Subgroups — #36564858

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028885	C₂	1	2	yes	maximal
2	#26021028	C₂	1	8	yes	maximal, complemented
3	#26020290	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151655	—	2	2	no	—
6	#19151655	—	2	2	no	—
7	#19151651	—	2	2	no	—
8	#19151651	—	2	2	no	—
9	C₂×#15667948	C₂²	1	8	yes	—
10	#19151622	C₂²	1	4	yes	—
11	#19151562	C₂²	1	4	yes	—
12	C₄×#13083558	C₂²	1	4	yes	—
13	#19151363	C₂²	1	8	yes	—
14	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₈₉	C₄	1	4	yes	—
17	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
20	C₂×#13083558	C₂³	1	8	yes	—
21	#15667948	—	4	8	no	—
22	C₄⋊#10623494	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₂²⋊#10623494	—	2	4	no	—
25	C₂²⋊#10623494	—	2	4	no	—
26	#15667923	—	2	4	no	—
27	C₄×#10623494	—	2	4	no	—
28	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₈₉	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	—	4	4	no	—
32	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
35	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₈₉	D₄	1	2	yes	—
40	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
42	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
43	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
44	C₄⋊C₈×C₈₉	C₂×C₄	1	8	yes	—
45	#13083558	—	2	8	no	—
46	#13083558	—	2	8	no	—
47	C₂×#10623494	—	2	8	no	—
48	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	—	2	4	no	—
50	C₂²×C₄×C₈₉	—	2	2	no	—
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	—	4	2	no	—
53	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₈₉	—	2	8	no	—
68	C₂×C₈×C₈₉	—	4	4	no	—
69	C₂×C₈×C₈₉	—	4	4	no	—
70	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
71	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
73	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
77	C₄²×C₈₉	C₂×D₄	1	2	yes	—
78	C₄²×C₈₉	C₂×D₄	1	2	yes	—
79	C₂³×C₈₉	—	2	2	no	—
80	C₂³×C₈₉	—	2	2	no	—
81	C₂³×C₈₉	—	2	2	no	—
82	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	—	2	2	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	—	2	4	no	—
95	C₂×C₄×C₈₉	—	2	4	no	—
96	C₂×C₄×C₈₉	—	4	1	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	—	4	2	no	—
99	C₂×C₄×C₈₉	#108	1	4	yes	complemented
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
109	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₈₉	—	2	8	no	—
111	#10623494	—	4	8	no	—
112	C₂²×C₈₉	—	2	8	no	—
113	C₂²×C₈₉	—	2	1	no	—
114	C₂²×C₈₉	—	2	1	no	—
115	C₂²×C₈₉	—	2	1	no	—
116	C₂²×C₈₉	—	2	1	no	—
117	C₂²×C₈₉	—	2	1	no	—
118	C₂²×C₈₉	—	2	1	no	—
119	C₂²×C₈₉	C₂×#108	1	2	yes	—
120	C₂²×C₈₉	#430	1	2	yes	—
121	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₈₉	#544	1	2	yes	—
123	C₄×C₈₉	—	2	1	no	—
124	C₄×C₈₉	—	2	1	no	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	2	no	—
128	C₄×C₈₉	C₂×#108	1	4	yes	—
129	C₄×C₈₉	#518	1	1	yes	characteristic
130	C₄×C₈₉	#519	1	1	yes	characteristic
131	C₄×C₈₉	#537	1	2	yes	—
132	#21969	—	89	1	no	maximal, Sylow
133	C₂×C₈₉	—	2	2	no	—
134	C₂×C₈₉	—	2	2	no	—
135	C₂×C₈₉	—	2	2	no	—
136	C₂×C₈₉	#2316	1	1	yes	characteristic
137	C₂×C₈₉	C₂²×#108	1	1	yes	characteristic
138	C₂×C₈₉	C₂×#430	1	1	yes	characteristic
139	#2920	—	89	2	no	—
140	C₂²×C₄⋊C₈	D₈₉	1	1	yes	characteristic
141	#1950	—	89	8	no	—
142	#1810	—	89	4	no	—
143	C₈₉	#21969	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₈	C₂×D₈₉	1	2	yes	—
145	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
146	#432	—	178	2	no	—
147	#432	—	178	2	no	—
148	#431	—	178	2	no	—
149	#431	—	178	2	no	—
150	C₂×#108	—	89	8	no	—
151	#423	—	89	4	no	—
152	C₂×C₄⋊C₈	Dic₈₉	1	4	yes	—
153	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
154	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	—
155	C₂×C₄⋊C₈	C₂×D₈₉	1	4	yes	—
156	#407	—	89	4	no	—
157	C₄×M₁₆	—	89	4	no	—
158	#341	—	89	8	no	—
159	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
160	C₂³×C₄	Dic₁₇₈	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
162	C₂×M₁₆	—	89	8	no	—
163	C₂²×C₈	—	4	4	no	—
164	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
165	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
166	C₂²×C₈	C₂²×D₈₉	1	4	yes	—
167	C₂×C₄²	Dic₁₇₈	1	2	yes	—
168	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
169	C₂×C₄²	D₃₅₆	1	2	yes	—
170	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
172	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
175	#108	—	356	8	no	—
176	C₄⋊C₈	—	178	4	no	—
177	C₄⋊C₈	—	178	4	no	—
178	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
179	C₄⋊C₈	C₂×Dic₈₉	1	8	yes	—
180	C₂²⋊C₈	—	178	4	no	—
181	C₂²⋊C₈	—	178	4	no	—
182	#97	—	178	4	no	—
183	C₄×C₈	—	178	4	no	—
184	C₂⁴	C₂×Dic₁₇₈	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
191	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
201	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
202	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
203	M₁₆	—	178	8	no	—
204	M₁₆	—	178	8	no	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	178	8	no	—
209	C₂×C₈	C₂²×Dic₈₉	1	8	yes	—
210	C₄²	C₄⋊Dic₈₉	1	2	yes	—
211	C₄²	C₄⋊Dic₈₉	1	2	yes	—
212	C₄²	C₄⋊Dic₈₉	1	2	yes	—
213	C₄²	C₄⋊Dic₈₉	1	2	yes	—
214	C₄²	C₂²×Dic₈₉	1	2	yes	—
215	C₄²	C₂²×Dic₈₉	1	2	yes	—
216	C₄²	C₂×D₃₅₆	1	2	yes	—
217	C₄²	C₂×D₃₅₆	1	2	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
222	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
223	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#15667948	1	4	yes	complemented
239	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
242	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
245	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
247	C₂×C₄	#15668057	1	2	yes	—
248	C₂×C₄	#15668060	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₈	—	356	8	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	C₂×#15667948	1	2	yes	—
259	C₂²	#19151647	1	2	yes	—
260	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
261	C₂²	#19152229	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	C₂×#15667948	1	4	yes	—
268	C₄	#19152073	1	1	yes	characteristic
269	C₄	#19152076	1	1	yes	characteristic
270	C₄	#19152179	1	2	yes	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	#26022870	1	1	yes	characteristic
275	C₂	C₂²×#15667948	1	1	yes	characteristic
276	C₂	C₂×#19151647	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄⋊C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#21969
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #26021028	C₂	semidirect
99. C₂×C₄×C₈₉	#108	semidirect
143. C₈₉	#21969	semidirect
238. C₂×C₄	#15667948	semidirect

Maximal subgroups

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

subgroup	index
#26028885	2
#26021028	2
#26020290	2
C₂²×C₄⋊C₈×C₈₉	2
#21969	89