Subgroups — C₂×C₈₉×#2153 #36411248

Order 22784.

698 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₈₉	C₂	1	2	yes	maximal
2	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2153	C₂	1	8	yes	maximal, direct product factor
4	C₂×#364×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₈₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#117×C₈₉	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₈₉	—	2	4	no	—
9	C₂×M₃₂×C₈₉	—	2	2	no	—
10	C₂×M₃₂×C₈₉	C₂²	1	4	yes	—
11	C₂×M₃₂×C₈₉	C₂²	1	8	yes	—
12	C₂²×C₁₆×C₈₉	—	2	2	no	—
13	#405×C₈₉	C₄	1	4	yes	—
14	#405×C₈₉	C₂²	1	4	yes	—
15	C₂×#97×C₈₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
17	#364×C₈₉	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₈₉	C₈	1	4	yes	—
20	C₂×M₁₆×C₈₉	C₈	1	4	yes	—
21	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
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	2	yes	—
25	C₂²×C₈×C₈₉	D₄	1	2	yes	—
26	C₂²×C₈×C₈₉	D₄	1	2	yes	—
27	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	#117×C₈₉	—	2	2	no	—
32	#117×C₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
36	M₃₂×C₈₉	—	2	8	no	—
37	M₃₂×C₈₉	—	2	4	no	—
38	M₃₂×C₈₉	C₂³	1	8	yes	—
39	C₂×C₁₆×C₈₉	—	2	8	no	—
40	C₂×C₁₆×C₈₉	—	2	4	no	—
41	#97×C₈₉	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₈₉	C₂³	1	4	yes	—
44	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
54	M₁₆×C₈₉	C₂×C₈	1	16	yes	—
55	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
57	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
59	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₈₉	—	2	4	no	—
72	C₂²⋊C₄×C₈₉	—	2	4	no	—
73	C₄²×C₈₉	—	2	2	no	—
74	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
75	C₁₆×C₈₉	—	2	8	no	—
76	C₂³×C₈₉	—	4	2	no	—
77	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
78	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
79	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
80	C₂³×C₈₉	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₈₉	—	2	4	no	—
82	C₂×C₄×C₈₉	—	4	4	no	—
83	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
85	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
86	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
87	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₈×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
98	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₈×C₈₉	C₂²×C₈	1	8	yes	—
100	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₈₉	—	2	2	no	—
103	C₂²×C₈₉	—	2	4	no	—
104	C₂²×C₈₉	—	4	2	no	—
105	C₂²×C₈₉	—	4	2	no	—
106	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₂²×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
108	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
110	C₄×C₈₉	—	4	4	no	—
111	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₈₉	C₂×C₄⋊C₈	1	4	yes	—
113	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
116	C₂×#2153	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₈₉	—	2	1	no	—
118	C₂×C₈₉	—	2	1	no	—
119	C₂×C₈₉	—	2	4	no	—
120	C₂×C₈₉	#2153	1	2	yes	direct product factor
121	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
122	C₂²×M₃₂	C₂×C₈₉	1	2	yes	—
123	C₂×#405	C₂×C₈₉	1	1	yes	characteristic
124	#2153	C₂×C₈₉	1	8	yes	direct product factor
125	C₂×#364	C₂×C₈₉	1	4	yes	complemented
126	C₈₉	C₂×#2153	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×M₁₆	C₄×C₈₉	1	1	yes	characteristic
128	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
129	C₂×#117	C₄×C₈₉	1	1	yes	characteristic
130	C₂×M₃₂	—	2	2	no	—
131	C₂×M₃₂	—	2	4	no	—
132	C₂×M₃₂	C₂²×C₈₉	1	4	yes	—
133	C₂×M₃₂	C₂²×C₈₉	1	8	yes	—
134	C₂²×C₁₆	—	2	2	no	—
135	#405	C₄×C₈₉	1	4	yes	—
136	#405	C₂²×C₈₉	1	4	yes	—
137	C₂×#97	C₄×C₈₉	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
139	#364	C₂²×C₈₉	1	16	yes	complemented
140	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
141	C₂×M₁₆	C₈×C₈₉	1	4	yes	—
142	C₂×M₁₆	C₈×C₈₉	1	4	yes	—
143	C₂×M₁₆	C₂×C₄×C₈₉	1	4	yes	—
144	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
145	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	—
146	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	—
147	C₂²×C₈	D₄×C₈₉	1	2	yes	—
148	C₂²×C₈	D₄×C₈₉	1	2	yes	—
149	C₂²×C₈	Q₈×C₈₉	1	2	yes	—
150	C₂²×C₈	Q₈×C₈₉	1	2	yes	—
151	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
153	#117	—	2	2	no	—
154	#117	C₂×C₄×C₈₉	1	4	yes	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₂²⋊C₄	—	2	1	no	—
157	C₂×C₄²	C₂×C₄×C₈₉	1	2	yes	—
158	M₃₂	—	2	8	no	—
159	M₃₂	—	2	4	no	—
160	M₃₂	C₂³×C₈₉	1	8	yes	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	—	2	8	no	—
163	#97	C₂×C₄×C₈₉	1	8	yes	—
164	C₄×C₈	C₂×C₄×C₈₉	1	4	yes	—
165	C₄×C₈	C₂³×C₈₉	1	4	yes	—
166	C₂⁴	M₁₆×C₈₉	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₈₉	1	2	yes	—
170	C₂²×C₄	C₂×C₈×C₈₉	1	2	yes	—
171	C₂²×C₄	C₂×C₈×C₈₉	1	2	yes	—
172	C₂²×C₄	M₁₆×C₈₉	1	1	yes	characteristic
173	C₂²×C₄	M₁₆×C₈₉	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₈₉	1	2	yes	—
176	M₁₆	C₂×C₈×C₈₉	1	16	yes	—
177	C₂×C₈	C₄⋊C₄×C₈₉	1	2	yes	—
178	C₂×C₈	C₄⋊C₄×C₈₉	1	2	yes	—
179	C₂×C₈	C₄⋊C₄×C₈₉	1	4	yes	—
180	C₂×C₈	C₄⋊C₄×C₈₉	1	4	yes	—
181	C₂×C₈	C₂×C₈×C₈₉	1	4	yes	—
182	C₂×C₈	C₂×C₈×C₈₉	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
184	C₂×C₈	C₂²×C₄×C₈₉	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
186	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
188	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
189	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
190	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
191	C₂×C₈	C₂×Q₈×C₈₉	1	4	yes	—
192	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
197	C₁₆	—	2	8	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₄⋊C₈×C₈₉	1	2	yes	—
200	C₂³	C₄⋊C₈×C₈₉	1	2	yes	—
201	C₂³	C₂×M₁₆×C₈₉	1	1	yes	characteristic
202	C₂³	C₂×M₁₆×C₈₉	1	2	yes	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₄⋊C₈×C₈₉	1	2	yes	—
206	C₂×C₄	C₄⋊C₈×C₈₉	1	4	yes	—
207	C₂×C₄	C₄⋊C₈×C₈₉	1	4	yes	—
208	C₂×C₄	C₄⋊C₈×C₈₉	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₈×C₈₉	1	4	yes	—
215	C₂×C₄	C₂×M₁₆×C₈₉	1	1	yes	characteristic
216	C₂×C₄	C₂×M₁₆×C₈₉	1	2	yes	—
217	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
218	C₈	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
219	C₈	C₂×C₄⋊C₄×C₈₉	1	4	yes	—
220	C₈	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
221	C₈	C₂²×C₈×C₈₉	1	8	yes	—
222	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
223	C₈	C₂²×Q₈×C₈₉	1	1	yes	characteristic
224	C₂²	—	2	2	no	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂×C₄⋊C₈×C₈₉	1	1	yes	characteristic
229	C₂²	C₂×C₄⋊C₈×C₈₉	1	4	yes	—
230	C₂²	C₂×C₄⋊C₈×C₈₉	1	1	yes	characteristic
231	C₂²	C₂²×M₁₆×C₈₉	1	1	yes	characteristic
232	C₄	—	4	4	no	—
233	C₄	C₂×C₄⋊C₈×C₈₉	1	1	yes	characteristic
234	C₄	C₂×C₄⋊C₈×C₈₉	1	1	yes	characteristic
235	C₄	C₂×C₄⋊C₈×C₈₉	1	4	yes	—
236	C₄	C₂²×C₄⋊C₄×C₈₉	1	1	yes	characteristic
237	C₄	C₂³×C₈×C₈₉	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	2	4	no	—
241	C₂	C₈₉×#2153	1	2	yes	direct product factor
242	C₂	C₂²×C₄⋊C₈×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂×D₄
commutator	C₄	C₂³×C₈×C₈₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂⁴×C₈₉	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ C₂³×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#2153	C₂	direct (N × H)
4. C₂×#364×C₈₉	C₂	semidirect
17. #364×C₈₉	C₂²	semidirect
116. C₂×#2153	C₈₉	direct (N × H)
120. C₂×C₈₉	#2153	direct (N × H)
124. #2153	C₂×C₈₉	direct (N × H)
125. C₂×#364	C₂×C₈₉	semidirect
126. C₈₉	C₂×#2153	direct (N × H)
139. #364	C₂²×C₈₉	semidirect
241. C₂	C₈₉×#2153	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×M₃₂×C₈₉	2
C₂×#405×C₈₉	2
C₈₉×#2153	2
C₂×#364×C₈₉	2
C₂×#2153	89

Subgroups — C2×C89×#2153 #36411248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2153 #36411248