Subgroups — #36466039

Order 22784.

246 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022051	C₂	1	1	yes	maximal, characteristic
2	C₈×#13083558	C₂	1	1	yes	maximal, characteristic
3	#26018321	C₂	1	2	yes	maximal, complemented
4	#26018315	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151651	C₄	1	1	yes	characteristic
7	C₄×#13083558	C₄	1	1	yes	characteristic
8	C₄⋊#13083549	C₂²	1	1	yes	characteristic
9	C₄⋊#13083549	C₂²	1	1	yes	characteristic
10	C₂²⋊#13083549	—	2	1	no	—
11	C₄×#13083549	—	2	1	no	—
12	#19151320	C₄	1	2	yes	—
13	C₈×#10623494	C₂²	1	2	yes	—
14	C₂²×C₁₆×C₈₉	—	2	1	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₈₉	—	2	1	no	—
17	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂×#13083558	C₈	1	1	yes	characteristic
19	C₂×#13083558	C₈	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	C₂×#13083549	—	2	1	no	—
22	C₄⋊#10623494	—	2	1	no	—
23	C₂²⋊#10623494	—	2	1	no	—
24	#15667923	C₈	1	2	yes	—
25	C₄×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	—	2	2	no	—
31	C₂×C₁₆×C₈₉	—	4	2	no	—
32	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
34	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
35	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	#13083558	—	2	2	no	—
37	#13083558	C₁₆	1	4	yes	complemented
38	C₂×#10623494	—	2	2	no	—
39	C₂×#10623494	C₂×C₈	1	2	yes	—
40	C₂×#10623494	C₂×C₈	1	2	yes	—
41	#13083549	—	4	1	no	—
42	#13083549	—	4	1	no	—
43	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₈₉	—	2	1	no	—
47	C₂×C₈×C₈₉	—	2	1	no	—
48	C₂×C₈×C₈₉	—	4	1	no	—
49	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
56	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₈₉	—	4	2	no	—
58	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	—	2	1	no	—
60	C₂×C₄×C₈₉	—	2	1	no	—
61	C₂×C₄×C₈₉	—	4	1	no	—
62	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
67	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
70	C₈×C₈₉	—	2	1	no	—
71	C₈×C₈₉	—	2	1	no	—
72	C₈×C₈₉	—	2	1	no	—
73	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₈×C₈₉	#137	1	1	yes	characteristic
75	#10623494	—	2	2	no	—
76	#10623494	—	4	2	no	—
77	#10623494	C₂×C₁₆	1	4	yes	complemented
78	C₂²×C₈₉	—	2	1	no	—
79	C₂²×C₈₉	—	2	1	no	—
80	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
81	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄×C₈₉	—	2	1	no	—
83	C₄×C₈₉	—	2	1	no	—
84	C₄×C₈₉	—	2	1	no	—
85	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
86	C₄×C₈₉	#428	1	1	yes	characteristic
87	C₄×C₈₉	#430	1	1	yes	characteristic
88	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
89	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
90	#12015	—	89	1	no	maximal, Sylow
91	C₂×C₈₉	—	2	2	no	—
92	C₂×C₈₉	#1565	1	1	yes	characteristic
93	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
94	C₂×C₈₉	#2153	1	1	yes	characteristic
95	#2162	—	89	1	no	—
96	C₂×C₄⋊C₁₆	D₈₉	1	1	yes	characteristic
97	C₈×M₁₆	—	89	1	no	—
98	#1370	—	89	2	no	—
99	#1367	—	89	2	no	—
100	C₈₉	#12015	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	2	1	no	—
102	#431	—	89	1	no	—
103	C₄×M₁₆	—	89	1	no	—
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	C₄⋊C₁₆	—	2	1	no	—
106	C₄⋊C₁₆	—	89	1	no	—
107	C₄⋊C₁₆	—	89	1	no	—
108	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
109	C₂²⋊C₁₆	—	178	1	no	—
110	C₄×C₁₆	—	178	1	no	—
111	#322	—	89	2	no	—
112	C₈²	—	89	2	no	—
113	C₂×M₁₆	—	89	1	no	—
114	C₂×M₁₆	—	89	1	no	—
115	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
116	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
117	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	4	2	no	—
120	C₂×C₁₆	—	178	1	no	—
121	C₂×C₁₆	—	178	1	no	—
122	C₄⋊C₈	—	178	1	no	—
123	C₂²⋊C₈	—	178	1	no	—
124	#97	—	89	2	no	—
125	C₄×C₈	—	89	2	no	—
126	C₄×C₈	—	89	2	no	—
127	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
128	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
130	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₄	#13083553	1	1	yes	characteristic
132	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
133	C₂²×C₄	#13083557	1	1	yes	characteristic
134	M₁₆	—	89	4	no	—
135	M₁₆	—	178	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	4	1	no	—
139	C₂×C₈	—	89	2	no	—
140	C₂×C₈	—	89	2	no	—
141	C₂×C₈	—	178	2	no	—
142	C₂×C₈	#13083553	1	1	yes	characteristic
143	C₂×C₈	#13083553	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₄²	#13083553	1	1	yes	characteristic
147	C₄²	C₈×D₈₉	1	1	yes	characteristic
148	C₄²	#13083557	1	1	yes	characteristic
149	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
150	C₁₆	—	4	2	no	—
151	C₁₆	—	356	1	no	—
152	C₁₆	—	356	1	no	—
153	C₂³	#15667941	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	#15667941	1	1	yes	characteristic
158	C₂×C₄	#15667941	1	1	yes	characteristic
159	C₂×C₄	#15667941	1	1	yes	characteristic
160	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
161	C₂×C₄	#15667951	1	2	yes	—
162	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
163	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
164	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	89	4	no	—
169	C₈	—	356	2	no	—
170	C₈	—	178	2	no	—
171	C₈	#15668038	1	1	yes	characteristic
172	C₈	#15668043	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	#19151416	1	2	yes	—
176	C₂²	C₂×#15667941	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#19151416	1	2	yes	—
181	C₄	#19151638	1	1	yes	characteristic
182	C₄	#19151646	1	1	yes	characteristic
183	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
184	C₄	C₂×#15667951	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#26019002	1	1	yes	characteristic
187	C₂	C₂×#19151416	1	1	yes	characteristic
188	C₂	#26022017	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₄⋊C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#12015
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26018321	C₂	semidirect
4. #26018315	C₂	semidirect
37. #13083558	C₁₆	semidirect
77. #10623494	C₂×C₁₆	semidirect
100. C₈₉	#12015	semidirect

Maximal subgroups

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

subgroup	index
#26022051	2
C₈×#13083558	2
#26018321	2
#26018315	2
C₂×C₄⋊C₁₆×C₈₉	2
#12015	89