Subgroups — C₄×#19127526 #35319804

Order 21248.

366 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	C₂×#19127526	C₂	1	2	yes	maximal
2	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#14581726	C₂	1	1	yes	maximal, characteristic
4	#24905554	C₂	1	2	yes	maximal
5	C₄×#107×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14581726	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₈₃	—	2	1	no	—
10	#19127526	C₄	1	8	yes	direct product factor
11	C₄×C₈₃⋊C₁₆	C₄	1	4	yes	complemented
12	#19127464	—	2	1	no	—
13	C₂×#107×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₈₃	—	2	1	no	—
16	#340×C₈₃	—	2	1	no	—
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₂×C₄×Dic₈₃	—	4	1	no	—
19	C₂×C₈₃⋊C₁₆	—	2	2	no	—
20	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
23	#14581726	—	2	2	no	—
24	#14581726	C₂×C₄	1	4	yes	—
25	C₈₃⋊#95	—	4	1	no	—
26	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₂×C₄²×C₈₃	—	4	1	no	—
30	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
31	#107×C₈₃	—	2	2	no	—
32	#107×C₈₃	C₂×C₄	1	4	yes	—
33	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₈₃	Q₈	1	2	yes	—
35	#95×C₈₃	—	4	1	no	—
36	C₂²×Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	2	4	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	C₄×Dic₈₃	—	8	2	no	—
40	C₈₃⋊C₁₆	—	2	2	no	—
41	C₈₃⋊C₁₆	C₄²	1	4	yes	complemented
42	C₂²×C₄×C₈₃	—	4	2	no	—
43	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	—	2	2	no	—
47	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₈₃	#41	1	2	yes	—
52	C₄⋊C₄×C₈₃	—	2	4	no	—
53	C₄⋊C₄×C₈₃	—	4	2	no	—
54	C₄²×C₈₃	—	8	2	no	—
55	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₈₃	Q₁₆	1	2	yes	—
57	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	4	4	no	—
60	C₂×C₄×C₈₃	—	8	2	no	—
61	C₂×C₄×C₈₃	—	8	2	no	—
62	C₂×C₄×C₈₃	#107	1	2	yes	—
63	C₂×C₄×C₈₃	#107	1	2	yes	—
64	C₂×C₄×C₈₃	D₁₆	1	2	yes	—
65	C₂×C₄×C₈₃	Q₃₂	1	2	yes	—
66	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₈₃	#135	1	2	yes	—
70	C₂×Dic₈₃	—	4	4	no	—
71	C₂×Dic₈₃	—	8	2	no	—
72	C₂×Dic₈₃	—	8	2	no	—
73	C₈×C₈₃	—	2	2	no	—
74	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₈₃	#117	1	1	yes	characteristic
76	C₈×C₈₃	C₄×Q₈	1	2	yes	—
77	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₈₃	C₄×D₈	1	2	yes	—
79	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
80	C₂²×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
81	C₂²×C₈₃	#508	1	2	yes	—
82	C₄×C₈₃	—	2	2	no	—
83	C₄×C₈₃	—	8	4	no	—
84	C₄×C₈₃	#366	1	4	yes	complemented
85	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
86	C₄×C₈₃	#427	1	1	yes	characteristic
87	C₄×C₈₃	C₄×Q₁₆	1	2	yes	—
88	Dic₈₃	—	8	4	no	—
89	C₄×#366	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	C₄×#107	1	1	yes	characteristic
91	C₂×C₈₃	C₂×#366	1	1	yes	characteristic
92	C₂×C₈₃	#2158	1	1	yes	characteristic
93	C₂×C₈₃	C₄×D₁₆	1	2	yes	—
94	C₂×C₈₃	C₄×Q₃₂	1	2	yes	—
95	C₂×#366	—	83	2	no	—
96	C₂×C₄×C₁₆	—	83	1	no	—
97	C₄×#107	—	83	1	no	—
98	C₄×#107	D₈₃	1	1	yes	characteristic
99	#1380	—	83	2	no	—
100	C₈₃	C₄×#366	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	83	1	no	—
102	C₂²×C₁₆	—	83	1	no	—
103	C₂×#107	—	83	2	no	—
104	C₂×#107	C₂×D₈₃	1	2	yes	—
105	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄×C₄⋊C₄	—	166	1	no	—
108	#366	—	83	8	no	—
109	C₄×C₁₆	—	83	4	no	—
110	#340	—	2	1	no	—
111	#340	—	166	1	no	—
112	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	166	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	—	332	1	no	—
118	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
119	C₂×C₁₆	—	83	2	no	—
120	C₂×C₁₆	—	83	4	no	—
121	C₂×C₁₆	—	83	2	no	—
122	C₂×C₁₆	—	166	2	no	—
123	#107	—	2	2	no	—
124	#107	—	83	4	no	—
125	#107	—	166	2	no	—
126	#107	C₄×D₈₃	1	4	yes	—
127	C₄×C₈	Dic₁₆₆	1	2	yes	—
128	C₄×C₈	C₄×D₈₃	1	2	yes	—
129	#95	—	4	1	no	—
130	#95	—	332	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	332	2	no	—
133	C₂²×C₄	#13050575	1	1	yes	characteristic
134	C₂²×C₄	#13050575	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
138	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
140	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
141	C₂×C₈	#13050590	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	332	2	no	—
145	C₄⋊C₄	—	166	4	no	—
146	C₄²	—	8	2	no	—
147	C₄²	—	664	2	no	—
148	C₄²	Dic₈₃⋊C₄	1	2	yes	—
149	C₄²	#13050580	1	2	yes	—
150	C₁₆	—	83	4	no	—
151	C₁₆	—	166	2	no	—
152	C₂³	C₂×#13050575	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	332	4	no	—
158	C₂×C₄	—	664	2	no	—
159	C₂×C₄	—	664	2	no	—
160	C₂×C₄	#14581725	1	2	yes	—
161	C₂×C₄	#14581725	1	2	yes	—
162	C₂×C₄	#14581733	1	2	yes	—
163	C₂×C₄	#14581738	1	2	yes	—
164	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
166	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
167	C₂×C₄	#14581812	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₄²×D₈₃	1	1	yes	characteristic
170	C₈	#14581749	1	1	yes	characteristic
171	C₈	C₄×Dic₁₆₆	1	2	yes	—
172	C₂²	C₂×#14581725	1	1	yes	characteristic
173	C₂²	C₄×#13050575	1	2	yes	—
174	C₂²	C₂×#14581733	1	1	yes	characteristic
175	C₂²	C₂×#14581738	1	1	yes	characteristic
176	C₂²	#19128131	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	8	4	no	—
179	C₄	—	664	4	no	—
180	C₄	#19127526	1	4	yes	direct product factor
181	C₄	C₄×Dic₈₃⋊C₄	1	1	yes	characteristic
182	C₄	#19127737	1	1	yes	characteristic
183	C₄	C₄×#13050580	1	2	yes	—
184	C₂	C₄×#14581725	1	1	yes	characteristic
185	C₂	C₂×#19127526	1	1	yes	characteristic
186	C₂	#24909245	1	1	yes	characteristic
187	C₂	C₄×#14581733	1	2	yes	—
188	C₂	C₄×#14581738	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#13050575
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₄×#107×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₄×#107	D₈₃
hypocenter	C₈₃	C₄×#366
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. #19127526	C₄	direct (N × H)
11. C₄×C₈₃⋊C₁₆	C₄	semidirect
41. C₈₃⋊C₁₆	C₄²	semidirect
84. C₄×C₈₃	#366	semidirect
100. C₈₃	C₄×#366	semidirect
180. C₄	#19127526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19127526	2
C₂×C₄×C₈₃⋊C₁₆	2
C₄×#14581726	2
#24905554	2
C₄×#107×C₈₃	2
C₄×#366	83

Subgroups — C4×#19127526 #35319804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19127526 #35319804