Subgroups — #36447553

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26020424	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018678	C₂	1	2	yes	maximal
4	#26018148	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₈₉	—	2	1	no	—
7	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
8	C₈×D₃₅₆	—	2	1	no	—
9	C₈×D₃₅₆	—	2	1	no	—
10	C₈×D₃₅₆	C₄	1	2	yes	—
11	C₈×D₃₅₆	C₄	1	2	yes	—
12	C₂×#15667948	—	2	1	no	—
13	C₂×#15667948	—	2	1	no	—
14	#19151622	C₂²	1	2	yes	—
15	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
16	C₂×#15667924	—	2	1	no	—
17	#19151327	—	2	4	no	—
18	#431×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#335×C₈₉	—	2	1	no	—
21	#334×C₈₉	—	2	1	no	—
22	C₂²×D₃₅₆	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₈₉	—	2	1	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂²×#10623494	—	2	1	no	—
26	C₂×C₈×D₈₉	—	2	2	no	—
27	C₂×C₈×D₈₉	—	2	2	no	—
28	C₂×C₈×D₈₉	—	4	2	no	—
29	C₂×C₈×D₈₉	—	4	2	no	—
30	C₄×D₃₅₆	—	2	1	no	—
31	C₄×D₃₅₆	—	2	1	no	—
32	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
33	C₄×D₃₅₆	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₈₉	Q₈	1	1	yes	characteristic
35	C₂×#13083551	—	2	1	no	—
36	#15667948	—	4	2	no	—
37	#15667948	—	4	2	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	—	2	2	no	—
41	C₄⋊#10623494	C₂×C₄	1	2	yes	—
42	#15667924	—	2	2	no	—
43	#15667924	—	2	2	no	—
44	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₂²⋊C₈×C₈₉	—	2	2	no	—
49	C₄×C₈×C₈₉	—	2	2	no	—
50	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂³×D₈₉	—	2	1	no	—
55	C₂×D₃₅₆	—	2	1	no	—
56	C₂×D₃₅₆	—	2	4	no	—
57	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
58	C₂²×Dic₈₉	—	2	1	no	—
59	C₂×C₄×D₈₉	—	2	2	no	—
60	C₂×C₄×D₈₉	—	2	2	no	—
61	C₂×C₄×D₈₉	—	4	1	no	—
62	C₂×C₄×D₈₉	—	4	1	no	—
63	C₂×C₄×D₈₉	—	4	1	no	—
64	C₂×C₄×D₈₉	—	4	1	no	—
65	#13083558	—	4	2	no	—
66	#13083558	—	4	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	4	1	no	—
69	C₂×#10623494	—	4	1	no	—
70	C₂×#10623494	—	4	2	no	—
71	C₈×D₈₉	—	4	4	no	—
72	C₈×D₈₉	—	4	4	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
75	#13083551	—	2	2	no	—
76	#13083551	—	2	2	no	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	—	2	1	no	—
82	C₂×C₈×C₈₉	—	4	2	no	—
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₈₉	D₈	1	2	yes	—
88	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₈₉	#41	1	1	yes	characteristic
93	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂²×D₈₉	—	2	4	no	—
95	C₂²×D₈₉	—	2	2	no	—
96	C₂²×D₈₉	—	4	2	no	—
97	C₂²×D₈₉	—	4	1	no	—
98	C₂²×D₈₉	—	4	1	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	4	1	no	—
113	C₂×Dic₈₉	—	4	1	no	—
114	D₃₅₆	—	2	4	no	—
115	D₃₅₆	—	2	2	no	—
116	D₃₅₆	#108	1	4	yes	complemented
117	C₄×D₈₉	—	4	2	no	—
118	C₄×D₈₉	—	4	2	no	—
119	C₄×D₈₉	—	4	2	no	—
120	C₄×D₈₉	—	4	2	no	—
121	C₈×C₈₉	—	2	1	no	—
122	C₈×C₈₉	—	2	1	no	—
123	C₈×C₈₉	—	8	2	no	—
124	C₈×C₈₉	#102	1	2	yes	—
125	C₈×C₈₉	#102	1	2	yes	—
126	#10623494	—	4	2	no	—
127	#10623494	—	8	2	no	—
128	C₂²×C₈₉	—	2	1	no	—
129	C₂²×C₈₉	—	2	1	no	—
130	C₂²×C₈₉	#380	1	1	yes	characteristic
131	C₂²×C₈₉	#474	1	1	yes	characteristic
132	C₂²×C₈₉	#476	1	1	yes	characteristic
133	C₂×D₈₉	—	2	4	no	—
134	C₂×D₈₉	—	4	4	no	—
135	C₂×D₈₉	—	4	4	no	—
136	C₂×D₈₉	—	4	2	no	—
137	C₂×D₈₉	—	4	2	no	—
138	C₄×C₈₉	—	2	1	no	—
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	2	1	no	—
141	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
142	C₄×C₈₉	#417	1	1	yes	characteristic
143	C₄×C₈₉	C₂×#108	1	2	yes	—
144	C₄×C₈₉	#459	1	1	yes	characteristic
145	C₄×C₈₉	#461	1	1	yes	characteristic
146	Dic₈₉	—	4	2	no	—
147	#8658	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	#1802	1	1	yes	characteristic
151	C₂×C₈₉	#1810	1	1	yes	characteristic
152	C₂×C₈₉	#1814	1	1	yes	characteristic
153	D₈₉	—	4	4	no	—
154	C₂×C₈×D₄	—	89	1	no	—
155	#1848	—	89	1	no	—
156	#1564	D₈₉	1	1	yes	characteristic, complemented
157	#1486	—	89	2	no	—
158	#1301	—	89	2	no	—
159	C₈₉	#8658	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	178	1	no	—
161	C₂×C₄×D₄	—	89	1	no	—
162	C₈×D₄	—	89	2	no	—
163	C₈×D₄	—	89	2	no	—
164	C₈×D₄	—	178	1	no	—
165	C₈×D₄	—	178	1	no	—
166	#431	C₂×D₈₉	1	2	yes	—
167	C₂×#108	—	178	1	no	—
168	C₂×#108	—	178	1	no	—
169	#423	—	89	2	no	—
170	C₂×C₄⋊C₈	—	89	1	no	—
171	C₂×C₂²⋊C₈	—	178	1	no	—
172	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
173	#335	—	2	1	no	—
174	#334	—	2	1	no	—
175	D₄⋊C₈	—	178	4	no	—
176	C₂²×D₄	—	89	1	no	—
177	C₂³×C₄	—	178	1	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂²×C₈	—	356	2	no	—
180	C₂²×C₈	—	356	2	no	—
181	C₂²×C₈	—	178	1	no	—
182	C₂²×C₈	—	178	2	no	—
183	C₂²×C₈	—	178	2	no	—
184	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
185	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
186	C₄×D₄	—	89	2	no	—
187	C₄×D₄	—	89	2	no	—
188	C₄×D₄	—	178	1	no	—
189	C₄×D₄	—	178	1	no	—
190	C₂×C₄⋊C₄	—	89	1	no	—
191	C₂×C₂²⋊C₄	—	178	1	no	—
192	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
193	#108	—	356	2	no	—
194	#108	—	356	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	89	2	no	—
197	C₄⋊C₈	—	178	2	no	—
198	C₄⋊C₈	—	178	1	no	—
199	C₄⋊C₈	—	178	2	no	—
200	C₂²⋊C₈	—	2	2	no	—
201	C₂²⋊C₈	—	178	2	no	—
202	C₂²⋊C₈	—	178	2	no	—
203	C₄×C₈	—	2	2	no	—
204	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
205	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
206	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
207	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂⁴	—	178	1	no	—
209	C₂×D₄	—	89	2	no	—
210	C₂×D₄	—	178	4	no	—
211	C₂×D₄	—	178	1	no	—
212	C₂²×C₄	—	356	1	no	—
213	C₂²×C₄	—	356	1	no	—
214	C₂²×C₄	—	356	1	no	—
215	C₂²×C₄	—	356	1	no	—
216	C₂²×C₄	—	178	1	no	—
217	C₂²×C₄	—	178	2	no	—
218	C₂²×C₄	—	178	2	no	—
219	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
221	C₂²×C₄	#13083573	1	1	yes	characteristic
222	M₁₆	—	356	2	no	—
223	M₁₆	—	356	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	356	1	no	—
228	C₂×C₈	—	356	1	no	—
229	C₂×C₈	—	356	4	no	—
230	C₂×C₈	—	356	4	no	—
231	C₂×C₈	—	356	2	no	—
232	C₂×C₈	—	178	2	no	—
233	C₂×C₈	#13083551	1	1	yes	characteristic
234	C₂×C₈	#13083551	1	1	yes	characteristic
235	C₂×C₈	#13083551	1	1	yes	characteristic
236	C₂×C₈	#13083551	1	1	yes	characteristic
237	C₂×C₈	D₇₁₂	1	2	yes	—
238	C₂×C₈	#13083563	1	2	yes	—
239	C₄⋊C₄	—	89	2	no	—
240	C₄⋊C₄	—	178	1	no	—
241	C₂²⋊C₄	—	178	2	no	—
242	C₂²⋊C₄	—	178	2	no	—
243	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₄²	D₄×D₈₉	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
246	C₄²	#13083575	1	1	yes	characteristic
247	C₂³	—	356	2	no	—
248	C₂³	—	356	1	no	—
249	C₂³	—	356	1	no	—
250	C₂³	—	178	4	no	—
251	C₂³	—	178	2	no	—
252	C₂³	#15667992	1	1	yes	characteristic
253	D₄	—	89	4	no	—
254	D₄	—	178	4	no	—
255	D₄	—	178	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	356	1	no	—
259	C₂×C₄	—	356	1	no	—
260	C₂×C₄	—	356	2	no	—
261	C₂×C₄	—	356	2	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	356	2	no	—
264	C₂×C₄	—	178	2	no	—
265	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₄	#15667970	1	1	yes	characteristic
268	C₂×C₄	#15667983	1	1	yes	characteristic
269	C₂×C₄	#15667987	1	1	yes	characteristic
270	C₂×C₄	#15667988	1	1	yes	characteristic
271	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
272	C₂×C₄	C₂×#13083563	1	1	yes	characteristic
273	C₂×C₄	#15668040	1	1	yes	characteristic
274	C₂×C₄	#15668045	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	356	2	no	—
279	C₈	—	712	2	no	—
280	C₈	#15667934	1	2	yes	—
281	C₈	#15667934	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	356	4	no	—
285	C₂²	—	356	4	no	—
286	C₂²	—	356	2	no	—
287	C₂²	—	356	2	no	—
288	C₂²	—	178	4	no	—
289	C₂²	#19151455	1	1	yes	characteristic
290	C₂²	#19151879	1	1	yes	characteristic
291	C₂²	#19151889	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	356	2	no	—
296	C₄	C₂×#15667934	1	1	yes	characteristic
297	C₄	#19151600	1	1	yes	characteristic
298	C₄	#108×D₈₉	1	2	yes	—
299	C₄	#19151794	1	1	yes	characteristic
300	C₄	#19151804	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	356	4	no	—
304	C₂	#26020253	1	1	yes	characteristic
305	C₂	#26020287	1	1	yes	characteristic
306	C₂	#26020303	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667987
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₃₅₆	Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#8658
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #26020424	C₂	semidirect
4. #26018148	C₂	semidirect
5. C₈₉×#1564	C₂	semidirect
116. D₃₅₆	#108	semidirect
156. #1564	D₈₉	semidirect
159. C₈₉	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₆	2
#26020424	2
#26018678	2
#26018148	2
C₈₉×#1564	2
#8658	89