Subgroups — #29596070

Order 15616.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839567	C₂	1	2	yes	maximal
3	#21445983	C₂	1	1	yes	maximal, characteristic
4	#21443911	C₂	1	2	yes	maximal
5	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₂	—	2	1	no	—
8	C₄×Dic₂₄₄	—	2	2	no	—
9	C₄×Dic₂₄₄	C₄	1	4	yes	complemented
10	C₂×#14519711	C₂²	1	2	yes	—
11	C₂×#14519711	C₂²	1	1	yes	characteristic
12	C₂×#14519709	—	2	1	no	—
13	C₂×#14519702	—	2	1	no	—
14	#16839628	—	2	1	no	—
15	#16839567	—	2	4	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×Dic₁₂₂	—	2	1	no	—
20	C₂×Dic₂₄₄	—	2	1	no	—
21	C₂×Dic₂₄₄	—	2	1	no	—
22	C₂×Dic₂₄₄	—	2	1	no	—
23	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
24	C₂×Dic₂₄₄	D₄	1	2	yes	—
25	C₂×Dic₂₄₄	Q₈	1	2	yes	—
26	C₄×Dic₁₂₂	—	2	4	no	—
27	C₄×Dic₁₂₂	—	4	2	no	—
28	C₂×C₄⋊Dic₆₁	—	2	1	no	—
29	C₂×C₄⋊Dic₆₁	—	2	2	no	—
30	C₂×#11758196	—	4	1	no	—
31	C₂×C₄×Dic₆₁	—	4	1	no	—
32	#14519711	—	2	1	no	—
33	#14519711	—	2	2	no	—
34	#14519711	—	2	2	no	—
35	#14519711	C₂×C₄	1	2	yes	—
36	#14519709	—	4	2	no	—
37	#14519702	—	2	4	no	—
38	#14519684	—	4	1	no	—
39	#14519684	—	4	1	no	—
40	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₆₁	—	2	2	no	—
44	C₂×C₁₆×C₆₁	—	2	2	no	—
45	C₂×C₁₆×C₆₁	—	2	2	no	—
46	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₆₁	D₄	1	2	yes	—
48	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₆₁	—	4	1	no	—
52	C₂²×Dic₆₁	—	4	1	no	—
53	C₂²×Dic₆₁	—	4	2	no	—
54	C₂×Dic₁₂₂	—	2	2	no	—
55	C₂×Dic₁₂₂	—	2	2	no	—
56	C₂×Dic₁₂₂	—	2	2	no	—
57	C₂×Dic₁₂₂	—	4	1	no	—
58	C₂×Dic₁₂₂	—	4	1	no	—
59	C₂×Dic₁₂₂	—	4	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	—	2	1	no	—
62	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
68	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
69	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₆₁	SD₁₆	1	2	yes	—
71	C₄⋊Dic₆₁	—	2	2	no	—
72	C₄⋊Dic₆₁	—	4	1	no	—
73	C₄⋊Dic₆₁	—	4	2	no	—
74	C₄⋊Dic₆₁	—	4	2	no	—
75	#11758196	—	4	4	no	—
76	C₄×Dic₆₁	—	4	4	no	—
77	Dic₂₄₄	—	2	2	no	—
78	Dic₂₄₄	—	2	2	no	—
79	Dic₂₄₄	—	2	2	no	—
80	Dic₂₄₄	C₄⋊C₄	1	4	yes	—
81	C₁₆×C₆₁	—	4	2	no	—
82	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	#102	1	2	yes	—
86	C₂×C₄×C₆₁	#102	1	2	yes	—
87	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
93	C₂×Dic₆₁	—	4	2	no	—
94	C₂×Dic₆₁	—	4	2	no	—
95	C₂×Dic₆₁	—	4	2	no	—
96	C₂×Dic₆₁	—	4	2	no	—
97	C₂×Dic₆₁	—	8	2	no	—
98	C₂×Dic₆₁	—	8	2	no	—
99	C₂×Dic₆₁	—	8	2	no	—
100	C₂×Dic₆₁	—	8	2	no	—
101	Dic₁₂₂	—	2	4	no	—
102	Dic₁₂₂	—	4	2	no	—
103	Dic₁₂₂	—	4	2	no	—
104	Dic₁₂₂	—	4	2	no	—
105	C₈×C₆₁	—	2	2	no	—
106	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₈×C₆₁	#117	1	1	yes	characteristic
108	C₈×C₆₁	#121	1	1	yes	characteristic
109	C₈×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
110	#12814	—	61	1	no	maximal, Sylow
111	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
112	C₂²×C₆₁	#459	1	1	yes	characteristic
113	C₂²×C₆₁	#474	1	1	yes	characteristic
114	C₂²×C₆₁	#508	1	2	yes	—
115	C₂²×C₆₁	#510	1	2	yes	—
116	C₄×C₆₁	—	2	2	no	—
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	#380	1	1	yes	characteristic
119	C₄×C₆₁	#417	1	1	yes	characteristic
120	C₄×C₆₁	#461	1	1	yes	characteristic
121	C₄×C₆₁	#476	1	1	yes	characteristic
122	Dic₆₁	—	4	4	no	—
123	Dic₆₁	—	8	2	no	—
124	Dic₆₁	—	16	2	no	—
125	C₂×C₄×Q₁₆	—	61	1	no	—
126	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
127	C₂×#358	—	61	2	no	—
128	#1846	—	61	1	no	—
129	#1380	—	61	2	no	—
130	C₂×C₆₁	#1802	1	1	yes	characteristic
131	C₂×C₆₁	#2138	1	1	yes	characteristic
132	C₂×C₆₁	#2141	1	1	yes	characteristic
133	C₂×C₆₁	#2211	1	2	yes	—
134	C₂×C₆₁	#2229	1	2	yes	—
135	C₂²×Q₁₆	—	61	1	no	—
136	C₂×C₄×Q₈	—	122	1	no	—
137	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
138	C₄×Q₁₆	—	61	4	no	—
139	C₄×Q₁₆	—	122	2	no	—
140	C₂×#107	—	61	1	no	—
141	C₂×#107	—	61	2	no	—
142	C₂×#106	—	122	1	no	—
143	C₂×#103	—	122	1	no	—
144	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
145	#382	—	122	1	no	—
146	C₄⋊C₁₆	—	2	2	no	—
147	#358	—	122	4	no	—
148	C₆₁	#12814	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	122	1	no	—
150	C₂×Q₁₆	—	61	2	no	—
151	C₂×Q₁₆	—	61	2	no	—
152	C₂×Q₁₆	—	61	2	no	—
153	C₂×Q₁₆	—	122	1	no	—
154	C₂×Q₁₆	—	122	1	no	—
155	C₂×Q₁₆	—	122	1	no	—
156	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
157	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
158	C₄×Q₈	—	122	4	no	—
159	C₄×Q₈	—	244	2	no	—
160	C₂×C₄⋊C₄	—	122	1	no	—
161	C₂×C₄⋊C₄	—	122	2	no	—
162	C₂×C₄⋊C₄	—	244	1	no	—
163	C₂×C₄²	—	244	1	no	—
164	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	#107	—	61	2	no	—
169	#107	—	122	1	no	—
170	#107	—	122	2	no	—
171	#107	—	122	2	no	—
172	#106	—	244	2	no	—
173	#103	—	122	4	no	—
174	C₄×C₈	C₄×D₆₁	1	2	yes	—
175	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
176	#95	—	244	1	no	—
177	#95	—	244	1	no	—
178	C₂×Q₈	—	122	2	no	—
179	C₂×Q₈	—	122	2	no	—
180	C₂×Q₈	—	122	2	no	—
181	C₂×Q₈	—	244	1	no	—
182	C₂×Q₈	—	244	1	no	—
183	C₂×Q₈	—	244	1	no	—
184	C₂²×C₄	—	244	2	no	—
185	C₂²×C₄	—	244	1	no	—
186	C₂²×C₄	—	244	1	no	—
187	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
188	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
190	Q₁₆	—	61	4	no	—
191	Q₁₆	—	122	2	no	—
192	Q₁₆	—	122	2	no	—
193	Q₁₆	—	122	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	#11758198	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
198	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
199	C₂×C₈	#11758224	1	1	yes	characteristic
200	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
201	C₂×C₈	#11758226	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
203	C₄⋊C₄	—	122	2	no	—
204	C₄⋊C₄	—	244	2	no	—
205	C₄⋊C₄	—	244	2	no	—
206	C₄⋊C₄	—	244	4	no	—
207	C₄⋊C₄	—	244	1	no	—
208	C₄²	—	244	4	no	—
209	C₄²	#11758190	1	2	yes	—
210	C₄²	#11758198	1	2	yes	—
211	C₁₆	—	4	2	no	—
212	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
213	Q₈	—	122	4	no	—
214	Q₈	—	244	2	no	—
215	Q₈	—	244	2	no	—
216	Q₈	—	244	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	488	2	no	—
220	C₂×C₄	—	488	2	no	—
221	C₂×C₄	—	488	2	no	—
222	C₂×C₄	—	488	2	no	—
223	C₂×C₄	—	244	2	no	—
224	C₂×C₄	—	244	2	no	—
225	C₂×C₄	—	244	2	no	—
226	C₂×C₄	—	244	2	no	—
227	C₂×C₄	#14519699	1	2	yes	—
228	C₂×C₄	#14519699	1	2	yes	—
229	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
230	C₂×C₄	#14519752	1	1	yes	characteristic
231	C₂×C₄	#14519757	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
233	C₂×C₄	#14519805	1	1	yes	characteristic
234	C₂×C₄	#14519810	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
237	C₈	#14519735	1	1	yes	characteristic
238	C₈	#14519748	1	1	yes	characteristic
239	C₈	#14519753	1	1	yes	characteristic
240	C₂²	C₂×#14519699	1	1	yes	characteristic
241	C₂²	#16839958	1	1	yes	characteristic
242	C₂²	#16840043	1	1	yes	characteristic
243	C₂²	#16840194	1	2	yes	—
244	C₂²	#16840205	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	976	2	no	—
248	C₄	—	488	2	no	—
249	C₄	—	244	4	no	—
250	C₄	#16839619	1	1	yes	characteristic
251	C₄	#16839764	1	1	yes	characteristic
252	C₄	#16839968	1	1	yes	characteristic
253	C₄	#16840053	1	1	yes	characteristic
254	C₂	#21445820	1	1	yes	characteristic
255	C₂	#21447527	1	1	yes	characteristic
256	C₂	#21447540	1	1	yes	characteristic
257	C₂	#21447882	1	2	yes	—
258	C₂	#21447984	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#12814
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₄₄	C₄	semidirect
148. C₆₁	#12814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
C₂×#16839567	2
#21445983	2
#21443911	2
C₂×C₆₁×C₄⋊C₁₆	2
#12814	61