Subgroups — #29578712

Order 15616.

298 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21446594	C₂	1	2	yes	maximal
2	#21444575	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	#21443715	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519712	—	2	2	no	—
7	#16839786	C₂²	1	2	yes	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	#16839526	—	2	2	no	—
10	#16839513	—	2	1	no	—
11	#16839513	—	2	1	no	—
12	#16839513	C₄	1	2	yes	—
13	#16839513	C₄	1	2	yes	—
14	C₆₁×#431	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#322	—	2	1	no	—
18	C₈²×C₆₁	—	2	1	no	—
19	C₂×#11758194	—	2	2	no	—
20	C₂²×#92418	—	2	2	no	—
21	#14519712	—	4	4	no	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
27	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	1	no	—
31	C₂²⋊C₈×C₆₁	Q₈	1	2	yes	—
32	#97×C₆₁	—	2	1	no	—
33	C₄×C₈×C₆₁	—	2	1	no	—
34	C₄×C₈×C₆₁	—	2	1	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	D₄	1	1	yes	characteristic
38	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₆₁	—	2	2	no	—
43	M₁₆×C₆₁	SD₁₆	1	4	yes	—
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	4	1	no	—
48	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
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	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	#41	1	1	yes	characteristic
55	C₄²×C₆₁	#41	1	1	yes	characteristic
56	#11758194	—	4	2	no	—
57	#11758194	—	4	2	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	4	2	no	—
60	C₂×#92418	—	4	2	no	—
61	C₂×#92418	—	4	2	no	—
62	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	1	no	—
64	C₂×C₄×C₆₁	—	2	1	no	—
65	C₂×C₄×C₆₁	—	4	1	no	—
66	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	#135	1	2	yes	—
74	C₈×C₆₁	—	2	1	no	—
75	C₈×C₆₁	—	2	1	no	—
76	C₈×C₆₁	—	2	1	no	—
77	C₈×C₆₁	—	2	4	no	—
78	C₈×C₆₁	—	4	2	no	—
79	C₈×C₆₁	#108	1	2	yes	complemented
80	#92418	—	4	4	no	—
81	#92418	—	8	2	no	—
82	#9623	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	1	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	#389	1	1	yes	characteristic
86	C₂²×C₆₁	#498	1	1	yes	characteristic
87	C₂²×C₆₁	#499	1	1	yes	characteristic
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	—	2	1	no	—
91	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
92	C₄×C₆₁	#430	1	1	yes	characteristic
93	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
94	C₄×C₆₁	#465	1	2	yes	—
95	#1950	—	61	2	no	—
96	#1566	—	61	1	no	—
97	C₂×#334	—	61	1	no	—
98	C₈×M₁₆	D₆₁	1	1	yes	characteristic
99	#1301	—	61	2	no	—
100	C₂×C₆₁	—	2	1	no	—
101	C₂×C₆₁	—	2	1	no	—
102	C₂×C₆₁	#1941	1	1	yes	characteristic
103	C₂×C₆₁	#1950	1	1	yes	characteristic
104	C₂×C₆₁	#1953	1	1	yes	characteristic
105	#431	C₂×D₆₁	1	1	yes	characteristic
106	C₂×#108	—	122	2	no	—
107	#423	—	61	2	no	—
108	C₂×C₄⋊C₈	—	61	2	no	—
109	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
111	#341	—	122	2	no	—
112	#334	—	61	2	no	—
113	#334	—	61	2	no	—
114	#334	—	122	1	no	—
115	#334	—	122	1	no	—
116	#322	—	2	1	no	—
117	C₈²	—	2	1	no	—
118	C₆₁	#9623	1	1	yes	characteristic, complemented, Sylow
119	C₂×M₁₆	—	122	2	no	—
120	C₂×M₁₆	D₂₄₄	1	2	yes	—
121	C₂²×C₈	—	122	2	no	—
122	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
123	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
125	#108	—	244	4	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	61	4	no	—
128	C₄⋊C₈	—	122	2	no	—
129	C₄⋊C₈	—	122	2	no	—
130	C₄⋊C₈	—	122	2	no	—
131	C₂²⋊C₈	Dic₁₂₂	1	2	yes	—
132	#97	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
136	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	244	2	no	—
144	M₁₆	—	244	2	no	—
145	M₁₆	#11758190	1	4	yes	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	122	4	no	—
151	C₂×C₈	—	244	2	no	—
152	C₂×C₈	—	244	2	no	—
153	C₂×C₈	—	244	2	no	—
154	C₂×C₈	#11758196	1	1	yes	characteristic
155	C₂×C₈	#11758196	1	1	yes	characteristic
156	C₂×C₈	#11758196	1	1	yes	characteristic
157	C₂×C₈	#11758196	1	1	yes	characteristic
158	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
159	C₄²	#11758222	1	1	yes	characteristic
160	C₄²	#11758222	1	1	yes	characteristic
161	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
162	C₂³	#14519777	1	1	yes	characteristic
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
167	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
169	C₂×C₄	#14519751	1	1	yes	characteristic
170	C₂×C₄	#14519756	1	1	yes	characteristic
171	C₂×C₄	#14519769	1	1	yes	characteristic
172	C₂×C₄	C₂×#11758190	1	2	yes	—
173	C₂×C₄	#14519799	1	2	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	488	2	no	—
180	C₈	—	244	4	no	—
181	C₈	#14519712	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#16839660	1	1	yes	characteristic
185	C₂²	#16840160	1	1	yes	characteristic
186	C₂²	#16840164	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#14519712	1	1	yes	characteristic
191	C₄	#16839810	1	1	yes	characteristic
192	C₄	C₄×#11758190	1	2	yes	—
193	C₄	#16839991	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#21446547	1	1	yes	characteristic
197	C₂	#21446594	1	1	yes	characteristic
198	C₂	#21446609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519751
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₈×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	#389
prime radical	C₂³×C₆₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₈×M₁₆	D₆₁
hypocenter	C₆₁	#9623
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
79. C₈×C₆₁	#108	semidirect
118. C₆₁	#9623	semidirect
181. C₈	#14519712	semidirect

Maximal subgroups

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

subgroup	index
#21446594	2
#21444575	2
C₂×#16839513	2
#21443715	2
C₈×M₁₆×C₆₁	2
#9623	61