Subgroups — #131×#10623494 #36520014

Order 22784.

922 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10623494	C₂	1	2	yes	maximal
2	#26018535	C₂	1	3	yes	maximal, complemented
3	M₁₆×#10623494	C₂	1	6	yes	maximal, complemented
4	C₂×C₈×#10623494	C₂	1	3	yes	maximal, complemented
5	C₄×#131×C₈₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10623494	C₄	1	4	yes	—
7	D₄×#10623494	C₄	1	12	yes	—
8	#19151651	C₂²	1	6	yes	—
9	C₂×C₄×#10623494	C₂²	1	6	yes	—
10	#19151319	C₂²	1	6	yes	—
11	C₈×#10623494	C₂²	1	1	yes	characteristic
12	C₈×#10623494	C₂²	1	6	yes	—
13	C₈×#10623494	C₂²	1	3	yes	—
14	C₂×#131×C₈₉	C₄	1	2	yes	—
15	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
16	#405×C₈₉	C₄	1	6	yes	—
17	C₄×M₁₆×C₈₉	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
19	C₂²×#10623494	C₂×C₄	1	12	yes	—
20	C₄⋊#10623494	C₂×C₄	1	12	yes	—
21	C₂²⋊#10623494	C₂×C₄	1	12	yes	—
22	C₄×#10623494	C₂×C₄	1	12	yes	—
23	C₄×#10623494	C₂³	1	6	yes	—
24	C₄×#10623494	C₂³	1	2	yes	—
25	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
26	#131×C₈₉	C₈	1	16	yes	complemented
27	C₂×M₁₆×C₈₉	C₂×C₄	1	6	yes	—
28	C₂²×C₈×C₈₉	C₂×C₄	1	6	yes	—
29	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
31	#117×C₈₉	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
33	#97×C₈₉	C₂×C₄	1	6	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	6	yes	—
35	C₄×C₈×C₈₉	C₂³	1	3	yes	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×#10623494	—	2	24	no	—
38	C₂×#10623494	C₂²×C₄	1	12	yes	—
39	C₂×#10623494	C₂²×C₄	1	4	yes	—
40	#41×C₈₉	C₂×C₈	1	8	yes	—
41	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
42	C₂×D₄×C₈₉	C₄²	1	3	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
44	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
45	M₁₆×C₈₉	C₂×C₈	1	24	yes	complemented
46	C₂×C₈×C₈₉	C₂×C₈	1	24	yes	complemented
47	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₈₉	C₄²	1	6	yes	—
50	C₂²⋊C₄×C₈₉	C₄²	1	6	yes	—
51	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
52	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
53	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
54	C₂³×C₈₉	C₂×C₄²	1	3	yes	—
55	Q₈×C₈₉	C₄×C₈	1	4	yes	—
56	D₄×C₈₉	C₄×C₈	1	12	yes	—
57	C₂×C₄×C₈₉	—	2	12	no	—
58	C₂×C₄×C₈₉	C₂×C₄²	1	6	yes	—
59	C₂×C₄×C₈₉	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₈₉	C₂²×C₈	1	12	yes	—
61	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₈₉	C₂²×C₈	1	4	yes	—
64	C₈×C₈₉	C₂²×C₈	1	12	yes	—
65	#10623494	—	2	12	no	—
66	#10623494	#131	1	8	yes	direct product factor
67	C₂²×C₈₉	—	2	6	no	—
68	C₂²×C₈₉	C₂×C₄×C₈	1	6	yes	—
69	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
70	C₄×C₈₉	—	2	6	no	—
71	C₄×C₈₉	C₂×C₄×C₈	1	6	yes	—
72	C₄×C₈₉	C₂³×C₈	1	2	yes	—
73	C₄×C₈₉	C₂×#131	1	4	yes	—
74	C₈×#131	—	89	1	no	maximal, Sylow
75	C₂×C₈₉	—	2	6	no	—
76	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
77	C₂×C₈₉	C₄×#131	1	2	yes	—
78	C₈×#41	—	89	2	no	—
79	C₄×#131	D₈₉	1	1	yes	characteristic
80	#1450	—	89	3	no	—
81	C₈×M₁₆	—	89	6	no	—
82	C₂×C₈²	—	89	3	no	—
83	C₈₉	C₈×#131	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	Dic₈₉	1	2	yes	—
85	C₄×#41	C₂×D₈₉	1	1	yes	characteristic
86	C₈×Q₈	—	89	4	no	—
87	C₈×D₄	—	89	12	no	—
88	#431	—	89	6	no	—
89	#405	Dic₈₉	1	6	yes	—
90	C₄×M₁₆	C₂×D₈₉	1	3	yes	—
91	C₂×C₄×C₈	—	89	6	no	—
92	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
93	#322	—	89	6	no	—
94	C₈²	—	89	6	no	—
95	C₈²	—	89	1	no	—
96	C₈²	—	89	3	no	—
97	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
98	#131	#10623494	1	16	yes	direct product factor
99	C₂×M₁₆	C₂×Dic₈₉	1	6	yes	—
100	C₂²×C₈	—	89	12	no	—
101	C₂²×C₈	C₂×Dic₈₉	1	6	yes	—
102	C₄×Q₈	C₄×D₈₉	1	2	yes	—
103	C₄×D₄	C₄×D₈₉	1	6	yes	—
104	#117	C₂×Dic₈₉	1	3	yes	—
105	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
106	C₄⋊C₈	—	89	12	no	—
107	C₂²⋊C₈	—	89	12	no	—
108	#97	C₂×Dic₈₉	1	6	yes	—
109	C₄×C₈	—	89	12	no	—
110	C₄×C₈	—	89	2	no	—
111	C₄×C₈	—	89	6	no	—
112	C₄×C₈	C₂×Dic₈₉	1	6	yes	—
113	C₄×C₈	C₂²×D₈₉	1	3	yes	—
114	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
115	#41	C₂×#10623494	1	8	yes	—
116	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
117	C₂×D₄	C₄×Dic₈₉	1	3	yes	—
118	C₂²×C₄	C₂×C₄×D₈₉	1	6	yes	—
119	C₂²×C₄	C₂²×Dic₈₉	1	3	yes	—
120	M₁₆	C₂×#10623494	1	24	yes	complemented
121	C₂×C₈	—	89	4	no	—
122	C₂×C₈	—	89	12	no	—
123	C₂×C₈	—	178	24	no	—
124	C₂×C₈	C₂×#10623494	1	24	yes	complemented
125	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
126	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
127	C₄⋊C₄	C₄×Dic₈₉	1	6	yes	—
128	C₂²⋊C₄	C₄×Dic₈₉	1	6	yes	—
129	C₄²	C₂×C₄×D₈₉	1	6	yes	—
130	C₄²	C₂²×Dic₈₉	1	3	yes	—
131	C₄²	C₂³×D₈₉	1	1	yes	characteristic
132	C₂³	C₂×C₄×Dic₈₉	1	3	yes	—
133	Q₈	C₄×#10623494	1	4	yes	—
134	D₄	C₄×#10623494	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂×C₄×Dic₈₉	1	6	yes	—
137	C₂×C₄	C₂×C₄×Dic₈₉	1	3	yes	—
138	C₂×C₄	C₂²×#10623494	1	12	yes	—
139	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
140	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
141	C₈	—	89	8	no	—
142	C₈	—	178	12	no	—
143	C₈	C₂²×#10623494	1	4	yes	—
144	C₈	C₂²×#10623494	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×#10623494	1	6	yes	—
147	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×#10623494	1	6	yes	—
150	C₄	C₂³×#10623494	1	2	yes	—
151	C₄	#131×D₈₉	1	4	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×#10623494	1	1	yes	characteristic
154	C₂	#131×Dic₈₉	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₄×#131×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄²
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₄×#131	D₈₉
hypocenter	C₈₉	C₈×#131
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #26018535	C₂	semidirect
3. M₁₆×#10623494	C₂	semidirect
4. C₂×C₈×#10623494	C₂	semidirect
26. #131×C₈₉	C₈	semidirect
45. M₁₆×C₈₉	C₂×C₈	semidirect
46. C₂×C₈×C₈₉	C₂×C₈	semidirect
66. #10623494	#131	direct (N × H)
83. C₈₉	C₈×#131	semidirect
98. #131	#10623494	direct (N × H)
120. M₁₆	C₂×#10623494	semidirect
124. C₂×C₈	C₂×#10623494	semidirect

Maximal subgroups

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

subgroup	index
#41×#10623494	2
#26018535	2
M₁₆×#10623494	2
C₂×C₈×#10623494	2
C₄×#131×C₈₉	2
C₈×#131	89