Subgroups — #26034747

Order 11392.

890 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	C₈₉⋊#564	C₂	1	4	yes	maximal, complemented
2	#19152327	C₂	1	4	yes	maximal, complemented
3	#19152326	C₂	1	4	yes	maximal, complemented
4	#19152323	C₂	1	4	yes	maximal, complemented
5	#19152210	C₂	1	4	yes	maximal
6	#19152208	C₂	1	8	yes	maximal
7	#19152155	C₂	1	2	yes	maximal
8	#533×C₈₉	C₂	1	1	yes	maximal, characteristic
9	#15668012	C₂²	1	2	yes	—
10	#15668012	C₂²	1	2	yes	—
11	#15668012	C₂²	1	4	yes	—
12	#15668010	C₂²	1	8	yes	—
13	#15668005	C₂²	1	16	yes	—
14	#15668004	C₂²	1	4	yes	—
15	#15668004	C₂²	1	4	yes	—
16	#15668003	C₂²	1	8	yes	—
17	#15668003	C₂²	1	16	yes	—
18	#15668002	C₂²	1	16	yes	—
19	#15668002	C₂²	1	16	yes	—
20	#15667991	C₂²	1	4	yes	—
21	#15667991	C₂²	1	4	yes	—
22	Q₈×Dic₈₉	C₂²	1	8	yes	—
23	C₄×Dic₁₇₈	C₂²	1	4	yes	—
24	C₄×Dic₁₇₈	C₂²	1	4	yes	—
25	C₄×Dic₁₇₈	C₂²	1	8	yes	—
26	#15667977	C₂²	1	8	yes	—
27	#15667971	C₂²	1	4	yes	—
28	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
29	#125×C₈₉	C₂²	1	2	yes	—
30	C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
31	C₄×Q₈×C₈₉	C₂²	1	4	yes	—
32	#117×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×Dic₁₇₈	C₂³	1	4	yes	—
35	C₂×Dic₁₇₈	C₂³	1	4	yes	—
36	C₄⋊Dic₈₉	C₂³	1	4	yes	—
37	C₄⋊Dic₈₉	C₂³	1	4	yes	—
38	C₄⋊Dic₈₉	C₂³	1	4	yes	—
39	C₄⋊Dic₈₉	C₂³	1	4	yes	—
40	C₄⋊Dic₈₉	C₂³	1	16	yes	—
41	#13083553	C₂³	1	16	yes	—
42	#13083553	C₂³	1	8	yes	—
43	#13083553	C₂³	1	8	yes	—
44	#13083553	C₂³	1	16	yes	—
45	C₂²⋊Dic₈₉	C₂³	1	8	yes	—
46	C₄×Dic₈₉	C₂³	1	8	yes	—
47	C₄×Dic₈₉	C₂³	1	16	yes	—
48	C₂×Q₈×C₈₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
52	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
53	C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
54	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
55	C₄²×C₈₉	C₂³	1	1	yes	characteristic
56	C₄²×C₈₉	C₂³	1	4	yes	—
57	C₄²×C₈₉	C₂³	1	2	yes	—
58	C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
60	Q₈×C₈₉	—	2	8	no	—
61	C₂×C₄×C₈₉	—	2	4	no	—
62	C₂×C₄×C₈₉	—	2	2	no	—
63	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
65	C₂×C₄×C₈₉	C₂⁴	1	8	yes	—
66	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
68	C₂×Dic₈₉	C₂⁴	1	8	yes	—
69	C₂×Dic₈₉	C₂⁴	1	8	yes	—
70	Dic₁₇₈	—	2	8	no	—
71	Dic₁₇₈	—	2	8	no	—
72	C₂²×C₈₉	—	2	2	no	—
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	2	1	no	—
77	C₄×C₈₉	—	2	8	no	—
78	C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
79	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
80	Dic₈₉	—	2	8	no	—
81	Dic₈₉	—	2	8	no	—
82	C₂×C₈₉	—	4	1	no	—
83	C₂×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
84	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
85	#3556	—	89	1	no	maximal, Sylow
86	C₈₉	#3556	1	1	yes	characteristic, complemented, Sylow
87	#564	—	89	4	no	—
88	#557	—	89	4	no	—
89	#557	—	89	4	no	—
90	#557	—	89	4	no	—
91	#541	—	89	8	no	—
92	#541	—	89	4	no	—
93	#533	—	89	2	no	—
94	#533	D₈₉	1	1	yes	characteristic
95	C₄⋊Q₈	—	89	8	no	—
96	C₄⋊Q₈	—	89	2	no	—
97	C₄⋊Q₈	—	89	2	no	—
98	C₄⋊Q₈	—	89	4	no	—
99	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
100	#126	—	89	16	no	—
101	#125	—	89	16	no	—
102	#125	—	89	16	no	—
103	#125	—	89	8	no	—
104	#125	—	89	16	no	—
105	#125	—	89	4	no	—
106	#125	—	89	4	no	—
107	#125	C₂×D₈₉	1	2	yes	—
108	C₂²⋊Q₈	—	89	4	no	—
109	C₂²⋊Q₈	—	89	4	no	—
110	C₂²⋊Q₈	C₂×D₈₉	1	4	yes	—
111	C₄×Q₈	—	89	8	no	—
112	C₄×Q₈	—	89	8	no	—
113	C₄×Q₈	—	89	4	no	—
114	C₄×Q₈	—	89	4	no	—
115	C₄×Q₈	—	89	8	no	—
116	C₄×Q₈	C₂×D₈₉	1	4	yes	—
117	#117	—	89	4	no	—
118	#117	C₂×D₈₉	1	2	yes	—
119	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
120	C₂×Q₈	—	89	4	no	—
121	C₂×Q₈	—	89	4	no	—
122	C₂×Q₈	C₂²×D₈₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
124	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
125	C₄⋊C₄	—	89	4	no	—
126	C₄⋊C₄	—	89	4	no	—
127	C₄⋊C₄	—	89	4	no	—
128	C₄⋊C₄	—	89	4	no	—
129	C₄⋊C₄	—	89	8	no	—
130	C₄⋊C₄	—	89	8	no	—
131	C₄⋊C₄	—	89	16	no	—
132	C₄⋊C₄	—	89	16	no	—
133	C₄⋊C₄	—	89	16	no	—
134	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
135	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
136	C₄⋊C₄	C₂²×D₈₉	1	8	yes	—
137	C₂²⋊C₄	—	89	8	no	—
138	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
139	C₄²	—	89	8	no	—
140	C₄²	—	89	16	no	—
141	C₄²	C₂²×D₈₉	1	1	yes	characteristic
142	C₄²	C₂²×D₈₉	1	2	yes	—
143	C₄²	C₂²×D₈₉	1	1	yes	characteristic
144	C₄²	C₂²×D₈₉	1	4	yes	—
145	C₂³	C₂³×D₈₉	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	178	8	no	—
148	Q₈	—	178	8	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	89	8	no	—
152	C₂×C₄	—	89	8	no	—
153	C₂×C₄	C₂³×D₈₉	1	8	yes	—
154	C₂×C₄	C₂³×D₈₉	1	2	yes	—
155	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₈₉	1	2	yes	—
157	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	—	2	1	no	—
160	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	2	8	no	—
163	C₄	—	2	1	no	—
164	C₄	—	178	8	no	—
165	C₄	—	178	8	no	—
166	C₄	#15668059	1	4	yes	—
167	C₄	#15668060	1	2	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#15668059	1	2	yes	—
170	C₂	C₂×#15668060	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#533×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂⁴
hypercenter	#533	D₈₉
hypocenter	C₈₉	#3556
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₈₉⋊#564	C₂	semidirect
2. #19152327	C₂	semidirect
3. #19152326	C₂	semidirect
4. #19152323	C₂	semidirect
86. C₈₉	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₈₉⋊#564	2
#19152327	2
#19152326	2
#19152323	2
#19152210	2
#19152208	2
#19152155	2
#533×C₈₉	2
#3556	89