Subgroups — C₂×Q₈×C₄⋊Dic₈₉ #36713731

Order 22784.

3246 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₈₉	C₂	1	2	yes	maximal
2	Q₈×C₄⋊Dic₈₉	C₂	1	16	yes	maximal, direct product factor
3	C₂×#19151497	C₂	1	6	yes	maximal
4	C₂×#19151468	C₂	1	3	yes	maximal
5	C₂×C₄×C₄⋊Dic₈₉	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₈₉	—	2	16	no	—
8	C₂×Q₈×Dic₈₉	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₈₉	C₂²	1	6	yes	—
10	C₂²×C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₂²×C₄⋊Dic₈₉	C₂²	1	6	yes	—
12	C₂²×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₈₉	C₂²	1	6	yes	—
14	#19151497	C₂²	1	48	yes	—
15	#19151468	C₂²	1	24	yes	—
16	C₄×C₄⋊Dic₈₉	C₂²	1	24	yes	—
17	C₂×#15667919	C₂²	1	6	yes	—
18	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	C₄	1	16	yes	complemented
20	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	3	yes	—
23	C₂³×Dic₈₉	C₂³	1	6	yes	—
24	C₂³×Dic₈₉	C₂³	1	2	yes	—
25	Q₈×Dic₈₉	—	2	64	no	—
26	C₂×C₄⋊Dic₈₉	—	2	24	no	—
27	C₂×C₄⋊Dic₈₉	—	2	12	no	—
28	C₂×C₄⋊Dic₈₉	—	4	12	no	—
29	C₂×C₄⋊Dic₈₉	Q₈	1	8	yes	direct product factor
30	C₂×C₄⋊Dic₈₉	C₂³	1	12	yes	—
31	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	24	yes	—
33	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
34	C₂×C₄×Dic₈₉	—	2	24	no	—
35	C₂×C₄×Dic₈₉	C₂³	1	24	yes	—
36	#15667919	C₂³	1	24	yes	—
37	C₂²×Q₈×C₈₉	C₂×C₄	1	8	yes	—
38	C₂²×Q₈×C₈₉	D₄	1	8	yes	—
39	C₂²×Q₈×C₈₉	Q₈	1	8	yes	—
40	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
41	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
42	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₈₉	C₂×C₄	1	64	yes	complemented
45	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	24	yes	—
46	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	24	yes	—
48	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
49	C₂²×Dic₈₉	—	2	4	no	—
50	C₂²×Dic₈₉	—	2	4	no	—
51	C₂²×Dic₈₉	—	2	12	no	—
52	C₂²×Dic₈₉	—	2	12	no	—
53	C₂²×Dic₈₉	—	2	12	no	—
54	C₂²×Dic₈₉	C₂×Q₈	1	8	yes	—
55	C₂²×Dic₈₉	C₂⁴	1	12	yes	—
56	C₂²×Dic₈₉	C₂⁴	1	4	yes	—
57	C₄⋊Dic₈₉	—	2	48	no	—
58	C₄⋊Dic₈₉	—	2	24	no	—
59	C₄⋊Dic₈₉	—	4	24	no	—
60	C₄⋊Dic₈₉	C₂×Q₈	1	16	yes	direct product factor
61	C₄×Dic₈₉	—	2	48	no	—
62	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₈₉	C₄⋊C₄	1	64	yes	complemented
64	C₂×Q₈×C₈₉	C₂²×C₄	1	16	yes	—
65	C₂×Q₈×C₈₉	C₂×D₄	1	16	yes	—
66	C₂×Q₈×C₈₉	C₂×Q₈	1	16	yes	—
67	C₂²×C₄×C₈₉	—	2	6	no	—
68	C₂²×C₄×C₈₉	—	2	6	no	—
69	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
73	C₂²×C₄×C₈₉	C₂×Q₈	1	12	yes	—
74	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₈₉	#41	1	4	yes	—
76	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
79	C₄⋊C₄×C₈₉	C₂²×C₄	1	48	yes	—
80	C₄²×C₈₉	C₂²×C₄	1	48	yes	—
81	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
82	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
83	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
84	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
85	C₂³×C₈₉	C₂×#41	1	2	yes	—
86	C₂³×C₈₉	H(5,𝔽₂)	1	2	yes	—
87	C₂³×C₈₉	2₋¹⁺⁴	1	2	yes	—
88	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
89	Q₈×C₈₉	C₂×C₄⋊C₄	1	64	yes	complemented
90	C₂×C₄×C₈₉	—	2	24	no	—
91	C₂×C₄×C₈₉	—	2	6	no	—
92	C₂×C₄×C₈₉	—	2	6	no	—
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	48	yes	—
94	C₂×C₄×C₈₉	C₄×Q₈	1	16	yes	complemented
95	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₈₉	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
99	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
100	C₂×C₄×C₈₉	C₂²×Q₈	1	12	yes	—
101	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
102	C₂×Dic₈₉	—	2	16	no	—
103	C₂×Dic₈₉	—	2	4	no	—
104	C₂×Dic₈₉	—	2	4	no	—
105	C₂×Dic₈₉	—	2	12	no	—
106	C₂×Dic₈₉	—	2	12	no	—
107	C₂×Dic₈₉	—	2	12	no	—
108	C₂×Dic₈₉	—	4	24	no	—
109	C₂×Dic₈₉	C₂²×Q₈	1	8	yes	—
110	C₂²×C₈₉	C₂²×C₄⋊C₄	1	4	yes	—
111	C₂²×C₈₉	C₂×C₄×Q₈	1	4	yes	—
112	C₂²×C₈₉	#520	1	4	yes	—
113	C₂²×C₈₉	D₄×Q₈	1	4	yes	—
114	C₂²×C₈₉	#550	1	4	yes	—
115	C₂²×C₈₉	#557	1	4	yes	—
116	C₂²×C₈₉	Q₈²	1	4	yes	—
117	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
118	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
119	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
120	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
121	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
122	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
123	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₈₉	—	2	12	no	—
125	C₄×C₈₉	C₂²×C₄⋊C₄	1	24	yes	—
126	C₄×C₈₉	C₂×C₄×Q₈	1	8	yes	complemented
127	Dic₈₉	—	2	8	no	—
128	Dic₈₉	—	4	12	no	—
129	C₂×Q₈×C₄⋊C₄	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	Q₈×C₄⋊C₄	1	8	yes	complemented
131	C₂×C₈₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₈₉	C₂²×C₄×Q₈	1	1	yes	characteristic
133	C₂×C₈₉	C₂×#520	1	1	yes	characteristic
134	C₂×C₈₉	C₂×D₄×Q₈	1	1	yes	characteristic
135	C₂×C₈₉	C₂×#550	1	1	yes	characteristic
136	C₂×C₈₉	C₂×#557	1	1	yes	characteristic
137	C₂×C₈₉	C₂×Q₈²	1	1	yes	characteristic
138	C₂²×C₄×Q₈	—	89	2	no	—
139	C₂²×C₄×Q₈	D₈₉	1	1	yes	characteristic
140	Q₈×C₄⋊C₄	—	89	16	no	—
141	C₂×#389	—	89	6	no	—
142	C₂×#384	—	89	3	no	—
143	C₂×C₄×C₄⋊C₄	—	89	3	no	—
144	C₈₉	C₂×Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
145	C₂³×Q₈	C₂×D₈₉	1	1	yes	characteristic
146	C₂×C₄×Q₈	—	89	16	no	—
147	C₂×C₄×Q₈	—	178	16	no	—
148	C₂×C₄×Q₈	Dic₈₉	1	16	yes	complemented
149	C₂×C₄×Q₈	C₂×D₈₉	1	8	yes	—
150	C₂²×C₄⋊C₄	—	89	1	no	—
151	C₂²×C₄⋊C₄	—	89	6	no	—
152	C₂²×C₄⋊C₄	—	89	6	no	—
153	C₂²×C₄⋊C₄	—	89	3	no	—
154	C₂²×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
155	C₂²×C₄²	—	89	6	no	—
156	C₂²×C₄²	C₂×D₈₉	1	3	yes	—
157	#389	—	89	48	no	—
158	#384	—	89	24	no	—
159	C₄×C₄⋊C₄	—	89	24	no	—
160	C₂×#95	—	89	6	no	—
161	C₂²×Q₈	Dic₁₇₈	1	8	yes	—
162	C₂²×Q₈	D₃₅₆	1	8	yes	—
163	C₂²×Q₈	C₂×Dic₈₉	1	8	yes	—
164	C₂²×Q₈	C₂²×D₈₉	1	4	yes	—
165	C₂³×C₄	—	89	2	no	—
166	C₂³×C₄	—	89	6	no	—
167	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
168	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
169	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
170	C₄×Q₈	—	178	64	no	—
171	C₄×Q₈	C₂×Dic₈₉	1	64	yes	complemented
172	C₂×C₄⋊C₄	—	89	8	no	—
173	C₂×C₄⋊C₄	—	89	4	no	—
174	C₂×C₄⋊C₄	—	89	24	no	—
175	C₂×C₄⋊C₄	—	89	24	no	—
176	C₂×C₄⋊C₄	—	89	12	no	—
177	C₂×C₄⋊C₄	—	356	12	no	—
178	C₂×C₄⋊C₄	—	178	24	no	—
179	C₂×C₄⋊C₄	—	178	12	no	—
180	C₂×C₄⋊C₄	C₂×Dic₈₉	1	24	yes	—
181	C₂×C₄⋊C₄	C₂²×D₈₉	1	12	yes	—
182	C₂×C₄²	—	89	24	no	—
183	C₂×C₄²	—	178	24	no	—
184	C₂×C₄²	C₂×Dic₈₉	1	24	yes	—
185	C₂×C₄²	C₂²×D₈₉	1	12	yes	—
186	#95	—	89	24	no	—
187	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
188	C₂×Q₈	C₄⋊Dic₈₉	1	64	yes	direct product factor
189	C₂×Q₈	C₂²×Dic₈₉	1	16	yes	—
190	C₂×Q₈	C₂×D₃₅₆	1	16	yes	—
191	C₂×Q₈	C₂×Dic₁₇₈	1	16	yes	—
192	C₂²×C₄	—	2	6	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	89	8	no	—
195	C₂²×C₄	—	89	4	no	—
196	C₂²×C₄	—	89	12	no	—
197	C₂²×C₄	—	178	12	no	—
198	C₂²×C₄	—	178	12	no	—
199	C₂²×C₄	—	178	12	no	—
200	C₂²×C₄	—	178	4	no	—
201	C₂²×C₄	—	178	4	no	—
202	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
203	C₂²×C₄	C₂²×Dic₈₉	1	4	yes	—
204	C₂²×C₄	C₂²×Dic₈₉	1	12	yes	—
205	C₂²×C₄	C₂×D₃₅₆	1	12	yes	—
206	C₂²×C₄	Q₈×D₈₉	1	4	yes	—
207	C₂²×C₄	C₂×Dic₁₇₈	1	12	yes	—
208	C₂²×C₄	#13083575	1	4	yes	—
209	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
210	C₂²×C₄	C₂³×D₈₉	1	6	yes	—
211	C₂²×C₄	C₂³×D₈₉	1	2	yes	—
212	C₄⋊C₄	—	89	16	no	—
213	C₄⋊C₄	—	356	24	no	—
214	C₄⋊C₄	—	178	48	no	—
215	C₄⋊C₄	—	178	24	no	—
216	C₄⋊C₄	C₂²×Dic₈₉	1	48	yes	—
217	C₄²	—	178	48	no	—
218	C₄²	C₂²×Dic₈₉	1	48	yes	—
219	C₂³	C₂³×Dic₈₉	1	2	yes	—
220	C₂³	C₂²×D₃₅₆	1	2	yes	—
221	C₂³	C₂×Q₈×D₈₉	1	2	yes	—
222	C₂³	C₂²×Dic₁₇₈	1	2	yes	—
223	C₂³	C₂×#13083575	1	2	yes	—
224	C₂³	#15668058	1	2	yes	—
225	C₂³	#15668059	1	2	yes	—
226	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
227	Q₈	C₂×C₄⋊Dic₈₉	1	64	yes	direct product factor
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	89	8	no	—
232	C₂×C₄	—	356	24	no	—
233	C₂×C₄	—	178	12	no	—
234	C₂×C₄	—	178	12	no	—
235	C₂×C₄	—	178	12	no	—
236	C₂×C₄	—	178	4	no	—
237	C₂×C₄	—	178	4	no	—
238	C₂×C₄	—	178	16	no	—
239	C₂×C₄	C₂×C₄⋊Dic₈₉	1	48	yes	—
240	C₂×C₄	Q₈×Dic₈₉	1	16	yes	complemented
241	C₂×C₄	C₂³×Dic₈₉	1	12	yes	—
242	C₂×C₄	C₂³×Dic₈₉	1	12	yes	—
243	C₂×C₄	C₂³×Dic₈₉	1	4	yes	—
244	C₂×C₄	C₂²×D₃₅₆	1	12	yes	—
245	C₂×C₄	C₂×Q₈×D₈₉	1	4	yes	—
246	C₂×C₄	C₂²×Dic₁₇₈	1	12	yes	—
247	C₂×C₄	C₂×#13083575	1	4	yes	—
248	C₂²	C₂²×C₄⋊Dic₈₉	1	4	yes	—
249	C₂²	C₂×Q₈×Dic₈₉	1	4	yes	—
250	C₂²	#19152081	1	4	yes	—
251	C₂²	Q₈×D₃₅₆	1	4	yes	—
252	C₂²	#19152270	1	4	yes	—
253	C₂²	#19152327	1	4	yes	—
254	C₂²	Q₈×Dic₁₇₈	1	4	yes	—
255	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
256	C₂²	C₂³×D₃₅₆	1	1	yes	characteristic
257	C₂²	C₂²×Q₈×D₈₉	1	1	yes	characteristic
258	C₂²	C₂³×Dic₁₇₈	1	1	yes	characteristic
259	C₂²	C₂²×#13083575	1	1	yes	characteristic
260	C₂²	C₂×#15668058	1	1	yes	characteristic
261	C₂²	C₂×#15668059	1	1	yes	characteristic
262	C₄	—	2	12	no	—
263	C₄	—	356	12	no	—
264	C₄	—	178	8	no	—
265	C₄	C₂²×C₄⋊Dic₈₉	1	24	yes	—
266	C₄	C₂×Q₈×Dic₈₉	1	8	yes	complemented
267	C₂	Q₈×C₄⋊Dic₈₉	1	8	yes	direct product factor
268	C₂	C₂³×C₄⋊Dic₈₉	1	1	yes	characteristic
269	C₂	C₂²×Q₈×Dic₈₉	1	1	yes	characteristic
270	C₂	C₂×#19152081	1	1	yes	characteristic
271	C₂	C₂×Q₈×D₃₅₆	1	1	yes	characteristic
272	C₂	C₂×#19152270	1	1	yes	characteristic
273	C₂	C₂×#19152327	1	1	yes	characteristic
274	C₂	C₂×Q₈×Dic₁₇₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂²×C₄×Q₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂⁴
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₈₉
hypocenter	C₈₉	C₂×Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊Dic₈₉	C₂	direct (N × H)
19. C₂×C₄×Q₈×C₈₉	C₄	semidirect
29. C₂×C₄⋊Dic₈₉	Q₈	direct (N × H)
44. C₄×Q₈×C₈₉	C₂×C₄	semidirect
60. C₄⋊Dic₈₉	C₂×Q₈	direct (N × H)
63. C₂×Q₈×C₈₉	C₄⋊C₄	semidirect
89. Q₈×C₈₉	C₂×C₄⋊C₄	semidirect
94. C₂×C₄×C₈₉	C₄×Q₈	semidirect
126. C₄×C₈₉	C₂×C₄×Q₈	semidirect
130. C₂×C₈₉	Q₈×C₄⋊C₄	semidirect
144. C₈₉	C₂×Q₈×C₄⋊C₄	semidirect
148. C₂×C₄×Q₈	Dic₈₉	semidirect
171. C₄×Q₈	C₂×Dic₈₉	semidirect
188. C₂×Q₈	C₄⋊Dic₈₉	direct (N × H)
227. Q₈	C₂×C₄⋊Dic₈₉	direct (N × H)
240. C₂×C₄	Q₈×Dic₈₉	semidirect
266. C₄	C₂×Q₈×Dic₈₉	semidirect
267. C₂	Q₈×C₄⋊Dic₈₉	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×Q₈×Dic₈₉	2
Q₈×C₄⋊Dic₈₉	2
C₂×#19151497	2
C₂×#19151468	2
C₂×C₄×C₄⋊Dic₈₉	2
C₂²×C₄×Q₈×C₈₉	2
C₂×Q₈×C₄⋊C₄	89

Subgroups — C2×Q8×C4⋊Dic89 #36713731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₄⋊Dic₈₉ #36713731