Subgroups — #29599082

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840167	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839566	C₂	1	2	yes	maximal
3	#21443910	C₂	1	2	yes	maximal
4	#21443897	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1969	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	—	2	1	no	—
7	C₂²×#11758185	C₂²	1	2	yes	—
8	#16840167	—	2	4	no	—
9	C₂×#14519711	—	2	1	no	—
10	C₂×#14519711	C₂²	1	2	yes	—
11	#16839566	—	2	4	no	—
12	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₆₁	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#391	—	2	1	no	—
17	C₂²×Dic₁₂₂	—	4	1	no	—
18	#14519777	—	2	4	no	—
19	#14519777	—	4	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	C₂×C₄⋊Dic₆₁	—	4	1	no	—
22	C₂×#11758185	—	2	2	no	—
23	C₂×#11758185	—	2	2	no	—
24	C₂×#11758185	—	2	2	no	—
25	#14519711	—	2	4	no	—
26	#14519711	—	2	2	no	—
27	#14519711	D₄	1	4	yes	—
28	C₂²×Q₈×C₆₁	—	2	1	no	—
29	C₂×Q₁₆×C₆₁	—	2	2	no	—
30	C₂×Q₁₆×C₆₁	—	2	2	no	—
31	C₂×Q₁₆×C₆₁	—	2	1	no	—
32	C₂×Q₁₆×C₆₁	Q₈	1	2	yes	complemented
33	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	#107×C₆₁	—	2	2	no	—
38	#103×C₆₁	—	4	2	no	—
39	C₄×C₈×C₆₁	—	2	2	no	—
40	#95×C₆₁	—	4	1	no	—
41	#95×C₆₁	—	4	1	no	—
42	C₂×Q₈×C₆₁	—	2	2	no	—
43	C₂×Q₈×C₆₁	—	4	1	no	—
44	C₂×Q₈×C₆₁	—	4	1	no	—
45	C₂×Q₈×C₆₁	—	4	1	no	—
46	C₂×Q₈×C₆₁	—	4	1	no	—
47	C₂×Q₈×C₆₁	—	4	1	no	—
48	C₂²×C₄×C₆₁	—	4	1	no	—
49	C₂²×C₄×C₆₁	—	4	1	no	—
50	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₆₁	—	4	2	no	—
54	C₂×Dic₁₂₂	—	4	4	no	—
55	C₂×Dic₁₂₂	—	8	2	no	—
56	C₂×Dic₁₂₂	—	8	2	no	—
57	Q₁₆×C₆₁	—	2	4	no	—
58	Q₁₆×C₆₁	—	4	2	no	—
59	C₂×C₈×C₆₁	—	2	1	no	—
60	C₂×C₈×C₆₁	—	2	1	no	—
61	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₆₁	#41	1	2	yes	—
65	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₆₁	—	4	1	no	—
67	C₄⋊C₄×C₆₁	—	4	1	no	—
68	C₄²×C₆₁	—	2	1	no	—
69	C₄²×C₆₁	—	2	1	no	—
70	C₄⋊Dic₆₁	—	2	4	no	—
71	C₄⋊Dic₆₁	—	4	4	no	—
72	C₄⋊Dic₆₁	—	4	2	no	—
73	#11758185	—	4	2	no	—
74	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₆₁	—	4	2	no	—
76	Q₈×C₆₁	—	4	2	no	—
77	Q₈×C₆₁	—	8	1	no	—
78	Q₈×C₆₁	—	8	1	no	—
79	C₂×C₄×C₆₁	—	2	1	no	—
80	C₂×C₄×C₆₁	—	2	1	no	—
81	C₂×C₄×C₆₁	—	2	1	no	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	4	2	no	—
84	C₂×C₄×C₆₁	—	8	1	no	—
85	C₂×C₄×C₆₁	—	8	1	no	—
86	C₂×C₄×C₆₁	—	8	1	no	—
87	C₂×C₄×C₆₁	—	8	1	no	—
88	C₂×C₄×C₆₁	—	8	1	no	—
89	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
90	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
91	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
92	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
97	C₂×Dic₆₁	—	4	4	no	—
98	C₂×Dic₆₁	—	8	2	no	—
99	C₂×Dic₆₁	—	8	2	no	—
100	Dic₁₂₂	—	8	8	no	—
101	C₈×C₆₁	—	2	2	no	—
102	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
103	C₈×C₆₁	#124	1	2	yes	—
104	#13364	—	61	1	no	maximal, Sylow
105	C₂²×C₆₁	#447	1	1	yes	characteristic
106	C₂²×C₆₁	#474	1	1	yes	characteristic
107	C₂²×C₆₁	#486	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
109	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
110	C₂²×C₆₁	#509	1	1	yes	characteristic
111	C₂²×C₆₁	#510	1	1	yes	characteristic
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	—	2	2	no	—
114	C₄×C₆₁	—	8	2	no	—
115	C₄×C₆₁	—	16	1	no	—
116	C₄×C₆₁	#393	1	1	yes	characteristic
117	C₄×C₆₁	#448	1	1	yes	characteristic
118	C₄×C₆₁	#476	1	1	yes	characteristic
119	C₄×C₆₁	#489	1	1	yes	characteristic
120	Dic₆₁	—	8	4	no	—
121	C₂×#500	—	61	1	no	—
122	C₂×#358	—	61	2	no	—
123	#1969	D₆₁	1	1	yes	characteristic
124	#1380	—	61	2	no	—
125	#1374	—	61	1	no	—
126	C₂×C₆₁	#2023	1	1	yes	characteristic
127	C₂×C₆₁	#2187	1	1	yes	characteristic
128	C₂×C₆₁	#2188	1	1	yes	characteristic
129	C₂×C₆₁	#2225	1	1	yes	characteristic
130	C₂×C₆₁	#2227	1	1	yes	characteristic
131	C₂×C₆₁	#2241	1	1	yes	characteristic
132	C₂×C₆₁	#2243	1	1	yes	characteristic
133	C₂²×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
134	C₂×C₄⋊Q₈	—	122	1	no	—
135	C₂²×C₁₆	—	61	2	no	—
136	#500	—	122	4	no	—
137	C₂×#107	—	61	2	no	—
138	C₂×#107	—	122	1	no	—
139	C₂×#107	C₂×D₆₁	1	1	yes	characteristic
140	C₂×#103	—	2	1	no	—
141	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
142	#391	—	2	1	no	—
143	#358	—	122	4	no	—
144	C₆₁	#13364	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	—	244	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×Q₁₆	—	2	2	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	Dic₁₂₂	1	2	yes	complemented
151	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
153	C₄⋊Q₈	—	122	4	no	—
154	C₄⋊Q₈	—	244	2	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	122	2	no	—
157	C₂×C₄⋊C₄	—	244	1	no	—
158	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
159	C₂×C₁₆	—	122	2	no	—
160	C₂×C₁₆	—	122	2	no	—
161	C₂×C₁₆	—	122	2	no	—
162	#107	—	2	2	no	—
163	#107	—	61	4	no	—
164	#107	—	122	4	no	—
165	#107	—	122	2	no	—
166	#103	—	4	2	no	—
167	C₄×C₈	—	2	2	no	—
168	#95	—	4	1	no	—
169	#95	—	4	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂×Q₈	—	488	2	no	—
177	C₂×Q₈	—	488	2	no	—
178	C₂×Q₈	—	244	4	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	244	2	no	—
182	C₂²×C₄	#11758199	1	1	yes	characteristic
183	C₂²×C₄	#11758199	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
185	Q₁₆	—	2	4	no	—
186	Q₁₆	—	4	2	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	#11758200	1	2	yes	—
190	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
191	C₂×C₈	#11758222	1	1	yes	characteristic
192	C₂×C₈	D₄×D₆₁	1	2	yes	—
193	C₂×C₈	#11758224	1	2	yes	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	122	4	no	—
197	C₄⋊C₄	—	244	4	no	—
198	C₄⋊C₄	—	244	2	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₁₆	—	244	2	no	—
202	C₂³	C₂×#11758199	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	1	no	—
206	Q₈	—	8	1	no	—
207	Q₈	—	488	8	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	1	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	488	2	no	—
219	C₂×C₄	—	488	2	no	—
220	C₂×C₄	—	244	4	no	—
221	C₂×C₄	#14519720	1	2	yes	—
222	C₂×C₄	#14519723	1	2	yes	—
223	C₂×C₄	#14519745	1	1	yes	characteristic
224	C₂×C₄	#14519756	1	1	yes	characteristic
225	C₂×C₄	#14519764	1	1	yes	characteristic
226	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
227	C₂×C₄	#14519802	1	1	yes	characteristic
228	C₂×C₄	#14519807	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	#14519754	1	2	yes	—
231	C₈	#14519763	1	2	yes	—
232	C₂²	#16839895	1	1	yes	characteristic
233	C₂²	#16840042	1	1	yes	characteristic
234	C₂²	#16840108	1	1	yes	characteristic
235	C₂²	C₂×#14519720	1	1	yes	characteristic
236	C₂²	C₂×#14519723	1	1	yes	characteristic
237	C₂²	#16840197	1	1	yes	characteristic
238	C₂²	#16840202	1	1	yes	characteristic
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	8	2	no	—
242	C₄	—	16	1	no	—
243	C₄	—	488	4	no	—
244	C₄	#16839674	1	1	yes	characteristic
245	C₄	#16839900	1	1	yes	characteristic
246	C₄	#16840052	1	1	yes	characteristic
247	C₄	#16840121	1	1	yes	characteristic
248	C₂	#21446955	1	1	yes	characteristic
249	C₂	#21447747	1	1	yes	characteristic
250	C₂	#21447752	1	1	yes	characteristic
251	C₂	#21447961	1	1	yes	characteristic
252	C₂	#21447971	1	1	yes	characteristic
253	C₂	#21448046	1	1	yes	characteristic
254	C₂	#21448055	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1969	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#1969	D₆₁
hypocenter	C₆₁	#13364
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1969

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₆₁	Q₈	semidirect
144. C₆₁	#13364	semidirect
150. C₂×Q₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
C₂×#16839566	2
#21443910	2
#21443897	2
C₆₁×#1969	2
#13364	61