Subgroups — #36707918

Order 22784.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029186	C₂	1	1	yes	maximal, characteristic
2	#26022157	C₂	1	4	yes	maximal, complemented
3	#26022154	C₂	1	4	yes	maximal, complemented
4	#26022012	C₂	1	2	yes	maximal
5	#26021851	C₂	1	2	yes	maximal
6	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#2150	C₂	1	1	yes	maximal, characteristic
8	#19152155	C₄	1	1	yes	characteristic
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	#19151728	C₄	1	4	yes	—
12	C₈×Dic₁₇₈	C₂²	1	4	yes	—
13	#19151624	C₄	1	2	yes	—
14	#19151622	C₄	1	1	yes	characteristic
15	#19151553	C₂²	1	2	yes	—
16	#19151416	C₂²	1	8	yes	—
17	#19151373	C₂²	1	4	yes	—
18	C₄×#13083549	C₂²	1	2	yes	—
19	C₄×#13083549	C₂²	1	2	yes	—
20	C₁₆×Dic₈₉	C₂²	1	4	yes	—
21	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	—
24	C₂×#13083558	C₂×C₄	1	2	yes	—
25	#15668012	C₈	1	2	yes	—
26	#15668004	C₈	1	2	yes	—
27	#15667991	C₈	1	4	yes	—
28	C₄×Dic₁₇₈	C₂×C₄	1	4	yes	—
29	#15667971	C₂×C₄	1	2	yes	—
30	C₂×#13083549	—	2	2	no	—
31	C₂×#13083549	Q₈	1	4	yes	complemented
32	C₂×#13083549	C₂³	1	2	yes	—
33	C₂×#13083549	C₂³	1	1	yes	characteristic
34	C₂×#13083549	C₂³	1	1	yes	characteristic
35	C₄⋊#10623494	C₂×C₄	1	4	yes	—
36	#15667941	C₂×C₄	1	8	yes	—
37	#15667922	C₂×C₄	1	4	yes	—
38	C₈×Dic₈₉	C₂³	1	4	yes	—
39	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	M₃₂×C₈₉	—	2	4	no	—
43	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
44	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
47	#13083558	—	2	4	no	—
48	C₂×#10623494	C₂²×C₄	1	4	yes	—
49	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
50	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
51	#13083553	C₂×C₈	1	8	yes	—
52	C₂²⋊Dic₈₉	C₂×C₈	1	4	yes	—
53	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
54	#13083549	—	2	2	no	—
55	#13083549	C₂×Q₈	1	2	yes	—
56	#13083549	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₈₉	#41	1	2	yes	—
64	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
65	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
66	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
68	C₁₆×C₈₉	—	2	4	no	—
69	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₈₉	#131	1	2	yes	—
76	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×Dic₈₉	C₂²×C₈	1	8	yes	—
78	Dic₁₇₈	—	2	8	no	—
79	C₈×C₈₉	—	2	2	no	—
80	C₈×C₈₉	C₄×Q₈	1	2	yes	—
81	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
82	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
83	#10623494	—	2	4	no	—
84	C₂²×C₈₉	—	2	2	no	—
85	C₂²×C₈₉	C₈×Q₈	1	2	yes	—
86	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
87	C₄×C₈₉	—	2	2	no	—
88	C₄×C₈₉	C₈×Q₈	1	2	yes	—
89	C₄×C₈₉	#504	1	4	yes	—
90	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
92	Dic₈₉	—	2	8	no	—
93	#33474	—	89	1	no	maximal, Sylow
94	C₂×C₈₉	—	2	2	no	—
95	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
96	C₂×C₈₉	C₂×#504	1	2	yes	—
97	#2960	—	89	1	no	—
98	#2183	—	89	4	no	—
99	Q₈×C₁₆	—	89	4	no	—
100	#2151	—	89	2	no	—
101	#2150	D₈₉	1	1	yes	characteristic
102	#2108	—	89	2	no	—
103	C₂×C₄×C₁₆	—	89	1	no	—
104	C₈₉	#33474	1	1	yes	characteristic, complemented, Sylow
105	#533	—	89	1	no	—
106	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
107	C₂²×C₁₆	—	89	1	no	—
108	C₂²×C₁₆	—	89	1	no	—
109	#446	—	89	4	no	—
110	C₈×Q₈	—	89	4	no	—
111	#424	—	89	2	no	—
112	#423	—	89	1	no	—
113	#405	—	89	2	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₄⋊C₁₆	—	89	8	no	—
116	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	—
117	#346	—	89	4	no	—
118	C₄×C₁₆	—	89	4	no	—
119	C₄×C₁₆	—	89	2	no	—
120	C₄×C₁₆	—	89	2	no	—
121	C₂×M₁₆	—	89	2	no	—
122	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
124	C₄⋊Q₈	—	89	2	no	—
125	#125	—	89	2	no	—
126	C₂²⋊Q₈	—	89	4	no	—
127	C₄×Q₈	—	89	4	no	—
128	#117	—	89	2	no	—
129	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
130	M₃₂	—	2	4	no	—
131	C₂×C₁₆	—	89	4	no	—
132	C₂×C₁₆	—	89	1	no	—
133	C₂×C₁₆	—	89	1	no	—
134	C₂×C₁₆	—	89	2	no	—
135	C₂×C₁₆	—	178	2	no	—
136	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
137	C₄⋊C₈	—	89	8	no	—
138	C₄⋊C₈	—	89	4	no	—
139	#97	—	89	4	no	—
140	C₄×C₈	—	89	4	no	—
141	C₄×C₈	C₄×D₈₉	1	2	yes	—
142	C₄×C₈	C₂²×D₈₉	1	2	yes	—
143	C₂×Q₈	—	89	4	no	—
144	C₂²×C₄	C₈×D₈₉	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
146	M₁₆	—	178	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	89	4	no	—
149	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₈₉	1	2	yes	—
152	C₂×C₈	#13083573	1	2	yes	—
153	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
154	C₄⋊C₄	—	89	4	no	—
155	C₄⋊C₄	—	89	8	no	—
156	C₄⋊C₄	—	89	4	no	—
157	C₂²⋊C₄	—	89	4	no	—
158	C₄²	—	89	4	no	—
159	C₄²	C₈×D₈₉	1	1	yes	characteristic
160	C₄²	C₈×D₈₉	1	1	yes	characteristic
161	C₄²	C₂×C₄×D₈₉	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	89	2	no	—
164	C₁₆	—	89	2	no	—
165	C₁₆	—	178	2	no	—
166	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
167	Q₈	—	178	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	89	8	no	—
170	C₂×C₄	#15667977	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
174	C₂×C₄	#15668020	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	178	4	no	—
178	C₈	#15667977	1	2	yes	—
179	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
180	C₈	C₂×#13083573	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	#19151722	1	2	yes	—
183	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	178	8	no	—
186	C₄	#19151722	1	2	yes	—
187	C₄	#19152016	1	4	yes	—
188	C₄	C₂×#15667977	1	1	yes	characteristic
189	C₄	C₂×#15668020	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×#19151722	1	1	yes	characteristic
192	C₂	C₂×#19152016	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2150	C₂
prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2150	D₈₉
hypocenter	C₈₉	#33474
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #26022157	C₂	semidirect
3. #26022154	C₂	semidirect
31. C₂×#13083549	Q₈	semidirect
104. C₈₉	#33474	semidirect

Maximal subgroups

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

subgroup	index
#26029186	2
#26022157	2
#26022154	2
#26022012	2
#26021851	2
C₂×C₄×#13083549	2
C₈₉×#2150	2
#33474	89