Subgroups — C₈₉×#33474 #36411322

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2183	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₆×C₈₉	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2151	C₂	1	2	yes	maximal
5	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2108	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#533×C₈₉	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	#446×C₈₉	C₄	1	4	yes	—
13	C₈×Q₈×C₈₉	C₂²	1	4	yes	—
14	#424×C₈₉	C₄	1	2	yes	—
15	#423×C₈₉	C₄	1	1	yes	characteristic
16	#405×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	—
19	C₄⋊C₁₆×C₈₉	C₂²	1	8	yes	—
20	#346×C₈₉	C₂²	1	4	yes	—
21	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
22	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
23	C₄×C₁₆×C₈₉	C₂²	1	4	yes	—
24	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₈₉	C₈	1	2	yes	—
28	#125×C₈₉	C₈	1	2	yes	—
29	C₂²⋊Q₈×C₈₉	C₈	1	4	yes	—
30	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
31	#117×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₈₉	—	2	4	no	—
34	C₂×C₁₆×C₈₉	—	2	2	no	—
35	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
36	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
37	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
38	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
42	#97×C₈₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₈₉	C₂³	1	2	yes	—
45	C₄×C₈×C₈₉	C₂³	1	4	yes	—
46	C₂×Q₈×C₈₉	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₈₉	—	2	4	no	—
50	C₂×C₈×C₈₉	—	2	2	no	—
51	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₈₉	#41	1	2	yes	—
56	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
61	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
64	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
65	C₁₆×C₈₉	—	2	2	no	—
66	C₁₆×C₈₉	—	2	4	no	—
67	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
68	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
69	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₈₉	—	2	8	no	—
71	C₂×C₄×C₈₉	—	2	4	no	—
72	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₈₉	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₈₉	#131	1	2	yes	—
78	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₈₉	—	2	2	no	—
80	C₈×C₈₉	—	2	4	no	—
81	C₈×C₈₉	C₄×Q₈	1	2	yes	—
82	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
86	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	—	2	8	no	—
89	C₄×C₈₉	C₈×Q₈	1	2	yes	—
90	C₄×C₈₉	#504	1	4	yes	—
91	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
92	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
93	#33474	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×C₈₉	—	2	2	no	—
95	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₈₉	C₂×#504	1	2	yes	—
97	#2960	C₂×C₈₉	1	1	yes	characteristic
98	#2183	C₂×C₈₉	1	4	yes	complemented
99	Q₈×C₁₆	C₂×C₈₉	1	4	yes	complemented
100	#2151	C₂×C₈₉	1	2	yes	—
101	#2150	C₂×C₈₉	1	1	yes	characteristic
102	#2108	C₂×C₈₉	1	2	yes	—
103	C₂×C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic
104	C₈₉	#33474	1	1	yes	characteristic, direct product factor, Sylow
105	#533	C₄×C₈₉	1	1	yes	characteristic
106	C₂×M₃₂	C₂²×C₈₉	1	2	yes	—
107	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
108	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
109	#446	C₄×C₈₉	1	4	yes	—
110	C₈×Q₈	C₂²×C₈₉	1	4	yes	—
111	#424	C₄×C₈₉	1	2	yes	—
112	#423	C₄×C₈₉	1	1	yes	characteristic
113	#405	C₂²×C₈₉	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
115	C₄⋊C₁₆	C₂²×C₈₉	1	4	yes	—
116	C₄⋊C₁₆	C₂²×C₈₉	1	8	yes	—
117	#346	C₂²×C₈₉	1	4	yes	—
118	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
119	C₄×C₁₆	C₂²×C₈₉	1	2	yes	—
120	C₄×C₁₆	C₂²×C₈₉	1	4	yes	—
121	C₂×M₁₆	C₂×C₄×C₈₉	1	2	yes	—
122	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
124	C₄⋊Q₈	C₈×C₈₉	1	2	yes	—
125	#125	C₈×C₈₉	1	2	yes	—
126	C₂²⋊Q₈	C₈×C₈₉	1	4	yes	—
127	C₄×Q₈	C₂×C₄×C₈₉	1	4	yes	—
128	#117	C₂×C₄×C₈₉	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
130	M₃₂	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	Q₈×C₈₉	1	4	yes	complemented
133	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
134	C₂×C₁₆	C₂³×C₈₉	1	4	yes	—
135	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
137	C₄⋊C₈	C₂×C₄×C₈₉	1	4	yes	—
138	C₄⋊C₈	C₂×C₄×C₈₉	1	8	yes	—
139	#97	C₂×C₄×C₈₉	1	4	yes	—
140	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
141	C₄×C₈	C₂³×C₈₉	1	2	yes	—
142	C₄×C₈	C₂³×C₈₉	1	4	yes	—
143	C₂×Q₈	C₂×C₈×C₈₉	1	4	yes	—
144	C₂²×C₄	C₂×C₈×C₈₉	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
150	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
152	C₂×C₈	#41×C₈₉	1	2	yes	—
153	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₈×C₈₉	1	4	yes	—
155	C₄⋊C₄	C₂×C₈×C₈₉	1	8	yes	—
156	C₄⋊C₄	C₂×C₈×C₈₉	1	4	yes	—
157	C₂²⋊C₄	C₂×C₈×C₈₉	1	4	yes	—
158	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
159	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
161	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	4	no	—
164	C₁₆	C₂×Q₈×C₈₉	1	2	yes	—
165	C₁₆	C₂×Q₈×C₈₉	1	2	yes	—
166	C₂³	C₂²×C₈×C₈₉	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₄×Q₈×C₈₉	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
171	C₂×C₄	C₂²×C₈×C₈₉	1	8	yes	—
172	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
173	C₂×C₄	C₂²×C₈×C₈₉	1	2	yes	—
174	C₂×C₄	#131×C₈₉	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	C₄×Q₈×C₈₉	1	2	yes	—
179	C₈	C₂²×Q₈×C₈₉	1	1	yes	characteristic
180	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Q₈×C₈₉	1	2	yes	—
183	C₂²	C₂³×C₈×C₈₉	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	C₈×Q₈×C₈₉	1	2	yes	—
187	C₄	#504×C₈₉	1	4	yes	—
188	C₄	C₂×C₄×Q₈×C₈₉	1	1	yes	characteristic
189	C₄	C₂×#131×C₈₉	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Q₈×C₈₉	1	1	yes	characteristic
192	C₂	C₂×#504×C₈₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2183	C₂	semidirect
3. Q₈×C₁₆×C₈₉	C₂	semidirect
35. C₂×C₁₆×C₈₉	Q₈	semidirect
93. #33474	C₈₉	direct (N × H)
98. #2183	C₂×C₈₉	semidirect
99. Q₈×C₁₆	C₂×C₈₉	semidirect
104. C₈₉	#33474	direct (N × H)
132. C₂×C₁₆	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2960	2
C₈₉×#2183	2
Q₈×C₁₆×C₈₉	2
C₈₉×#2151	2
C₈₉×#2150	2
C₈₉×#2108	2
C₂×C₄×C₁₆×C₈₉	2
#33474	89

Subgroups — C89×#33474 #36411322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33474 #36411322