Subgroups — #36444517

Order 22784.

706 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#26018135	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10623494	—	2	1	no	—
6	#19151651	—	2	2	no	—
7	#19151651	C₄	1	4	yes	complemented
8	C₂×C₄⋊#10623494	—	2	2	no	—
9	C₂×C₄⋊#10623494	—	2	1	no	—
10	C₂×C₄⋊#10623494	C₄	1	4	yes	complemented
11	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×C₂²⋊#10623494	—	2	1	no	—
14	C₂×C₄×#10623494	—	2	1	no	—
15	C₂×#117×C₈₉	—	2	1	no	—
16	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	#384×C₈₉	C₂²	1	2	yes	—
18	#382×C₈₉	—	2	1	no	—
19	#380×C₈₉	—	2	1	no	—
20	#380×C₈₉	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
22	C₄×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
23	C₂²×#10623494	—	2	4	no	—
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	2	2	no	—
26	C₂²×#10623494	—	4	2	no	—
27	C₂²×#10623494	—	4	2	no	—
28	C₄⋊#10623494	—	2	4	no	—
29	C₄⋊#10623494	—	2	2	no	—
30	C₄⋊#10623494	—	2	2	no	—
31	C₄⋊#10623494	C₂×C₄	1	4	yes	—
32	C₄⋊#10623494	C₂×C₄	1	4	yes	—
33	C₂²⋊#10623494	—	2	4	no	—
34	C₄×#10623494	—	2	4	no	—
35	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
38	#117×C₈₉	—	4	2	no	—
39	#117×C₈₉	—	4	2	no	—
40	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
44	C₂×C₄²×C₈₉	—	2	2	no	—
45	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	D₄	1	2	yes	—
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	#95×C₈₉	—	2	2	no	—
54	#95×C₈₉	—	2	2	no	—
55	C₂×#10623494	—	2	4	no	—
56	C₂×#10623494	—	2	4	no	—
57	C₂×#10623494	—	4	8	no	—
58	C₂×#10623494	—	4	2	no	—
59	C₂×#10623494	—	4	2	no	—
60	C₂×#10623494	—	4	4	no	—
61	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	—	2	4	no	—
63	C₂²×C₄×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	—	4	2	no	—
65	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
66	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₉	C₂²⋊C₄	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	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	D₈	1	2	yes	—
75	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₈₉	SD₁₆	1	2	yes	—
77	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₈₉	—	4	4	no	—
82	C₄⋊C₄×C₈₉	—	4	2	no	—
83	C₄⋊C₄×C₈₉	—	4	2	no	—
84	C₄⋊C₄×C₈₉	—	4	4	no	—
85	C₂²⋊C₄×C₈₉	—	4	4	no	—
86	C₄²×C₈₉	—	4	4	no	—
87	C₄²×C₈₉	C₄²	1	2	yes	—
88	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₈₉	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
91	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
92	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
93	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
97	C₂³×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	—	2	2	no	—
99	C₂³×C₈₉	#95	1	2	yes	—
100	C₂³×C₈₉	#102	1	2	yes	—
101	C₂³×C₈₉	#103	1	2	yes	—
102	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	4	no	—
108	C₂×C₄×C₈₉	—	4	2	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	4	4	no	—
111	C₂×C₄×C₈₉	—	4	4	no	—
112	C₂×C₄×C₈₉	—	4	4	no	—
113	C₂×C₄×C₈₉	#95	1	2	yes	—
114	C₂×C₄×C₈₉	#95	1	2	yes	—
115	C₂×C₄×C₈₉	#95	1	2	yes	—
116	C₂×C₄×C₈₉	#102	1	4	yes	—
117	C₂×C₄×C₈₉	#102	1	2	yes	—
118	C₂×C₄×C₈₉	#103	1	4	yes	—
119	C₂×C₄×C₈₉	#103	1	2	yes	—
120	C₂×C₄×C₈₉	#106	1	4	yes	—
121	C₂×C₄×C₈₉	#107	1	4	yes	—
122	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#135	1	4	yes	—
133	#10623494	—	4	4	no	—
134	#10623494	—	4	4	no	—
135	C₂²×C₈₉	—	2	4	no	—
136	C₂²×C₈₉	—	2	4	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	#340	1	4	yes	—
140	C₂²×C₈₉	C₂×#95	1	1	yes	characteristic
141	C₂²×C₈₉	#413	1	2	yes	—
142	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
143	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
144	C₂²×C₈₉	#416	1	2	yes	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	8	4	no	—
150	C₄×C₈₉	#340	1	4	yes	—
151	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
153	C₄×C₈₉	#416	1	2	yes	—
154	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
155	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
156	C₄×C₈₉	#427	1	2	yes	—
157	#8039	—	89	1	no	maximal, Sylow
158	C₂×C₈₉	—	2	4	no	—
159	C₂×C₈₉	#1540	1	2	yes	—
160	C₂×C₈₉	#1542	1	2	yes	—
161	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
162	C₂×C₈₉	#1735	1	1	yes	characteristic
163	C₂×C₈₉	#1742	1	1	yes	characteristic
164	C₂×#431	—	89	1	no	—
165	C₂²×C₄⋊C₈	—	89	1	no	—
166	#2285	D₈₉	1	1	yes	characteristic
167	#1296	—	89	4	no	—
168	C₈₉	#8039	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	178	1	no	—
170	C₂×#117	—	2	1	no	—
171	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
172	#431	—	89	4	no	—
173	#431	—	178	2	no	—
174	C₂×C₄⋊C₈	—	89	4	no	—
175	C₂×C₄⋊C₈	—	89	2	no	—
176	C₂×C₄⋊C₈	—	89	2	no	—
177	C₂×C₄⋊C₈	—	178	2	no	—
178	C₂×C₄⋊C₈	—	178	1	no	—
179	C₂×C₂²⋊C₈	—	178	1	no	—
180	C₂×C₄×C₈	—	178	1	no	—
181	#384	C₂×D₈₉	1	2	yes	—
182	#382	—	2	1	no	—
183	#380	—	2	1	no	—
184	#380	Dic₈₉	1	2	yes	—
185	C₄×C₄⋊C₄	—	2	1	no	—
186	C₄×C₄⋊C₄	Dic₈₉	1	2	yes	—
187	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
188	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
189	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
190	C₂²×C₈	—	356	2	no	—
191	C₂²×C₈	—	356	2	no	—
192	C₂²×C₈	—	178	4	no	—
193	C₂²×C₈	—	178	2	no	—
194	C₂²×C₈	—	178	2	no	—
195	#117	—	4	2	no	—
196	#117	—	4	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	Dic₁₇₈	1	2	yes	—
203	C₂×C₄²	C₄×D₈₉	1	2	yes	—
204	C₂×C₄²	C₄×D₈₉	1	2	yes	—
205	C₂×C₄²	D₃₅₆	1	2	yes	—
206	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
210	C₄⋊C₈	—	89	4	no	—
211	C₄⋊C₈	—	89	4	no	—
212	C₄⋊C₈	—	178	4	no	—
213	C₄⋊C₈	—	178	2	no	—
214	C₄⋊C₈	—	178	2	no	—
215	C₂²⋊C₈	—	178	4	no	—
216	C₄×C₈	—	178	4	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂⁴	C₂²⋊Dic₈₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
224	C₂²×C₄	#13083551	1	2	yes	—
225	C₂²×C₄	#13083551	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
229	C₂²×C₄	#13083553	1	2	yes	—
230	C₂²×C₄	#13083553	1	2	yes	—
231	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
232	C₂²×C₄	#13083559	1	2	yes	—
233	C₂²×C₄	#13083561	1	2	yes	—
234	C₂²×C₄	#13083562	1	2	yes	—
235	C₂²×C₄	#13083564	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
239	C₂×C₈	—	356	2	no	—
240	C₂×C₈	—	356	2	no	—
241	C₂×C₈	—	356	4	no	—
242	C₂×C₈	—	356	8	no	—
243	C₂×C₈	—	178	4	no	—
244	C₂×C₈	—	178	4	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	4	4	no	—
250	C₄²	—	4	4	no	—
251	C₄²	C₄×Dic₈₉	1	2	yes	—
252	C₄²	#13083551	1	2	yes	—
253	C₄²	#13083551	1	2	yes	—
254	C₄²	#13083553	1	2	yes	—
255	C₄²	#13083553	1	2	yes	—
256	C₄²	C₄⋊Dic₈₉	1	2	yes	—
257	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
258	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
259	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
260	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	#15667919	1	2	yes	—
264	C₂³	#15667933	1	2	yes	—
265	C₂³	#15667936	1	2	yes	—
266	C₂³	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	#15667919	1	2	yes	—
278	C₂×C₄	#15667919	1	2	yes	—
279	C₂×C₄	#15667919	1	2	yes	—
280	C₂×C₄	#15667932	1	4	yes	—
281	C₂×C₄	#15667933	1	2	yes	—
282	C₂×C₄	#15667935	1	4	yes	—
283	C₂×C₄	#15667936	1	2	yes	—
284	C₂×C₄	#15667943	1	4	yes	—
285	C₂×C₄	#15667945	1	4	yes	—
286	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
288	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
289	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
290	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
293	C₂×C₄	C₂×#13083561	1	1	yes	characteristic
294	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
295	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
296	C₂×C₄	#15668032	1	4	yes	—
297	C₈	—	356	4	no	—
298	C₈	—	356	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	#19151360	1	4	yes	—
304	C₂²	C₂×#15667919	1	1	yes	characteristic
305	C₂²	#19151582	1	2	yes	—
306	C₂²	C₂×#15667933	1	1	yes	characteristic
307	C₂²	C₂×#15667936	1	1	yes	characteristic
308	C₂²	#19151594	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	4	no	—
314	C₄	#19151360	1	4	yes	—
315	C₄	C₂×#15667932	1	1	yes	characteristic
316	C₄	C₂×#15667935	1	1	yes	characteristic
317	C₄	#19151592	1	2	yes	—
318	C₄	C₂×#15667943	1	1	yes	characteristic
319	C₄	C₂×#15667945	1	1	yes	characteristic
320	C₄	#19151634	1	2	yes	—
321	C₂	—	2	4	no	—
322	C₂	#26018901	1	2	yes	—
323	C₂	#26018909	1	2	yes	—
324	C₂	C₂×#19151360	1	1	yes	characteristic
325	C₂	#26019968	1	1	yes	characteristic
326	C₂	#26019991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2285	C₂
socle	C₂³×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2285	D₈₉
hypocenter	C₈₉	#8039
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
3. #26018135	C₂	semidirect
7. #19151651	C₄	semidirect
10. C₂×C₄⋊#10623494	C₄	semidirect
168. C₈₉	#8039	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151651	2
C₂²×C₄⋊#10623494	2
#26018135	2
C₈₉×#2285	2
#8039	89