Subgroups — #36469277

Order 22784.

586 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	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151553	C₂	1	1	yes	maximal, characteristic
3	#26018331	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	C₄	1	1	yes	characteristic
6	C₂×#15667971	C₄	1	1	yes	characteristic
7	C₂×#15667952	—	2	2	no	—
8	C₂×#15667952	—	2	2	no	—
9	C₂×#15667952	C₄	1	2	yes	complemented
10	C₂×#15667952	C₂²	1	2	yes	—
11	C₂²×#13083549	C₄	1	2	yes	complemented
12	#19151553	—	2	2	no	—
13	#19151553	—	2	2	no	—
14	C₂×#15667922	C₂²	1	2	yes	—
15	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₈₉	—	2	2	no	—
18	C₂×M₃₂×C₈₉	—	2	2	no	—
19	C₂×M₃₂×C₈₉	C₄	1	2	yes	complemented
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₈₉	C₄	1	2	yes	complemented
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	1	no	—
24	C₂×#13083558	—	2	1	no	—
25	C₂×#13083558	C₈	1	4	yes	—
26	C₂²×#10623494	C₂×C₄	1	2	yes	—
27	#15667971	—	2	2	no	—
28	#15667971	C₈	1	4	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
32	#15667952	—	2	4	no	—
33	#15667952	—	2	2	no	—
34	#15667952	—	2	2	no	—
35	C₂×#13083549	—	2	4	no	—
36	C₂×#13083549	C₂×C₄	1	4	yes	—
37	#15667922	—	2	2	no	—
38	#15667922	—	2	2	no	—
39	C₈×Dic₈₉	—	2	4	no	—
40	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	2	yes	—
47	C₂²×C₈×C₈₉	D₄	1	2	yes	—
48	C₂²×C₈×C₈₉	D₄	1	2	yes	—
49	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	M₃₂×C₈₉	—	2	4	no	—
52	M₃₂×C₈₉	—	2	2	no	—
53	M₃₂×C₈₉	—	2	2	no	—
54	C₂×C₁₆×C₈₉	—	2	4	no	—
55	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
56	C₂²×Dic₈₉	—	2	2	no	—
57	#13083558	—	2	4	no	—
58	#13083558	—	2	4	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	C₂×C₈	1	4	yes	—
62	C₄⋊Dic₈₉	—	2	4	no	—
63	C₂²⋊Dic₈₉	—	2	4	no	—
64	C₄×Dic₈₉	—	2	2	no	—
65	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
66	#13083549	—	2	4	no	—
67	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
96	C₁₆×C₈₉	—	2	4	no	—
97	C₂³×C₈₉	—	4	2	no	—
98	C₂³×C₈₉	#97	1	1	yes	characteristic
99	C₂³×C₈₉	#97	1	1	yes	characteristic
100	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
101	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
102	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#95	1	2	yes	—
104	C₂×C₄×C₈₉	#95	1	2	yes	—
105	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
107	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
113	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	4	4	no	—
123	C₈×C₈₉	#95	1	2	yes	—
124	C₈×C₈₉	#95	1	2	yes	—
125	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	#10623494	—	2	4	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	—	2	2	no	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	4	2	no	—
134	C₂²×C₈₉	—	4	2	no	—
135	C₂²×C₈₉	#336	1	2	yes	—
136	C₂²×C₈₉	#336	1	2	yes	—
137	C₂²×C₈₉	C₂×#97	1	1	yes	characteristic
138	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
140	C₄×C₈₉	#336	1	2	yes	—
141	C₄×C₈₉	#336	1	2	yes	—
142	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
143	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
144	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
146	Dic₈₉	—	4	4	no	—
147	#12547	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	—	2	1	no	—
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	C₂×#336	1	1	yes	characteristic
153	C₂×C₈₉	#2110	1	2	yes	—
154	C₂²×M₃₂	—	89	1	no	—
155	C₂²×M₃₂	D₈₉	1	1	yes	characteristic
156	C₂×#405	—	89	1	no	—
157	#1375	—	89	4	no	—
158	C₈₉	#12547	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	89	1	no	—
160	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
161	C₂×#117	—	89	1	no	—
162	C₂×M₃₂	—	2	2	no	—
163	C₂×M₃₂	—	2	2	no	—
164	C₂×M₃₂	—	89	2	no	—
165	C₂×M₃₂	—	89	2	no	—
166	C₂×M₃₂	—	178	2	no	—
167	C₂×M₃₂	—	178	2	no	—
168	C₂×M₃₂	Dic₈₉	1	2	yes	complemented
169	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
170	C₂²×C₁₆	—	89	2	no	—
171	C₂²×C₁₆	Dic₈₉	1	2	yes	complemented
172	#405	—	178	2	no	—
173	#405	—	178	2	no	—
174	C₂×#97	—	89	2	no	—
175	C₂×C₄×C₈	—	89	2	no	—
176	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
177	C₂×M₁₆	—	89	4	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂×M₁₆	—	178	1	no	—
180	C₂×M₁₆	—	178	1	no	—
181	C₂²×C₈	—	89	2	no	—
182	C₂²×C₈	Dic₁₇₈	1	2	yes	—
183	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
185	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
186	C₂²×C₈	D₃₅₆	1	2	yes	—
187	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
190	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
191	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
192	#117	—	89	4	no	—
193	#117	—	178	2	no	—
194	C₂×C₄⋊C₄	—	178	1	no	—
195	C₂×C₂²⋊C₄	—	178	1	no	—
196	C₂×C₄²	—	89	2	no	—
197	M₃₂	—	2	4	no	—
198	M₃₂	—	2	2	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	178	4	no	—
201	M₃₂	—	178	2	no	—
202	M₃₂	—	178	2	no	—
203	C₂×C₁₆	—	2	4	no	—
204	C₂×C₁₆	—	89	4	no	—
205	C₂×C₁₆	—	178	4	no	—
206	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
207	#97	—	178	2	no	—
208	#97	—	178	2	no	—
209	C₄×C₈	—	178	4	no	—
210	C₂⁴	#13083557	1	1	yes	characteristic
211	C₂²×C₄	—	178	2	no	—
212	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
213	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
214	C₂²×C₄	#13083551	1	2	yes	—
215	C₂²×C₄	#13083553	1	2	yes	—
216	C₂²×C₄	C₈×D₈₉	1	2	yes	—
217	C₂²×C₄	C₈×D₈₉	1	2	yes	—
218	C₂²×C₄	#13083557	1	1	yes	characteristic
219	C₂²×C₄	#13083557	1	1	yes	characteristic
220	C₂²×C₄	#13083557	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
222	M₁₆	—	178	4	no	—
223	M₁₆	—	178	4	no	—
224	C₂×C₈	—	89	4	no	—
225	C₂×C₈	—	178	2	no	—
226	C₂×C₈	—	178	2	no	—
227	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
228	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
229	C₂×C₈	#13083551	1	2	yes	—
230	C₂×C₈	#13083551	1	2	yes	—
231	C₂×C₈	#13083551	1	2	yes	—
232	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
234	C₂×C₈	#13083553	1	2	yes	—
235	C₂×C₈	#13083553	1	2	yes	—
236	C₂×C₈	#13083553	1	2	yes	—
237	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
238	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
244	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
245	C₄⋊C₄	—	178	4	no	—
246	C₂²⋊C₄	—	178	4	no	—
247	C₄²	—	89	4	no	—
248	C₄²	—	178	2	no	—
249	C₁₆	—	2	4	no	—
250	C₁₆	—	178	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	#15667922	1	1	yes	characteristic
253	C₂³	#15667922	1	1	yes	characteristic
254	C₂³	#15667924	1	2	yes	—
255	C₂³	#15667941	1	2	yes	—
256	C₂³	C₂×#13083557	1	1	yes	characteristic
257	C₂×C₄	—	356	4	no	—
258	C₂×C₄	—	178	4	no	—
259	C₂×C₄	#15667919	1	2	yes	—
260	C₂×C₄	#15667919	1	2	yes	—
261	C₂×C₄	C₈×Dic₈₉	1	2	yes	—
262	C₂×C₄	C₈×Dic₈₉	1	2	yes	—
263	C₂×C₄	#15667922	1	1	yes	characteristic
264	C₂×C₄	#15667922	1	1	yes	characteristic
265	C₂×C₄	#15667924	1	2	yes	—
266	C₂×C₄	#15667924	1	2	yes	—
267	C₂×C₄	#15667924	1	2	yes	—
268	C₂×C₄	#15667941	1	2	yes	—
269	C₂×C₄	#15667941	1	2	yes	—
270	C₂×C₄	#15667941	1	2	yes	—
271	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
272	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
273	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
277	C₈	—	178	4	no	—
278	C₈	#15667919	1	2	yes	—
279	C₈	#15667919	1	2	yes	—
280	C₈	C₂×#13083551	1	1	yes	characteristic
281	C₈	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
282	C₈	C₂×#13083553	1	1	yes	characteristic
283	C₈	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	#19151352	1	2	yes	—
290	C₂²	#19151352	1	2	yes	—
291	C₂²	C₂×#15667922	1	1	yes	characteristic
292	C₂²	C₂×#15667924	1	1	yes	characteristic
293	C₂²	C₂×#15667941	1	1	yes	characteristic
294	C₄	—	356	4	no	—
295	C₄	#19151352	1	2	yes	—
296	C₄	#19151352	1	2	yes	—
297	C₄	C₂×#15667919	1	1	yes	characteristic
298	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
299	C₄	C₂×#15667924	1	1	yes	characteristic
300	C₄	C₂×#15667941	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	C₂×#19151352	1	1	yes	characteristic
306	C₂	#26021856	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#97
prime radical	C₂⁴×C₈₉	M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	#12547
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #26018331	C₂	semidirect
9. C₂×#15667952	C₄	semidirect
11. C₂²×#13083549	C₄	semidirect
19. C₂×M₃₂×C₈₉	C₄	semidirect
21. C₂²×C₁₆×C₈₉	C₄	semidirect
158. C₈₉	#12547	semidirect
168. C₂×M₃₂	Dic₈₉	semidirect
171. C₂²×C₁₆	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667952	2
C₂×#19151553	2
#26018331	2
C₂²×M₃₂×C₈₉	2
#12547	89