Subgroups — #36470351

Order 22784.

478 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151416	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic
4	#26018329	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₄	1	1	yes	characteristic
7	C₂×#15667971	C₄	1	1	yes	characteristic
8	C₂²×#13083549	C₂²	1	2	yes	—
9	#19151553	—	2	4	no	—
10	C₂×#15667922	C₄	1	1	yes	characteristic
11	C₂×#15667922	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151416	—	2	4	no	—
15	C₂²⋊#13083549	—	2	2	no	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₈₉	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
24	#15667971	—	2	2	no	—
25	#15667971	—	2	2	no	—
26	C₂×C₄⋊Dic₈₉	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₈₉	C₈	1	2	yes	—
28	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×#13083549	—	2	2	no	—
31	C₂×#13083549	—	2	2	no	—
32	C₂×#13083549	—	2	2	no	—
33	#15667922	—	2	2	no	—
34	#15667922	—	2	2	no	—
35	C₈×Dic₈₉	—	2	2	no	—
36	C₈×Dic₈₉	D₄	1	4	yes	—
37	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	—	2	2	no	—
40	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	D₄	1	2	yes	—
42	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₈₉	—	2	2	no	—
45	C₂×C₁₆×C₈₉	—	2	2	no	—
46	C₂×C₁₆×C₈₉	—	2	2	no	—
47	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
48	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
49	#13083558	—	2	8	no	—
50	C₂×#10623494	—	2	1	no	—
51	C₂×#10623494	—	2	1	no	—
52	C₂×#10623494	—	2	1	no	—
53	C₂×#10623494	—	2	1	no	—
54	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
55	C₂×#10623494	M₁₆	1	2	yes	—
56	C₄⋊Dic₈₉	—	2	4	no	—
57	C₂²⋊Dic₈₉	—	2	4	no	—
58	C₄×Dic₈₉	—	2	1	no	—
59	C₄×Dic₈₉	—	2	1	no	—
60	C₄×Dic₈₉	C₂²⋊C₄	1	2	yes	—
61	C₄×Dic₈₉	M₁₆	1	2	yes	—
62	#13083549	—	4	2	no	—
63	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	—	2	1	no	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	—	2	1	no	—
67	C₂²×C₄×C₈₉	—	2	1	no	—
68	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	—	2	1	no	—
73	C₂×C₈×C₈₉	—	2	1	no	—
74	C₂×C₈×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	—	2	4	no	—
76	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₈×C₈₉	#41	1	2	yes	—
81	C₁₆×C₈₉	—	4	2	no	—
82	C₂³×C₈₉	—	2	1	no	—
83	C₂³×C₈₉	—	2	1	no	—
84	C₂³×C₈₉	—	2	2	no	—
85	C₂³×C₈₉	—	2	2	no	—
86	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
87	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	—	2	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	2	2	no	—
94	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×Dic₈₉	—	2	2	no	—
102	C₂×Dic₈₉	—	2	2	no	—
103	C₂×Dic₈₉	—	2	2	no	—
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	C₂²⋊C₈	1	4	yes	—
107	C₈×C₈₉	—	2	2	no	—
108	C₈×C₈₉	#121	1	2	yes	—
109	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
110	#10623494	—	2	2	no	—
111	#10623494	—	2	2	no	—
112	C₂²×C₈₉	—	2	2	no	—
113	C₂²×C₈₉	—	2	2	no	—
114	C₂²×C₈₉	—	2	2	no	—
115	C₂²×C₈₉	—	4	2	no	—
116	C₂²×C₈₉	—	4	2	no	—
117	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₈₉	#431	1	1	yes	characteristic
120	C₂²×C₈₉	#504	1	2	yes	—
121	C₂²×C₈₉	#504	1	2	yes	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	#380	1	1	yes	characteristic
124	C₄×C₈₉	#407	1	1	yes	characteristic
125	C₄×C₈₉	#423	1	1	yes	characteristic
126	C₄×C₈₉	#432	1	1	yes	characteristic
127	Dic₈₉	—	2	4	no	—
128	Dic₈₉	—	4	2	no	—
129	#12784	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	4	2	no	—
131	C₂×C₈₉	#1797	1	1	yes	characteristic
132	C₂×C₈₉	#2113	1	2	yes	—
133	C₂×C₈₉	#2151	1	1	yes	characteristic
134	C₂×C₈₉	#2151	1	1	yes	characteristic
135	C₂×C₈₉	#2164	1	2	yes	—
136	C₂×#405	—	89	1	no	—
137	C₂×C₄⋊C₁₆	—	89	2	no	—
138	C₂×C₂²⋊C₁₆	—	89	1	no	—
139	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
140	#1374	—	89	2	no	—
141	C₈₉	#12784	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	—	89	1	no	—
143	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
144	C₂×#117	—	89	1	no	—
145	C₂²×C₁₆	—	89	2	no	—
146	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
147	#405	—	178	4	no	—
148	C₂×#97	—	89	1	no	—
149	C₂×#97	—	89	1	no	—
150	C₂×C₄×C₈	—	89	1	no	—
151	C₂×C₄×C₈	—	89	1	no	—
152	C₄⋊C₁₆	—	178	4	no	—
153	C₂²⋊C₁₆	—	2	2	no	—
154	C₂²⋊C₁₆	—	178	2	no	—
155	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
156	C₂×M₁₆	—	178	2	no	—
157	C₂×M₁₆	—	178	2	no	—
158	C₂×M₁₆	—	178	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	89	1	no	—
162	C₂²×C₈	—	89	1	no	—
163	C₂²×C₈	Dic₁₇₈	1	2	yes	—
164	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
165	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
166	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
167	#117	—	178	2	no	—
168	#117	—	178	2	no	—
169	C₂×C₄⋊C₄	—	89	2	no	—
170	C₂×C₂²⋊C₄	—	89	2	no	—
171	C₂×C₄²	—	89	1	no	—
172	C₂×C₄²	—	89	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	178	2	no	—
177	C₂×C₁₆	—	178	2	no	—
178	C₂×C₁₆	—	178	2	no	—
179	#97	—	178	2	no	—
180	#97	—	178	2	no	—
181	C₄×C₈	—	89	4	no	—
182	C₄×C₈	—	178	2	no	—
183	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	89	2	no	—
189	C₂²×C₄	—	89	2	no	—
190	C₂²×C₄	#13083553	1	2	yes	—
191	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
192	C₂²×C₄	#13083557	1	2	yes	—
193	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
194	M₁₆	—	178	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	89	2	no	—
200	C₂×C₈	—	89	2	no	—
201	C₂×C₈	—	178	1	no	—
202	C₂×C₈	—	178	1	no	—
203	C₂×C₈	—	178	1	no	—
204	C₂×C₈	—	178	1	no	—
205	C₂×C₈	#13083553	1	2	yes	—
206	C₂×C₈	D₄×D₈₉	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
209	C₂×C₈	#13083573	1	2	yes	—
210	C₄⋊C₄	—	178	4	no	—
211	C₂²⋊C₄	—	178	4	no	—
212	C₄²	—	89	2	no	—
213	C₄²	—	89	2	no	—
214	C₄²	—	178	1	no	—
215	C₄²	—	178	1	no	—
216	C₁₆	—	4	2	no	—
217	C₁₆	—	356	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	#15667941	1	2	yes	—
223	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	89	4	no	—
231	C₂×C₄	—	178	2	no	—
232	C₂×C₄	—	178	2	no	—
233	C₂×C₄	—	178	2	no	—
234	C₂×C₄	—	178	2	no	—
235	C₂×C₄	—	178	2	no	—
236	C₂×C₄	#15667941	1	2	yes	—
237	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
239	C₂×C₄	#15667968	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	178	2	no	—
245	C₈	—	178	2	no	—
246	C₈	#15667984	1	2	yes	—
247	C₈	#15667989	1	2	yes	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
254	C₂²	C₂×#15667941	1	1	yes	characteristic
255	C₂²	#19151648	1	1	yes	characteristic
256	C₂²	#19152014	1	2	yes	—
257	C₂²	#19152016	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	356	2	no	—
260	C₄	—	178	4	no	—
261	C₄	#19151456	1	1	yes	characteristic
262	C₄	#19151560	1	1	yes	characteristic
263	C₄	#19151621	1	1	yes	characteristic
264	C₄	#19151652	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	#26020230	1	1	yes	characteristic
267	C₂	#26021864	1	2	yes	—
268	C₂	#26022010	1	1	yes	characteristic
269	C₂	#26022012	1	1	yes	characteristic
270	C₂	#26022056	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂²×C₈
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	#12784
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19151416	C₂	semidirect
4. #26018329	C₂	semidirect
141. C₈₉	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151553	2
C₂×#19151416	2
C₂×C₂²⋊#13083549	2
#26018329	2
C₂×C₂²⋊C₁₆×C₈₉	2
#12784	89