Subgroups — #2868×D₈₉ #36760523

Order 22784.

3774 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×C₄⋊Q₈×D₈₉	C₂	1	6	yes	maximal
2	#26028650	C₂	1	1	yes	maximal, characteristic, complemented
3	#26028649	C₂	1	8	yes	maximal, complemented
4	#26028648	C₂	1	6	yes	maximal, complemented
5	#384×D₈₉	C₂	1	8	yes	maximal
6	C₄³×D₈₉	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×D₈₉	C₂²	1	4	yes	—
9	C₂×#15668012	C₂²	1	6	yes	—
10	C₂×#15668011	C₂²	1	24	yes	—
11	C₂×#15668010	C₂²	1	12	yes	—
12	C₄⋊Q₈×D₈₉	—	2	48	no	—
13	C₂×C₄⋊C₄×D₈₉	C₂²	1	24	yes	—
14	C₂×C₄²×D₈₉	C₂²	1	3	yes	—
15	C₂×C₄²×D₈₉	C₂²	1	4	yes	—
16	#19151468	C₂²	1	8	yes	—
17	#19151467	C₂²	1	48	yes	—
18	C₄²×Dic₈₉	C₂²	1	3	yes	—
19	C₄²×Dic₈₉	C₂²	1	4	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	6	yes	—
21	#384×C₈₉	C₂²	1	8	yes	—
22	C₄³×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₇₈	C₂³	1	4	yes	—
24	C₂²×Dic₁₇₈	C₂³	1	8	yes	—
25	C₂×Q₈×D₈₉	—	2	48	no	—
26	C₂²×C₄×D₈₉	C₂³	1	8	yes	—
27	C₂²×C₄×D₈₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×D₈₉	C₂³	1	6	yes	—
29	#15668012	—	2	24	no	—
30	#15668011	—	2	96	no	—
31	#15668010	—	2	48	no	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
33	C₂×#13083553	C₂³	1	48	yes	—
34	C₄⋊C₄×D₈₉	—	2	96	no	—
35	C₂×C₄×Dic₈₉	C₂³	1	3	yes	—
36	C₂×C₄×Dic₈₉	C₂³	1	6	yes	—
37	C₂×C₄×Dic₈₉	C₂³	1	12	yes	—
38	C₄²×D₈₉	D₄	1	24	yes	—
39	C₄²×D₈₉	Q₈	1	32	yes	complemented
40	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
41	C₄⋊Q₈×C₈₉	—	2	24	no	—
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
43	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
45	C₂³×D₈₉	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₇₈	—	2	24	no	—
47	C₂×Dic₁₇₈	—	2	48	no	—
48	Q₈×D₈₉	—	4	64	no	—
49	C₂²×Dic₈₉	C₂⁴	1	8	yes	—
50	C₂²×Dic₈₉	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₈₉	C₂⁴	1	6	yes	—
52	C₂×C₄×D₈₉	—	2	48	no	—
53	C₂×C₄×D₈₉	C₂×D₄	1	12	yes	—
54	C₂×C₄×D₈₉	C₂×D₄	1	24	yes	—
55	C₂×C₄×D₈₉	C₂×Q₈	1	48	yes	—
56	C₄⋊Dic₈₉	—	2	48	no	—
57	#13083553	—	2	96	no	—
58	C₄×Dic₈₉	C₂×D₄	1	12	yes	—
59	C₄×Dic₈₉	C₂×D₄	1	24	yes	—
60	C₄×Dic₈₉	C₂×Q₈	1	48	yes	complemented
61	C₂×Q₈×C₈₉	—	2	24	no	—
62	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
65	C₄⋊C₄×C₈₉	—	2	48	no	—
66	C₄²×C₈₉	C₂×D₄	1	12	yes	—
67	C₄²×C₈₉	C₂×Q₈	1	16	yes	complemented
68	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
69	C₂²×D₈₉	C₂²×D₄	1	6	yes	—
70	C₂²×D₈₉	C₂²×Q₈	1	8	yes	—
71	Q₈×C₈₉	—	4	16	no	—
72	C₂×C₄×C₈₉	—	2	24	no	—
73	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
74	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
75	C₂×C₄×C₈₉	C₂²×Q₈	1	24	yes	—
76	C₂×Dic₈₉	—	2	24	no	—
77	C₂×Dic₈₉	C₂²×D₄	1	6	yes	—
78	C₂×Dic₈₉	C₂²×D₄	1	12	yes	—
79	C₂×Dic₈₉	C₂²×Q₈	1	24	yes	—
80	C₄×D₈₉	—	4	32	no	—
81	C₄×D₈₉	#127	1	16	yes	—
82	C₄×D₈₉	C₄⋊Q₈	1	96	yes	complemented
83	Dic₁₇₈	—	4	16	no	—
84	Dic₁₇₈	—	4	32	no	—
85	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
86	C₂²×C₈₉	C₂³×Q₈	1	4	yes	—
87	C₂×D₈₉	C₂×#127	1	4	yes	—
88	C₂×D₈₉	C₂×C₄⋊Q₈	1	24	yes	—
89	C₄×C₈₉	—	4	8	no	—
90	C₄×C₈₉	C₂×#127	1	4	yes	—
91	C₄×C₈₉	C₂×C₄⋊Q₈	1	24	yes	complemented
92	Dic₈₉	—	4	8	no	—
93	Dic₈₉	C₂×#127	1	4	yes	—
94	Dic₈₉	C₂×C₄⋊Q₈	1	24	yes	complemented
95	C₂×#2868	—	89	1	no	maximal, Sylow
96	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
97	C₂×C₈₉	C₂²×C₄⋊Q₈	1	6	yes	—
98	D₈₉	#2868	1	8	yes	direct product factor
99	C₂²×C₄⋊Q₈	—	89	6	no	—
100	#2868	—	89	8	no	—
101	#2868	—	89	1	no	—
102	#2868	—	89	6	no	—
103	#2868	D₈₉	1	1	yes	characteristic, direct product factor
104	C₂×#384	—	89	8	no	—
105	C₂×C₄³	—	89	1	no	—
106	C₈₉	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
107	C₂³×Q₈	—	89	4	no	—
108	C₂×C₄⋊Q₈	—	89	12	no	—
109	C₂×C₄⋊Q₈	—	89	24	no	—
110	C₂×C₄⋊Q₈	—	89	6	no	—
111	C₂×C₄⋊Q₈	—	178	48	no	—
112	C₂×C₄⋊Q₈	C₂×D₈₉	1	6	yes	—
113	C₂²×C₄⋊C₄	—	89	24	no	—
114	C₂²×C₄²	—	89	3	no	—
115	C₂²×C₄²	—	89	4	no	—
116	#384	—	89	48	no	—
117	#384	—	89	8	no	—
118	#384	C₂×D₈₉	1	8	yes	—
119	C₄³	—	89	4	no	—
120	C₄³	—	89	3	no	—
121	C₄³	C₂×D₈₉	1	1	yes	characteristic
122	C₂²×Q₈	—	89	8	no	—
123	C₂²×Q₈	—	89	4	no	—
124	C₂²×Q₈	—	178	48	no	—
125	C₂²×Q₈	C₂²×D₈₉	1	4	yes	—
126	C₂³×C₄	—	89	6	no	—
127	C₂³×C₄	—	89	1	no	—
128	C₂³×C₄	—	89	8	no	—
129	C₄⋊Q₈	—	2	24	no	—
130	C₄⋊Q₈	—	178	96	no	—
131	C₄⋊Q₈	—	178	48	no	—
132	C₄⋊Q₈	—	178	24	no	—
133	C₂×C₄⋊C₄	—	89	48	no	—
134	C₂×C₄⋊C₄	—	89	24	no	—
135	C₂×C₄⋊C₄	—	178	96	no	—
136	C₂×C₄⋊C₄	C₂²×D₈₉	1	24	yes	—
137	C₂×C₄²	—	89	3	no	—
138	C₂×C₄²	—	89	6	no	—
139	C₂×C₄²	—	89	12	no	—
140	C₂×C₄²	—	89	24	no	—
141	C₂×C₄²	—	89	32	no	—
142	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
143	C₂×C₄²	C₂²×D₈₉	1	4	yes	—
144	C₂⁴	—	89	1	no	—
145	C₂×Q₈	—	2	24	no	—
146	C₂×Q₈	—	356	64	no	—
147	C₂×Q₈	—	178	48	no	—
148	C₂×Q₈	—	178	24	no	—
149	C₂²×C₄	—	89	6	no	—
150	C₂²×C₄	—	89	1	no	—
151	C₂²×C₄	—	89	24	no	—
152	C₂²×C₄	—	89	12	no	—
153	C₂²×C₄	—	89	48	no	—
154	C₂²×C₄	—	89	8	no	—
155	C₂²×C₄	—	178	48	no	—
156	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
157	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
158	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
159	C₄⋊C₄	—	2	48	no	—
160	C₄⋊C₄	—	178	96	no	—
161	C₄⋊C₄	—	178	48	no	—
162	C₄²	—	89	12	no	—
163	C₄²	—	89	24	no	—
164	C₄²	—	89	48	no	—
165	C₄²	D₄×D₈₉	1	12	yes	—
166	C₄²	Q₈×D₈₉	1	16	yes	complemented
167	C₂³	—	89	6	no	—
168	C₂³	—	89	8	no	—
169	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
170	Q₈	—	4	16	no	—
171	Q₈	—	356	32	no	—
172	Q₈	—	356	16	no	—
173	C₂×C₄	—	2	24	no	—
174	C₂×C₄	—	89	12	no	—
175	C₂×C₄	—	89	6	no	—
176	C₂×C₄	—	89	24	no	—
177	C₂×C₄	—	89	16	no	—
178	C₂×C₄	—	89	96	no	—
179	C₂×C₄	—	356	32	no	—
180	C₂×C₄	—	178	24	no	—
181	C₂×C₄	C₂×D₄×D₈₉	1	6	yes	—
182	C₂×C₄	C₂×D₄×D₈₉	1	12	yes	—
183	C₂×C₄	C₂×Q₈×D₈₉	1	24	yes	—
184	C₂²	—	89	4	no	—
185	C₂²	—	89	24	no	—
186	C₂²	C₂²×D₄×D₈₉	1	3	yes	—
187	C₂²	C₂²×Q₈×D₈₉	1	4	yes	—
188	C₄	—	4	8	no	—
189	C₄	—	89	4	no	—
190	C₄	—	89	24	no	—
191	C₄	—	356	8	no	—
192	C₄	#127×D₈₉	1	4	yes	—
193	C₄	C₄⋊Q₈×D₈₉	1	24	yes	complemented
194	C₂	—	89	8	no	—
195	C₂	C₂×#127×D₈₉	1	1	yes	characteristic
196	C₂	C₂×C₄⋊Q₈×D₈₉	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁵
Fitting	C₈₉×#2868	C₂
prime radical	C₂³×D₈₉	C₂⁴
hypercenter	#2868	D₈₉
hypocenter	C₈₉	C₂×#2868
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #2868

Complements

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

N	H ≅ G/N	type
2. #26028650	C₂	semidirect
3. #26028649	C₂	semidirect
4. #26028648	C₂	semidirect
7. C₈₉×#2868	C₂	semidirect
39. C₄²×D₈₉	Q₈	semidirect
60. C₄×Dic₈₉	C₂×Q₈	semidirect
67. C₄²×C₈₉	C₂×Q₈	semidirect
82. C₄×D₈₉	C₄⋊Q₈	semidirect
91. C₄×C₈₉	C₂×C₄⋊Q₈	semidirect
94. Dic₈₉	C₂×C₄⋊Q₈	semidirect
98. D₈₉	#2868	direct (N × H)
103. #2868	D₈₉	direct (N × H)
106. C₈₉	C₂×#2868	semidirect
166. C₄²	Q₈×D₈₉	semidirect
193. C₄	C₄⋊Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Q₈×D₈₉	2
#26028650	2
#26028649	2
#26028648	2
#384×D₈₉	2
C₄³×D₈₉	2
C₈₉×#2868	2
C₂×#2868	89

Subgroups — #2868×D89 #36760523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2868×D₈₉ #36760523