Subgroups — C₂×C₈₉×#2616 #36412458

Order 22784.

5390 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Q₈×C₈₉	C₂	1	3	yes	maximal, complemented
3	C₈₉×#2616	C₂	1	16	yes	maximal, direct product factor
4	C₂×#395×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#394×C₈₉	C₂	1	6	yes	maximal, complemented
6	C₂×#380×C₈₉	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₈₉	—	2	24	no	—
8	C₂²×#41×C₈₉	C₂²	1	8	yes	—
9	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
12	C₂⁴×C₄×C₈₉	C₂²	1	3	yes	—
13	C₂×C₂²⋊Q₈×C₈₉	—	2	24	no	—
14	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	24	yes	complemented
15	C₂²×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
16	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
17	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	—
19	#395×C₈₉	C₂²	1	16	yes	complemented
20	#394×C₈₉	C₂²	1	48	yes	complemented
21	#380×C₈₉	C₂²	1	24	yes	complemented
22	C₂×#95×C₈₉	C₂²	1	6	yes	—
23	C₂⁵×C₈₉	C₂³	1	3	yes	—
24	C₂×#41×C₈₉	—	2	128	no	—
25	C₂×#41×C₈₉	—	2	96	no	—
26	C₂²×Q₈×C₈₉	—	2	12	no	—
27	C₂²×Q₈×C₈₉	—	2	12	no	—
28	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
29	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
30	C₂²×D₄×C₈₉	—	2	24	no	—
31	C₂²×D₄×C₈₉	—	2	12	no	—
32	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
33	C₂³×C₄×C₈₉	—	2	24	no	—
34	C₂³×C₄×C₈₉	D₄	1	24	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
38	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²⋊Q₈×C₈₉	—	2	96	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	12	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	24	no	—
43	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
44	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
45	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
46	C₂×C₂²⋊C₄×C₈₉	—	4	12	no	—
47	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
48	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
49	#95×C₈₉	C₂³	1	24	yes	—
50	C₂⁴×C₈₉	—	2	12	no	—
51	C₂⁴×C₈₉	—	2	6	no	—
52	C₂⁴×C₈₉	—	2	12	no	—
53	C₂⁴×C₈₉	—	2	6	no	—
54	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
55	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
56	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
57	#41×C₈₉	—	2	256	no	—
58	C₂×Q₈×C₈₉	—	2	32	no	—
59	C₂×Q₈×C₈₉	—	2	24	no	—
60	C₂×Q₈×C₈₉	—	2	24	no	—
61	C₂×Q₈×C₈₉	—	4	16	no	—
62	C₂×D₄×C₈₉	—	2	96	no	—
63	C₂×D₄×C₈₉	—	2	48	no	—
64	C₂×D₄×C₈₉	—	2	24	no	—
65	C₂²×C₄×C₈₉	—	2	12	no	—
66	C₂²×C₄×C₈₉	—	2	96	no	—
67	C₂²×C₄×C₈₉	—	2	48	no	—
68	C₂²×C₄×C₈₉	—	2	12	no	—
69	C₂²×C₄×C₈₉	—	2	12	no	—
70	C₂²×C₄×C₈₉	—	2	12	no	—
71	C₂²×C₄×C₈₉	—	2	12	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₈₉	C₂×D₄	1	48	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
75	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
76	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
79	C₄⋊C₄×C₈₉	—	2	24	no	—
80	C₄⋊C₄×C₈₉	—	2	48	no	—
81	C₂²⋊C₄×C₈₉	—	2	48	no	—
82	C₂²⋊C₄×C₈₉	—	4	24	no	—
83	C₂³×C₈₉	—	2	24	no	—
84	C₂³×C₈₉	—	2	24	no	—
85	C₂³×C₈₉	—	2	24	no	—
86	C₂³×C₈₉	—	2	12	no	—
87	C₂³×C₈₉	—	2	6	no	—
88	C₂³×C₈₉	—	2	12	no	—
89	C₂³×C₈₉	—	2	6	no	—
90	C₂³×C₈₉	—	4	24	no	—
91	C₂³×C₈₉	—	4	24	no	—
92	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
93	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
94	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
95	C₂³×C₈₉	2₋¹⁺⁴	1	6	yes	—
96	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
97	Q₈×C₈₉	—	2	32	no	—
98	Q₈×C₈₉	—	4	16	no	—
99	D₄×C₈₉	—	2	96	no	—
100	C₂×C₄×C₈₉	—	2	24	no	—
101	C₂×C₄×C₈₉	—	2	12	no	—
102	C₂×C₄×C₈₉	—	2	96	no	—
103	C₂×C₄×C₈₉	—	2	12	no	—
104	C₂×C₄×C₈₉	—	2	12	no	—
105	C₂×C₄×C₈₉	—	2	12	no	—
106	C₂×C₄×C₈₉	—	2	12	no	—
107	C₂×C₄×C₈₉	—	4	8	no	—
108	C₂×C₄×C₈₉	—	4	24	no	—
109	C₂×C₄×C₈₉	#120	1	16	yes	—
110	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
111	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
112	C₂²×C₈₉	—	2	12	no	—
113	C₂²×C₈₉	—	2	12	no	—
114	C₂²×C₈₉	—	2	12	no	—
115	C₂²×C₈₉	—	4	48	no	—
116	C₂²×C₈₉	—	4	12	no	—
117	C₂²×C₈₉	—	4	12	no	—
118	C₂²×C₈₉	C₂×#120	1	4	yes	—
119	C₂²×C₈₉	#536	1	12	yes	—
120	C₂²×C₈₉	#537	1	12	yes	—
121	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
122	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	3	yes	—
124	C₄×C₈₉	—	2	12	no	—
125	C₄×C₈₉	—	4	4	no	—
126	C₄×C₈₉	—	4	12	no	—
127	C₄×C₈₉	C₂×#120	1	8	yes	—
128	C₂×#2616	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂×C₈₉	—	4	12	no	—
130	C₂×C₈₉	#2616	1	8	yes	direct product factor
131	C₂×C₈₉	C₂²×#120	1	1	yes	characteristic
132	C₂×C₈₉	C₂×#536	1	3	yes	—
133	C₂×C₈₉	C₂×#537	1	3	yes	—
134	C₂³×#41	C₂×C₈₉	1	1	yes	characteristic
135	C₂²×C₂²⋊Q₈	C₂×C₈₉	1	3	yes	complemented
136	#2616	C₂×C₈₉	1	16	yes	direct product factor
137	C₂×#395	C₂×C₈₉	1	2	yes	complemented
138	C₂×#394	C₂×C₈₉	1	6	yes	complemented
139	C₂×#380	C₂×C₈₉	1	3	yes	complemented
140	C₈₉	C₂×#2616	1	1	yes	characteristic, direct product factor, Sylow
141	C₂²×#41	—	2	24	no	—
142	C₂²×#41	C₂²×C₈₉	1	8	yes	—
143	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
144	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
145	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
146	C₂⁴×C₄	C₂²×C₈₉	1	3	yes	—
147	C₂×C₂²⋊Q₈	—	2	24	no	—
148	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	24	yes	complemented
149	C₂²×C₄⋊C₄	C₂²×C₈₉	1	6	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
151	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	—
153	#395	C₂²×C₈₉	1	16	yes	complemented
154	#394	C₂²×C₈₉	1	48	yes	complemented
155	#380	C₂²×C₈₉	1	24	yes	complemented
156	C₂×#95	C₂²×C₈₉	1	6	yes	—
157	C₂⁵	C₂³×C₈₉	1	3	yes	—
158	C₂×#41	—	2	128	no	—
159	C₂×#41	—	2	96	no	—
160	C₂²×Q₈	—	2	12	no	—
161	C₂²×Q₈	—	2	12	no	—
162	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
163	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
164	C₂²×D₄	—	2	24	no	—
165	C₂²×D₄	—	2	12	no	—
166	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
167	C₂³×C₄	—	2	24	no	—
168	C₂³×C₄	D₄×C₈₉	1	24	yes	—
169	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
170	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
171	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
172	C₂³×C₄	C₂³×C₈₉	1	12	yes	—
173	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
174	C₂²⋊Q₈	—	2	96	no	—
175	C₂×C₄⋊C₄	—	2	12	no	—
176	C₂×C₄⋊C₄	—	2	24	no	—
177	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
178	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
179	C₂×C₂²⋊C₄	—	2	24	no	—
180	C₂×C₂²⋊C₄	—	4	12	no	—
181	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
183	#95	C₂³×C₈₉	1	24	yes	—
184	C₂⁴	—	2	12	no	—
185	C₂⁴	—	2	6	no	—
186	C₂⁴	—	2	12	no	—
187	C₂⁴	—	2	6	no	—
188	C₂⁴	C₂×D₄×C₈₉	1	12	yes	—
189	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
190	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
191	#41	—	2	256	no	—
192	C₂×Q₈	—	2	32	no	—
193	C₂×Q₈	—	2	24	no	—
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	4	16	no	—
196	C₂×D₄	—	2	96	no	—
197	C₂×D₄	—	2	48	no	—
198	C₂×D₄	—	2	24	no	—
199	C₂²×C₄	—	2	12	no	—
200	C₂²×C₄	—	2	96	no	—
201	C₂²×C₄	—	2	48	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	2	12	no	—
205	C₂²×C₄	—	2	12	no	—
206	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	—
207	C₂²×C₄	C₂×D₄×C₈₉	1	48	yes	—
208	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	—
209	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
210	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
211	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
212	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
213	C₄⋊C₄	—	2	24	no	—
214	C₄⋊C₄	—	2	48	no	—
215	C₂²⋊C₄	—	2	48	no	—
216	C₂²⋊C₄	—	4	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	—	2	24	no	—
220	C₂³	—	2	12	no	—
221	C₂³	—	2	6	no	—
222	C₂³	—	2	12	no	—
223	C₂³	—	2	6	no	—
224	C₂³	—	4	24	no	—
225	C₂³	—	4	24	no	—
226	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
227	C₂³	C₂²×D₄×C₈₉	1	6	yes	—
228	C₂³	H(5,𝔽₂)×C₈₉	1	2	yes	—
229	C₂³	2₋¹⁺⁴×C₈₉	1	6	yes	—
230	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
231	Q₈	—	2	32	no	—
232	Q₈	—	4	16	no	—
233	D₄	—	2	96	no	—
234	C₂×C₄	—	2	24	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	96	no	—
237	C₂×C₄	—	2	12	no	—
238	C₂×C₄	—	2	12	no	—
239	C₂×C₄	—	2	12	no	—
240	C₂×C₄	—	2	12	no	—
241	C₂×C₄	—	4	8	no	—
242	C₂×C₄	—	4	24	no	—
243	C₂×C₄	#120×C₈₉	1	16	yes	—
244	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
245	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
246	C₂²	—	2	12	no	—
247	C₂²	—	2	12	no	—
248	C₂²	—	2	12	no	—
249	C₂²	—	4	48	no	—
250	C₂²	—	4	12	no	—
251	C₂²	—	4	12	no	—
252	C₂²	C₂×#120×C₈₉	1	4	yes	—
253	C₂²	#536×C₈₉	1	12	yes	—
254	C₂²	#537×C₈₉	1	12	yes	—
255	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
256	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
257	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	3	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	12	no	—
261	C₄	C₂×#120×C₈₉	1	8	yes	—
262	C₂	—	4	12	no	—
263	C₂	C₈₉×#2616	1	8	yes	direct product factor
264	C₂	C₂²×#120×C₈₉	1	1	yes	characteristic
265	C₂	C₂×#536×C₈₉	1	3	yes	—
266	C₂	C₂×#537×C₈₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂³×#41×C₈₉	C₂
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. C₂²×C₂²⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2616	C₂	direct (N × H)
4. C₂×#395×C₈₉	C₂	semidirect
5. C₂×#394×C₈₉	C₂	semidirect
6. C₂×#380×C₈₉	C₂	semidirect
14. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
19. #395×C₈₉	C₂²	semidirect
20. #394×C₈₉	C₂²	semidirect
21. #380×C₈₉	C₂²	semidirect
128. C₂×#2616	C₈₉	direct (N × H)
130. C₂×C₈₉	#2616	direct (N × H)
135. C₂²×C₂²⋊Q₈	C₂×C₈₉	semidirect
136. #2616	C₂×C₈₉	direct (N × H)
137. C₂×#395	C₂×C₈₉	semidirect
138. C₂×#394	C₂×C₈₉	semidirect
139. C₂×#380	C₂×C₈₉	semidirect
140. C₈₉	C₂×#2616	direct (N × H)
148. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
153. #395	C₂²×C₈₉	semidirect
154. #394	C₂²×C₈₉	semidirect
155. #380	C₂²×C₈₉	semidirect
263. C₂	C₈₉×#2616	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₈₉	2
C₂²×C₂²⋊Q₈×C₈₉	2
C₈₉×#2616	2
C₂×#395×C₈₉	2
C₂×#394×C₈₉	2
C₂×#380×C₈₉	2
C₂×#2616	89

Subgroups — C2×C89×#2616 #36412458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2616 #36412458