Subgroups — #29578563

Order 15616.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446535	C₂	1	2	yes	maximal
2	C₄×#14519709	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
4	#21443658	C₂	1	2	yes	maximal
5	C₂×C₆₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519709	C₄	1	1	yes	characteristic
7	C₂×#14519709	C₄	1	1	yes	characteristic
8	C₂×#14519706	—	2	2	no	—
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	#16839525	C₄	1	2	yes	—
13	#16839516	—	2	2	no	—
14	#16839483	—	2	2	no	—
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#335	—	2	2	no	—
18	C₂²×#92418	—	2	1	no	—
19	C₂²×#92418	D₄	1	1	yes	characteristic
20	C₂²×#92418	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₆₁	—	2	2	no	—
24	#14519709	—	2	2	no	—
25	#14519709	C₈	1	4	yes	complemented
26	#14519706	—	4	4	no	—
27	C₄×#92418	—	2	1	no	—
28	C₄×#92418	—	2	1	no	—
29	C₄×#92418	—	2	1	no	—
30	C₄×#92418	Q₈	1	2	yes	—
31	#14519684	—	2	2	no	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₆₁	—	2	2	no	—
37	C₄⋊C₈×C₆₁	—	2	2	no	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₄×C₈×C₆₁	Q₈	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	2	2	no	—
44	C₂²×Dic₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	2	1	no	—
47	C₂×C₈×C₆₁	—	4	2	no	—
48	C₂×C₈×C₆₁	—	4	2	no	—
49	C₂×C₈×C₆₁	—	4	2	no	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
54	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
57	C₄²×C₆₁	#41	1	1	yes	characteristic
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄⋊Dic₆₁	—	2	2	no	—
61	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
62	C₄×Dic₆₁	—	4	2	no	—
63	C₄×Dic₆₁	—	4	2	no	—
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	—	2	2	no	—
68	C₂×#92418	—	2	2	no	—
69	C₂×#92418	—	4	2	no	—
70	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	—	2	2	no	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
82	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₆₁	#137	1	2	yes	—
84	C₂×Dic₆₁	—	2	4	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	—	4	2	no	—
87	C₂×Dic₆₁	—	4	2	no	—
88	C₂×Dic₆₁	—	4	2	no	—
89	C₂×Dic₆₁	—	4	2	no	—
90	C₈×C₆₁	—	2	2	no	—
91	C₈×C₆₁	—	8	2	no	—
92	C₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
93	#92418	—	4	2	no	—
94	#92418	—	4	1	no	—
95	#92418	—	4	1	no	—
96	#9596	—	61	1	no	maximal, Sylow
97	C₂²×C₆₁	#389	1	1	yes	characteristic
98	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
99	C₂²×C₆₁	#436	1	1	yes	characteristic
100	C₂²×C₆₁	#440	1	2	yes	—
101	C₂²×C₆₁	#444	1	2	yes	—
102	C₄×C₆₁	—	2	1	no	—
103	C₄×C₆₁	—	2	1	no	—
104	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₆₁	#424	1	1	yes	characteristic
106	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
107	C₄×C₆₁	#446	1	1	yes	characteristic
108	C₄×C₆₁	#469	1	1	yes	characteristic
109	C₄×C₆₁	#470	1	1	yes	characteristic
110	C₄×C₆₁	#501	1	2	yes	—
111	Dic₆₁	—	4	4	no	—
112	Dic₆₁	—	8	2	no	—
113	#1939	—	61	2	no	—
114	C₄×#106	—	61	1	no	—
115	C₂×#335	D₆₁	1	1	yes	characteristic
116	C₂×#322	—	61	1	no	—
117	#1279	—	61	2	no	—
118	C₂×C₆₁	SD₁₆⋊C₈	1	2	yes	—
119	C₂×C₆₁	#1592	1	2	yes	—
120	C₂×C₆₁	#1939	1	1	yes	characteristic
121	C₂×C₆₁	#1944	1	1	yes	characteristic
122	C₂×C₆₁	#1946	1	1	yes	characteristic
123	C₂×#106	—	61	1	no	—
124	C₂×#106	—	61	1	no	—
125	C₂×C₄⋊C₈	—	122	2	no	—
126	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
127	C₂×C₄×C₈	—	61	1	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	61	2	no	—
131	#340	—	61	2	no	—
132	#336	—	122	2	no	—
133	#335	—	2	2	no	—
134	#322	—	122	2	no	—
135	C₆₁	#9596	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	61	1	no	—
138	C₂²×C₈	—	61	1	no	—
139	C₂²×C₈	—	122	1	no	—
140	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
141	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	61	2	no	—
143	C₂×C₄⋊C₄	—	61	2	no	—
144	C₂×C₄²	—	122	2	no	—
145	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
146	#106	—	61	4	no	—
147	#106	—	122	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	244	4	no	—
151	C₄×C₈	—	61	2	no	—
152	C₄×C₈	—	122	1	no	—
153	C₄×C₈	—	122	1	no	—
154	C₄×C₈	—	122	1	no	—
155	C₄×C₈	Dic₁₂₂	1	2	yes	—
156	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
157	#95	—	122	2	no	—
158	C₂²×C₄	—	122	2	no	—
159	C₂²×C₄	—	122	2	no	—
160	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
161	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
162	C₂²×C₄	#11758226	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	122	1	no	—
170	C₂×C₈	—	122	1	no	—
171	C₂×C₈	—	122	2	no	—
172	C₂×C₈	—	122	2	no	—
173	C₂×C₈	—	244	2	no	—
174	C₂×C₈	C₈×D₆₁	1	2	yes	—
175	C₂×C₈	#11758189	1	2	yes	—
176	C₂×C₈	#11758196	1	2	yes	—
177	C₂×C₈	#11758196	1	2	yes	—
178	C₄⋊C₄	—	61	4	no	—
179	C₄⋊C₄	—	122	2	no	—
180	C₄⋊C₄	—	122	2	no	—
181	C₄⋊C₄	—	122	2	no	—
182	C₄²	—	244	2	no	—
183	C₄²	—	244	2	no	—
184	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
185	C₄²	#11758224	1	1	yes	characteristic
186	C₄²	Q₈×D₆₁	1	1	yes	characteristic
187	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
188	C₂³	#14519739	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	122	4	no	—
191	C₂×C₄	—	244	2	no	—
192	C₂×C₄	—	244	2	no	—
193	C₂×C₄	—	244	2	no	—
194	C₂×C₄	—	244	2	no	—
195	C₂×C₄	—	244	2	no	—
196	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
197	C₂×C₄	#14519742	1	1	yes	characteristic
198	C₂×C₄	#14519748	1	1	yes	characteristic
199	C₂×C₄	#14519753	1	1	yes	characteristic
200	C₂×C₄	#14519768	1	1	yes	characteristic
201	C₂×C₄	#14519776	1	1	yes	characteristic
202	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
203	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
204	C₂×C₄	#14519785	1	1	yes	characteristic
205	C₂×C₄	#14519785	1	1	yes	characteristic
206	C₂×C₄	#14519801	1	2	yes	—
207	C₂×C₄	#14519809	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	8	2	no	—
210	C₈	—	244	2	no	—
211	C₈	—	244	1	no	—
212	C₈	—	244	1	no	—
213	C₈	#14519706	1	4	yes	complemented
214	C₂²	#16839658	1	1	yes	characteristic
215	C₂²	#16839827	1	1	yes	characteristic
216	C₂²	#16839839	1	1	yes	characteristic
217	C₂²	#16839861	1	2	yes	—
218	C₂²	#16839883	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	488	2	no	—
222	C₄	—	244	4	no	—
223	C₄	C₂×#14519706	1	1	yes	characteristic
224	C₄	#16839790	1	1	yes	characteristic
225	C₄	#16839886	1	1	yes	characteristic
226	C₄	#16839889	1	1	yes	characteristic
227	C₄	#16840012	1	1	yes	characteristic
228	C₄	#16840017	1	1	yes	characteristic
229	C₄	#16840170	1	2	yes	—
230	C₂	#21444630	1	2	yes	—
231	C₂	#21444720	1	2	yes	—
232	C₂	#21446535	1	1	yes	characteristic
233	C₂	#21446560	1	1	yes	characteristic
234	C₂	#21446570	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#335	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#335	D₆₁
hypocenter	C₆₁	#9596
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
25. #14519709	C₈	semidirect
92. C₈×C₆₁	C₄⋊C₈	semidirect
135. C₆₁	#9596	semidirect
213. C₈	#14519706	semidirect

Maximal subgroups

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

subgroup	index
#21446535	2
C₄×#14519709	2
C₂×#16839483	2
#21443658	2
C₂×C₆₁×#335	2
#9596	61