Subgroups — C₂²×#19127521 #35557399

Order 21248.

782 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	C₂²×#14581732	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127521	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₈₃	C₄	1	1	yes	characteristic
7	C₂×#14581732	—	2	3	no	—
8	C₂×#14581732	C₂²	1	6	yes	—
9	C₂²×C₈₃⋊C₁₆	—	2	1	no	—
10	C₂×C₈×Dic₈₃	C₄	1	6	yes	—
11	C₂×C₈×Dic₈₃	C₂²	1	6	yes	—
12	#19127521	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₈₃	—	2	3	no	—
15	C₂×M₃₂×C₈₃	C₂²	1	6	yes	—
16	C₂²×C₁₆×C₈₃	—	2	1	no	—
17	C₂³×Dic₈₃	—	2	1	no	—
18	C₂²×C₈₃⋊C₈	C₈	1	6	yes	—
19	C₂²×C₈₃⋊C₈	C₈	1	2	yes	—
20	C₂²×C₈₃⋊C₈	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₈₃	—	2	3	no	—
22	C₂×C₄×Dic₈₃	C₂×C₄	1	6	yes	—
23	#14581732	—	2	6	no	—
24	#14581732	C₂³	1	4	yes	—
25	C₂×C₈₃⋊C₁₆	—	2	6	no	—
26	C₈×Dic₈₃	C₂×C₄	1	12	yes	—
27	C₈×Dic₈₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₈₃	D₄	1	3	yes	—
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	Q₈	1	3	yes	—
34	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
35	M₃₂×C₈₃	—	2	6	no	—
36	M₃₂×C₈₃	C₂³	1	4	yes	—
37	C₂×C₁₆×C₈₃	—	2	6	no	—
38	C₂²×Dic₈₃	—	2	8	no	—
39	C₂²×Dic₈₃	—	2	6	no	—
40	C₂×C₈₃⋊C₈	C₂×C₈	1	12	yes	—
41	C₂×C₈₃⋊C₈	C₂×C₈	1	12	yes	—
42	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
43	C₄×Dic₈₃	—	2	6	no	—
44	C₄×Dic₈₃	C₂²×C₄	1	4	yes	—
45	C₈₃⋊C₁₆	—	2	4	no	—
46	C₂⁴×C₈₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₈₃	C₄⋊C₄	1	3	yes	—
48	C₂²×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	M₁₆	1	3	yes	—
52	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₈₃	C₄⋊C₄	1	6	yes	—
54	C₂×C₈×C₈₃	C₄⋊C₄	1	3	yes	—
55	C₂×C₈×C₈₃	C₄⋊C₄	1	3	yes	—
56	C₂×C₈×C₈₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₈₃	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₈₃	C₂×D₄	1	3	yes	—
59	C₂×C₈×C₈₃	C₂×Q₈	1	3	yes	—
60	C₂×C₈×C₈₃	C₂×Q₈	1	3	yes	—
61	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
62	C₁₆×C₈₃	—	2	4	no	—
63	C₂³×C₈₃	—	2	4	no	—
64	C₂³×C₈₃	C₄⋊C₈	1	3	yes	—
65	C₂³×C₈₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂³×C₈₃	C₂×M₁₆	1	3	yes	—
67	C₂×C₄×C₈₃	C₄⋊C₈	1	6	yes	—
68	C₂×C₄×C₈₃	C₄⋊C₈	1	3	yes	—
69	C₂×C₄×C₈₃	C₄⋊C₈	1	3	yes	—
70	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₈₃	C₂²×C₈	1	3	yes	—
73	C₂×C₄×C₈₃	C₂²×C₈	1	3	yes	—
74	C₂×C₄×C₈₃	C₂×M₁₆	1	3	yes	—
75	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×Dic₈₃	—	2	24	no	—
77	C₂×Dic₈₃	—	2	4	no	—
78	C₈×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
79	C₈×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
80	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
81	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
82	C₈₃⋊C₈	C₂²×C₈	1	8	yes	—
83	C₂²×C₈₃	—	2	6	no	—
84	C₂²×C₈₃	—	2	6	no	—
85	C₂²×C₈₃	#364	1	4	yes	complemented
86	C₂²×C₈₃	C₂×C₄⋊C₈	1	3	yes	—
87	C₂²×C₈₃	C₂×C₄⋊C₈	1	3	yes	—
88	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
89	C₄×C₈₃	C₂×C₄⋊C₈	1	3	yes	—
90	C₄×C₈₃	C₂×C₄⋊C₈	1	3	yes	—
91	C₄×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₈₃	C₂³×C₈	1	1	yes	characteristic
93	Dic₈₃	—	2	8	no	—
94	C₂²×#364	—	83	1	no	maximal, Sylow
95	C₂×C₈₃	—	2	1	no	—
96	C₂×C₈₃	—	2	3	no	—
97	C₂×C₈₃	C₂×#364	1	6	yes	complemented
98	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
99	C₂²×M₃₂	—	83	1	no	—
100	C₂²×M₃₂	D₈₃	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	83	1	no	—
102	C₂×#364	—	83	12	no	—
103	C₈₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₈	—	83	1	no	—
105	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
106	C₂²×C₄²	—	83	1	no	—
107	C₂×M₃₂	—	2	3	no	—
108	C₂×M₃₂	—	83	6	no	—
109	C₂×M₃₂	—	166	3	no	—
110	C₂×M₃₂	C₂×D₈₃	1	6	yes	—
111	C₂²×C₁₆	—	2	1	no	—
112	C₂²×C₁₆	—	166	1	no	—
113	C₂×C₄×C₈	—	83	6	no	—
114	C₂×C₄×C₈	—	83	6	no	—
115	#364	—	83	16	no	—
116	C₂³×C₄	—	166	1	no	—
117	C₂³×C₄	C₄×D₈₃	1	1	yes	characteristic
118	C₂²×C₈	—	83	6	no	—
119	C₂²×C₈	—	83	2	no	—
120	C₂²×C₈	—	83	6	no	—
121	C₂²×C₈	Dic₁₆₆	1	1	yes	characteristic
122	C₂²×C₈	Dic₁₆₆	1	3	yes	—
123	C₂²×C₈	C₄×D₈₃	1	3	yes	—
124	C₂²×C₈	C₂²⋊D₈₃	1	3	yes	—
125	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
127	C₂×C₄²	—	83	6	no	—
128	C₂×C₄²	—	166	3	no	—
129	M₃₂	—	2	6	no	—
130	M₃₂	—	83	4	no	—
131	M₃₂	—	166	6	no	—
132	M₃₂	C₂²×D₈₃	1	4	yes	—
133	C₂×C₁₆	—	2	6	no	—
134	C₂×C₁₆	—	166	6	no	—
135	C₄×C₈	—	83	12	no	—
136	C₄×C₈	—	83	4	no	—
137	C₂⁴	#13050573	1	1	yes	characteristic
138	C₂²×C₄	—	166	8	no	—
139	C₂²×C₄	—	166	6	no	—
140	C₂²×C₄	Dic₈₃⋊C₄	1	3	yes	—
141	C₂²×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
142	C₂²×C₄	C₈×D₈₃	1	3	yes	—
143	C₂²×C₄	C₈×D₈₃	1	1	yes	characteristic
144	C₂²×C₄	#13050573	1	3	yes	—
145	C₂²×C₄	C₂×C₄×D₈₃	1	3	yes	—
146	C₂×C₈	—	83	12	no	—
147	C₂×C₈	—	83	12	no	—
148	C₂×C₈	—	83	4	no	—
149	C₂×C₈	Dic₈₃⋊C₄	1	3	yes	—
150	C₂×C₈	Dic₈₃⋊C₄	1	6	yes	—
151	C₂×C₈	Dic₈₃⋊C₄	1	3	yes	—
152	C₂×C₈	C₂×C₄×D₈₃	1	3	yes	—
153	C₂×C₈	C₂×C₂²⋊D₈₃	1	3	yes	—
154	C₂×C₈	C₂×C₂²⋊D₈₃	1	3	yes	—
155	C₂×C₈	C₂×Dic₁₆₆	1	3	yes	—
156	C₂×C₈	C₂×Dic₁₆₆	1	3	yes	—
157	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
158	C₄²	—	83	4	no	—
159	C₄²	—	166	6	no	—
160	C₁₆	—	2	4	no	—
161	C₁₆	—	166	4	no	—
162	C₂³	—	2	4	no	—
163	C₂³	Dic₈₃⋊C₈	1	3	yes	—
164	C₂³	Dic₈₃⋊C₈	1	1	yes	characteristic
165	C₂³	C₂×#13050573	1	3	yes	—
166	C₂×C₄	—	166	24	no	—
167	C₂×C₄	—	166	4	no	—
168	C₂×C₄	Dic₈₃⋊C₈	1	6	yes	—
169	C₂×C₄	Dic₈₃⋊C₈	1	3	yes	—
170	C₂×C₄	Dic₈₃⋊C₈	1	3	yes	—
171	C₂×C₄	C₂×Dic₈₃⋊C₄	1	3	yes	—
172	C₂×C₄	C₂×Dic₈₃⋊C₄	1	3	yes	—
173	C₂×C₄	C₂×C₈×D₈₃	1	3	yes	—
174	C₂×C₄	C₂×C₈×D₈₃	1	3	yes	—
175	C₂×C₄	C₂×#13050573	1	3	yes	—
176	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
177	C₈	—	83	8	no	—
178	C₈	C₂×Dic₈₃⋊C₄	1	3	yes	—
179	C₈	C₂×Dic₈₃⋊C₄	1	3	yes	—
180	C₈	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
181	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#19127521	1	4	yes	direct product factor
185	C₂²	C₂×Dic₈₃⋊C₈	1	3	yes	—
186	C₂²	C₂×Dic₈₃⋊C₈	1	3	yes	—
187	C₂²	C₂²×#13050573	1	1	yes	characteristic
188	C₄	—	166	8	no	—
189	C₄	C₂×Dic₈₃⋊C₈	1	3	yes	—
190	C₄	C₂×Dic₈₃⋊C₈	1	3	yes	—
191	C₄	C₂²×Dic₈₃⋊C₄	1	1	yes	characteristic
192	C₄	C₂²×C₈×D₈₃	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#19127521	1	6	yes	direct product factor
196	C₂	C₂²×Dic₈₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₈₃
commutator	C₄×C₈₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂²×M₃₂×C₈₃	C₂
socle	C₂³×C₈₃	C₄⋊C₈
prime radical	C₂⁴×C₈₃	M₁₆
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂²×M₃₂	D₈₃
hypocenter	C₈₃	C₂²×#364
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. C₂×#19127521	C₂	direct (N × H)
12. #19127521	C₂²	direct (N × H)
85. C₂²×C₈₃	#364	semidirect
97. C₂×C₈₃	C₂×#364	semidirect
103. C₈₃	C₂²×#364	semidirect
184. C₂²	#19127521	direct (N × H)
195. C₂	C₂×#19127521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14581732	2
C₂²×C₈×Dic₈₃	2
C₂×#19127521	2
C₂²×M₃₂×C₈₃	2
C₂²×#364	83

Subgroups — C22×#19127521 #35557399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19127521 #35557399