808	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₅₉	#547	1	1	yes	characteristic
138	C₂²×C₅₉	#550	1	1	yes	characteristic
139	C₂²×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₅₉	C₂×#135	1	1	yes	characteristic
141	C₂²×C₅₉	#576	1	1	yes	characteristic
142	C₂²×C₅₉	#578	1	1	yes	characteristic
143	C₄×C₅₉	—	2	4	no	—
144	C₄×C₅₉	—	2	2	no	—
145	C₄×C₅₉	—	4	2	no	—
146	C₄×C₅₉	C₄×Q₁₆	1	4	yes	—
147	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
148	C₄×C₅₉	#520	1	1	yes	characteristic
149	C₄×C₅₉	#555	1	1	yes	characteristic
150	C₄×C₅₉	#556	1	1	yes	characteristic
151	Dic₅₉	—	4	4	no	—
152	Dic₅₉	—	4	4	no	—
153	Dic₅₉	—	8	4	no	—
154	C₂×C₄×Q₁₆	—	59	1	no	—
155	#2355	—	59	2	no	—
156	#1969	—	59	2	no	—
157	#1928	—	59	4	no	—
158	#1845	—	59	1	no	—
159	#1803	—	59	2	no	—
160	C₄×#107	—	59	2	no	—
161	C₈×C₄⋊C₄	D₅₉	1	1	yes	characteristic
162	C₂×C₅₉	#2354	1	1	yes	characteristic
163	C₂×C₅₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
164	C₂×C₅₉	#2951	1	1	yes	characteristic
165	C₂×C₅₉	#3338	1	1	yes	characteristic
166	C₂×C₅₉	#3342	1	1	yes	characteristic
167	C₂×C₅₉	#3395	1	1	yes	characteristic
168	C₂×C₅₉	#3396	1	1	yes	characteristic
169	C₂²×Q₁₆	—	59	1	no	—
170	C₂×C₄×Q₈	—	59	2	no	—
171	C₄×Q₁₆	—	59	8	no	—
172	C₂×#107	—	59	1	no	—
173	C₂×#107	—	59	2	no	—
174	C₂×#107	—	59	2	no	—
175	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
176	C₂×#103	—	59	2	no	—
177	C₂×#103	—	59	4	no	—
178	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
180	#391	—	59	4	no	—
181	#387	—	118	4	no	—
182	#384	—	59	2	no	—
183	#382	—	118	2	no	—
184	C₄×C₄⋊C₄	—	59	4	no	—
185	C₄×C₄⋊C₄	—	118	2	no	—
186	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
187	#340	—	59	4	no	—
188	#336	C₂×D₅₉	1	2	yes	—
189	C₅₉	#22808	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	59	2	no	—
191	C₂×Q₁₆	—	59	4	no	—
192	C₂×Q₁₆	—	118	2	no	—
193	C₂×Q₁₆	—	118	2	no	—
194	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
195	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
197	C₄×Q₈	—	59	8	no	—
198	C₄×Q₈	—	118	4	no	—
199	C₂×C₄⋊C₄	—	59	2	no	—
200	C₂×C₄⋊C₄	—	59	4	no	—
201	C₂×C₄⋊C₄	—	118	2	no	—
202	C₂×C₄⋊C₄	—	118	2	no	—
203	C₂×C₄⋊C₄	—	118	2	no	—
204	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
205	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
206	C₂×C₄²	—	118	2	no	—
207	C₂×C₄²	—	118	4	no	—
208	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
209	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
210	#107	—	59	4	no	—
211	#107	—	118	4	no	—
212	#107	—	236	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#103	—	59	8	no	—
215	#103	—	236	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₄×D₅₉	1	4	yes	—
218	#95	—	118	4	no	—
219	#95	—	118	4	no	—
220	#95	—	118	4	no	—
221	#95	D₂₃₆	1	2	yes	—
222	C₂×Q₈	—	59	4	no	—
223	C₂×Q₈	—	118	2	no	—
224	C₂×Q₈	—	118	2	no	—
225	C₂×Q₈	—	118	2	no	—
226	C₂×Q₈	—	118	2	no	—
227	C₂×Q₈	—	118	2	no	—
228	C₂²×C₄	—	118	2	no	—
229	C₂²×C₄	—	118	2	no	—
230	C₂²×C₄	—	118	4	no	—
231	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
232	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
234	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
236	Q₁₆	—	118	8	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
243	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
244	C₂×C₈	#11745687	1	2	yes	—
245	C₂×C₈	#11745689	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	59	4	no	—
248	C₄⋊C₄	—	118	8	no	—
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	118	4	no	—
251	C₄⋊C₄	—	118	4	no	—
252	C₄⋊C₄	—	472	2	no	—
253	C₄⋊C₄	—	236	4	no	—
254	C₄⋊C₄	Dic₂₃₆	1	4	yes	complemented
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	118	8	no	—
258	C₄²	—	472	4	no	—
259	C₄²	C₂×C₄×D₅₉	1	2	yes	—
260	C₄²	#11745685	1	2	yes	—
261	C₂³	C₂²×D₂₃₆	1	1	yes	characteristic
262	Q₈	—	118	4	no	—
263	Q₈	—	118	4	no	—
264	Q₈	—	236	2	no	—
265	Q₈	—	236	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	118	4	no	—
274	C₂×C₄	—	118	4	no	—
275	C₂×C₄	—	236	4	no	—
276	C₂×C₄	—	236	4	no	—
277	C₂×C₄	—	236	4	no	—
278	C₂×C₄	—	236	2	no	—
279	C₂×C₄	—	236	2	no	—
280	C₂×C₄	—	236	2	no	—
281	C₂×C₄	—	236	2	no	—
282	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
283	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
284	C₂×C₄	C₂×Dic₂₃₆	1	4	yes	—
285	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
286	C₂×C₄	#14273749	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
289	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
290	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
291	C₂×C₄	#14273773	1	1	yes	characteristic
292	C₂×C₄	#14273774	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	#14273685	1	4	yes	—
296	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
297	C₂²	#16838056	1	1	yes	characteristic
298	C₂²	#16838074	1	1	yes	characteristic
299	C₂²	C₂²×Dic₂₃₆	1	1	yes	characteristic
300	C₂²	C₂×#14273749	1	1	yes	characteristic
301	C₂²	#16838207	1	1	yes	characteristic
302	C₂²	#16838219	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	472	4	no	—
307	C₄	—	236	4	no	—
308	C₄	—	236	4	no	—
309	C₄	C₄×Dic₂₃₆	1	4	yes	—
310	C₄	C₂×#14273685	1	1	yes	characteristic
311	C₄	#16837882	1	1	yes	characteristic
312	C₄	#16838117	1	1	yes	characteristic
313	C₄	#16838125	1	1	yes	characteristic
314	C₂	#21427043	1	1	yes	characteristic
315	C₂	C₂×C₄×Dic₂₃₆	1	1	yes	characteristic
316	C₂	#21432973	1	1	yes	characteristic
317	C₂	#21436489	1	1	yes	characteristic
318	C₂	#21436527	1	1	yes	characteristic
319	C₂	#21437102	1	1	yes	characteristic
320	C₂	#21437109	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₈×C₄⋊C₄×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂²×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₉
hypocenter	C₅₉	#22808
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₄×Dic₂₃₆	C₄	semidirect
85. C₄⋊C₄×C₅₉	Q₁₆	semidirect
189. C₅₉	#22808	semidirect
254. C₄⋊C₄	Dic₂₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₃₆	2
#21427051	2
#21424957	2
#21424738	2
#21424256	2
#21424102	2
C₄×#14273661	2
C₈×C₄⋊C₄×C₅₉	2
#22808	59