Subgroups — C₂×C₈₉×#2864 #36414482

Order 22784.

2078 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	C₈₉×#2864	C₂	1	16	yes	maximal, direct product factor
2	C₂×#389×C₈₉	C₂	1	4	yes	maximal
3	C₂×#387×C₈₉	C₂	1	8	yes	maximal
4	C₂×#382×C₈₉	C₂	1	2	yes	maximal
5	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
7	C₂²×C₄⋊C₄×C₈₉	C₂²	1	8	yes	—
8	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
9	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	#389×C₈₉	C₂²	1	32	yes	—
12	#387×C₈₉	C₂²	1	64	yes	—
13	#382×C₈₉	C₂²	1	16	yes	—
14	C₂×#95×C₈₉	C₂²	1	8	yes	—
15	C₂×#95×C₈₉	C₂²	1	8	yes	—
16	C₄³×C₈₉	C₂²	1	8	yes	—
17	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
18	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
19	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
20	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₈₉	—	2	32	no	—
22	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
23	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
24	C₂×C₄⋊C₄×C₈₉	C₂³	1	32	yes	—
25	C₂×C₄²×C₈₉	—	2	16	no	—
26	C₂×C₄²×C₈₉	—	2	8	no	—
27	C₂×C₄²×C₈₉	Q₈	1	8	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
29	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
31	#95×C₈₉	C₂³	1	32	yes	—
32	#95×C₈₉	C₂³	1	32	yes	—
33	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	—	2	8	no	—
35	C₂²×C₄×C₈₉	—	2	8	no	—
36	C₂²×C₄×C₈₉	—	2	8	no	—
37	C₂²×C₄×C₈₉	—	2	16	no	—
38	C₂²×C₄×C₈₉	—	2	32	no	—
39	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
40	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₈₉	#41	1	16	yes	—
42	C₂²×C₄×C₈₉	#41	1	8	yes	—
43	C₂²×C₄×C₈₉	C₂⁴	1	16	yes	—
44	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
47	C₄⋊C₄×C₈₉	—	2	64	no	—
48	C₄⋊C₄×C₈₉	—	2	32	no	—
49	C₄²×C₈₉	—	2	32	no	—
50	C₄²×C₈₉	—	2	16	no	—
51	C₄²×C₈₉	C₂×Q₈	1	16	yes	—
52	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
53	C₂³×C₈₉	C₂×#41	1	4	yes	—
54	C₂³×C₈₉	C₂×#41	1	8	yes	—
55	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	32	no	—
57	C₂×C₄×C₈₉	—	2	16	no	—
58	C₂×C₄×C₈₉	—	2	8	no	—
59	C₂×C₄×C₈₉	—	2	8	no	—
60	C₂×C₄×C₈₉	—	2	8	no	—
61	C₂×C₄×C₈₉	—	2	16	no	—
62	C₂×C₄×C₈₉	—	2	32	no	—
63	C₂×C₄×C₈₉	—	4	32	no	—
64	C₂×C₄×C₈₉	#125	1	16	yes	—
65	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
66	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
67	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
68	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
69	C₂²×C₈₉	C₂×#125	1	4	yes	—
70	C₂²×C₈₉	#529	1	16	yes	—
71	C₂²×C₈₉	#533	1	8	yes	—
72	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
73	C₂²×C₈₉	C₂²×#41	1	2	yes	—
74	C₂²×C₈₉	C₂²×#41	1	4	yes	—
75	C₄×C₈₉	—	2	16	no	—
76	C₄×C₈₉	—	2	8	no	—
77	C₄×C₈₉	—	4	16	no	—
78	C₄×C₈₉	C₂×#125	1	8	yes	—
79	C₂×#2864	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₈₉	#2864	1	8	yes	direct product factor
81	C₂×C₈₉	C₂²×#125	1	1	yes	characteristic
82	C₂×C₈₉	C₂×#529	1	4	yes	—
83	C₂×C₈₉	C₂×#533	1	2	yes	—
84	#2864	C₂×C₈₉	1	16	yes	direct product factor
85	C₂×#389	C₂×C₈₉	1	4	yes	—
86	C₂×#387	C₂×C₈₉	1	8	yes	—
87	C₂×#382	C₂×C₈₉	1	2	yes	—
88	C₂×C₄³	C₂×C₈₉	1	1	yes	characteristic
89	C₈₉	C₂×#2864	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
91	C₂²×C₄⋊C₄	C₂²×C₈₉	1	8	yes	—
92	C₂²×C₄²	C₂²×C₈₉	1	4	yes	—
93	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
94	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
95	#389	C₂²×C₈₉	1	32	yes	—
96	#387	C₂²×C₈₉	1	64	yes	—
97	#382	C₂²×C₈₉	1	16	yes	—
98	C₂×#95	C₂²×C₈₉	1	8	yes	—
99	C₂×#95	C₂²×C₈₉	1	8	yes	—
100	C₄³	C₂²×C₈₉	1	8	yes	—
101	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
102	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
103	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
104	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	—	2	32	no	—
106	C₂×C₄⋊C₄	—	2	16	no	—
107	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
108	C₂×C₄⋊C₄	C₂³×C₈₉	1	32	yes	—
109	C₂×C₄²	—	2	16	no	—
110	C₂×C₄²	—	2	8	no	—
111	C₂×C₄²	Q₈×C₈₉	1	8	yes	—
112	C₂×C₄²	C₂³×C₈₉	1	16	yes	—
113	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
114	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
115	#95	C₂³×C₈₉	1	32	yes	—
116	#95	C₂³×C₈₉	1	32	yes	—
117	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
118	C₂²×C₄	—	2	8	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	2	8	no	—
121	C₂²×C₄	—	2	16	no	—
122	C₂²×C₄	—	2	32	no	—
123	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
124	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
125	C₂²×C₄	#41×C₈₉	1	16	yes	—
126	C₂²×C₄	#41×C₈₉	1	8	yes	—
127	C₂²×C₄	C₂⁴×C₈₉	1	16	yes	—
128	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
129	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
130	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
131	C₄⋊C₄	—	2	64	no	—
132	C₄⋊C₄	—	2	32	no	—
133	C₄²	—	2	32	no	—
134	C₄²	—	2	16	no	—
135	C₄²	C₂×Q₈×C₈₉	1	16	yes	—
136	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
137	C₂³	C₂×#41×C₈₉	1	4	yes	—
138	C₂³	C₂×#41×C₈₉	1	8	yes	—
139	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
140	C₂×C₄	—	2	32	no	—
141	C₂×C₄	—	2	16	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	16	no	—
146	C₂×C₄	—	2	32	no	—
147	C₂×C₄	—	4	32	no	—
148	C₂×C₄	#125×C₈₉	1	16	yes	—
149	C₂×C₄	C₂²×Q₈×C₈₉	1	8	yes	—
150	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
151	C₂×C₄	C₂×#41×C₈₉	1	16	yes	—
152	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
153	C₂²	C₂×#125×C₈₉	1	4	yes	—
154	C₂²	#529×C₈₉	1	16	yes	—
155	C₂²	#533×C₈₉	1	8	yes	—
156	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
157	C₂²	C₂²×#41×C₈₉	1	2	yes	—
158	C₂²	C₂²×#41×C₈₉	1	4	yes	—
159	C₄	—	2	16	no	—
160	C₄	—	2	8	no	—
161	C₄	—	4	16	no	—
162	C₄	C₂×#125×C₈₉	1	8	yes	—
163	C₂	C₈₉×#2864	1	8	yes	direct product factor
164	C₂	C₂²×#125×C₈₉	1	1	yes	characteristic
165	C₂	C₂×#529×C₈₉	1	4	yes	—
166	C₂	C₂×#533×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2864	C₂	direct (N × H)
79. C₂×#2864	C₈₉	direct (N × H)
80. C₂×C₈₉	#2864	direct (N × H)
84. #2864	C₂×C₈₉	direct (N × H)
89. C₈₉	C₂×#2864	direct (N × H)
163. C₂	C₈₉×#2864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2864	2
C₂×#389×C₈₉	2
C₂×#387×C₈₉	2
C₂×#382×C₈₉	2
C₂×C₄³×C₈₉	2
C₂×#2864	89

Subgroups — C2×C89×#2864 #36414482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2864 #36414482