Subgroups — C₂×C₈₉×#2835 #36414505

Order 22784.

2022 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	C₈₉×#2835	C₂	1	16	yes	maximal, direct product factor
2	C₂×#400×C₈₉	C₂	1	6	yes	maximal, complemented
3	C₂×#397×C₈₉	C₂	1	3	yes	maximal
4	C₂×#395×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#393×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#385×C₈₉	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
10	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
11	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
13	#400×C₈₉	C₂²	1	48	yes	complemented
14	#397×C₈₉	C₂²	1	24	yes	—
15	#395×C₈₉	C₂²	1	16	yes	complemented
16	#393×C₈₉	C₂²	1	8	yes	—
17	#385×C₈₉	C₂²	1	24	yes	—
18	C₂×#95×C₈₉	C₂²	1	3	yes	—
19	C₂×#95×C₈₉	C₂²	1	6	yes	—
20	C₂×#95×C₈₉	C₂²	1	3	yes	—
21	C₂×#95×C₈₉	C₂²	1	6	yes	—
22	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
23	C₂²×Q₈×C₈₉	—	4	6	no	—
24	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
25	C₂³×C₄×C₈₉	—	4	6	no	—
26	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
27	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
28	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
29	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₈₉	—	4	6	no	—
32	C₂×C₄⋊C₄×C₈₉	—	4	12	no	—
33	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
34	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
35	C₂×C₂²⋊C₄×C₈₉	—	2	12	no	—
36	C₂×C₂²⋊C₄×C₈₉	—	4	6	no	—
37	C₂×C₂²⋊C₄×C₈₉	C₂³	1	12	yes	—
38	C₂×C₂²⋊C₄×C₈₉	C₂³	1	12	yes	—
39	#95×C₈₉	C₂³	1	24	yes	—
40	#95×C₈₉	C₂³	1	12	yes	—
41	#95×C₈₉	C₂³	1	24	yes	—
42	#95×C₈₉	C₂³	1	12	yes	—
43	C₂⁴×C₈₉	—	2	6	no	—
44	C₂⁴×C₈₉	—	2	6	no	—
45	C₂⁴×C₈₉	#41	1	4	yes	—
46	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
47	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₈₉	—	4	12	no	—
49	C₂×Q₈×C₈₉	—	4	16	no	—
50	C₂²×C₄×C₈₉	—	2	12	no	—
51	C₂²×C₄×C₈₉	—	2	6	no	—
52	C₂²×C₄×C₈₉	—	2	6	no	—
53	C₂²×C₄×C₈₉	—	2	12	no	—
54	C₂²×C₄×C₈₉	—	2	12	no	—
55	C₂²×C₄×C₈₉	—	2	12	no	—
56	C₂²×C₄×C₈₉	—	2	12	no	—
57	C₂²×C₄×C₈₉	—	2	12	no	—
58	C₂²×C₄×C₈₉	—	4	12	no	—
59	C₂²×C₄×C₈₉	—	4	16	no	—
60	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
62	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
64	C₄⋊C₄×C₈₉	—	4	12	no	—
65	C₄⋊C₄×C₈₉	—	4	24	no	—
66	C₂²⋊C₄×C₈₉	—	2	24	no	—
67	C₂²⋊C₄×C₈₉	—	4	12	no	—
68	C₂³×C₈₉	—	2	24	no	—
69	C₂³×C₈₉	—	2	6	no	—
70	C₂³×C₈₉	—	2	6	no	—
71	C₂³×C₈₉	—	4	4	no	—
72	C₂³×C₈₉	—	4	12	no	—
73	C₂³×C₈₉	C₂×#41	1	4	yes	—
74	C₂³×C₈₉	C₂×#41	1	2	yes	—
75	C₂³×C₈₉	H(5,𝔽₂)	1	6	yes	—
76	C₂³×C₈₉	2₋¹⁺⁴	1	6	yes	—
77	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
78	Q₈×C₈₉	—	4	16	no	—
79	C₂×C₄×C₈₉	—	2	12	no	—
80	C₂×C₄×C₈₉	—	2	6	no	—
81	C₂×C₄×C₈₉	—	2	6	no	—
82	C₂×C₄×C₈₉	—	2	12	no	—
83	C₂×C₄×C₈₉	—	2	12	no	—
84	C₂×C₄×C₈₉	—	2	12	no	—
85	C₂×C₄×C₈₉	—	2	12	no	—
86	C₂×C₄×C₈₉	—	2	12	no	—
87	C₂×C₄×C₈₉	—	4	16	no	—
88	C₂×C₄×C₈₉	—	4	24	no	—
89	C₂×C₄×C₈₉	—	4	8	no	—
90	C₂×C₄×C₈₉	—	4	12	no	—
91	C₂×C₄×C₈₉	—	4	12	no	—
92	C₂²×C₈₉	—	2	12	no	—
93	C₂²×C₈₉	—	4	16	no	—
94	C₂²×C₈₉	—	4	2	no	—
95	C₂²×C₈₉	—	4	6	no	—
96	C₂²×C₈₉	#539	1	12	yes	—
97	C₂²×C₈₉	H(3,𝔽₄)	1	4	yes	—
98	C₂²×C₈₉	#563	1	12	yes	—
99	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
100	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	3	yes	—
101	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	3	yes	—
102	C₄×C₈₉	—	4	12	no	—
103	C₄×C₈₉	—	4	4	no	—
104	C₄×C₈₉	—	4	6	no	—
105	C₄×C₈₉	—	4	6	no	—
106	C₂×#2835	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₈₉	—	4	4	no	—
108	C₂×C₈₉	#2835	1	8	yes	direct product factor
109	C₂×C₈₉	C₂×#539	1	3	yes	—
110	C₂×C₈₉	C₂×H(3,𝔽₄)	1	1	yes	characteristic
111	C₂×C₈₉	C₂×#563	1	3	yes	—
112	#2835	C₂×C₈₉	1	16	yes	direct product factor
113	C₂×#400	C₂×C₈₉	1	6	yes	complemented
114	C₂×#397	C₂×C₈₉	1	3	yes	—
115	C₂×#395	C₂×C₈₉	1	2	yes	complemented
116	C₂×#393	C₂×C₈₉	1	1	yes	characteristic
117	C₂×#385	C₂×C₈₉	1	3	yes	—
118	C₈₉	C₂×#2835	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
120	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	C₂²×C₈₉	1	6	yes	—
122	C₂²×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
123	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
124	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
125	#400	C₂²×C₈₉	1	48	yes	complemented
126	#397	C₂²×C₈₉	1	24	yes	—
127	#395	C₂²×C₈₉	1	16	yes	complemented
128	#393	C₂²×C₈₉	1	8	yes	—
129	#385	C₂²×C₈₉	1	24	yes	—
130	C₂×#95	C₂²×C₈₉	1	3	yes	—
131	C₂×#95	C₂²×C₈₉	1	6	yes	—
132	C₂×#95	C₂²×C₈₉	1	3	yes	—
133	C₂×#95	C₂²×C₈₉	1	6	yes	—
134	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
135	C₂²×Q₈	—	4	6	no	—
136	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
137	C₂³×C₄	—	4	6	no	—
138	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
139	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
140	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
141	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
142	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
143	C₂×C₄⋊C₄	—	4	6	no	—
144	C₂×C₄⋊C₄	—	4	12	no	—
145	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
146	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
147	C₂×C₂²⋊C₄	—	2	12	no	—
148	C₂×C₂²⋊C₄	—	4	6	no	—
149	C₂×C₂²⋊C₄	C₂³×C₈₉	1	12	yes	—
150	C₂×C₂²⋊C₄	C₂³×C₈₉	1	12	yes	—
151	#95	C₂³×C₈₉	1	24	yes	—
152	#95	C₂³×C₈₉	1	12	yes	—
153	#95	C₂³×C₈₉	1	24	yes	—
154	#95	C₂³×C₈₉	1	12	yes	—
155	C₂⁴	—	2	6	no	—
156	C₂⁴	—	2	6	no	—
157	C₂⁴	#41×C₈₉	1	4	yes	—
158	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
159	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
160	C₂×Q₈	—	4	12	no	—
161	C₂×Q₈	—	4	16	no	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	2	6	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	12	no	—
166	C₂²×C₄	—	2	12	no	—
167	C₂²×C₄	—	2	12	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	12	no	—
170	C₂²×C₄	—	4	12	no	—
171	C₂²×C₄	—	4	16	no	—
172	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
173	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
174	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
175	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
176	C₄⋊C₄	—	4	12	no	—
177	C₄⋊C₄	—	4	24	no	—
178	C₂²⋊C₄	—	2	24	no	—
179	C₂²⋊C₄	—	4	12	no	—
180	C₂³	—	2	24	no	—
181	C₂³	—	2	6	no	—
182	C₂³	—	2	6	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	12	no	—
185	C₂³	C₂×#41×C₈₉	1	4	yes	—
186	C₂³	C₂×#41×C₈₉	1	2	yes	—
187	C₂³	H(5,𝔽₂)×C₈₉	1	6	yes	—
188	C₂³	2₋¹⁺⁴×C₈₉	1	6	yes	—
189	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
190	Q₈	—	4	16	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	6	no	—
193	C₂×C₄	—	2	6	no	—
194	C₂×C₄	—	2	12	no	—
195	C₂×C₄	—	2	12	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	12	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	4	16	no	—
200	C₂×C₄	—	4	24	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	4	12	no	—
203	C₂×C₄	—	4	12	no	—
204	C₂²	—	2	12	no	—
205	C₂²	—	4	16	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	6	no	—
208	C₂²	#539×C₈₉	1	12	yes	—
209	C₂²	H(3,𝔽₄)×C₈₉	1	4	yes	—
210	C₂²	#563×C₈₉	1	12	yes	—
211	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
212	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	3	yes	—
213	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	3	yes	—
214	C₄	—	4	12	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	6	no	—
217	C₄	—	4	6	no	—
218	C₂	—	4	4	no	—
219	C₂	C₈₉×#2835	1	8	yes	direct product factor
220	C₂	C₂×#539×C₈₉	1	3	yes	—
221	C₂	C₂×H(3,𝔽₄)×C₈₉	1	1	yes	characteristic
222	C₂	C₂×#563×C₈₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁵×C₈₉	C₂³
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
1. C₈₉×#2835	C₂	direct (N × H)
2. C₂×#400×C₈₉	C₂	semidirect
4. C₂×#395×C₈₉	C₂	semidirect
13. #400×C₈₉	C₂²	semidirect
15. #395×C₈₉	C₂²	semidirect
106. C₂×#2835	C₈₉	direct (N × H)
108. C₂×C₈₉	#2835	direct (N × H)
112. #2835	C₂×C₈₉	direct (N × H)
113. C₂×#400	C₂×C₈₉	semidirect
115. C₂×#395	C₂×C₈₉	semidirect
118. C₈₉	C₂×#2835	direct (N × H)
125. #400	C₂²×C₈₉	semidirect
127. #395	C₂²×C₈₉	semidirect
219. C₂	C₈₉×#2835	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2835	2
C₂×#400×C₈₉	2
C₂×#397×C₈₉	2
C₂×#395×C₈₉	2
C₂×#393×C₈₉	2
C₂×#385×C₈₉	2
C₂×#2835	89

Subgroups — C2×C89×#2835 #36414505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2835 #36414505