Subgroups — C₂²×#19151416 #36707402

Order 22784.

862 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#13083549	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151416	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10623494	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₈₉	C₄	1	1	yes	characteristic
7	C₂²×#13083549	—	2	4	no	—
8	C₂²×#13083549	C₂²	1	6	yes	—
9	C₂×C₈×Dic₈₉	C₄	1	6	yes	—
10	C₂×C₈×Dic₈₉	C₂²	1	6	yes	—
11	#19151416	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₈₉	—	2	4	no	—
14	C₂²×C₁₆×C₈₉	C₂²	1	6	yes	—
15	C₂³×Dic₈₉	C₈	1	1	yes	characteristic
16	C₂³×Dic₈₉	C₈	1	1	yes	characteristic
17	C₂²×#10623494	—	2	4	no	—
18	C₂²×#10623494	C₂×C₄	1	6	yes	—
19	C₂×C₄×Dic₈₉	C₈	1	6	yes	—
20	C₂×C₄×Dic₈₉	C₂×C₄	1	6	yes	—
21	C₂×#13083549	—	2	12	no	—
22	C₂×#13083549	C₂³	1	4	yes	—
23	C₈×Dic₈₉	C₂×C₄	1	12	yes	—
24	C₈×Dic₈₉	C₂³	1	4	yes	—
25	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₈₉	D₄	1	4	yes	—
28	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
30	C₂×C₁₆×C₈₉	—	2	12	no	—
31	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
32	C₂²×Dic₈₉	—	2	4	no	—
33	C₂²×Dic₈₉	C₁₆	1	8	yes	complemented
34	C₂²×Dic₈₉	C₂×C₈	1	6	yes	—
35	C₂²×Dic₈₉	C₂×C₈	1	6	yes	—
36	C₂×#10623494	—	2	12	no	—
37	C₂×#10623494	C₂²×C₄	1	4	yes	—
38	C₄×Dic₈₉	C₂×C₈	1	12	yes	—
39	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
40	#13083549	—	2	4	no	—
41	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₈₉	M₁₆	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
46	C₂×C₈×C₈₉	C₄⋊C₄	1	12	yes	—
47	C₂×C₈×C₈₉	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₈₉	C₂×Q₈	1	6	yes	—
50	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
51	C₁₆×C₈₉	—	2	4	no	—
52	C₂³×C₈₉	C₄⋊C₈	1	4	yes	—
53	C₂³×C₈₉	C₂×C₁₆	1	4	yes	—
54	C₂³×C₈₉	M₃₂	1	4	yes	—
55	C₂³×C₈₉	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₈₉	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₈₉	C₂²×C₈	1	3	yes	—
59	C₂×C₄×C₈₉	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₈₉	—	2	12	no	—
62	C₂×Dic₈₉	C₂×C₁₆	1	24	yes	complemented
63	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
64	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
65	C₈×C₈₉	C₂×C₄⋊C₄	1	6	yes	—
66	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
67	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
68	#10623494	—	2	4	no	—
69	C₂²×C₈₉	C₄⋊C₁₆	1	16	yes	complemented
70	C₂²×C₈₉	C₂×C₄⋊C₈	1	6	yes	—
71	C₂²×C₈₉	C₂²×C₁₆	1	6	yes	—
72	C₂²×C₈₉	C₂×M₃₂	1	6	yes	—
73	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
74	C₄×C₈₉	C₂×C₄⋊C₈	1	6	yes	—
75	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
77	Dic₈₉	—	2	4	no	—
78	Dic₈₉	C₂²×C₁₆	1	8	yes	complemented
79	C₂²×C₄⋊C₁₆	—	89	1	no	maximal, Sylow
80	C₂×C₈₉	C₂×C₄⋊C₁₆	1	12	yes	complemented
81	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
82	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
83	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
84	C₂³×C₁₆	—	89	1	no	—
85	C₂³×C₁₆	D₈₉	1	1	yes	characteristic
86	C₂²×C₄×C₈	—	89	1	no	—
87	C₂×C₄⋊C₁₆	—	89	12	no	—
88	C₈₉	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₈	—	89	1	no	—
90	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
91	C₂²×C₄²	—	89	1	no	—
92	C₂²×C₁₆	—	2	4	no	—
93	C₂²×C₁₆	—	89	6	no	—
94	C₂²×C₁₆	—	178	4	no	—
95	C₂²×C₁₆	C₂×D₈₉	1	6	yes	—
96	C₂×C₄×C₈	—	89	6	no	—
97	C₂×C₄×C₈	—	89	6	no	—
98	C₄⋊C₁₆	—	89	16	no	—
99	C₂³×C₄	—	89	1	no	—
100	C₂³×C₄	—	89	1	no	—
101	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
102	C₂²×C₈	—	89	6	no	—
103	C₂²×C₈	—	178	4	no	—
104	C₂²×C₈	Dic₁₇₈	1	4	yes	—
105	C₂²×C₈	C₄×D₈₉	1	3	yes	—
106	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
107	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
108	C₂×C₄²	—	89	6	no	—
109	C₂×C₄²	—	89	6	no	—
110	C₂×C₁₆	—	2	12	no	—
111	C₂×C₁₆	—	89	4	no	—
112	C₂×C₁₆	—	178	12	no	—
113	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
114	C₄×C₈	—	89	12	no	—
115	C₄×C₈	—	89	4	no	—
116	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
117	C₂²×C₄	—	89	6	no	—
118	C₂²×C₄	—	89	8	no	—
119	C₂²×C₄	—	89	6	no	—
120	C₂²×C₄	—	178	4	no	—
121	C₂²×C₄	#13083553	1	4	yes	—
122	C₂²×C₄	C₈×D₈₉	1	3	yes	—
123	C₂²×C₄	#13083557	1	4	yes	—
124	C₂²×C₄	C₂×C₄×D₈₉	1	3	yes	—
125	C₂×C₈	—	89	4	no	—
126	C₂×C₈	—	178	12	no	—
127	C₂×C₈	#13083553	1	12	yes	—
128	C₂×C₈	C₂×C₄×D₈₉	1	3	yes	—
129	C₂×C₈	C₂×C₂²⋊D₈₉	1	6	yes	—
130	C₂×C₈	C₂×Dic₁₇₈	1	6	yes	—
131	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
132	C₄²	—	89	12	no	—
133	C₄²	—	89	4	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	—	178	4	no	—
136	C₂³	#15667941	1	4	yes	—
137	C₂³	C₁₆×D₈₉	1	4	yes	—
138	C₂³	#15667951	1	4	yes	—
139	C₂³	C₂×C₈×D₈₉	1	3	yes	—
140	C₂×C₄	—	89	24	no	—
141	C₂×C₄	—	89	4	no	—
142	C₂×C₄	—	89	4	no	—
143	C₂×C₄	—	178	12	no	—
144	C₂×C₄	#15667941	1	12	yes	—
145	C₂×C₄	C₂×#13083553	1	6	yes	—
146	C₂×C₄	C₂×C₈×D₈₉	1	3	yes	—
147	C₂×C₄	C₂×#13083557	1	6	yes	—
148	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
149	C₈	—	178	4	no	—
150	C₈	C₂×#13083553	1	6	yes	—
151	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
152	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
153	C₂²	#19151416	1	16	yes	direct product factor
154	C₂²	C₂×#15667941	1	6	yes	—
155	C₂²	C₂×C₁₆×D₈₉	1	6	yes	—
156	C₂²	C₂×#15667951	1	6	yes	—
157	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
158	C₄	—	89	8	no	—
159	C₄	—	178	4	no	—
160	C₄	C₂×#15667941	1	6	yes	—
161	C₄	C₂²×#13083553	1	1	yes	characteristic
162	C₄	C₂²×#13083557	1	1	yes	characteristic
163	C₂	C₂×#19151416	1	12	yes	direct product factor
164	C₂	C₂²×#15667941	1	1	yes	characteristic
165	C₂	C₂²×C₁₆×D₈₉	1	1	yes	characteristic
166	C₂	C₂²×#15667951	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×#19151416	C₂	direct (N × H)
11. #19151416	C₂²	direct (N × H)
33. C₂²×Dic₈₉	C₁₆	semidirect
62. C₂×Dic₈₉	C₂×C₁₆	semidirect
69. C₂²×C₈₉	C₄⋊C₁₆	semidirect
78. Dic₈₉	C₂²×C₁₆	semidirect
80. C₂×C₈₉	C₂×C₄⋊C₁₆	semidirect
88. C₈₉	C₂²×C₄⋊C₁₆	semidirect
153. C₂²	#19151416	direct (N × H)
163. C₂	C₂×#19151416	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#13083549	2
C₂²×C₈×Dic₈₉	2
C₂×#19151416	2
C₂³×C₁₆×C₈₉	2
C₂²×C₄⋊C₁₆	89

Subgroups — C22×#19151416 #36707402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151416 #36707402