Subgroups — #35295643

Order 21248.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14581724	C₂	1	1	yes	maximal, characteristic
2	#24907330	C₂	1	2	yes	maximal
3	C₂×#19127452	C₂	1	1	yes	maximal, characteristic
4	#24905303	C₂	1	2	yes	maximal
5	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581724	C₄	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14581702	—	2	2	no	—
9	C₄×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	#19127464	C₄	1	2	yes	—
11	#19127455	—	2	2	no	—
12	#19127452	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₈²×C₈₃	C₄	1	4	yes	complemented
17	C₂²×C₈₃⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₈₃	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₈₃	—	2	2	no	—
20	#14581724	—	2	4	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
23	#14581702	—	4	4	no	—
24	C₈₃⋊#95	—	2	2	no	—
25	C₂²×C₈×C₈₃	—	2	1	no	—
26	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₃	—	2	1	no	—
32	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₃	Q₈	1	2	yes	—
37	C₂²×Dic₈₃	—	2	4	no	—
38	C₂×C₈₃⋊C₈	—	2	4	no	—
39	C₂×C₈₃⋊C₈	—	4	2	no	—
40	C₂×C₈₃⋊C₈	—	4	2	no	—
41	C₄⋊Dic₈₃	—	2	4	no	—
42	C₄⋊Dic₈₃	—	2	4	no	—
43	C₄×Dic₈₃	—	4	2	no	—
44	C₄×Dic₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
48	C₂×C₈×C₈₃	—	2	4	no	—
49	C₂×C₈×C₈₃	—	2	2	no	—
50	C₂×C₈×C₈₃	—	2	2	no	—
51	C₂×C₈×C₈₃	—	2	2	no	—
52	C₂×C₈×C₈₃	C₄²	1	2	yes	—
53	C₂×C₈×C₈₃	C₄²	1	2	yes	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₃	M₁₆	1	4	yes	—
57	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
58	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₈₃	#41	1	1	yes	characteristic
62	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#117	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	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₈₃	#131	1	2	yes	—
72	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
73	C₂×C₄×C₈₃	#135	1	2	yes	—
74	C₂×Dic₈₃	—	4	4	no	—
75	C₂×Dic₈₃	—	4	4	no	—
76	C₂×Dic₈₃	—	4	4	no	—
77	C₈×C₈₃	—	2	4	no	—
78	C₈×C₈₃	—	2	2	no	—
79	C₈×C₈₃	—	2	2	no	—
80	C₈×C₈₃	#97	1	4	yes	complemented
81	C₈×C₈₃	#106	1	4	yes	complemented
82	C₈₃⋊C₈	—	4	4	no	—
83	C₂²×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈₃	#436	1	2	yes	—
85	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₈₃	#443	1	2	yes	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	2	1	no	—
89	C₄×C₈₃	C₂×#97	1	1	yes	characteristic
90	C₄×C₈₃	#405	1	1	yes	characteristic
91	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
92	C₄×C₈₃	#427	1	1	yes	characteristic
93	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
94	C₄×C₈₃	#446	1	2	yes	—
95	Dic₈₃	—	8	4	no	—
96	#8283	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	#1590	1	4	yes	—
98	C₂×C₈₃	#1770	1	1	yes	characteristic
99	C₂×C₈₃	#1772	1	1	yes	characteristic
100	C₂×C₈₃	C₄×#106	1	1	yes	characteristic
101	C₄×#106	—	83	1	no	—
102	#1770	—	83	2	no	—
103	C₂×#334	—	83	1	no	—
104	C₂×C₈²	D₈₃	1	1	yes	characteristic
105	#1279	—	83	2	no	—
106	C₈₃	#8283	1	1	yes	characteristic, complemented, Sylow
107	C₂×#106	—	83	2	no	—
108	C₂×C₄⋊C₈	—	83	2	no	—
109	C₂×#97	—	166	2	no	—
110	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	83	2	no	—
114	#340	—	83	2	no	—
115	#336	—	166	2	no	—
116	#334	—	83	4	no	—
117	C₈²	Dic₈₃	1	4	yes	complemented
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	166	2	no	—
120	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
122	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
123	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
124	C₂×C₄⋊C₄	—	83	4	no	—
125	C₂×C₄²	—	166	2	no	—
126	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
127	#106	—	166	4	no	—
128	C₄⋊C₈	—	83	4	no	—
129	C₄⋊C₈	—	166	2	no	—
130	#97	—	332	4	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	Dic₁₆₆	1	2	yes	—
133	C₄×C₈	C₄×D₈₃	1	2	yes	—
134	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
135	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
136	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
137	#95	—	166	2	no	—
138	C₂²×C₄	—	166	4	no	—
139	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
141	C₂²×C₄	#13050590	1	1	yes	characteristic
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	332	2	no	—
147	C₂×C₈	—	332	2	no	—
148	C₂×C₈	—	166	4	no	—
149	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
150	C₂×C₈	C₄×Dic₈₃	1	2	yes	—
151	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
152	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
153	C₂×C₈	#13050573	1	4	yes	—
154	C₂×C₈	#13050579	1	4	yes	—
155	C₄⋊C₄	—	166	4	no	—
156	C₄⋊C₄	—	166	4	no	—
157	C₄²	—	332	2	no	—
158	C₄²	—	332	2	no	—
159	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
160	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
161	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
162	C₄²	#13050590	1	1	yes	characteristic
163	C₂³	C₄×D₃₃₂	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	332	4	no	—
166	C₂×C₄	—	332	4	no	—
167	C₂×C₄	—	332	4	no	—
168	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
170	C₂×C₄	#14581751	1	1	yes	characteristic
171	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
172	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
173	C₂×C₄	C₄×Dic₁₆₆	1	1	yes	characteristic
174	C₂×C₄	C₂×#13050573	1	2	yes	—
175	C₂×C₄	#14581798	1	2	yes	—
176	C₂×C₄	C₂×#13050579	1	2	yes	—
177	C₂×C₄	#14581814	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	—	332	4	no	—
182	C₈	#14581702	1	4	yes	complemented
183	C₈	#14581724	1	4	yes	complemented
184	C₂²	C₄×C₄⋊Dic₈₃	1	1	yes	characteristic
185	C₂²	#19127780	1	2	yes	—
186	C₂²	C₄×#13050579	1	2	yes	—
187	C₂²	#19127819	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	664	4	no	—
191	C₄	C₂×#14581702	1	1	yes	characteristic
192	C₄	#19127656	1	1	yes	characteristic
193	C₄	C₂×#14581724	1	1	yes	characteristic
194	C₄	#19127739	1	1	yes	characteristic
195	C₄	C₄×#13050579	1	2	yes	—
196	C₄	#19127831	1	2	yes	—
197	C₂	#24906355	1	4	yes	—
198	C₂	#24907330	1	1	yes	characteristic
199	C₂	#24907342	1	1	yes	characteristic
200	C₂	C₄×#14581724	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₃₂
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₄×D₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#8283
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. #19127452	C₄	semidirect
16. C₈²×C₈₃	C₄	semidirect
80. C₈×C₈₃	#97	semidirect
81. C₈×C₈₃	#106	semidirect
106. C₈₃	#8283	semidirect
117. C₈²	Dic₈₃	semidirect
182. C₈	#14581702	semidirect
183. C₈	#14581724	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14581724	2
#24907330	2
C₂×#19127452	2
#24905303	2
C₂×C₈²×C₈₃	2
#8283	83