Subgroups — #36708029

Order 22784.

614 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022084	C₂	1	8	yes	maximal, complemented
2	#26022051	C₂	1	2	yes	maximal, complemented
3	#26021863	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667952	C₂²	1	8	yes	complemented
6	C₄⋊#13083549	C₂²	1	4	yes	complemented
7	C₂²⋊#13083549	C₂²	1	4	yes	—
8	C₂²⋊#13083549	C₂²	1	8	yes	complemented
9	C₄×#13083549	C₂²	1	4	yes	complemented
10	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
12	C₈×D₄×C₈₉	C₄	1	4	yes	—
13	C₈×D₄×C₈₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#15667952	—	4	8	no	—
18	C₂×#13083549	C₂³	1	8	yes	—
19	C₂²×D₄×C₈₉	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₈₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₈₉	—	4	4	no	—
22	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
27	C₄×D₄×C₈₉	C₈	1	4	yes	—
28	C₄×D₄×C₈₉	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₈₉	C₈	1	2	yes	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
33	C₂²⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₈₉	C₂³	1	2	yes	—
36	#13083549	—	2	8	no	—
37	C₂⁴×C₈₉	C₂×C₈	1	2	yes	—
38	C₂×D₄×C₈₉	—	2	4	no	—
39	C₂×D₄×C₈₉	C₂×C₈	1	4	yes	—
40	C₂²×C₄×C₈₉	—	4	2	no	—
41	C₂²×C₄×C₈₉	—	4	2	no	—
42	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₉	—	2	2	no	—
49	C₂×C₈×C₈₉	—	2	8	no	—
50	C₂×C₈×C₈₉	—	4	4	no	—
51	C₂×C₈×C₈₉	—	4	2	no	—
52	C₂×C₈×C₈₉	—	4	1	no	—
53	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₈₉	C₂×C₈	1	4	yes	—
58	C₂²⋊C₄×C₈₉	C₂×C₈	1	8	yes	—
59	C₄²×C₈₉	C₂×C₈	1	2	yes	—
60	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
61	C₂³×C₈₉	—	2	4	no	—
62	C₂³×C₈₉	—	4	2	no	—
63	C₂³×C₈₉	—	4	2	no	—
64	C₂³×C₈₉	C₂²×C₈	1	4	yes	—
65	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
66	D₄×C₈₉	—	2	8	no	—
67	C₂×C₄×C₈₉	—	2	1	no	—
68	C₂×C₄×C₈₉	—	2	4	no	—
69	C₂×C₄×C₈₉	—	2	1	no	—
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	—	4	4	no	—
72	C₂×C₄×C₈₉	—	4	2	no	—
73	C₂×C₄×C₈₉	—	4	1	no	—
74	C₂×C₄×C₈₉	M₃₂	1	4	yes	complemented
75	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
78	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₈₉	—	2	1	no	—
81	C₈×C₈₉	—	2	4	no	—
82	C₈×C₈₉	—	2	2	no	—
83	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₈₉	—	2	1	no	—
85	C₂²×C₈₉	—	2	1	no	—
86	C₂²×C₈₉	—	2	4	no	—
87	C₂²×C₈₉	—	2	4	no	—
88	C₂²×C₈₉	—	2	4	no	—
89	C₂²×C₈₉	—	4	8	no	—
90	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
91	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
92	C₄×C₈₉	—	2	2	no	—
93	C₄×C₈₉	—	2	1	no	—
94	C₄×C₈₉	—	2	4	no	—
95	C₄×C₈₉	C₂×M₃₂	1	4	yes	—
96	C₄×C₈₉	#518	1	1	yes	characteristic
97	C₄×C₈₉	#568	1	1	yes	characteristic
98	#33489	—	89	1	no	maximal, Sylow
99	C₂×C₈₉	—	2	2	no	—
100	C₂×C₈₉	—	4	4	no	—
101	C₂×C₈₉	#2972	1	1	yes	characteristic
102	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
103	C₂×C₈₉	#3407	1	1	yes	characteristic
104	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
105	#2169	—	89	8	no	—
106	#2162	—	89	2	no	—
107	#2112	—	89	4	no	—
108	C₈₉	#33489	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₈	C₂×D₈₉	1	2	yes	—
110	C₂×C₄×D₄	Dic₈₉	1	1	yes	characteristic
111	C₂×M₃₂	—	89	8	no	—
112	C₈×D₄	Dic₈₉	1	4	yes	—
113	C₈×D₄	C₂×D₈₉	1	4	yes	—
114	C₂×C₄⋊C₈	Dic₈₉	1	1	yes	characteristic
115	C₂×C₂²⋊C₈	Dic₈₉	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
117	C₄⋊C₁₆	—	89	4	no	—
118	C₂²⋊C₁₆	—	89	4	no	—
119	C₂²⋊C₁₆	—	89	8	no	—
120	C₄×C₁₆	—	89	4	no	—
121	C₂²×D₄	#10623494	1	1	yes	characteristic
122	C₂³×C₄	C₂×Dic₈₉	1	2	yes	—
123	C₂²×C₈	—	4	4	no	—
124	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
125	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
126	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
129	C₄×D₄	#10623494	1	4	yes	—
130	C₄×D₄	C₂×Dic₈₉	1	4	yes	—
131	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	#10623494	1	2	yes	—
133	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
134	M₃₂	—	356	8	no	—
135	C₂×C₁₆	—	89	8	no	—
136	C₄⋊C₈	C₂×Dic₈₉	1	4	yes	—
137	C₂²⋊C₈	C₂×Dic₈₉	1	8	yes	—
138	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
139	C₄×C₈	C₂²×D₈₉	1	2	yes	—
140	C₂⁴	C₂×#10623494	1	2	yes	—
141	C₂×D₄	—	2	4	no	—
142	C₂×D₄	C₂×#10623494	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
146	C₂²×C₄	C₂×#10623494	1	2	yes	—
147	C₂²×C₄	C₂×#10623494	1	4	yes	—
148	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
158	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
159	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
160	C₄⋊C₄	C₂×#10623494	1	4	yes	—
161	C₂²⋊C₄	C₂×#10623494	1	8	yes	—
162	C₄²	C₂×#10623494	1	2	yes	—
163	C₄²	C₂²×Dic₈₉	1	2	yes	—
164	C₁₆	—	178	8	no	—
165	C₂³	—	2	4	no	—
166	C₂³	—	4	2	no	—
167	C₂³	—	4	2	no	—
168	C₂³	C₂²×#10623494	1	4	yes	—
169	C₂³	C₂²×#10623494	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	#15667952	1	4	yes	complemented
179	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
180	C₂×C₄	C₂²×#10623494	1	2	yes	—
181	C₂×C₄	C₂²×#10623494	1	4	yes	—
182	C₂×C₄	C₂²×#10623494	1	2	yes	—
183	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	4	no	—
186	C₈	—	2	2	no	—
187	C₈	#15668057	1	2	yes	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	4	8	no	—
194	C₂²	C₂×#15667952	1	2	yes	—
195	C₂²	C₂³×#10623494	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂×#15667952	1	4	yes	—
200	C₄	#19152073	1	1	yes	characteristic
201	C₄	#19152365	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	#26029276	1	1	yes	characteristic
205	C₂	C₂²×#15667952	1	1	yes	characteristic
206	C₂	#26033355	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₂×C₈×D₄×C₈₉	C₂
prime radical	C₂²×D₄×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#33489
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #26022084	C₂	semidirect
2. #26022051	C₂	semidirect
3. #26021863	C₂	semidirect
5. C₂×#15667952	C₂²	semidirect
6. C₄⋊#13083549	C₂²	semidirect
8. C₂²⋊#13083549	C₂²	semidirect
9. C₄×#13083549	C₂²	semidirect
74. C₂×C₄×C₈₉	M₃₂	semidirect
108. C₈₉	#33489	semidirect
178. C₂×C₄	#15667952	semidirect

Maximal subgroups

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

subgroup	index
#26022084	2
#26022051	2
#26021863	2
C₂×C₈×D₄×C₈₉	2
#33489	89