Subgroups — #36447251

Order 22784.

476 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#15668012	C₂	1	1	yes	maximal, characteristic
2	#26020369	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151343	C₂	1	2	yes	maximal
4	#26018136	C₂	1	2	yes	maximal
5	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668012	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₇₈	—	2	1	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	#19151504	—	2	1	no	—
10	#19151496	—	2	1	no	—
11	#19151496	C₄	1	2	yes	—
12	#19151468	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₈₉	—	2	1	no	—
14	C₄×C₄⋊Dic₈₉	C₄	1	2	yes	—
15	#19151353	—	2	1	no	—
16	#19151353	—	2	1	no	—
17	#19151343	—	2	4	no	—
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
19	C₄³×C₈₉	C₂²	1	1	yes	characteristic
20	#336×C₈₉	—	2	1	no	—
21	#336×C₈₉	—	2	1	no	—
22	C₂²×Dic₁₇₈	—	2	1	no	—
23	C₂²×#10623494	—	2	2	no	—
24	#15668012	—	2	1	no	—
25	#15668012	—	2	1	no	—
26	#15668012	D₄	1	2	yes	—
27	#15668012	Q₈	1	2	yes	—
28	C₄×Dic₁₇₈	—	4	4	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
34	C₂×#13083553	—	2	2	no	—
35	C₂×C₄×Dic₈₉	—	2	2	no	—
36	C₄⋊#10623494	—	2	2	no	—
37	C₄⋊#10623494	—	2	2	no	—
38	#15667919	—	2	2	no	—
39	#15667919	—	2	2	no	—
40	C₂²×C₈×C₈₉	—	2	2	no	—
41	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₈₉	—	2	2	no	—
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₂×Dic₁₇₈	—	2	2	no	—
51	C₂×Dic₁₇₈	—	2	2	no	—
52	C₂×Dic₁₇₈	—	4	2	no	—
53	C₂×Dic₁₇₈	—	4	2	no	—
54	C₂²×Dic₈₉	—	2	2	no	—
55	C₂²×Dic₈₉	—	2	2	no	—
56	C₂×#10623494	—	4	2	no	—
57	C₂×#10623494	—	4	2	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₄⋊Dic₈₉	—	2	2	no	—
60	C₄⋊Dic₈₉	—	2	2	no	—
61	C₄⋊Dic₈₉	—	2	2	no	—
62	C₄⋊Dic₈₉	—	4	2	no	—
63	C₄⋊Dic₈₉	—	4	2	no	—
64	C₄⋊Dic₈₉	—	4	2	no	—
65	C₄⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
66	#13083553	—	4	4	no	—
67	C₄×Dic₈₉	—	4	4	no	—
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₂×C₈×C₈₉	—	4	2	no	—
77	C₂×C₈×C₈₉	—	4	2	no	—
78	C₄²×C₈₉	—	2	2	no	—
79	C₄²×C₈₉	—	2	2	no	—
80	C₄²×C₈₉	—	2	2	no	—
81	C₄²×C₈₉	—	2	2	no	—
82	C₄²×C₈₉	—	2	2	no	—
83	C₄²×C₈₉	—	4	1	no	—
84	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₈₉	#41	1	1	yes	characteristic
87	C₄²×C₈₉	#41	1	1	yes	characteristic
88	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	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	#101	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#135	1	2	yes	—
110	C₂×C₄×C₈₉	#135	1	2	yes	—
111	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
112	C₂×C₄×C₈₉	#137	1	2	yes	—
113	C₂×Dic₈₉	—	2	4	no	—
114	C₂×Dic₈₉	—	4	2	no	—
115	C₂×Dic₈₉	—	4	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	Dic₁₇₈	—	4	4	no	—
120	Dic₁₇₈	—	4	4	no	—
121	C₈×C₈₉	—	8	2	no	—
122	#10623494	—	8	2	no	—
123	C₂²×C₈₉	#380	1	1	yes	characteristic
124	C₂²×C₈₉	C₂×#101	1	1	yes	characteristic
125	C₂²×C₈₉	#413	1	1	yes	characteristic
126	C₂²×C₈₉	#416	1	1	yes	characteristic
127	C₂²×C₈₉	#416	1	1	yes	characteristic
128	C₂²×C₈₉	#418	1	1	yes	characteristic
129	C₂²×C₈₉	#419	1	1	yes	characteristic
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	—	4	1	no	—
134	C₄×C₈₉	—	4	1	no	—
135	C₄×C₈₉	—	4	1	no	—
136	C₄×C₈₉	#463	1	2	yes	—
137	C₄×C₈₉	#464	1	2	yes	—
138	C₄×C₈₉	#478	1	2	yes	—
139	C₄×C₈₉	#479	1	2	yes	—
140	Dic₈₉	—	4	4	no	—
141	Dic₈₉	—	8	2	no	—
142	#8593	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	#1544	1	1	yes	characteristic
144	C₂×C₈₉	#1553	1	1	yes	characteristic
145	C₂×C₈₉	#1554	1	1	yes	characteristic
146	C₂×C₈₉	#1556	1	1	yes	characteristic
147	C₂×C₈₉	#1801	1	1	yes	characteristic
148	C₂×C₈₉	#1804	1	1	yes	characteristic
149	C₂×C₈₉	#1805	1	1	yes	characteristic
150	C₄×C₄⋊Q₈	—	89	1	no	—
151	#1831	—	89	1	no	—
152	#1831	D₈₉	1	1	yes	characteristic
153	C₂×#332	—	89	2	no	—
154	#1296	—	89	2	no	—
155	C₈₉	#8593	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊Q₈	—	89	1	no	—
157	C₂×C₄×Q₈	—	178	1	no	—
158	C₂×C₄⋊C₈	—	89	2	no	—
159	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
160	#391	—	178	1	no	—
161	#389	—	89	2	no	—
162	#389	—	178	1	no	—
163	#384	—	89	1	no	—
164	C₄×C₄⋊C₄	—	89	2	no	—
165	C₄×C₄⋊C₄	—	178	1	no	—
166	C₄³	C₂×D₈₉	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#336	—	2	1	no	—
169	#336	—	178	1	no	—
170	#336	—	178	1	no	—
171	#332	—	178	4	no	—
172	C₂²×Q₈	—	178	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	178	2	no	—
175	C₄⋊Q₈	—	89	2	no	—
176	C₄⋊Q₈	—	89	2	no	—
177	C₄⋊Q₈	—	178	1	no	—
178	C₄⋊Q₈	—	178	1	no	—
179	C₄×Q₈	—	356	4	no	—
180	C₂×C₄⋊C₄	—	89	2	no	—
181	C₂×C₄⋊C₄	—	89	2	no	—
182	C₂×C₄⋊C₄	—	178	2	no	—
183	C₂×C₄⋊C₄	—	178	1	no	—
184	C₂×C₄⋊C₄	—	178	1	no	—
185	C₂×C₄⋊C₄	—	178	1	no	—
186	C₂×C₄²	—	178	2	no	—
187	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
189	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
190	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	178	2	no	—
197	C₄⋊C₈	—	178	2	no	—
198	#95	—	178	2	no	—
199	#95	—	178	2	no	—
200	C₂×Q₈	—	356	2	no	—
201	C₂×Q₈	—	356	2	no	—
202	C₂×Q₈	—	178	2	no	—
203	C₂×Q₈	—	178	2	no	—
204	C₂²×C₄	—	178	2	no	—
205	C₂²×C₄	—	178	2	no	—
206	C₂²×C₄	#13083551	1	1	yes	characteristic
207	C₂²×C₄	#13083551	1	1	yes	characteristic
208	C₂²×C₄	#13083551	1	1	yes	characteristic
209	C₂²×C₄	#13083551	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	356	2	no	—
217	C₂×C₈	—	356	2	no	—
218	C₂×C₈	—	356	2	no	—
219	C₄⋊C₄	—	89	4	no	—
220	C₄⋊C₄	—	356	2	no	—
221	C₄⋊C₄	—	356	4	no	—
222	C₄⋊C₄	—	356	2	no	—
223	C₄⋊C₄	—	356	2	no	—
224	C₄⋊C₄	—	178	2	no	—
225	C₄⋊C₄	—	178	2	no	—
226	C₄⋊C₄	—	178	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	356	4	no	—
234	C₄²	D₄×D₈₉	1	1	yes	characteristic
235	C₄²	Q₈×D₈₉	1	1	yes	characteristic
236	C₄²	#13083573	1	1	yes	characteristic
237	C₄²	#13083575	1	1	yes	characteristic
238	C₂³	C₂×#13083551	1	1	yes	characteristic
239	Q₈	—	356	4	no	—
240	Q₈	—	356	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	356	2	no	—
255	C₂×C₄	—	356	2	no	—
256	C₂×C₄	—	356	2	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	178	4	no	—
260	C₂×C₄	#15667930	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
262	C₂×C₄	#15667970	1	1	yes	characteristic
263	C₂×C₄	#15667983	1	1	yes	characteristic
264	C₂×C₄	#15667987	1	1	yes	characteristic
265	C₂×C₄	#15667988	1	1	yes	characteristic
266	C₂×C₄	#15667992	1	1	yes	characteristic
267	C₂×C₄	#15668032	1	2	yes	—
268	C₂×C₄	#15668034	1	2	yes	—
269	C₂×C₄	#15668040	1	2	yes	—
270	C₂×C₄	#15668042	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	712	2	no	—
273	C₂²	#19151455	1	1	yes	characteristic
274	C₂²	C₂×#15667930	1	1	yes	characteristic
275	C₂²	#19151580	1	1	yes	characteristic
276	C₂²	#19151592	1	1	yes	characteristic
277	C₂²	#19151595	1	1	yes	characteristic
278	C₂²	#19151604	1	1	yes	characteristic
279	C₂²	#19151606	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	356	4	no	—
287	C₄	—	712	2	no	—
288	C₄	#19151814	1	2	yes	—
289	C₄	#19151817	1	2	yes	—
290	C₄	#19151899	1	2	yes	—
291	C₄	#19151902	1	2	yes	—
292	C₂	#26018916	1	1	yes	characteristic
293	C₂	#26018958	1	1	yes	characteristic
294	C₂	#26018960	1	1	yes	characteristic
295	C₂	#26018968	1	1	yes	characteristic
296	C₂	#26020245	1	1	yes	characteristic
297	C₂	#26020263	1	1	yes	characteristic
298	C₂	#26020264	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083551
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8593
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
155. C₈₉	#8593	semidirect

Maximal subgroups

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

subgroup	index
C₄×#15668012	2
#26020369	2
C₂×#19151343	2
#26018136	2
C₈₉×#1831	2
#8593	89