Subgroups — #35303104

Order 21248.

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	#24908239	C₂	1	2	yes	maximal
2	#24906219	C₂	1	1	yes	maximal, characteristic
3	C₈×#13050574	C₂	1	1	yes	maximal, characteristic
4	#24905360	C₂	1	2	yes	maximal
5	C₂×#334×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#19127754	C₂²	1	1	yes	characteristic
7	C₂×#14581728	—	2	2	no	—
8	#19127725	C₂²	1	2	yes	—
9	C₄×#13050574	C₂²	1	1	yes	characteristic
10	#19127466	—	2	2	no	—
11	#19127452	—	2	1	no	—
12	#19127452	—	2	1	no	—
13	#19127423	—	2	1	no	—
14	C₈×C₈₃⋊C₈	—	2	1	no	—
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#334×C₈₃	C₄	1	2	yes	—
18	#334×C₈₃	C₄	1	2	yes	—
19	C₂×#13050574	—	2	2	no	—
20	C₂×#13050574	D₄	1	2	yes	—
21	#14581728	—	4	4	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₈₃⋊C₈)	Q₈	1	2	yes	—
26	#14581703	—	2	1	no	—
27	C₄×C₈₃⋊C₈	—	2	1	no	—
28	C₄×C₈₃⋊C₈	—	2	1	no	—
29	C₂²×C₈×C₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₃	—	2	2	no	—
34	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
39	#13050574	—	2	2	no	—
40	#13050574	—	4	2	no	—
41	#13050574	—	4	2	no	—
42	#13050574	SD₁₆	1	4	yes	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	2	2	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	4	no	—
51	C₂×C₈×C₈₃	—	4	1	no	—
52	C₂×C₈×C₈₃	—	4	2	no	—
53	C₂×C₈×C₈₃	—	4	2	no	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₈₃	#41	1	1	yes	characteristic
61	C₄²×C₈₃	#41	1	1	yes	characteristic
62	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	1	no	—
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	—	4	1	no	—
66	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₈₃	#135	1	2	yes	—
74	C₈×C₈₃	—	2	1	no	—
75	C₈×C₈₃	—	2	1	no	—
76	C₈×C₈₃	—	2	1	no	—
77	C₈×C₈₃	—	4	4	no	—
78	C₈×C₈₃	#108	1	2	yes	complemented
79	C₈₃⋊C₈	—	2	4	no	—
80	C₈₃⋊C₈	—	4	2	no	—
81	C₈₃⋊C₈	—	8	2	no	—
82	C₂²×C₈₃	—	2	1	no	—
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	#389	1	1	yes	characteristic
85	C₂²×C₈₃	#498	1	1	yes	characteristic
86	C₂²×C₈₃	#499	1	1	yes	characteristic
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	2	1	no	—
89	C₄×C₈₃	—	2	1	no	—
90	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
91	C₄×C₈₃	#430	1	1	yes	characteristic
92	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
93	C₄×C₈₃	#465	1	2	yes	—
94	#9623	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	—	2	1	no	—
96	C₂×C₈₃	—	2	1	no	—
97	C₂×C₈₃	#1941	1	1	yes	characteristic
98	C₂×C₈₃	#1950	1	1	yes	characteristic
99	C₂×C₈₃	#1953	1	1	yes	characteristic
100	#1950	—	83	2	no	—
101	#1566	—	83	1	no	—
102	C₂×#334	D₈₃	1	1	yes	characteristic
103	C₈×M₁₆	—	83	1	no	—
104	#1301	—	83	2	no	—
105	C₈₃	#9623	1	1	yes	characteristic, complemented, Sylow
106	#431	—	83	1	no	—
107	C₂×#108	—	166	2	no	—
108	#423	—	83	2	no	—
109	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
110	C₄×M₁₆	—	83	1	no	—
111	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
112	#341	—	166	2	no	—
113	#334	—	166	1	no	—
114	#334	—	166	1	no	—
115	#334	Dic₈₃	1	2	yes	—
116	#334	Dic₈₃	1	2	yes	—
117	#322	—	166	1	no	—
118	C₈²	—	166	1	no	—
119	C₂×M₁₆	—	83	2	no	—
120	C₂×M₁₆	—	166	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
123	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
125	#108	—	332	4	no	—
126	C₄⋊C₈	—	2	2	no	—
127	C₄⋊C₈	—	166	2	no	—
128	C₄⋊C₈	—	166	1	no	—
129	C₄⋊C₈	—	166	2	no	—
130	C₄⋊C₈	C₂×Dic₈₃	1	4	yes	—
131	C₂²⋊C₈	—	83	2	no	—
132	#97	—	166	1	no	—
133	C₄×C₈	—	166	1	no	—
134	C₄×C₈	—	166	1	no	—
135	C₄×C₈	D₃₃₂	1	1	yes	characteristic
136	C₄×C₈	D₃₃₂	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
139	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
141	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
142	M₁₆	—	332	2	no	—
143	M₁₆	—	332	2	no	—
144	M₁₆	—	83	4	no	—
145	M₁₆	—	166	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	332	2	no	—
152	C₂×C₈	—	166	2	no	—
153	C₂×C₈	—	166	2	no	—
154	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
155	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
156	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
157	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
158	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
159	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
160	C₄²	#13050589	1	1	yes	characteristic
161	C₄²	#13050591	1	1	yes	characteristic
162	C₂³	#14581791	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×C₄⋊Dic₈₃	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
169	C₂×C₄	#14581767	1	1	yes	characteristic
170	C₂×C₄	#14581772	1	1	yes	characteristic
171	C₂×C₄	#14581783	1	1	yes	characteristic
172	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
173	C₂×C₄	#14581813	1	2	yes	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	4	no	—
178	C₈	—	332	2	no	—
179	C₈	—	664	2	no	—
180	C₈	—	166	4	no	—
181	C₈	#14581728	1	2	yes	complemented
182	C₂²	—	2	1	no	—
183	C₂²	—	2	1	no	—
184	C₂²	#19127600	1	1	yes	characteristic
185	C₂²	#19128098	1	1	yes	characteristic
186	C₂²	#19128101	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	C₂×#14581728	1	1	yes	characteristic
191	C₄	#19127750	1	1	yes	characteristic
192	C₄	SD₁₆×Dic₈₃	1	2	yes	—
193	C₄	#19127931	1	2	yes	—
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	#24908192	1	1	yes	characteristic
197	C₂	#24908239	1	1	yes	characteristic
198	C₂	#24908254	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#334×C₈₃	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#334	D₈₃
hypocenter	C₈₃	#9623
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
78. C₈×C₈₃	#108	semidirect
105. C₈₃	#9623	semidirect
181. C₈	#14581728	semidirect

Maximal subgroups

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

subgroup	index
#24908239	2
#24906219	2
C₈×#13050574	2
#24905360	2
C₂×#334×C₈₃	2
#9623	83