Subgroups — #35291343

Order 21248.

274 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127522	C₂	1	1	yes	maximal, characteristic
2	#24907342	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581732	—	2	1	no	—
5	C₂×#14581728	C₄	1	1	yes	characteristic
6	#19127739	C₄	1	1	yes	characteristic
7	#19127656	—	2	1	no	—
8	#19127522	—	2	1	no	—
9	#19127522	C₄	1	2	yes	complemented
10	#19127461	—	2	1	no	—
11	C₂×C₄×C₈×C₈₃	—	2	1	no	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₈²×C₈₃	—	2	1	no	—
14	C₈²×C₈₃	C₄	1	2	yes	complemented
15	C₂×#13050574	—	2	1	no	—
16	#14581751	—	2	1	no	—
17	#14581732	—	2	2	no	—
18	#14581732	—	4	1	no	—
19	C₂×C₈₃⋊C₁₆	—	4	1	no	—
20	#14581728	—	2	1	no	—
21	#14581728	C₈	1	2	yes	complemented
22	#14581726	C₈	1	2	yes	complemented
23	#14581724	—	2	1	no	—
24	#14581702	—	4	1	no	—
25	C₈×Dic₈₃	—	4	1	no	—
26	C₂²×C₈×C₈₃	—	2	2	no	—
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	—	2	2	no	—
31	C₄×C₈×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
36	#13050574	—	2	2	no	—
37	#13050574	—	4	1	no	—
38	C₂×C₈₃⋊C₈	—	4	1	no	—
39	C₄⋊Dic₈₃	—	2	2	no	—
40	C₂²⋊Dic₈₃	—	4	1	no	—
41	C₄×Dic₈₃	—	4	1	no	—
42	C₈₃⋊C₁₆	—	4	2	no	—
43	C₂²×C₄×C₈₃	—	2	1	no	—
44	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	—	2	4	no	—
46	C₂×C₈×C₈₃	—	2	8	no	—
47	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
56	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
58	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
59	C₄²×C₈₃	—	2	1	no	—
60	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₈₃	#104	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	4	no	—
64	C₂×C₄×C₈₃	—	2	1	no	—
65	C₂×C₄×C₈₃	—	2	1	no	—
66	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
70	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
71	C₂×C₄×C₈₃	#104	1	1	yes	characteristic
72	C₂×Dic₈₃	—	4	2	no	—
73	C₈×C₈₃	—	2	8	no	—
74	C₈×C₈₃	C₄×C₈	1	1	yes	characteristic
75	C₈×C₈₃	#97	1	1	yes	characteristic
76	C₈×C₈₃	#102	1	1	yes	characteristic
77	C₈×C₈₃	#103	1	1	yes	characteristic
78	C₈×C₈₃	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₈₃	C₄⋊C₈	1	1	yes	characteristic
80	C₈×C₈₃	#106	1	1	yes	characteristic
81	C₈×C₈₃	#107	1	1	yes	characteristic
82	C₈₃⋊C₈	—	4	2	no	—
83	C₂²×C₈₃	—	2	2	no	—
84	C₂²×C₈₃	D₄⋊C₈	1	1	yes	characteristic
85	C₂²×C₈₃	Q₈⋊C₈	1	1	yes	characteristic
86	C₂²×C₈₃	#339	1	1	yes	characteristic
87	C₄×C₈₃	—	2	2	no	—
88	C₄×C₈₃	—	2	2	no	—
89	C₄×C₈₃	D₄⋊C₈	1	1	yes	characteristic
90	C₄×C₈₃	Q₈⋊C₈	1	1	yes	characteristic
91	C₄×C₈₃	#336	1	1	yes	characteristic
92	C₄×C₈₃	#340	1	1	yes	characteristic
93	Dic₈₃	—	8	2	no	—
94	#7179	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	—	2	1	no	—
96	C₂×C₈₃	—	2	1	no	—
97	C₂×C₈₃	#1279	1	1	yes	characteristic
98	C₂×C₈₃	#1335	1	2	yes	—
99	C₂×#364	—	83	1	no	—
100	#1772	—	83	1	no	—
101	C₂×C₈²	D₈₃	1	1	yes	characteristic
102	C₈₃	#7179	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₃₂	—	166	1	no	—
104	C₂×#108	—	83	1	no	—
105	#427	—	83	1	no	—
106	#405	—	166	1	no	—
107	C₂×C₄×C₈	—	2	1	no	—
108	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
109	#364	—	83	2	no	—
110	#364	—	166	1	no	—
111	#339	—	166	1	no	—
112	C₈²	—	2	1	no	—
113	C₈²	Dic₈₃	1	2	yes	complemented
114	C₂×M₁₆	—	166	1	no	—
115	C₂²×C₈	—	2	2	no	—
116	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
117	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
118	#117	—	166	1	no	—
119	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
120	M₃₂	—	332	1	no	—
121	M₃₂	—	166	2	no	—
122	C₂×C₁₆	—	332	1	no	—
123	#108	—	83	2	no	—
124	#108	—	166	1	no	—
125	#107	—	83	2	no	—
126	#106	—	166	1	no	—
127	#97	—	332	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	332	1	no	—
131	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
132	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
137	M₁₆	—	332	1	no	—
138	M₁₆	—	166	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	—	332	1	no	—
142	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
143	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
144	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
145	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
146	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
147	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
148	C₂×C₈	#13050573	1	1	yes	characteristic
149	C₂×C₈	#13050573	1	1	yes	characteristic
150	C₂×C₈	D₆₆₄	1	1	yes	characteristic
151	C₂×C₈	#13050579	1	1	yes	characteristic
152	C₂×C₈	#13050579	1	1	yes	characteristic
153	C₂×C₈	Dic₃₃₂	1	1	yes	characteristic
154	C₄⋊C₄	—	166	2	no	—
155	C₂²⋊C₄	—	332	1	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	332	1	no	—
158	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
159	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
160	C₁₆	—	332	2	no	—
161	C₂³	#14581720	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	332	2	no	—
166	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
167	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
168	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
169	C₂×C₄	#14581714	1	1	yes	characteristic
170	C₂×C₄	#14581717	1	1	yes	characteristic
171	C₂×C₄	#14581720	1	1	yes	characteristic
172	C₈	—	2	8	no	—
173	C₈	—	332	2	no	—
174	C₈	C₈×Dic₈₃	1	1	yes	characteristic
175	C₈	#14581702	1	1	yes	characteristic
176	C₈	#14581714	1	1	yes	characteristic
177	C₈	#14581717	1	1	yes	characteristic
178	C₈	Dic₈₃⋊C₈	1	1	yes	characteristic
179	C₈	Dic₈₃⋊C₈	1	1	yes	characteristic
180	C₈	#14581724	1	1	yes	characteristic
181	C₈	#14581726	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#19127430	1	1	yes	characteristic
184	C₂²	#19127433	1	1	yes	characteristic
185	C₂²	#19127461	1	1	yes	characteristic
186	C₄	—	2	2	no	—
187	C₄	—	2	2	no	—
188	C₄	—	664	2	no	—
189	C₄	#19127430	1	1	yes	characteristic
190	C₄	#19127433	1	1	yes	characteristic
191	C₄	#19127455	1	1	yes	characteristic
192	C₄	#19127464	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#24905303	1	1	yes	characteristic
196	C₂	#24905437	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₆₆₄
commutator	C₈×C₈₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	D₄⋊C₈
prime radical	C₂³×C₈₃	#104
cosocle	C₂×C₄×C₈×C₈₃	C₂²
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#7179
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
9. #19127522	C₄	semidirect
14. C₈²×C₈₃	C₄	semidirect
21. #14581728	C₈	semidirect
22. #14581726	C₈	semidirect
102. C₈₃	#7179	semidirect
113. C₈²	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127522	2
#24907342	2
C₂×C₈²×C₈₃	2
#7179	83