Subgroups — #36466046

Order 22784.

238 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022055	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	#19151374	—	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	—	2	1	no	—
23	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	C₈	1	2	yes	—
27	C₂²⋊C₈×C₈₉	C₈	1	2	yes	—
28	#97×C₈₉	—	2	1	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	#13083549	—	4	1	no	—
36	#13083549	—	4	2	no	—
37	#13083549	—	4	1	no	—
38	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
41	M₁₆×C₈₉	—	2	4	no	—
42	C₂×C₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	—	2	1	no	—
44	C₂×C₈×C₈₉	—	2	1	no	—
45	C₂×C₈×C₈₉	—	4	1	no	—
46	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
49	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
54	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	1	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₈₉	—	2	4	no	—
67	C₈×C₈₉	—	2	1	no	—
68	C₈×C₈₉	—	2	1	no	—
69	C₈×C₈₉	—	2	1	no	—
70	C₈×C₈₉	—	2	4	no	—
71	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₈×C₈₉	#137	1	1	yes	characteristic
73	C₂²×C₈₉	—	2	1	no	—
74	C₂²×C₈₉	—	2	1	no	—
75	C₂²×C₈₉	#364	1	2	yes	—
76	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₄×C₈₉	—	2	1	no	—
78	C₄×C₈₉	—	2	1	no	—
79	C₄×C₈₉	—	2	1	no	—
80	C₄×C₈₉	—	2	1	no	—
81	C₄×C₈₉	#428	1	1	yes	characteristic
82	C₄×C₈₉	#430	1	1	yes	characteristic
83	C₄×C₈₉	#504	1	2	yes	—
84	#12016	—	89	1	no	maximal, Sylow
85	C₂×C₈₉	—	2	2	no	—
86	C₂×C₈₉	#1565	1	1	yes	characteristic
87	C₂×C₈₉	#2151	1	1	yes	characteristic
88	C₂×C₈₉	C₂×#364	1	1	yes	characteristic
89	#2163	—	89	1	no	—
90	C₂×C₄⋊C₁₆	—	89	1	no	—
91	C₈×M₁₆	D₈₉	1	1	yes	characteristic
92	#1370	—	89	2	no	—
93	#1367	—	89	2	no	—
94	C₈₉	#12016	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	178	1	no	—
96	#431	Dic₈₉	1	1	yes	characteristic
97	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₁₆	—	89	1	no	—
100	C₄⋊C₁₆	—	89	2	no	—
101	C₄⋊C₁₆	—	89	1	no	—
102	C₄⋊C₁₆	—	178	1	no	—
103	C₂²⋊C₁₆	—	178	1	no	—
104	#346	—	178	1	no	—
105	#322	Dic₈₉	1	2	yes	—
106	C₈²	C₂×D₈₉	1	2	yes	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
109	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
111	C₂×C₁₆	—	356	2	no	—
112	C₂×C₁₆	—	178	1	no	—
113	C₂×C₁₆	—	178	1	no	—
114	C₂×C₁₆	—	178	2	no	—
115	C₄⋊C₈	#10623494	1	2	yes	—
116	C₂²⋊C₈	#10623494	1	2	yes	—
117	#97	—	2	1	no	—
118	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
119	C₄×C₈	D₃₅₆	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
121	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
122	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
123	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
125	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
126	C₂²×C₄	#13083558	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
133	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
134	C₂×C₈	C₂×#10623494	1	4	yes	—
135	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
136	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
138	C₄²	C₂×#10623494	1	1	yes	characteristic
139	C₄²	#13083558	1	1	yes	characteristic
140	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
141	C₁₆	—	356	2	no	—
142	C₁₆	—	356	1	no	—
143	C₁₆	—	356	1	no	—
144	C₂³	C₄⋊#10623494	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
150	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
151	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
153	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
154	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	4	no	—
160	C₈	#15668040	1	1	yes	characteristic
161	C₈	#15668045	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	#19151419	1	2	yes	—
165	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#19151640	1	1	yes	characteristic
171	C₄	#19151647	1	1	yes	characteristic
172	C₄	#19152015	1	2	yes	—
173	C₂	—	2	2	no	—
174	C₂	#26019004	1	1	yes	characteristic
175	C₂	#26022011	1	1	yes	characteristic
176	C₂	C₂×#19151419	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
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	#12016
LCS	G ▷ C₂×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
94. C₈₉	#12016	semidirect

Maximal subgroups

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

subgroup	index
#26022055	2
C₂×C₄⋊#13083549	2
#26018322	2
#26018316	2
C₈×M₁₆×C₈₉	2
#12016	89