Subgroups — #35316008

Order 21248.

250 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24909266	C₂	1	2	yes	maximal
2	#24905533	C₂	1	2	yes	maximal, complemented
3	#24905527	C₂	1	2	yes	maximal, complemented
4	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	4	yes	—
6	C₂²⋊(C₈₃⋊C₁₆)	—	2	2	no	—
7	#19127477	—	2	2	no	—
8	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	C₈²×C₈₃	C₄	1	1	yes	characteristic
12	C₈²×C₈₃	C₄	1	1	yes	characteristic
13	C₈²×C₈₃	C₂²	1	1	yes	characteristic
14	C₈²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₈₃⋊C₁₆	—	2	4	no	—
16	C₂²×C₈×C₈₃	—	2	1	no	—
17	C₂²×C₈×C₈₃	C₈	1	1	yes	characteristic
18	C₂²×C₈×C₈₃	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₃	—	2	1	no	—
23	C₄×C₈×C₈₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₈₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₈₃⋊C₁₆	—	4	4	no	—
34	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₈₃	—	2	4	no	—
38	C₂×C₈×C₈₃	—	2	1	no	—
39	C₂×C₈×C₈₃	—	2	1	no	—
40	C₂×C₈×C₈₃	—	2	2	no	—
41	C₂×C₈×C₈₃	—	2	2	no	—
42	C₂×C₈×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	—	2	2	no	—
44	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
55	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₂³×C₈₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₈₃	—	2	2	no	—
58	C₂×C₄×C₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	—	2	1	no	—
60	C₂×C₄×C₈₃	—	2	1	no	—
61	C₂×C₄×C₈₃	—	2	1	no	—
62	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
68	C₈×C₈₃	—	2	2	no	—
69	C₈×C₈₃	—	2	2	no	—
70	C₈×C₈₃	—	2	1	no	—
71	C₈×C₈₃	—	2	1	no	—
72	C₈×C₈₃	—	2	1	no	—
73	C₈×C₈₃	—	2	1	no	—
74	C₈×C₈₃	#108	1	2	yes	—
75	C₈×C₈₃	M₃₂	1	2	yes	—
76	C₈×C₈₃	M₃₂	1	2	yes	—
77	C₈×C₈₃	#135	1	2	yes	—
78	C₂²×C₈₃	—	2	1	no	—
79	C₂²×C₈₃	—	2	1	no	—
80	C₂²×C₈₃	#364	1	2	yes	—
81	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₈₃	—	2	1	no	—
83	C₄×C₈₃	—	2	1	no	—
84	C₄×C₈₃	—	2	1	no	—
85	C₄×C₈₃	—	2	1	no	—
86	C₄×C₈₃	#427	1	1	yes	characteristic
87	C₄×C₈₃	C₂×#108	1	1	yes	characteristic
88	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
89	C₄×C₈₃	C₂×M₃₂	1	1	yes	characteristic
90	#12013	—	83	1	no	maximal, Sylow
91	C₂×C₈₃	—	2	2	no	—
92	C₂×C₈₃	#1564	1	1	yes	characteristic
93	C₂×C₈₃	#2150	1	1	yes	characteristic
94	C₂×C₈₃	C₂×#364	1	1	yes	characteristic
95	#2163	—	83	2	no	—
96	C₂×C₈²	D₈₃	1	1	yes	characteristic
97	#1370	—	83	2	no	—
98	#1367	—	83	2	no	—
99	C₈₃	#12013	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
101	C₂×C₄×C₈	Dic₈₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
103	C₄⋊C₁₆	—	83	4	no	—
104	C₂²⋊C₁₆	—	166	2	no	—
105	#346	—	166	2	no	—
106	C₈²	Dic₈₃	1	1	yes	characteristic
107	C₈²	Dic₈₃	1	1	yes	characteristic
108	C₈²	C₂×D₈₃	1	1	yes	characteristic
109	C₈²	C₂×D₈₃	1	1	yes	characteristic
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	C₈₃⋊C₈	1	1	yes	characteristic
112	C₂²×C₈	C₈₃⋊C₈	1	1	yes	characteristic
113	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
114	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
115	C₂×C₄²	C₂×Dic₈₃	1	1	yes	characteristic
116	C₂×C₁₆	—	166	4	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	C₈₃⋊C₈	1	1	yes	characteristic
119	C₄×C₈	C₈₃⋊C₈	1	1	yes	characteristic
120	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
121	C₄×C₈	D₃₃₂	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
123	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
124	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
125	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
127	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊Dic₈₃	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₈₃⋊C₈	1	1	yes	characteristic
130	C₂²×C₄	#13050574	1	1	yes	characteristic
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
141	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
142	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
143	C₂×C₈	C₂×C₈₃⋊C₈	1	1	yes	characteristic
144	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
146	C₄²	C₄⋊Dic₈₃	1	1	yes	characteristic
147	C₄²	C₂×C₈₃⋊C₈	1	1	yes	characteristic
148	C₄²	#13050574	1	1	yes	characteristic
149	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
150	C₁₆	—	332	4	no	—
151	C₂³	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
158	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
159	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₈₃⋊C₈	1	1	yes	characteristic
162	C₂×C₄	C₂×#13050574	1	1	yes	characteristic
163	C₈	—	2	2	no	—
164	C₈	—	2	2	no	—
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	#14581728	1	2	yes	—
170	C₈	#14581732	1	2	yes	—
171	C₈	#14581732	1	2	yes	—
172	C₈	#14581814	1	2	yes	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#19127522	1	2	yes	—
176	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	#19127739	1	1	yes	characteristic
182	C₄	C₂×#14581728	1	1	yes	characteristic
183	C₄	C₂×#14581732	1	1	yes	characteristic
184	C₄	C₂×#14581732	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#24906211	1	1	yes	characteristic
187	C₂	#24909219	1	1	yes	characteristic
188	C₂	C₂×#19127522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₄⋊C₈
prime radical	C₂³×C₈₃	C₄⋊C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#12013
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #24905533	C₂	semidirect
3. #24905527	C₂	semidirect
99. C₈₃	#12013	semidirect

Maximal subgroups

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

subgroup	index
#24909266	2
#24905533	2
#24905527	2
C₂×C₈²×C₈₃	2
#12013	83