Subgroups — Q₈×#127×C₈₉ #36414405

Order 22784.

3498 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₈₉	C₂	1	6	yes	maximal, complemented
2	C₈₉×#2868	C₂	1	3	yes	maximal, complemented
3	C₈₉×#2542	C₂	1	18	yes	maximal, complemented
4	C₄×#127×C₈₉	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₈₉	C₂²	1	4	yes	—
7	D₄×Q₈×C₈₉	—	2	48	no	—
8	C₂×C₄⋊Q₈×C₈₉	C₂²	1	18	yes	—
9	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	36	yes	—
11	C₂×C₄×Q₈×C₈₉	C₂²	1	3	yes	—
12	C₂×C₄×D₄×C₈₉	C₂²	1	18	yes	—
13	#399×C₈₉	C₂²	1	24	yes	—
14	#390×C₈₉	C₂²	1	18	yes	—
15	#384×C₈₉	C₂²	1	3	yes	—
16	#384×C₈₉	C₂²	1	18	yes	—
17	C₄×C₄⋊C₄×C₈₉	C₂²	1	9	yes	—
18	C₄³×C₈₉	C₂²	1	3	yes	—
19	C₂²×Q₈×C₈₉	—	2	48	no	—
20	C₂²×Q₈×C₈₉	C₂³	1	12	yes	—
21	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₂²×D₄×C₈₉	C₂³	1	6	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
24	C₄⋊Q₈×C₈₉	—	2	72	no	—
25	#127×C₈₉	Q₈	1	8	yes	direct product factor
26	C₂²⋊Q₈×C₈₉	—	2	144	no	—
27	C₄×Q₈×C₈₉	D₄	1	24	yes	complemented
28	C₄×D₄×C₈₉	—	2	72	no	—
29	C₂×C₄⋊C₄×C₈₉	C₂³	1	18	yes	—
30	C₂×C₄⋊C₄×C₈₉	C₂³	1	36	yes	—
31	C₂×C₄⋊C₄×C₈₉	C₂³	1	9	yes	—
32	C₂×C₂²⋊C₄×C₈₉	C₂³	1	36	yes	—
33	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
36	#95×C₈₉	C₂³	1	6	yes	—
37	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
38	C₂×Q₈×C₈₉	—	2	72	no	—
39	C₂×Q₈×C₈₉	—	4	64	no	—
40	C₂×Q₈×C₈₉	C₂×D₄	1	12	yes	—
41	C₂×D₄×C₈₉	—	2	12	no	—
42	C₂×D₄×C₈₉	—	2	12	no	—
43	C₂×D₄×C₈₉	C₂×Q₈	1	24	yes	complemented
44	C₂²×C₄×C₈₉	—	2	72	no	—
45	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
46	C₂²×C₄×C₈₉	C₂⁴	1	9	yes	—
47	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
48	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
49	C₄⋊C₄×C₈₉	—	2	36	no	—
50	C₄⋊C₄×C₈₉	—	2	72	no	—
51	C₄⋊C₄×C₈₉	C₂×D₄	1	36	yes	—
52	C₂²⋊C₄×C₈₉	—	2	72	no	—
53	C₄²×C₈₉	—	2	18	no	—
54	C₄²×C₈₉	C₂×D₄	1	36	yes	—
55	C₄²×C₈₉	C₂×Q₈	1	4	yes	complemented
56	C₂³×C₈₉	—	2	12	no	—
57	C₂³×C₈₉	—	2	12	no	—
58	C₂³×C₈₉	C₂²×Q₈	1	8	yes	—
59	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
60	Q₈×C₈₉	—	4	48	no	—
61	Q₈×C₈₉	#127	1	16	yes	direct product factor
62	D₄×C₈₉	—	2	48	no	—
63	C₂×C₄×C₈₉	—	2	18	no	—
64	C₂×C₄×C₈₉	—	2	36	no	—
65	C₂×C₄×C₈₉	—	4	48	no	—
66	C₂×C₄×C₈₉	C₂²×D₄	1	18	yes	—
67	C₂×C₄×C₈₉	C₂²×D₄	1	18	yes	—
68	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
69	C₂×C₄×C₈₉	C₂²×Q₈	1	6	yes	—
70	C₂×C₄×C₈₉	2₋¹⁺⁴	1	6	yes	—
71	C₂²×C₈₉	—	2	24	no	—
72	C₂²×C₈₉	—	4	4	no	—
73	C₂²×C₈₉	—	4	12	no	—
74	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
75	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
76	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	3	yes	—
77	C₄×C₈₉	—	2	18	no	—
78	C₄×C₈₉	—	4	12	no	—
79	C₄×C₈₉	C₂×#127	1	12	yes	—
80	C₄×C₈₉	D₄×Q₈	1	12	yes	complemented
81	Q₈×#127	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₈₉	—	4	8	no	—
83	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
84	C₂×C₈₉	C₂×D₄×Q₈	1	3	yes	—
85	C₂×C₈₉	#3499	1	3	yes	—
86	C₂×D₄×Q₈	C₂×C₈₉	1	6	yes	complemented
87	#2868	C₂×C₈₉	1	3	yes	complemented
88	#2542	C₂×C₈₉	1	18	yes	complemented
89	C₄×#127	C₂×C₈₉	1	3	yes	—
90	C₄²×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
91	C₈₉	Q₈×#127	1	1	yes	characteristic, direct product factor, Sylow
92	C₂³×Q₈	C₂²×C₈₉	1	4	yes	—
93	D₄×Q₈	—	2	48	no	—
94	C₂×C₄⋊Q₈	C₂²×C₈₉	1	18	yes	—
95	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
96	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	36	yes	—
97	C₂×C₄×Q₈	C₂²×C₈₉	1	3	yes	—
98	C₂×C₄×D₄	C₂²×C₈₉	1	18	yes	—
99	#399	C₂²×C₈₉	1	24	yes	—
100	#390	C₂²×C₈₉	1	18	yes	—
101	#384	C₂²×C₈₉	1	3	yes	—
102	#384	C₂²×C₈₉	1	18	yes	—
103	C₄×C₄⋊C₄	C₂²×C₈₉	1	9	yes	—
104	C₄³	C₂²×C₈₉	1	3	yes	—
105	C₂²×Q₈	—	2	48	no	—
106	C₂²×Q₈	C₂³×C₈₉	1	12	yes	—
107	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
108	C₂²×D₄	C₂³×C₈₉	1	6	yes	—
109	C₂³×C₄	C₂³×C₈₉	1	12	yes	—
110	C₄⋊Q₈	—	2	72	no	—
111	#127	Q₈×C₈₉	1	8	yes	direct product factor
112	C₂²⋊Q₈	—	2	144	no	—
113	C₄×Q₈	D₄×C₈₉	1	24	yes	complemented
114	C₄×D₄	—	2	72	no	—
115	C₂×C₄⋊C₄	C₂³×C₈₉	1	18	yes	—
116	C₂×C₄⋊C₄	C₂³×C₈₉	1	36	yes	—
117	C₂×C₄⋊C₄	C₂³×C₈₉	1	9	yes	—
118	C₂×C₂²⋊C₄	C₂³×C₈₉	1	36	yes	—
119	C₂×C₄²	C₂³×C₈₉	1	9	yes	—
120	C₂×C₄²	C₂³×C₈₉	1	9	yes	—
121	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
122	#95	C₂³×C₈₉	1	6	yes	—
123	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
124	C₂×Q₈	—	2	72	no	—
125	C₂×Q₈	—	4	64	no	—
126	C₂×Q₈	C₂×D₄×C₈₉	1	12	yes	—
127	C₂×D₄	—	2	12	no	—
128	C₂×D₄	—	2	12	no	—
129	C₂×D₄	C₂×Q₈×C₈₉	1	24	yes	complemented
130	C₂²×C₄	—	2	72	no	—
131	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
132	C₂²×C₄	C₂⁴×C₈₉	1	9	yes	—
133	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
135	C₄⋊C₄	—	2	36	no	—
136	C₄⋊C₄	—	2	72	no	—
137	C₄⋊C₄	C₂×D₄×C₈₉	1	36	yes	—
138	C₂²⋊C₄	—	2	72	no	—
139	C₄²	—	2	18	no	—
140	C₄²	C₂×D₄×C₈₉	1	36	yes	—
141	C₄²	C₂×Q₈×C₈₉	1	4	yes	complemented
142	C₂³	—	2	12	no	—
143	C₂³	—	2	12	no	—
144	C₂³	C₂²×Q₈×C₈₉	1	8	yes	—
145	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
146	Q₈	—	4	48	no	—
147	Q₈	#127×C₈₉	1	16	yes	direct product factor
148	D₄	—	2	48	no	—
149	C₂×C₄	—	2	18	no	—
150	C₂×C₄	—	2	36	no	—
151	C₂×C₄	—	4	48	no	—
152	C₂×C₄	C₂²×D₄×C₈₉	1	18	yes	—
153	C₂×C₄	C₂²×D₄×C₈₉	1	18	yes	—
154	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
155	C₂×C₄	C₂²×Q₈×C₈₉	1	6	yes	—
156	C₂×C₄	2₋¹⁺⁴×C₈₉	1	6	yes	—
157	C₂²	—	2	24	no	—
158	C₂²	—	4	4	no	—
159	C₂²	—	4	12	no	—
160	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
161	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
162	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	3	yes	—
163	C₄	—	2	18	no	—
164	C₄	—	4	12	no	—
165	C₄	C₂×#127×C₈₉	1	12	yes	—
166	C₄	D₄×Q₈×C₈₉	1	12	yes	complemented
167	C₂	—	4	8	no	—
168	C₂	C₂²×#127×C₈₉	1	1	yes	characteristic
169	C₂	C₂×D₄×Q₈×C₈₉	1	3	yes	—
170	C₂	C₈₉×#3499	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈×C₈₉	C₂	semidirect
2. C₈₉×#2868	C₂	semidirect
3. C₈₉×#2542	C₂	semidirect
5. C₄²×Q₈×C₈₉	C₂	semidirect
25. #127×C₈₉	Q₈	direct (N × H)
27. C₄×Q₈×C₈₉	D₄	semidirect
43. C₂×D₄×C₈₉	C₂×Q₈	semidirect
55. C₄²×C₈₉	C₂×Q₈	semidirect
61. Q₈×C₈₉	#127	direct (N × H)
80. C₄×C₈₉	D₄×Q₈	semidirect
81. Q₈×#127	C₈₉	direct (N × H)
86. C₂×D₄×Q₈	C₂×C₈₉	semidirect
87. #2868	C₂×C₈₉	semidirect
88. #2542	C₂×C₈₉	semidirect
90. C₄²×Q₈	C₂×C₈₉	semidirect
91. C₈₉	Q₈×#127	direct (N × H)
111. #127	Q₈×C₈₉	direct (N × H)
113. C₄×Q₈	D₄×C₈₉	semidirect
129. C₂×D₄	C₂×Q₈×C₈₉	semidirect
141. C₄²	C₂×Q₈×C₈₉	semidirect
147. Q₈	#127×C₈₉	direct (N × H)
166. C₄	D₄×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₈₉	2
C₈₉×#2868	2
C₈₉×#2542	2
C₄×#127×C₈₉	2
C₄²×Q₈×C₈₉	2
Q₈×#127	89

Subgroups — Q8×#127×C89 #36414405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#127×C₈₉ #36414405