Subgroups — #29578754

Order 15616.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445991	C₂	1	1	yes	maximal, characteristic
2	#21444589	C₂	1	2	yes	maximal
3	#21444567	C₂	1	1	yes	maximal, characteristic
4	#21443722	C₂	1	2	yes	maximal
5	C₆₁×#1770	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519713	—	2	1	no	—
8	C₂×#14519713	—	2	1	no	—
9	#16839786	C₂²	1	2	yes	—
10	#16839515	C₄	1	2	yes	complemented
11	#16839513	C₄	1	2	yes	complemented
12	#16839512	—	2	4	no	—
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#97×C₆₁	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#336	—	2	1	no	—
18	C₂×#11758194	—	2	2	no	—
19	#14519713	—	4	2	no	—
20	#14519713	—	4	2	no	—
21	C₄⋊#92418	—	2	2	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	C₂×C₄	1	4	yes	—
24	C₂²⋊#92418	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₂²×C₈×C₆₁	—	2	1	no	—
27	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₂×C₄²×C₆₁	—	2	1	no	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₆₁	—	2	1	no	—
33	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
34	#97×C₆₁	—	4	2	no	—
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
39	#95×C₆₁	Q₈	1	2	yes	—
40	C₂²×C₄×C₆₁	—	2	2	no	—
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	4	1	no	—
48	C₂×C₈×C₆₁	—	4	1	no	—
49	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
55	C₄⋊C₄×C₆₁	—	4	2	no	—
56	C₄²×C₆₁	—	4	1	no	—
57	C₄²×C₆₁	—	4	1	no	—
58	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₆₁	#41	1	1	yes	characteristic
62	#11758194	—	4	2	no	—
63	#11758194	—	4	2	no	—
64	C₂×#92418	—	2	4	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	4	1	no	—
69	C₂×C₄×C₆₁	—	4	2	no	—
70	C₂×C₄×C₆₁	—	4	2	no	—
71	C₂×C₄×C₆₁	—	4	2	no	—
72	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	#121	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	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	4	2	no	—
85	C₈×C₆₁	#106	1	2	yes	—
86	C₈×C₆₁	#106	1	2	yes	—
87	#92418	—	4	4	no	—
88	#92418	—	8	2	no	—
89	#9631	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	2	1	no	—
91	C₂²×C₆₁	—	2	1	no	—
92	C₂²×C₆₁	#389	1	1	yes	characteristic
93	C₂²×C₆₁	#475	1	1	yes	characteristic
94	C₂²×C₆₁	#476	1	1	yes	characteristic
95	C₄×C₆₁	—	2	1	no	—
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	—	8	2	no	—
99	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
100	C₄×C₆₁	#428	1	1	yes	characteristic
101	C₄×C₆₁	#444	1	2	yes	—
102	C₄×C₆₁	#460	1	1	yes	characteristic
103	C₄×C₆₁	#461	1	1	yes	characteristic
104	#1848	—	61	1	no	—
105	#1770	D₆₁	1	1	yes	characteristic
106	#1570	—	61	2	no	—
107	#1564	—	61	1	no	—
108	#1303	—	61	2	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	—	2	1	no	—
111	C₂×C₆₁	#1946	1	1	yes	characteristic
112	C₂×C₆₁	#1948	1	1	yes	characteristic
113	C₂×C₆₁	#1953	1	1	yes	characteristic
114	#431	—	61	2	no	—
115	C₂×#108	—	122	1	no	—
116	C₂×#108	—	122	1	no	—
117	#423	—	61	2	no	—
118	C₂×C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
119	C₂×#97	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#335	—	61	2	no	—
124	#334	—	61	2	no	—
125	#334	—	122	4	no	—
126	C₆₁	#9631	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	122	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
134	#108	—	244	2	no	—
135	#108	—	244	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	61	4	no	—
138	C₄⋊C₈	—	122	2	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₄⋊C₈	D₂₄₄	1	2	yes	—
141	C₂²⋊C₈	—	122	2	no	—
142	#97	—	4	2	no	—
143	C₄×C₈	—	122	2	no	—
144	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
145	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
146	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
147	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
148	#95	Dic₁₂₂	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
151	C₂²×C₄	#11758222	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
153	M₁₆	—	244	2	no	—
154	M₁₆	—	244	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	122	4	no	—
161	C₂×C₈	—	244	2	no	—
162	C₂×C₈	#11758196	1	1	yes	characteristic
163	C₂×C₈	#11758196	1	1	yes	characteristic
164	C₂×C₈	#11758196	1	1	yes	characteristic
165	C₂×C₈	#11758196	1	1	yes	characteristic
166	C₂×C₈	#11758200	1	2	yes	—
167	C₂×C₈	#11758201	1	2	yes	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
172	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
173	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
174	C₄²	#11758222	1	1	yes	characteristic
175	C₂³	#14519756	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
182	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
184	C₂×C₄	#14519751	1	1	yes	characteristic
185	C₂×C₄	#14519769	1	1	yes	characteristic
186	C₂×C₄	#14519777	1	1	yes	characteristic
187	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
188	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
189	C₂×C₄	#14519802	1	1	yes	characteristic
190	C₂×C₄	#14519807	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	—	488	2	no	—
195	C₈	—	244	4	no	—
196	C₈	#14519708	1	2	yes	—
197	C₈	#14519708	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16839660	1	1	yes	characteristic
201	C₂²	#16840047	1	1	yes	characteristic
202	C₂²	#16840052	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	8	2	no	—
207	C₄	C₂×#14519708	1	1	yes	characteristic
208	C₄	#16839805	1	1	yes	characteristic
209	C₄	#16839885	1	2	yes	—
210	C₄	#16839962	1	1	yes	characteristic
211	C₄	#16839967	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#21446572	1	1	yes	characteristic
215	C₂	#21446582	1	1	yes	characteristic
216	C₂	#21446609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1770	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1770	D₆₁
hypocenter	C₆₁	#9631
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
10. #16839515	C₄	semidirect
11. #16839513	C₄	semidirect
126. C₆₁	#9631	semidirect

Maximal subgroups

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

subgroup	index
#21445991	2
#21444589	2
#21444567	2
#21443722	2
C₆₁×#1770	2
#9631	61