Subgroups — C₂²×#19127452 #35373370

Order 21248.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#19127452	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈₃⋊C₈	—	2	2	no	—
5	C₂×C₄⋊(C₈₃⋊C₈)	—	2	6	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	12	yes	—
7	#19127452	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₈₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₈₃	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₈₃	C₄	1	6	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	6	yes	—
13	C₂²×C₈₃⋊C₈	—	2	12	no	—
14	C₂²×C₈₃⋊C₈	—	4	8	no	—
15	C₄⋊(C₈₃⋊C₈)	—	2	12	no	—
16	C₄⋊(C₈₃⋊C₈)	C₂³	1	8	yes	—
17	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₈₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₈₃	—	2	4	no	—
21	C₂²×C₈×C₈₃	C₈	1	8	yes	complemented
22	C₂²×C₈×C₈₃	C₂×C₄	1	6	yes	—
23	C₂²×C₈×C₈₃	C₂×C₄	1	6	yes	—
24	C₂×C₄²×C₈₃	C₂×C₄	1	3	yes	—
25	C₂×C₄²×C₈₃	D₄	1	3	yes	—
26	C₂×C₄²×C₈₃	Q₈	1	3	yes	—
27	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
28	C₄×C₈×C₈₃	C₂×C₄	1	12	yes	—
29	C₄×C₈×C₈₃	C₂³	1	4	yes	—
30	C₂×C₈₃⋊C₈	—	2	8	no	—
31	C₂×C₈₃⋊C₈	—	4	24	no	—
32	C₂⁴×C₈₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₃	—	2	4	no	—
34	C₂²×C₄×C₈₃	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₈₃	C₄⋊C₄	1	3	yes	—
36	C₂²×C₄×C₈₃	C₄⋊C₄	1	3	yes	—
37	C₂²×C₄×C₈₃	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₈₃	M₁₆	1	4	yes	—
39	C₂²×C₄×C₈₃	SD₁₆	1	8	yes	—
40	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₈₃	C₂×Q₈	1	3	yes	—
43	C₂×C₈×C₈₃	—	2	12	no	—
44	C₂×C₈×C₈₃	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
47	C₄²×C₈₃	C₄⋊C₄	1	6	yes	—
48	C₄²×C₈₃	C₂²×C₄	1	3	yes	—
49	C₄²×C₈₃	C₂×D₄	1	3	yes	—
50	C₄²×C₈₃	C₂×Q₈	1	3	yes	—
51	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
52	C₂³×C₈₃	C₄⋊C₈	1	4	yes	—
53	C₂³×C₈₃	#106	1	4	yes	—
54	C₂³×C₈₃	#108	1	4	yes	—
55	C₂³×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₈₃	—	2	6	no	—
57	C₂×C₄×C₈₃	—	2	6	no	—
58	C₂×C₄×C₈₃	C₄⋊C₈	1	12	yes	—
59	C₂×C₄×C₈₃	#106	1	12	yes	—
60	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
61	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₈₃	C₂²×C₈	1	6	yes	—
64	C₂×C₄×C₈₃	C₂×M₁₆	1	6	yes	—
65	C₂×C₄×C₈₃	C₂×SD₁₆	1	12	yes	—
66	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
69	C₈×C₈₃	—	2	4	no	—
70	C₈×C₈₃	C₂²×C₈	1	8	yes	complemented
71	C₈₃⋊C₈	—	4	8	no	—
72	C₂²×C₈₃	#334	1	16	yes	complemented
73	C₂²×C₈₃	C₂×C₄⋊C₈	1	6	yes	—
74	C₂²×C₈₃	C₂×#106	1	6	yes	—
75	C₂²×C₈₃	C₂×#108	1	6	yes	—
76	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₈₃	—	2	3	no	—
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	C₂×C₄⋊C₈	1	6	yes	—
80	C₄×C₈₃	C₂×#106	1	6	yes	—
81	C₄×C₈₃	C₂³×C₈	1	1	yes	characteristic
82	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
83	C₄×C₈₃	C₂²×SD₁₆	1	2	yes	—
84	C₂²×#334	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	C₂×#334	1	12	yes	complemented
86	C₂×C₈₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
87	C₂×C₈₃	C₂²×#106	1	1	yes	characteristic
88	C₂×C₈₃	C₂²×#108	1	1	yes	characteristic
89	C₂²×C₄⋊C₈	—	83	2	no	—
90	C₂²×C₄×C₈	D₈₃	1	1	yes	characteristic
91	C₂×#334	—	83	12	no	—
92	C₈₃	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
93	C₂³×C₈	—	166	2	no	—
94	C₂³×C₈	Dic₈₃	1	1	yes	characteristic
95	C₂³×C₈	Dic₈₃	1	1	yes	characteristic
96	C₂²×C₄²	C₂×D₈₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	83	12	no	—
98	C₂×C₄⋊C₈	—	166	6	no	—
99	C₂×C₄×C₈	Dic₈₃	1	6	yes	—
100	C₂×C₄×C₈	C₂×D₈₃	1	6	yes	—
101	#334	—	83	16	no	—
102	C₂³×C₄	Dic₁₆₆	1	1	yes	characteristic
103	C₂³×C₄	D₃₃₂	1	1	yes	characteristic
104	C₂³×C₄	C₂×Dic₈₃	1	1	yes	characteristic
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	332	8	no	—
107	C₂²×C₈	—	166	12	no	—
108	C₂²×C₈	C₈₃⋊C₈	1	8	yes	complemented
109	C₂²×C₈	C₂×Dic₈₃	1	6	yes	—
110	C₂²×C₈	C₂×Dic₈₃	1	6	yes	—
111	C₂×C₄²	Dic₁₆₆	1	3	yes	—
112	C₂×C₄²	D₃₃₂	1	3	yes	—
113	C₂×C₄²	C₂×Dic₈₃	1	3	yes	—
114	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
115	C₄⋊C₈	—	83	8	no	—
116	C₄⋊C₈	—	166	12	no	—
117	C₄×C₈	C₂×Dic₈₃	1	12	yes	—
118	C₄×C₈	C₂²×D₈₃	1	4	yes	—
119	C₂⁴	C₄⋊Dic₈₃	1	1	yes	characteristic
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	C₄⋊Dic₈₃	1	3	yes	—
122	C₂²×C₄	C₄⋊Dic₈₃	1	3	yes	—
123	C₂²×C₄	C₄⋊Dic₈₃	1	3	yes	—
124	C₂²×C₄	C₂×C₈₃⋊C₈	1	4	yes	—
125	C₂²×C₄	#13050574	1	4	yes	—
126	C₂²×C₄	#13050579	1	8	yes	—
127	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
128	C₂²×C₄	C₂×D₃₃₂	1	3	yes	—
129	C₂²×C₄	C₂×Dic₁₆₆	1	3	yes	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	—	332	24	no	—
132	C₂×C₈	—	166	8	no	—
133	C₂×C₈	C₂×C₈₃⋊C₈	1	24	yes	complemented
134	C₂×C₈	C₂²×Dic₈₃	1	4	yes	—
135	C₂×C₈	C₂²×Dic₈₃	1	4	yes	—
136	C₄²	C₄⋊Dic₈₃	1	6	yes	—
137	C₄²	C₂²×Dic₈₃	1	3	yes	—
138	C₄²	C₂×D₃₃₂	1	3	yes	—
139	C₄²	C₂×Dic₁₆₆	1	3	yes	—
140	C₄²	C₂³×D₈₃	1	1	yes	characteristic
141	C₂³	C₄⋊(C₈₃⋊C₈)	1	4	yes	—
142	C₂³	#14581724	1	4	yes	—
143	C₂³	#14581728	1	4	yes	—
144	C₂³	C₂×C₄⋊Dic₈₃	1	3	yes	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	C₄⋊(C₈₃⋊C₈)	1	12	yes	—
148	C₂×C₄	#14581724	1	12	yes	—
149	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
150	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
152	C₂×C₄	C₂²×C₈₃⋊C₈	1	6	yes	—
153	C₂×C₄	C₂×#13050574	1	6	yes	—
154	C₂×C₄	C₂×#13050579	1	12	yes	—
155	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	332	8	no	—
160	C₈	C₂²×C₈₃⋊C₈	1	8	yes	complemented
161	C₂²	#19127452	1	16	yes	direct product factor
162	C₂²	C₂×C₄⋊(C₈₃⋊C₈)	1	6	yes	—
163	C₂²	C₂×#14581724	1	6	yes	—
164	C₂²	C₂×#14581728	1	6	yes	—
165	C₂²	C₂²×C₄⋊Dic₈₃	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	3	no	—
168	C₄	C₂×C₄⋊(C₈₃⋊C₈)	1	6	yes	—
169	C₄	C₂×#14581724	1	6	yes	—
170	C₄	C₂³×C₈₃⋊C₈	1	1	yes	characteristic
171	C₄	C₂²×#13050574	1	1	yes	characteristic
172	C₄	C₂²×#13050579	1	2	yes	—
173	C₂	C₂×#19127452	1	12	yes	direct product factor
174	C₂	C₂²×C₄⋊(C₈₃⋊C₈)	1	1	yes	characteristic
175	C₂	C₂²×#14581724	1	1	yes	characteristic
176	C₂	C₂²×#14581728	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₃₃₂
commutator	C₄×C₈₃	C₂³×C₈
Frattini	C₄²	C₂³×D₈₃
Fitting	C₂²×C₄×C₈×C₈₃	C₂
socle, prime radical	C₂⁴×C₈₃	C₄⋊C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₃
hypocenter	C₈₃	C₂²×#334
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19127452	C₂	direct (N × H)
7. #19127452	C₂²	direct (N × H)
21. C₂²×C₈×C₈₃	C₈	semidirect
44. C₂×C₈×C₈₃	C₂×C₈	semidirect
70. C₈×C₈₃	C₂²×C₈	semidirect
72. C₂²×C₈₃	#334	semidirect
85. C₂×C₈₃	C₂×#334	semidirect
92. C₈₃	C₂²×#334	semidirect
108. C₂²×C₈	C₈₃⋊C₈	semidirect
133. C₂×C₈	C₂×C₈₃⋊C₈	semidirect
160. C₈	C₂²×C₈₃⋊C₈	semidirect
161. C₂²	#19127452	direct (N × H)
173. C₂	C₂×#19127452	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×C₄⋊(C₈₃⋊C₈)	2
C₂×#19127452	2
C₂²×C₄×C₈×C₈₃	2
C₂²×#334	83

Subgroups — C22×#19127452 #35373370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19127452 #35373370