Subgroups — C₂×C₈₉×#2822 #36414531

Order 22784.

2046 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#2822	C₂	1	16	yes	maximal, direct product factor
3	C₂×#400×C₈₉	C₂	1	4	yes	maximal
4	C₂×#398×C₈₉	C₂	1	4	yes	maximal
5	C₂×#398×C₈₉	C₂	1	4	yes	maximal
6	C₂×#382×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₈₉	—	2	16	no	—
8	C₂×#125×C₈₉	C₂²	1	16	yes	—
9	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
14	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#400×C₈₉	C₂²	1	32	yes	—
17	#398×C₈₉	C₂²	1	32	yes	—
18	#398×C₈₉	C₂²	1	32	yes	—
19	#382×C₈₉	C₂²	1	8	yes	—
20	C₂×#95×C₈₉	C₂²	1	8	yes	—
21	C₂×#95×C₈₉	C₂²	1	4	yes	—
22	C₂×#95×C₈₉	C₂²	1	4	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
24	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
27	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	#125×C₈₉	—	2	64	no	—
29	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
30	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
31	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
32	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
33	C₂×C₄⋊C₄×C₈₉	—	4	4	no	—
34	C₂×C₄⋊C₄×C₈₉	—	4	8	no	—
35	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
36	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
38	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
39	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
40	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
41	C₂×C₄²×C₈₉	D₄	1	8	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
43	#95×C₈₉	C₂³	1	16	yes	—
44	#95×C₈₉	C₂³	1	32	yes	—
45	#95×C₈₉	C₂³	1	16	yes	—
46	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
47	C₂²×C₄×C₈₉	—	2	2	no	—
48	C₂²×C₄×C₈₉	—	2	8	no	—
49	C₂²×C₄×C₈₉	—	2	2	no	—
50	C₂²×C₄×C₈₉	—	2	8	no	—
51	C₂²×C₄×C₈₉	—	2	8	no	—
52	C₂²×C₄×C₈₉	—	2	8	no	—
53	C₂²×C₄×C₈₉	—	2	8	no	—
54	C₂²×C₄×C₈₉	—	2	16	no	—
55	C₂²×C₄×C₈₉	—	2	8	no	—
56	C₂²×C₄×C₈₉	—	2	16	no	—
57	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
62	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
64	C₄⋊C₄×C₈₉	—	2	32	no	—
65	C₄⋊C₄×C₈₉	—	2	16	no	—
66	C₄⋊C₄×C₈₉	—	2	32	no	—
67	C₄⋊C₄×C₈₉	—	2	16	no	—
68	C₄⋊C₄×C₈₉	—	4	8	no	—
69	C₄⋊C₄×C₈₉	—	4	16	no	—
70	C₄²×C₈₉	C₂×D₄	1	16	yes	—
71	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
72	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
73	C₂³×C₈₉	2₋¹⁺⁴	1	8	yes	—
74	C₂³×C₈₉	2₋¹⁺⁴	1	2	yes	—
75	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
76	C₂×C₄×C₈₉	—	2	16	no	—
77	C₂×C₄×C₈₉	—	2	8	no	—
78	C₂×C₄×C₈₉	—	2	2	no	—
79	C₂×C₄×C₈₉	—	2	8	no	—
80	C₂×C₄×C₈₉	—	2	2	no	—
81	C₂×C₄×C₈₉	—	2	8	no	—
82	C₂×C₄×C₈₉	—	2	8	no	—
83	C₂×C₄×C₈₉	—	2	8	no	—
84	C₂×C₄×C₈₉	—	2	8	no	—
85	C₂×C₄×C₈₉	—	2	16	no	—
86	C₂×C₄×C₈₉	—	2	8	no	—
87	C₂×C₄×C₈₉	—	2	16	no	—
88	C₂×C₄×C₈₉	—	4	16	no	—
89	C₂×C₄×C₈₉	—	4	16	no	—
90	C₂×C₄×C₈₉	—	4	16	no	—
91	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
92	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
93	C₂²×C₈₉	#536	1	8	yes	—
94	C₂²×C₈₉	#537	1	4	yes	—
95	C₂²×C₈₉	#563	1	8	yes	—
96	C₂²×C₈₉	#564	1	8	yes	—
97	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
98	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	4	yes	—
100	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
101	C₄×C₈₉	—	2	8	no	—
102	C₄×C₈₉	—	2	4	no	—
103	C₄×C₈₉	—	4	8	no	—
104	C₄×C₈₉	—	4	8	no	—
105	C₄×C₈₉	—	4	8	no	—
106	C₂×#2822	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₈₉	#2822	1	8	yes	direct product factor
108	C₂×C₈₉	C₂×#536	1	2	yes	—
109	C₂×C₈₉	C₂×#537	1	1	yes	characteristic
110	C₂×C₈₉	C₂×#563	1	2	yes	—
111	C₂×C₈₉	C₂×#564	1	2	yes	—
112	C₂²×#125	C₂×C₈₉	1	2	yes	—
113	#2822	C₂×C₈₉	1	16	yes	direct product factor
114	C₂×#400	C₂×C₈₉	1	4	yes	—
115	C₂×#398	C₂×C₈₉	1	4	yes	—
116	C₂×#398	C₂×C₈₉	1	4	yes	—
117	C₂×#382	C₂×C₈₉	1	1	yes	characteristic
118	C₈₉	C₂×#2822	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×#125	—	2	16	no	—
120	C₂×#125	C₂²×C₈₉	1	16	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
122	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
123	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
127	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
128	#400	C₂²×C₈₉	1	32	yes	—
129	#398	C₂²×C₈₉	1	32	yes	—
130	#398	C₂²×C₈₉	1	32	yes	—
131	#382	C₂²×C₈₉	1	8	yes	—
132	C₂×#95	C₂²×C₈₉	1	8	yes	—
133	C₂×#95	C₂²×C₈₉	1	4	yes	—
134	C₂×#95	C₂²×C₈₉	1	4	yes	—
135	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
136	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
137	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
138	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
139	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
140	#125	—	2	64	no	—
141	C₂×C₄⋊C₄	—	2	16	no	—
142	C₂×C₄⋊C₄	—	2	8	no	—
143	C₂×C₄⋊C₄	—	2	16	no	—
144	C₂×C₄⋊C₄	—	2	8	no	—
145	C₂×C₄⋊C₄	—	4	4	no	—
146	C₂×C₄⋊C₄	—	4	8	no	—
147	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
148	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
149	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
150	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
151	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
152	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
153	C₂×C₄²	D₄×C₈₉	1	8	yes	—
154	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
155	#95	C₂³×C₈₉	1	16	yes	—
156	#95	C₂³×C₈₉	1	32	yes	—
157	#95	C₂³×C₈₉	1	16	yes	—
158	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	8	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	2	16	no	—
169	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
170	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
171	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
172	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
173	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
174	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
176	C₄⋊C₄	—	2	32	no	—
177	C₄⋊C₄	—	2	16	no	—
178	C₄⋊C₄	—	2	32	no	—
179	C₄⋊C₄	—	2	16	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	16	no	—
182	C₄²	C₂×D₄×C₈₉	1	16	yes	—
183	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
184	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
185	C₂³	2₋¹⁺⁴×C₈₉	1	8	yes	—
186	C₂³	2₋¹⁺⁴×C₈₉	1	2	yes	—
187	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
188	C₂×C₄	—	2	16	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	8	no	—
197	C₂×C₄	—	2	16	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	16	no	—
200	C₂×C₄	—	4	16	no	—
201	C₂×C₄	—	4	16	no	—
202	C₂×C₄	—	4	16	no	—
203	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
204	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
205	C₂²	#536×C₈₉	1	8	yes	—
206	C₂²	#537×C₈₉	1	4	yes	—
207	C₂²	#563×C₈₉	1	8	yes	—
208	C₂²	#564×C₈₉	1	8	yes	—
209	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
210	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
211	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	4	yes	—
212	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	2	4	no	—
215	C₄	—	4	8	no	—
216	C₄	—	4	8	no	—
217	C₄	—	4	8	no	—
218	C₂	C₈₉×#2822	1	8	yes	direct product factor
219	C₂	C₂×#536×C₈₉	1	2	yes	—
220	C₂	C₂×#537×C₈₉	1	1	yes	characteristic
221	C₂	C₂×#563×C₈₉	1	2	yes	—
222	C₂	C₂×#564×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2822	C₂	direct (N × H)
106. C₂×#2822	C₈₉	direct (N × H)
107. C₂×C₈₉	#2822	direct (N × H)
113. #2822	C₂×C₈₉	direct (N × H)
118. C₈₉	C₂×#2822	direct (N × H)
218. C₂	C₈₉×#2822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₈₉	2
C₈₉×#2822	2
C₂×#400×C₈₉	2
C₂×#398×C₈₉	2
C₂×#382×C₈₉	2
C₂×#2822	89

Subgroups — C2×C89×#2822 #36414531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2822 #36414531