Subgroups — #36447529

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151677	C₂	1	1	yes	maximal, characteristic
2	#26020249	C₂	1	2	yes	maximal
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018135	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	—	2	1	no	—
7	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
8	#19151677	—	2	2	no	—
9	#19151677	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×#15667932	—	2	2	no	—
13	C₂×#15667924	—	2	1	no	—
14	#19151360	—	2	2	no	—
15	#19151351	—	2	2	no	—
16	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#384×C₈₉	C₂²	1	2	yes	—
18	#340×C₈₉	—	2	1	no	—
19	#340×C₈₉	—	2	1	no	—
20	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₈₉	—	2	1	no	—
22	C₂×#13083557	—	2	4	no	—
23	C₂×#13083557	—	4	4	no	—
24	C₂²×#10623494	—	2	1	no	—
25	C₂²×#10623494	—	2	2	no	—
26	C₄×D₃₅₆	—	2	1	no	—
27	C₄×D₃₅₆	—	2	1	no	—
28	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
29	C₄×D₃₅₆	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
31	C₂×#13083551	—	2	1	no	—
32	C₄⋊#10623494	—	2	1	no	—
33	C₄⋊#10623494	—	2	2	no	—
34	C₄⋊#10623494	—	2	2	no	—
35	C₄⋊#10623494	C₂×C₄	1	2	yes	—
36	#15667932	—	4	4	no	—
37	#15667924	—	2	4	no	—
38	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₈₉	D₄	1	2	yes	—
46	C₂³×D₈₉	—	2	1	no	—
47	C₂×D₃₅₆	—	2	2	no	—
48	C₂×D₃₅₆	—	2	2	no	—
49	C₂×D₃₅₆	—	2	1	no	—
50	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₈₉	—	2	1	no	—
52	C₂×C₄×D₈₉	—	2	2	no	—
53	C₂×C₄×D₈₉	—	2	2	no	—
54	C₂×C₄×D₈₉	—	4	2	no	—
55	C₂×C₄×D₈₉	—	4	2	no	—
56	#13083557	—	4	8	no	—
57	C₂×#10623494	—	2	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₄⋊Dic₈₉	—	2	1	no	—
63	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
64	#13083551	—	2	2	no	—
65	#13083551	—	2	2	no	—
66	C₂²×C₄×C₈₉	—	4	2	no	—
67	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	—	2	2	no	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₄⋊C₄×C₈₉	—	4	2	no	—
75	C₄⋊C₄×C₈₉	—	4	2	no	—
76	C₄⋊C₄×C₈₉	—	4	2	no	—
77	C₄⋊C₄×C₈₉	—	4	2	no	—
78	C₄⋊C₄×C₈₉	—	8	2	no	—
79	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₈₉	#41	1	1	yes	characteristic
83	C₂³×C₈₉	#121	1	1	yes	characteristic
84	C₂²×D₈₉	—	2	2	no	—
85	C₂²×D₈₉	—	4	2	no	—
86	C₂²×D₈₉	—	4	2	no	—
87	C₂²×D₈₉	—	4	1	no	—
88	C₂²×D₈₉	—	4	1	no	—
89	C₂×C₄×C₈₉	—	2	1	no	—
90	C₂×C₄×C₈₉	—	2	1	no	—
91	C₂×C₄×C₈₉	—	8	2	no	—
92	C₂×C₄×C₈₉	—	8	2	no	—
93	C₂×C₄×C₈₉	—	8	2	no	—
94	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#135	1	2	yes	—
101	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
103	C₂×C₄×C₈₉	#137	1	2	yes	—
104	C₂×Dic₈₉	—	2	2	no	—
105	C₂×Dic₈₉	—	4	2	no	—
106	D₃₅₆	—	2	4	no	—
107	D₃₅₆	—	4	1	no	—
108	D₃₅₆	—	4	1	no	—
109	C₄×D₈₉	—	4	2	no	—
110	C₄×D₈₉	—	4	2	no	—
111	C₄×D₈₉	—	4	4	no	—
112	C₈×C₈₉	—	2	2	no	—
113	C₈×C₈₉	—	2	2	no	—
114	#10623494	—	4	2	no	—
115	#10623494	—	8	2	no	—
116	C₂²×C₈₉	#380	1	1	yes	characteristic
117	C₂²×C₈₉	#463	1	2	yes	—
118	C₂²×C₈₉	#468	1	2	yes	—
119	C₂²×C₈₉	#472	1	1	yes	characteristic
120	C₂²×C₈₉	#473	1	1	yes	characteristic
121	C₂×D₈₉	—	4	2	no	—
122	C₂×D₈₉	—	4	2	no	—
123	C₂×D₈₉	—	4	2	no	—
124	C₂×D₈₉	—	8	2	no	—
125	C₂×D₈₉	—	8	2	no	—
126	C₄×C₈₉	—	2	1	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	—	16	2	no	—
129	C₄×C₈₉	#416	1	1	yes	characteristic
130	C₄×C₈₉	#418	1	1	yes	characteristic
131	C₄×C₈₉	#428	1	2	yes	—
132	C₄×C₈₉	#469	1	1	yes	characteristic
133	C₄×C₈₉	#470	1	1	yes	characteristic
134	C₄×C₈₉	#478	1	2	yes	—
135	Dic₈₉	—	4	2	no	—
136	#8654	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	#1697	1	2	yes	—
138	C₂×C₈₉	#1699	1	2	yes	—
139	C₂×C₈₉	#1804	1	1	yes	characteristic
140	C₂×C₈₉	#1809	1	1	yes	characteristic
141	C₂×C₈₉	#1816	1	1	yes	characteristic
142	D₈₉	—	8	2	no	—
143	C₂×#436	—	89	1	no	—
144	#1834	D₈₉	1	1	yes	characteristic, complemented
145	#1802	—	89	2	no	—
146	C₂×#335	—	89	1	no	—
147	#1296	—	89	2	no	—
148	C₈₉	#8654	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	178	1	no	—
150	C₂×C₄×D₄	—	89	1	no	—
151	#436	—	89	4	no	—
152	#436	—	178	2	no	—
153	C₂×C₄⋊C₈	—	89	1	no	—
154	C₂×C₄⋊C₈	—	89	2	no	—
155	C₂×#102	—	178	2	no	—
156	C₂×C₂²⋊C₈	—	178	1	no	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	#384	C₂×D₈₉	1	2	yes	—
159	#340	—	2	1	no	—
160	#340	—	2	1	no	—
161	#340	—	178	2	no	—
162	#335	—	178	2	no	—
163	C₂²×D₄	—	89	1	no	—
164	C₂³×C₄	—	178	1	no	—
165	C₂×M₁₆	—	356	4	no	—
166	C₂×M₁₆	—	178	4	no	—
167	C₂²×C₈	—	178	2	no	—
168	C₂²×C₈	—	178	1	no	—
169	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
171	C₄×D₄	—	89	2	no	—
172	C₄×D₄	—	89	2	no	—
173	C₄×D₄	—	178	1	no	—
174	C₄×D₄	—	178	1	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	89	1	no	—
179	C₂×C₂²⋊C₄	—	178	1	no	—
180	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
181	C₄⋊C₈	—	89	2	no	—
182	C₄⋊C₈	—	178	2	no	—
183	C₄⋊C₈	—	178	1	no	—
184	C₄⋊C₈	—	178	2	no	—
185	#102	—	356	4	no	—
186	C₂²⋊C₈	—	178	4	no	—
187	C₄×C₈	C₄×D₈₉	1	2	yes	—
188	C₄×C₈	D₃₅₆	1	2	yes	—
189	C₂⁴	—	178	1	no	—
190	C₂×D₄	—	89	2	no	—
191	C₂×D₄	—	178	1	no	—
192	C₂×D₄	—	178	2	no	—
193	C₂×D₄	—	178	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	356	2	no	—
196	C₂²×C₄	—	356	2	no	—
197	C₂²×C₄	—	178	1	no	—
198	C₂²×C₄	—	178	2	no	—
199	C₂²×C₄	—	178	2	no	—
200	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂²×C₄	#13083573	1	1	yes	characteristic
203	M₁₆	—	356	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	356	2	no	—
207	C₂×C₈	—	356	2	no	—
208	C₂×C₈	—	356	2	no	—
209	C₂×C₈	—	356	2	no	—
210	C₂×C₈	—	178	2	no	—
211	C₂×C₈	#13083551	1	2	yes	—
212	C₂×C₈	#13083551	1	2	yes	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	8	2	no	—
218	C₄⋊C₄	—	89	2	no	—
219	C₄⋊C₄	—	178	1	no	—
220	C₂²⋊C₄	—	178	2	no	—
221	C₂²⋊C₄	—	178	2	no	—
222	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
223	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
224	C₄²	Q₈×D₈₉	1	1	yes	characteristic
225	C₄²	#13083575	1	1	yes	characteristic
226	C₂³	—	356	2	no	—
227	C₂³	—	356	2	no	—
228	C₂³	—	356	1	no	—
229	C₂³	—	356	1	no	—
230	C₂³	—	178	2	no	—
231	C₂³	#15667983	1	1	yes	characteristic
232	D₄	—	356	1	no	—
233	D₄	—	356	1	no	—
234	D₄	—	178	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	356	2	no	—
241	C₂×C₄	—	356	4	no	—
242	C₂×C₄	—	356	2	no	—
243	C₂×C₄	—	356	2	no	—
244	C₂×C₄	—	178	2	no	—
245	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
246	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
247	C₂×C₄	#15667970	1	1	yes	characteristic
248	C₂×C₄	#15667987	1	1	yes	characteristic
249	C₂×C₄	#15667988	1	1	yes	characteristic
250	C₂×C₄	#15667992	1	1	yes	characteristic
251	C₂×C₄	#15668032	1	2	yes	—
252	C₂×C₄	#15668037	1	2	yes	—
253	C₂×C₄	#15668039	1	2	yes	—
254	C₂×C₄	#15668041	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	356	2	no	—
258	C₈	—	712	2	no	—
259	C₂²	—	356	2	no	—
260	C₂²	—	356	2	no	—
261	C₂²	—	356	2	no	—
262	C₂²	—	712	2	no	—
263	C₂²	—	712	2	no	—
264	C₂²	#19151455	1	1	yes	characteristic
265	C₂²	#19151810	1	2	yes	—
266	C₂²	#19151841	1	2	yes	—
267	C₂²	#19151865	1	1	yes	characteristic
268	C₂²	#19151870	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	16	2	no	—
272	C₄	—	356	2	no	—
273	C₄	#19151592	1	1	yes	characteristic
274	C₄	#19151602	1	1	yes	characteristic
275	C₄	#19151637	1	2	yes	—
276	C₄	#19151852	1	1	yes	characteristic
277	C₄	#19151857	1	1	yes	characteristic
278	C₄	#19151895	1	2	yes	—
279	C₂	—	712	2	no	—
280	C₂	#26019782	1	2	yes	—
281	C₂	#26019796	1	2	yes	—
282	C₂	#26020259	1	1	yes	characteristic
283	C₂	#26020282	1	1	yes	characteristic
284	C₂	#26020317	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1834	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1834	D₈₉
hypocenter	C₈₉	#8654
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
3. C₂×#19151351	C₂	semidirect
4. #26018135	C₂	semidirect
5. C₈₉×#1834	C₂	semidirect
9. #19151677	C₄	semidirect
144. #1834	D₈₉	semidirect
148. C₈₉	#8654	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151677	2
#26020249	2
C₂×#19151351	2
#26018135	2
C₈₉×#1834	2
#8654	89