Subgroups — C₂²×#19151428 #36707480

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667948	C₂	1	2	yes	maximal
2	C₂×#19151428	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083558	—	2	2	no	—
5	C₂×#15667948	—	2	6	no	—
6	C₂×#15667948	C₂²	1	12	yes	—
7	#19151428	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₈₉	C₄	1	6	yes	—
10	C₂²×C₁₆×C₈₉	C₄	1	2	yes	—
11	C₂²×C₁₆×C₈₉	C₂²	1	6	yes	—
12	C₂×#13083558	—	2	12	no	—
13	C₂×#13083558	—	4	6	no	—
14	C₂²×#10623494	—	4	2	no	—
15	#15667948	—	2	12	no	—
16	#15667948	C₂³	1	8	yes	—
17	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	D₄	1	3	yes	—
21	C₂²×C₈×C₈₉	Q₈	1	3	yes	—
22	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
24	C₂×C₁₆×C₈₉	C₂×C₄	1	12	yes	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	12	yes	—
26	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
27	#13083558	—	2	8	no	—
28	#13083558	—	4	12	no	—
29	C₂×#10623494	—	4	12	no	—
30	C₂⁴×C₈₉	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	D₈	1	3	yes	—
34	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₈₉	Q₁₆	1	3	yes	—
36	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
37	C₂×C₈×C₈₉	C₄⋊C₄	1	6	yes	—
38	C₂×C₈×C₈₉	C₄⋊C₄	1	3	yes	—
39	C₂×C₈×C₈₉	C₄⋊C₄	1	3	yes	—
40	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
41	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
43	C₂×C₈×C₈₉	C₂×Q₈	1	3	yes	—
44	C₂×C₈×C₈₉	C₂×Q₈	1	3	yes	—
45	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₁₆×C₈₉	C₂²×C₄	1	8	yes	—
47	C₂³×C₈₉	—	2	4	no	—
48	C₂³×C₈₉	#107	1	3	yes	—
49	C₂³×C₈₉	#107	1	1	yes	characteristic
50	C₂³×C₈₉	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄×C₈₉	#107	1	6	yes	—
52	C₂×C₄×C₈₉	#107	1	3	yes	—
53	C₂×C₄×C₈₉	#107	1	3	yes	—
54	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₈₉	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
60	C₈×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
61	C₈×C₈₉	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
64	#10623494	—	4	8	no	—
65	C₂²×C₈₉	—	2	6	no	—
66	C₂²×C₈₉	—	2	6	no	—
67	C₂²×C₈₉	#368	1	4	yes	complemented
68	C₂²×C₈₉	C₂×#107	1	3	yes	—
69	C₂²×C₈₉	C₂×#107	1	3	yes	—
70	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
71	C₄×C₈₉	C₂×#107	1	3	yes	—
72	C₄×C₈₉	C₂×#107	1	3	yes	—
73	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
75	C₂²×#368	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	—	2	1	no	—
77	C₂×C₈₉	—	2	3	no	—
78	C₂×C₈₉	C₂×#368	1	6	yes	complemented
79	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
80	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
81	C₂²×#108	—	89	2	no	—
82	C₂×#368	—	89	12	no	—
83	C₈₉	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	—	178	2	no	—
85	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₈₉	1	2	yes	—
87	C₂²×C₁₆	Dic₈₉	1	6	yes	—
88	C₂²×C₁₆	C₂×D₈₉	1	6	yes	—
89	C₂×#108	—	89	12	no	—
90	C₂×#108	—	178	6	no	—
91	#368	—	89	16	no	—
92	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
93	C₂×M₁₆	—	356	6	no	—
94	C₂×M₁₆	—	178	12	no	—
95	C₂²×C₈	—	356	2	no	—
96	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
97	C₂²×C₈	Dic₁₇₈	1	3	yes	—
98	C₂²×C₈	D₃₅₆	1	3	yes	—
99	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
100	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
101	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
102	C₂×C₁₆	C₂×Dic₈₉	1	12	yes	—
103	C₂×C₁₆	C₂×Dic₈₉	1	12	yes	—
104	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
105	#108	—	89	8	no	—
106	#108	—	178	12	no	—
107	C₂⁴	Dic₃₅₆	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
109	C₂²×C₄	C₄⋊Dic₈₉	1	3	yes	—
110	C₂²×C₄	D₇₁₂	1	3	yes	—
111	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
112	C₂²×C₄	Dic₃₅₆	1	3	yes	—
113	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
114	M₁₆	—	356	12	no	—
115	M₁₆	—	178	8	no	—
116	C₂×C₈	—	356	12	no	—
117	C₂×C₈	C₄⋊Dic₈₉	1	6	yes	—
118	C₂×C₈	C₄⋊Dic₈₉	1	3	yes	—
119	C₂×C₈	C₄⋊Dic₈₉	1	3	yes	—
120	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
121	C₂×C₈	C₂²×Dic₈₉	1	3	yes	—
122	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
123	C₂×C₈	C₂×Dic₁₇₈	1	3	yes	—
124	C₂×C₈	C₂×Dic₁₇₈	1	3	yes	—
125	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₈₉	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#15667946	1	3	yes	—
129	C₂³	#15667946	1	1	yes	characteristic
130	C₂³	C₂×Dic₃₅₆	1	3	yes	—
131	C₂×C₄	#15667946	1	6	yes	—
132	C₂×C₄	#15667946	1	3	yes	—
133	C₂×C₄	#15667946	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₈₉	1	3	yes	—
136	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
137	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
138	C₂×C₄	C₂×Dic₃₅₆	1	3	yes	—
139	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
140	C₈	—	356	8	no	—
141	C₈	C₂×C₄⋊Dic₈₉	1	3	yes	—
142	C₈	C₂×C₄⋊Dic₈₉	1	3	yes	—
143	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
144	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#19151428	1	4	yes	direct product factor
148	C₂²	C₂×#15667946	1	3	yes	—
149	C₂²	C₂×#15667946	1	3	yes	—
150	C₂²	C₂²×Dic₃₅₆	1	1	yes	characteristic
151	C₄	C₂×#15667946	1	3	yes	—
152	C₄	C₂×#15667946	1	3	yes	—
153	C₄	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
154	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	1	no	—
157	C₂	C₂×#19151428	1	6	yes	direct product factor
158	C₂	C₂²×#15667946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₇₁₂
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	#107
prime radical	C₂⁴×C₈₉	Q₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	C₂²×#368
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151428	C₂	direct (N × H)
7. #19151428	C₂²	direct (N × H)
67. C₂²×C₈₉	#368	semidirect
78. C₂×C₈₉	C₂×#368	semidirect
83. C₈₉	C₂²×#368	semidirect
147. C₂²	#19151428	direct (N × H)
157. C₂	C₂×#19151428	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×#15667948	2
C₂×#19151428	2
C₂³×C₁₆×C₈₉	2
C₂²×#368	89

Subgroups — C22×#19151428 #36707480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151428 #36707480