Subgroups — #36463647

Order 22784.

310 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018194	C₂	1	2	yes	maximal, complemented
4	#26018178	C₂	1	2	yes	maximal
5	C₈₉×#2922	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#13083549	—	2	1	no	—
7	C₄⋊#13083549	—	2	1	no	—
8	C₂²⋊#13083549	C₂²	1	2	yes	—
9	C₂²⋊#13083549	C₂²	1	2	yes	—
10	#19151374	—	2	1	no	—
11	C₄×#13083549	—	2	1	no	—
12	C₄×#41×C₈₉	C₄	1	1	yes	characteristic
13	#433×C₈₉	—	2	1	no	—
14	#433×C₈₉	—	2	1	no	—
15	#431×C₈₉	—	2	1	no	—
16	#431×C₈₉	—	2	1	no	—
17	#424×C₈₉	C₄	1	1	yes	characteristic
18	#424×C₈₉	C₂²	1	1	yes	characteristic
19	#408×C₈₉	C₄	1	1	yes	characteristic
20	#408×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₈₉	C₄	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	—	2	2	no	—
25	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
31	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
32	C₄×D₄×C₈₉	—	2	1	no	—
33	C₄×D₄×C₈₉	—	2	1	no	—
34	C₄×D₄×C₈₉	C₈	1	1	yes	characteristic
35	C₄×D₄×C₈₉	C₈	1	1	yes	characteristic
36	#117×C₈₉	—	2	1	no	—
37	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	—	2	1	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₈₉	—	2	1	no	—
41	C₄⋊C₈×C₈₉	—	2	1	no	—
42	C₄⋊C₈×C₈₉	—	2	1	no	—
43	C₄⋊C₈×C₈₉	—	2	1	no	—
44	C₂²⋊C₈×C₈₉	—	2	1	no	—
45	C₂²⋊C₈×C₈₉	—	2	1	no	—
46	C₂²⋊C₈×C₈₉	—	2	1	no	—
47	C₂²⋊C₈×C₈₉	—	2	1	no	—
48	#97×C₈₉	—	2	1	no	—
49	C₄×C₈×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
54	#13083549	—	4	2	no	—
55	#13083549	—	4	2	no	—
56	#41×C₈₉	—	4	1	no	—
57	#41×C₈₉	—	4	1	no	—
58	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
59	C₂×D₄×C₈₉	—	2	1	no	—
60	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	—	2	1	no	—
62	C₂²×C₄×C₈₉	—	2	1	no	—
63	C₂²×C₄×C₈₉	—	2	1	no	—
64	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₈₉	—	4	1	no	—
68	M₁₆×C₈₉	—	4	1	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	—	4	1	no	—
72	C₂×C₈×C₈₉	—	4	1	no	—
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
80	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
81	C₂²⋊C₄×C₈₉	—	2	1	no	—
82	C₂²⋊C₄×C₈₉	—	2	1	no	—
83	C₂²⋊C₄×C₈₉	—	2	1	no	—
84	C₄²×C₈₉	—	2	1	no	—
85	C₄²×C₈₉	—	2	1	no	—
86	C₄²×C₈₉	—	2	1	no	—
87	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
91	C₂³×C₈₉	—	2	1	no	—
92	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
93	Q₈×C₈₉	—	4	1	no	—
94	D₄×C₈₉	—	4	1	no	—
95	D₄×C₈₉	—	4	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	1	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	2	1	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	4	2	no	—
104	C₂×C₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	4	1	no	—
106	C₂×C₄×C₈₉	—	4	1	no	—
107	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₈×C₈₉	—	2	1	no	—
114	C₈×C₈₉	—	2	1	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	4	1	no	—
118	C₈×C₈₉	—	4	1	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	2	1	no	—
122	C₂²×C₈₉	—	4	1	no	—
123	C₂²×C₈₉	—	4	1	no	—
124	C₂²×C₈₉	—	4	1	no	—
125	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	2	1	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	4	1	no	—
131	C₄×C₈₉	—	4	1	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	#350	1	2	yes	—
134	C₄×C₈₉	#410	1	1	yes	characteristic
135	C₄×C₈₉	#413	1	1	yes	characteristic
136	#11464	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	—	4	1	no	—
138	C₂×C₈₉	—	8	1	no	—
139	C₂×C₈₉	#1458	1	1	yes	characteristic
140	C₂×C₈₉	C₂×#350	1	1	yes	characteristic
141	C₂×C₈₉	#2117	1	1	yes	characteristic
142	#2922	D₈₉	1	1	yes	characteristic
143	#2163	—	89	1	no	—
144	#2162	—	89	1	no	—
145	#1323	—	89	2	no	—
146	#1315	—	89	2	no	—
147	C₈₉	#11464	1	1	yes	characteristic, complemented, Sylow
148	C₄×#41	Dic₈₉	1	1	yes	characteristic
149	#433	—	2	1	no	—
150	#433	—	2	1	no	—
151	#431	—	2	1	no	—
152	#431	—	2	1	no	—
153	#424	Dic₈₉	1	1	yes	characteristic
154	#424	C₂×D₈₉	1	1	yes	characteristic
155	#408	Dic₈₉	1	1	yes	characteristic
156	#408	C₂×D₈₉	1	1	yes	characteristic
157	C₄×M₁₆	Dic₈₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
159	C₄⋊C₁₆	—	178	1	no	—
160	C₄⋊C₁₆	—	178	1	no	—
161	C₂²⋊C₁₆	—	89	2	no	—
162	C₂²⋊C₁₆	—	89	2	no	—
163	#346	—	178	1	no	—
164	C₄×C₁₆	—	178	1	no	—
165	C₂×#41	C₂×Dic₈₉	1	1	yes	characteristic
166	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
167	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
168	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₄×Q₈	#10623494	1	1	yes	characteristic
171	C₄×Q₈	#10623494	1	1	yes	characteristic
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	#10623494	1	1	yes	characteristic
175	C₄×D₄	#10623494	1	1	yes	characteristic
176	#117	—	2	1	no	—
177	#117	C₂×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄²	—	2	1	no	—
179	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
180	C₂×C₁₆	—	178	2	no	—
181	C₂×C₁₆	—	178	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	2	1	no	—
185	C₄⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	—	2	1	no	—
187	C₂²⋊C₈	—	2	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	2	1	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	#13083558	1	1	yes	characteristic
205	C₂²×C₄	#13083558	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
207	M₁₆	—	4	1	no	—
208	M₁₆	—	4	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
214	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
216	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
228	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
229	C₄²	C₂×#10623494	1	1	yes	characteristic
230	C₄²	C₂×#10623494	1	1	yes	characteristic
231	C₁₆	—	356	2	no	—
232	C₁₆	—	356	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₂×#13083558	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
251	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
254	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	1	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	#19151382	1	2	yes	—
276	C₄	#19151572	1	1	yes	characteristic
277	C₄	#19151582	1	1	yes	characteristic
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	#26018566	1	1	yes	characteristic
281	C₂	C₂×#19151382	1	1	yes	characteristic
282	C₂	#26021880	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#2922	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#2922	D₈₉
hypocenter	C₈₉	#11464
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #26022055	C₂	semidirect
2. #26022051	C₂	semidirect
3. #26018194	C₂	semidirect
147. C₈₉	#11464	semidirect

Maximal subgroups

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

subgroup	index
#26022055	2
#26022051	2
#26018194	2
#26018178	2
C₈₉×#2922	2
#11464	89