Subgroups — #29649324

Order 15616.

612 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454951	C₂	1	2	yes	maximal
2	#21444608	C₂	1	6	yes	maximal, complemented
3	#21444578	C₂	1	3	yes	maximal, complemented
4	#21444567	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×#92418	—	2	2	no	—
7	#16839840	—	2	6	no	—
8	#16839815	C₂²	1	6	yes	—
9	#16839786	C₂²	1	6	yes	—
10	#16839515	C₂²	1	6	yes	—
11	#16839515	C₂²	1	1	yes	characteristic
12	#16839515	C₂²	1	3	yes	—
13	#16839513	C₂²	1	6	yes	—
14	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
17	C₈×D₄×C₆₁	C₄	1	3	yes	—
18	C₈×D₄×C₆₁	C₄	1	3	yes	—
19	C₆₁×#431	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
21	C₂×#11758194	—	2	6	no	—
22	C₄⋊#92418	—	2	6	no	—
23	C₄⋊#92418	C₂³	1	2	yes	—
24	C₄⋊#92418	C₂³	1	6	yes	—
25	C₂²⋊#92418	—	2	6	no	—
26	C₄×#92418	—	2	6	no	—
27	C₂×#41×C₆₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
32	C₄×D₄×C₆₁	C₂×C₄	1	3	yes	—
33	C₄×D₄×C₆₁	D₄	1	3	yes	—
34	#117×C₆₁	Q₈	1	3	yes	—
35	C₂×C₄²×C₆₁	C₂³	1	3	yes	—
36	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
37	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
38	C₂²⋊C₈×C₆₁	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₆₁	C₂³	1	3	yes	—
42	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	#41×C₆₁	—	2	4	no	—
44	C₂×Q₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×D₄×C₆₁	C₄⋊C₄	1	3	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₆₁	C₂×Q₈	1	3	yes	—
49	C₂×C₈×C₆₁	—	2	6	no	—
50	C₂×C₈×C₆₁	—	4	3	no	—
51	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
56	C₄⋊C₄×C₆₁	C₄⋊C₄	1	3	yes	—
57	C₂²⋊C₄×C₆₁	C₄⋊C₄	1	6	yes	—
58	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
59	C₄²×C₆₁	C₂×D₄	1	3	yes	—
60	C₄²×C₆₁	C₂×Q₈	1	3	yes	—
61	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
62	#11758194	—	8	6	no	—
63	C₂×#92418	—	2	2	no	—
64	C₂×#92418	—	2	6	no	—
65	C₂³×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₆₁	#108	1	4	yes	—
67	D₄×C₆₁	—	2	6	no	—
68	C₂×C₄×C₆₁	—	2	3	no	—
69	C₂×C₄×C₆₁	—	2	3	no	—
70	C₂×C₄×C₆₁	—	2	3	no	—
71	C₂×C₄×C₆₁	—	2	3	no	—
72	C₂×C₄×C₆₁	—	2	6	no	—
73	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
76	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
79	C₈×C₆₁	—	2	3	no	—
80	C₈×C₆₁	—	2	1	no	—
81	C₈×C₆₁	—	2	3	no	—
82	C₈×C₆₁	#131	1	2	yes	—
83	#92418	—	4	6	no	—
84	#92418	—	4	2	no	—
85	#17837	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	3	no	—
87	C₂²×C₆₁	—	2	3	no	—
88	C₂²×C₆₁	—	2	3	no	—
89	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₆₁	—	2	3	no	—
91	C₄×C₆₁	—	2	3	no	—
92	C₄×C₆₁	—	2	1	no	—
93	C₄×C₆₁	C₂×#108	1	6	yes	—
94	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
95	C₄×C₆₁	#568	1	1	yes	characteristic
96	C₄×C₆₁	#576	1	1	yes	characteristic
97	C₄×C₆₁	#578	1	1	yes	characteristic
98	#2991	—	61	2	no	—
99	C₈×#41	D₆₁	1	1	yes	characteristic
100	#1574	—	61	6	no	—
101	#1567	—	61	3	no	—
102	#1564	—	61	3	no	—
103	C₂×C₆₁	—	4	3	no	—
104	C₂×C₆₁	#2906	1	1	yes	characteristic
105	C₂×C₆₁	#2913	1	1	yes	characteristic
106	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
107	C₄×#41	C₂×D₆₁	1	1	yes	characteristic
108	C₈×Q₈	—	122	2	no	—
109	C₈×Q₈	Dic₆₁	1	1	yes	characteristic
110	C₈×Q₈	Dic₆₁	1	1	yes	characteristic
111	#436	—	122	6	no	—
112	C₈×D₄	Dic₆₁	1	3	yes	—
113	C₈×D₄	Dic₆₁	1	3	yes	—
114	#431	—	61	6	no	—
115	#431	C₂×D₆₁	1	3	yes	—
116	#423	—	61	6	no	—
117	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
118	#335	—	61	1	no	—
119	#335	—	61	3	no	—
120	#335	—	61	6	no	—
121	#334	—	61	6	no	—
122	C₆₁	#17837	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	Dic₁₂₂	1	1	yes	characteristic
124	C₂×M₁₆	—	122	6	no	—
125	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
126	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
127	C₄×Q₈	D₂₄₄	1	1	yes	characteristic
128	C₄×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
129	C₄×D₄	D₂₄₄	1	3	yes	—
130	C₄×D₄	C₂×Dic₆₁	1	3	yes	—
131	#117	Dic₁₂₂	1	3	yes	—
132	C₂×C₄²	C₂²×D₆₁	1	3	yes	—
133	C₄⋊C₈	—	61	2	no	—
134	C₄⋊C₈	—	61	6	no	—
135	C₄⋊C₈	—	122	6	no	—
136	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
137	C₄⋊C₈	C₂×Dic₆₁	1	3	yes	—
138	C₂²⋊C₈	—	122	6	no	—
139	C₂²⋊C₈	C₂×Dic₆₁	1	6	yes	—
140	C₄×C₈	—	122	6	no	—
141	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
142	C₄×C₈	C₂×Dic₆₁	1	3	yes	—
143	C₄×C₈	C₂²×D₆₁	1	3	yes	—
144	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
145	#41	—	2	4	no	—
146	C₂×Q₈	C₄⋊Dic₆₁	1	1	yes	characteristic
147	C₂×D₄	C₄⋊Dic₆₁	1	3	yes	—
148	C₂²×C₄	C₂×Dic₁₂₂	1	3	yes	—
149	C₂²×C₄	C₂×D₂₄₄	1	3	yes	—
150	C₂²×C₄	C₂²×Dic₆₁	1	3	yes	—
151	M₁₆	—	488	6	no	—
152	C₂×C₈	—	2	6	no	—
153	C₂×C₈	—	4	3	no	—
154	C₂×C₈	—	122	2	no	—
155	C₂×C₈	—	122	6	no	—
156	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
158	C₂×C₈	C₂²×Dic₆₁	1	3	yes	—
159	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
160	C₄⋊C₄	C₄⋊Dic₆₁	1	3	yes	—
161	C₄⋊C₄	C₄⋊Dic₆₁	1	3	yes	—
162	C₂²⋊C₄	C₄⋊Dic₆₁	1	6	yes	—
163	C₄²	C₂×Dic₁₂₂	1	3	yes	—
164	C₄²	C₂×D₂₄₄	1	3	yes	—
165	C₄²	C₂²×Dic₆₁	1	3	yes	—
166	C₄²	C₂³×D₆₁	1	1	yes	characteristic
167	C₂³	C₂×C₄⋊Dic₆₁	1	3	yes	—
168	Q₈	#14519713	1	4	yes	—
169	D₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	—	2	3	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
178	C₂×C₄	C₂³×Dic₆₁	1	1	yes	characteristic
179	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
180	C₂×C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
181	C₈	—	2	3	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	3	no	—
184	C₈	—	244	6	no	—
185	C₈	—	244	2	no	—
186	C₈	#14519784	1	2	yes	—
187	C₂²	—	2	3	no	—
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
191	C₄	—	2	3	no	—
192	C₄	—	2	3	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×#14519713	1	6	yes	—
195	C₄	C₂×#14519784	1	1	yes	characteristic
196	C₄	#16840529	1	1	yes	characteristic
197	C₄	#16840567	1	1	yes	characteristic
198	C₄	#16840579	1	1	yes	characteristic
199	C₂	—	4	3	no	—
200	C₂	#21454387	1	1	yes	characteristic
201	C₂	#21454419	1	1	yes	characteristic
202	C₂	C₂²×#14519713	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₈×#41×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₆₁	Q₈
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	C₈×#41	D₆₁
hypocenter	C₆₁	#17837
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #21444608	C₂	semidirect
3. #21444578	C₂	semidirect
4. #21444567	C₂	semidirect
122. C₆₁	#17837	semidirect

Maximal subgroups

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

subgroup	index
#21454951	2
#21444608	2
#21444578	2
#21444567	2
C₈×#41×C₆₁	2
#17837	61