Subgroups — C₄×H(3,𝔽₄)×C₈₉ #36412201

Order 22784.

2664 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(3,𝔽₄)×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2379	C₂	1	9	yes	maximal, complemented
3	C₈₉×#2375	C₂	1	6	yes	maximal, complemented
4	C₄×#124×C₈₉	C₂	1	9	yes	maximal, complemented
5	C₄×#120×C₈₉	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	complemented
7	H(3,𝔽₄)×C₈₉	C₄	1	16	yes	direct product factor
8	C₂×#124×C₈₉	C₂²	1	9	yes	—
9	C₂×#120×C₈₉	C₂²	1	6	yes	—
10	C₂×C₄×Q₈×C₈₉	C₂²	1	3	yes	complemented
11	C₂×C₄×D₄×C₈₉	C₂²	1	9	yes	—
12	#391×C₈₉	C₂²	1	9	yes	complemented
13	#390×C₈₉	C₂²	1	9	yes	—
14	#388×C₈₉	C₂²	1	36	yes	complemented
15	#386×C₈₉	C₂²	1	18	yes	—
16	#385×C₈₉	C₂²	1	18	yes	complemented
17	#382×C₈₉	C₂²	1	9	yes	complemented
18	#379×C₈₉	C₂²	1	6	yes	complemented
19	C₄×C₂²⋊C₄×C₈₉	C₂²	1	18	yes	complemented
20	C₄³×C₈₉	C₂²	1	3	yes	complemented
21	C₂⁵×C₈₉	C₂³	1	2	yes	—
22	C₂²×Q₈×C₈₉	C₂³	1	3	yes	—
23	C₂²×D₄×C₈₉	C₂³	1	9	yes	—
24	C₂³×C₄×C₈₉	—	4	12	no	—
25	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
27	#124×C₈₉	C₂×C₄	1	72	yes	complemented
28	#120×C₈₉	C₂×C₄	1	48	yes	complemented
29	C₄×Q₈×C₈₉	—	4	6	no	—
30	C₄×D₄×C₈₉	—	4	18	no	—
31	C₂×C₄⋊C₄×C₈₉	C₂³	1	9	yes	—
32	C₂×C₄⋊C₄×C₈₉	C₂³	1	9	yes	—
33	C₂×C₂²⋊C₄×C₈₉	C₂³	1	18	yes	—
34	C₂×C₂²⋊C₄×C₈₉	C₂³	1	18	yes	—
35	C₂×C₂²⋊C₄×C₈₉	C₂³	1	18	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
37	C₂×C₄²×C₈₉	C₂³	1	9	yes	—
38	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
39	#95×C₈₉	C₂³	1	36	yes	—
40	C₂⁴×C₈₉	—	4	6	no	—
41	C₂⁴×C₈₉	—	4	6	no	—
42	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	complemented
43	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
44	C₂×Q₈×C₈₉	—	4	3	no	—
45	C₂×Q₈×C₈₉	—	4	3	no	—
46	C₂×Q₈×C₈₉	C₂²×C₄	1	12	yes	complemented
47	C₂×D₄×C₈₉	—	4	9	no	—
48	C₂×D₄×C₈₉	—	4	9	no	—
49	C₂×D₄×C₈₉	C₂²×C₄	1	36	yes	—
50	C₂²×C₄×C₈₉	—	2	36	no	—
51	C₂²×C₄×C₈₉	—	4	18	no	—
52	C₂²×C₄×C₈₉	—	4	32	no	—
53	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
54	C₂²×C₄×C₈₉	C₂⁴	1	9	yes	—
55	C₂²×C₄×C₈₉	C₂⁴	1	9	yes	—
56	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₉	—	4	9	no	—
58	C₄⋊C₄×C₈₉	—	4	9	no	—
59	C₂²⋊C₄×C₈₉	—	4	18	no	—
60	C₂²⋊C₄×C₈₉	—	4	18	no	—
61	C₂²⋊C₄×C₈₉	C₂²×C₄	1	72	yes	complemented
62	C₄²×C₈₉	—	2	18	no	—
63	C₄²×C₈₉	—	4	9	no	—
64	C₄²×C₈₉	C₂²×C₄	1	12	yes	complemented
65	C₂³×C₈₉	—	2	18	no	—
66	C₂³×C₈₉	—	2	18	no	—
67	C₂³×C₈₉	—	4	24	no	—
68	C₂³×C₈₉	—	4	8	no	—
69	C₂³×C₈₉	—	4	24	no	—
70	C₂³×C₈₉	C₂³×C₄	1	12	yes	—
71	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
72	Q₈×C₈₉	—	4	12	no	—
73	D₄×C₈₉	—	4	36	no	—
74	C₂×C₄×C₈₉	—	2	9	no	—
75	C₂×C₄×C₈₉	—	2	9	no	—
76	C₂×C₄×C₈₉	—	2	9	no	—
77	C₂×C₄×C₈₉	—	2	18	no	—
78	C₂×C₄×C₈₉	—	2	18	no	—
79	C₂×C₄×C₈₉	—	4	24	no	—
80	C₂×C₄×C₈₉	—	4	24	no	—
81	C₂×C₄×C₈₉	C₂³×C₄	1	18	yes	—
82	C₂×C₄×C₈₉	H(5,𝔽₂)	1	6	yes	—
83	C₂²×C₈₉	—	2	36	no	—
84	C₂²×C₈₉	—	4	32	no	—
85	C₂²×C₈₉	—	4	6	no	—
86	C₂²×C₈₉	—	4	18	no	—
87	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
88	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	3	yes	—
89	C₂²×C₈₉	#585	1	3	yes	—
90	C₄×C₈₉	—	2	18	no	—
91	C₄×C₈₉	—	4	6	no	—
92	C₄×C₈₉	—	4	9	no	—
93	C₄×C₈₉	H(3,𝔽₄)	1	4	yes	direct product factor
94	C₄×H(3,𝔽₄)	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₈₉	—	4	12	no	—
96	C₂×C₈₉	C₄×H(5,𝔽₂)	1	3	yes	—
97	C₂×C₈₉	C₂×H(3,𝔽₄)	1	1	yes	characteristic
98	C₂×C₈₉	#3529	1	3	yes	—
99	C₂×H(3,𝔽₄)	C₂×C₈₉	1	1	yes	characteristic
100	#2379	C₂×C₈₉	1	9	yes	complemented
101	#2375	C₂×C₈₉	1	6	yes	complemented
102	C₄×#124	C₂×C₈₉	1	9	yes	complemented
103	C₄×#120	C₂×C₈₉	1	6	yes	complemented
104	C₈₉	C₄×H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
105	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	complemented
106	H(3,𝔽₄)	C₄×C₈₉	1	16	yes	direct product factor
107	C₂×#124	C₂²×C₈₉	1	9	yes	—
108	C₂×#120	C₂²×C₈₉	1	6	yes	—
109	C₂×C₄×Q₈	C₂²×C₈₉	1	3	yes	complemented
110	C₂×C₄×D₄	C₂²×C₈₉	1	9	yes	—
111	#391	C₂²×C₈₉	1	9	yes	complemented
112	#390	C₂²×C₈₉	1	9	yes	—
113	#388	C₂²×C₈₉	1	36	yes	complemented
114	#386	C₂²×C₈₉	1	18	yes	—
115	#385	C₂²×C₈₉	1	18	yes	complemented
116	#382	C₂²×C₈₉	1	9	yes	complemented
117	#379	C₂²×C₈₉	1	6	yes	complemented
118	C₄×C₂²⋊C₄	C₂²×C₈₉	1	18	yes	complemented
119	C₄³	C₂²×C₈₉	1	3	yes	complemented
120	C₂⁵	C₂³×C₈₉	1	2	yes	—
121	C₂²×Q₈	C₂³×C₈₉	1	3	yes	—
122	C₂²×D₄	C₂³×C₈₉	1	9	yes	—
123	C₂³×C₄	—	4	12	no	—
124	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
125	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
126	#124	C₂×C₄×C₈₉	1	72	yes	complemented
127	#120	C₂×C₄×C₈₉	1	48	yes	complemented
128	C₄×Q₈	—	4	6	no	—
129	C₄×D₄	—	4	18	no	—
130	C₂×C₄⋊C₄	C₂³×C₈₉	1	9	yes	—
131	C₂×C₄⋊C₄	C₂³×C₈₉	1	9	yes	—
132	C₂×C₂²⋊C₄	C₂³×C₈₉	1	18	yes	—
133	C₂×C₂²⋊C₄	C₂³×C₈₉	1	18	yes	—
134	C₂×C₂²⋊C₄	C₂³×C₈₉	1	18	yes	—
135	C₂×C₄²	C₂³×C₈₉	1	9	yes	—
136	C₂×C₄²	C₂³×C₈₉	1	9	yes	—
137	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
138	#95	C₂³×C₈₉	1	36	yes	—
139	C₂⁴	—	4	6	no	—
140	C₂⁴	—	4	6	no	—
141	C₂⁴	C₂²×C₄×C₈₉	1	8	yes	complemented
142	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
143	C₂×Q₈	—	4	3	no	—
144	C₂×Q₈	—	4	3	no	—
145	C₂×Q₈	C₂²×C₄×C₈₉	1	12	yes	complemented
146	C₂×D₄	—	4	9	no	—
147	C₂×D₄	—	4	9	no	—
148	C₂×D₄	C₂²×C₄×C₈₉	1	36	yes	—
149	C₂²×C₄	—	2	36	no	—
150	C₂²×C₄	—	4	18	no	—
151	C₂²×C₄	—	4	32	no	—
152	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
153	C₂²×C₄	C₂⁴×C₈₉	1	9	yes	—
154	C₂²×C₄	C₂⁴×C₈₉	1	9	yes	—
155	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
156	C₄⋊C₄	—	4	9	no	—
157	C₄⋊C₄	—	4	9	no	—
158	C₂²⋊C₄	—	4	18	no	—
159	C₂²⋊C₄	—	4	18	no	—
160	C₂²⋊C₄	C₂²×C₄×C₈₉	1	72	yes	complemented
161	C₄²	—	2	18	no	—
162	C₄²	—	4	9	no	—
163	C₄²	C₂²×C₄×C₈₉	1	12	yes	complemented
164	C₂³	—	2	18	no	—
165	C₂³	—	2	18	no	—
166	C₂³	—	4	24	no	—
167	C₂³	—	4	8	no	—
168	C₂³	—	4	24	no	—
169	C₂³	C₂³×C₄×C₈₉	1	12	yes	—
170	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
171	Q₈	—	4	12	no	—
172	D₄	—	4	36	no	—
173	C₂×C₄	—	2	9	no	—
174	C₂×C₄	—	2	9	no	—
175	C₂×C₄	—	2	9	no	—
176	C₂×C₄	—	2	18	no	—
177	C₂×C₄	—	2	18	no	—
178	C₂×C₄	—	4	24	no	—
179	C₂×C₄	—	4	24	no	—
180	C₂×C₄	C₂³×C₄×C₈₉	1	18	yes	—
181	C₂×C₄	H(5,𝔽₂)×C₈₉	1	6	yes	—
182	C₂²	—	2	36	no	—
183	C₂²	—	4	32	no	—
184	C₂²	—	4	6	no	—
185	C₂²	—	4	18	no	—
186	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
187	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	3	yes	—
188	C₂²	#585×C₈₉	1	3	yes	—
189	C₄	—	2	18	no	—
190	C₄	—	4	6	no	—
191	C₄	—	4	9	no	—
192	C₄	H(3,𝔽₄)×C₈₉	1	4	yes	direct product factor
193	C₂	—	4	12	no	—
194	C₂	C₄×H(5,𝔽₂)×C₈₉	1	3	yes	—
195	C₂	C₂×H(3,𝔽₄)×C₈₉	1	1	yes	characteristic
196	C₂	C₈₉×#3529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂³	C₂⁵×C₈₉
socle	C₂³×C₈₉	C₂⁵
prime radical	C₂×H(3,𝔽₄)×C₈₉	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2379	C₂	semidirect
3. C₈₉×#2375	C₂	semidirect
4. C₄×#124×C₈₉	C₂	semidirect
5. C₄×#120×C₈₉	C₂	semidirect
6. C₂⁴×C₄×C₈₉	C₂²	semidirect
7. H(3,𝔽₄)×C₈₉	C₄	direct (N × H)
10. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
12. #391×C₈₉	C₂²	semidirect
14. #388×C₈₉	C₂²	semidirect
16. #385×C₈₉	C₂²	semidirect
17. #382×C₈₉	C₂²	semidirect
18. #379×C₈₉	C₂²	semidirect
19. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
20. C₄³×C₈₉	C₂²	semidirect
27. #124×C₈₉	C₂×C₄	semidirect
28. #120×C₈₉	C₂×C₄	semidirect
42. C₂⁴×C₈₉	C₂²×C₄	semidirect
46. C₂×Q₈×C₈₉	C₂²×C₄	semidirect
61. C₂²⋊C₄×C₈₉	C₂²×C₄	semidirect
64. C₄²×C₈₉	C₂²×C₄	semidirect
93. C₄×C₈₉	H(3,𝔽₄)	direct (N × H)
94. C₄×H(3,𝔽₄)	C₈₉	direct (N × H)
100. #2379	C₂×C₈₉	semidirect
101. #2375	C₂×C₈₉	semidirect
102. C₄×#124	C₂×C₈₉	semidirect
103. C₄×#120	C₂×C₈₉	semidirect
104. C₈₉	C₄×H(3,𝔽₄)	direct (N × H)
105. C₂⁴×C₄	C₂²×C₈₉	semidirect
106. H(3,𝔽₄)	C₄×C₈₉	direct (N × H)
109. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
111. #391	C₂²×C₈₉	semidirect
113. #388	C₂²×C₈₉	semidirect
115. #385	C₂²×C₈₉	semidirect
116. #382	C₂²×C₈₉	semidirect
117. #379	C₂²×C₈₉	semidirect
118. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
119. C₄³	C₂²×C₈₉	semidirect
126. #124	C₂×C₄×C₈₉	semidirect
127. #120	C₂×C₄×C₈₉	semidirect
141. C₂⁴	C₂²×C₄×C₈₉	semidirect
145. C₂×Q₈	C₂²×C₄×C₈₉	semidirect
160. C₂²⋊C₄	C₂²×C₄×C₈₉	semidirect
163. C₄²	C₂²×C₄×C₈₉	semidirect
192. C₄	H(3,𝔽₄)×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×H(3,𝔽₄)×C₈₉	2
C₈₉×#2379	2
C₈₉×#2375	2
C₄×#124×C₈₉	2
C₄×#120×C₈₉	2
C₄×H(3,𝔽₄)	89

Subgroups — C4×H(3,𝔽4)×C89 #36412201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×H(3,𝔽₄)×C₈₉ #36412201