Subgroups — #36466041

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	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
3	#26018322	C₂	1	2	yes	maximal, complemented
4	#26018316	C₂	1	2	yes	maximal, complemented
5	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083549	—	2	1	no	—
7	C₄⋊#13083549	—	2	1	no	—
8	C₄⋊#13083549	C₂²	1	1	yes	characteristic
9	C₄⋊#13083549	C₂²	1	2	yes	—
10	C₄⋊#13083549	C₂²	1	1	yes	characteristic
11	C₂²⋊#13083549	—	2	1	no	—
12	C₄×#13083549	—	2	1	no	—
13	#431×C₈₉	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	#322×C₈₉	C₄	1	2	yes	—
17	C₈²×C₈₉	C₂²	1	2	yes	—
18	C₂×#13083549	—	2	1	no	—
19	C₂×#13083549	—	2	1	no	—
20	C₂×#13083549	—	2	2	no	—
21	C₂×#13083549	—	4	2	no	—
22	C₂×M₁₆×C₈₉	C₈	1	1	yes	characteristic
23	C₂×M₁₆×C₈₉	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	1	no	—
28	C₂²⋊C₈×C₈₉	—	2	1	no	—
29	#97×C₈₉	C₈	1	2	yes	—
30	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
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	#13083549	—	4	1	no	—
37	#13083549	—	4	2	no	—
38	#13083549	—	4	1	no	—
39	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
42	M₁₆×C₈₉	—	2	2	no	—
43	M₁₆×C₈₉	C₁₆	1	4	yes	complemented
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	—	2	1	no	—
46	C₂×C₈×C₈₉	—	2	2	no	—
47	C₂×C₈×C₈₉	—	4	1	no	—
48	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
56	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
57	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
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	—	2	2	no	—
74	C₈×C₈₉	—	4	2	no	—
75	C₈×C₈₉	C₂×C₁₆	1	4	yes	complemented
76	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₈×C₈₉	#137	1	1	yes	characteristic
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	—	89	1	no	—
97	C₈×M₁₆	D₈₉	1	1	yes	characteristic
98	#1370	—	89	2	no	—
99	#1367	—	89	2	no	—
100	C₈₉	#12015	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	178	1	no	—
102	#431	Dic₈₉	1	1	yes	characteristic
103	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	C₄⋊C₁₆	—	89	1	no	—
106	C₄⋊C₁₆	—	89	2	no	—
107	C₄⋊C₁₆	—	89	1	no	—
108	C₄⋊C₁₆	—	178	1	no	—
109	C₂²⋊C₁₆	—	178	1	no	—
110	C₄×C₁₆	—	178	1	no	—
111	#322	Dic₈₉	1	2	yes	—
112	C₈²	C₂×D₈₉	1	2	yes	—
113	C₂×M₁₆	#10623494	1	1	yes	characteristic
114	C₂×M₁₆	#10623494	1	1	yes	characteristic
115	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
116	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
117	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
118	C₂×C₁₆	—	356	2	no	—
119	C₂×C₁₆	—	178	1	no	—
120	C₂×C₁₆	—	178	1	no	—
121	C₂×C₁₆	—	178	2	no	—
122	C₄⋊C₈	—	2	1	no	—
123	C₂²⋊C₈	—	2	1	no	—
124	#97	#10623494	1	2	yes	—
125	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
126	C₄×C₈	D₃₅₆	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
128	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
130	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
131	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
132	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
133	C₂²×C₄	#13083558	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	#13083549	1	4	yes	complemented
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
141	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
142	C₂×C₈	C₂×#10623494	1	2	yes	—
143	C₂×C₈	C₂×#10623494	1	2	yes	—
144	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
146	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
147	C₄²	C₂×#10623494	1	1	yes	characteristic
148	C₄²	#13083558	1	1	yes	characteristic
149	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
150	C₁₆	—	356	2	no	—
151	C₁₆	—	356	1	no	—
152	C₁₆	—	356	1	no	—
153	C₂³	C₄⋊#10623494	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	C₄⋊#10623494	1	1	yes	characteristic
158	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
159	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
160	C₂×C₄	C₂×#13083549	1	2	yes	—
161	C₂×C₄	#15667952	1	2	yes	—
162	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
163	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
164	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	2	no	—
169	C₈	—	4	2	no	—
170	C₈	C₂×#13083549	1	4	yes	complemented
171	C₈	#15668040	1	1	yes	characteristic
172	C₈	#15668045	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	C₄⋊#13083549	1	2	yes	—
176	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₄⋊#13083549	1	2	yes	—
181	C₄	#19151640	1	1	yes	characteristic
182	C₄	#19151647	1	1	yes	characteristic
183	C₄	C₂²×#13083549	1	1	yes	characteristic
184	C₄	C₂×#15667952	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	#26019004	1	1	yes	characteristic
187	C₂	C₂×C₄⋊#13083549	1	1	yes	characteristic
188	C₂	#26022018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#12015
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #26018322	C₂	semidirect
4. #26018316	C₂	semidirect
43. M₁₆×C₈₉	C₁₆	semidirect
75. C₈×C₈₉	C₂×C₁₆	semidirect
100. C₈₉	#12015	semidirect
135. M₁₆	#13083549	semidirect
170. C₈	C₂×#13083549	semidirect

Maximal subgroups

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

subgroup	index
#26022051	2
C₂×C₄⋊#13083549	2
#26018322	2
#26018316	2
C₈×M₁₆×C₈₉	2
#12015	89