Subgroups — #29599631

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#16839590	C₂	1	2	yes	maximal
3	#21446552	C₂	1	1	yes	maximal, characteristic
4	#21443911	C₂	1	2	yes	maximal
5	C₆₁×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	—	2	1	no	—
7	#16840167	—	2	4	no	—
8	C₂×#14519711	—	2	1	no	—
9	C₂×#14519711	C₂²	1	1	yes	characteristic
10	C₂×#14519711	C₂²	1	2	yes	—
11	C₂×#14519711	C₂²	1	1	yes	characteristic
12	#16839660	—	2	1	no	—
13	#16839590	—	2	4	no	—
14	#16839525	—	2	1	no	—
15	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₂₂	—	4	1	no	—
18	#14519777	—	2	4	no	—
19	#14519777	—	4	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	1	no	—
21	C₂×C₄⋊Dic₆₁	—	2	2	no	—
22	C₂×C₄⋊Dic₆₁	—	2	1	no	—
23	C₂×C₄⋊Dic₆₁	—	4	1	no	—
24	C₂×#11758196	—	4	1	no	—
25	#14519711	—	2	4	no	—
26	#14519711	—	2	2	no	—
27	#14519711	—	2	1	no	—
28	#14519711	—	2	2	no	—
29	#14519711	D₄	1	4	yes	—
30	#14519711	Q₈	1	2	yes	complemented
31	#14519684	—	4	1	no	—
32	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₆₁	—	2	2	no	—
36	C₂×C₁₆×C₆₁	—	2	2	no	—
37	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
38	C₄×C₈×C₆₁	—	2	2	no	—
39	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	4	2	no	—
43	C₂²×Dic₆₁	—	4	1	no	—
44	C₂²×Dic₆₁	—	4	1	no	—
45	C₂×Dic₁₂₂	—	4	4	no	—
46	C₂×Dic₁₂₂	—	8	2	no	—
47	C₂×Dic₁₂₂	—	8	2	no	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
55	C₄²×C₆₁	—	2	1	no	—
56	C₄²×C₆₁	—	2	1	no	—
57	C₄⋊Dic₆₁	—	2	4	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	4	4	no	—
60	C₄⋊Dic₆₁	—	4	2	no	—
61	C₄⋊Dic₆₁	—	4	1	no	—
62	C₄⋊Dic₆₁	—	4	1	no	—
63	C₄⋊Dic₆₁	—	4	1	no	—
64	#11758196	—	8	2	no	—
65	C₁₆×C₆₁	—	2	4	no	—
66	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	—	2	1	no	—
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
72	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
73	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
79	C₂×Dic₆₁	—	4	4	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	—	8	1	no	—
82	C₂×Dic₆₁	—	8	2	no	—
83	C₂×Dic₆₁	—	8	2	no	—
84	C₂×Dic₆₁	—	8	1	no	—
85	C₂×Dic₆₁	—	8	1	no	—
86	C₂×Dic₆₁	—	8	1	no	—
87	C₂×Dic₆₁	—	8	1	no	—
88	Dic₁₂₂	—	8	8	no	—
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
92	#13452	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	#447	1	1	yes	characteristic
94	C₂²×C₆₁	#474	1	1	yes	characteristic
95	C₂²×C₆₁	#500	1	1	yes	characteristic
96	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
98	C₂²×C₆₁	#509	1	1	yes	characteristic
99	C₂²×C₆₁	#510	1	1	yes	characteristic
100	C₄×C₆₁	—	2	2	no	—
101	C₄×C₆₁	—	2	2	no	—
102	C₄×C₆₁	#395	1	1	yes	characteristic
103	C₄×C₆₁	#451	1	1	yes	characteristic
104	C₄×C₆₁	#477	1	1	yes	characteristic
105	C₄×C₆₁	#501	1	1	yes	characteristic
106	Dic₆₁	—	8	4	no	—
107	Dic₆₁	—	8	2	no	—
108	Dic₆₁	—	16	1	no	—
109	C₂×#500	—	61	1	no	—
110	C₂×#367	—	61	2	no	—
111	#1942	—	61	1	no	—
112	#1380	—	61	2	no	—
113	#1374	D₆₁	1	1	yes	characteristic
114	C₂×C₆₁	#2057	1	1	yes	characteristic
115	C₂×C₆₁	#2185	1	1	yes	characteristic
116	C₂×C₆₁	#2187	1	1	yes	characteristic
117	C₂×C₆₁	#2225	1	1	yes	characteristic
118	C₂×C₆₁	#2226	1	1	yes	characteristic
119	C₂×C₆₁	#2254	1	1	yes	characteristic
120	C₂×C₆₁	#2255	1	1	yes	characteristic
121	C₂×C₄⋊Q₈	—	122	1	no	—
122	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
123	#500	—	122	4	no	—
124	C₂×#107	—	61	2	no	—
125	C₂×#107	—	61	1	no	—
126	C₂×#107	—	61	1	no	—
127	C₂×#107	—	122	1	no	—
128	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
129	#389	—	122	1	no	—
130	#367	—	122	4	no	—
131	#340	—	122	1	no	—
132	C₆₁	#13452	1	1	yes	characteristic, complemented, Sylow
133	C₂²×Q₈	—	244	1	no	—
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	122	4	no	—
137	C₄⋊Q₈	—	244	2	no	—
138	C₂×C₄⋊C₄	—	122	2	no	—
139	C₂×C₄⋊C₄	—	122	1	no	—
140	C₂×C₄⋊C₄	—	122	1	no	—
141	C₂×C₄⋊C₄	—	244	1	no	—
142	C₂×C₄⋊C₄	—	244	1	no	—
143	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
147	#107	—	61	4	no	—
148	#107	—	61	2	no	—
149	#107	—	122	4	no	—
150	#107	—	122	2	no	—
151	#107	—	122	1	no	—
152	#107	—	122	2	no	—
153	C₄×C₈	—	2	2	no	—
154	#95	—	244	1	no	—
155	C₂×Q₈	—	488	2	no	—
156	C₂×Q₈	—	488	2	no	—
157	C₂×Q₈	—	244	4	no	—
158	C₂²×C₄	—	244	1	no	—
159	C₂²×C₄	—	244	1	no	—
160	C₂²×C₄	—	244	2	no	—
161	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
162	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	Dic₂₄₄	1	2	yes	—
167	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
168	C₂×C₈	D₄×D₆₁	1	2	yes	—
169	C₂×C₈	#11758224	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	122	4	no	—
172	C₄⋊C₄	—	122	2	no	—
173	C₄⋊C₄	—	488	2	no	—
174	C₄⋊C₄	—	244	4	no	—
175	C₄⋊C₄	—	244	1	no	—
176	C₄⋊C₄	—	244	1	no	—
177	C₄⋊C₄	—	244	1	no	—
178	C₄⋊C₄	—	244	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₁₆	—	2	4	no	—
182	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
183	Q₈	—	488	8	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	488	1	no	—
189	C₂×C₄	—	488	1	no	—
190	C₂×C₄	—	488	1	no	—
191	C₂×C₄	—	488	1	no	—
192	C₂×C₄	—	488	2	no	—
193	C₂×C₄	—	488	1	no	—
194	C₂×C₄	—	488	2	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	—	244	4	no	—
197	C₂×C₄	#14519722	1	2	yes	—
198	C₂×C₄	#14519722	1	2	yes	—
199	C₂×C₄	#14519747	1	1	yes	characteristic
200	C₂×C₄	#14519757	1	1	yes	characteristic
201	C₂×C₄	#14519777	1	1	yes	characteristic
202	C₂×C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
203	C₂×C₄	#14519805	1	1	yes	characteristic
204	C₂×C₄	#14519810	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	#14519754	1	2	yes	—
207	C₈	#14519776	1	2	yes	—
208	C₂²	#16839898	1	1	yes	characteristic
209	C₂²	#16840043	1	1	yes	characteristic
210	C₂²	#16840167	1	1	yes	characteristic
211	C₂²	C₂×#14519722	1	1	yes	characteristic
212	C₂²	C₂×#14519722	1	1	yes	characteristic
213	C₂²	#16840200	1	1	yes	characteristic
214	C₂²	#16840205	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	—	976	1	no	—
218	C₄	—	488	2	no	—
219	C₄	—	488	4	no	—
220	C₄	#16839683	1	1	yes	characteristic
221	C₄	#16839920	1	1	yes	characteristic
222	C₄	#16840058	1	1	yes	characteristic
223	C₄	#16840171	1	1	yes	characteristic
224	C₂	#21447148	1	1	yes	characteristic
225	C₂	#21447738	1	1	yes	characteristic
226	C₂	#21447750	1	1	yes	characteristic
227	C₂	#21447962	1	1	yes	characteristic
228	C₂	#21447967	1	1	yes	characteristic
229	C₂	#21448101	1	1	yes	characteristic
230	C₂	#21448105	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₈₈
commutator, cosocle	C₂²×C₈×C₆₁	C₂³
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1374	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	#1374	D₆₁
hypocenter	C₆₁	#13452
LCS	G ▷ C₂²×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
30. #14519711	Q₈	semidirect
37. C₂×C₁₆×C₆₁	Q₈	semidirect
132. C₆₁	#13452	semidirect
146. C₂×C₁₆	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×#16839590	2
#21446552	2
#21443911	2
C₆₁×#1374	2
#13452	61