Subgroups — #36707644

Order 22784.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152014	C₂	1	2	yes	maximal, complemented
2	#26028959	C₂	1	1	yes	maximal, characteristic
3	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022080	C₂	1	2	yes	maximal, complemented
6	#26022077	C₂	1	2	yes	maximal, complemented
7	#26022071	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021864	C₂	1	2	yes	maximal, complemented
10	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668018	C₂²	1	2	yes	—
13	#19152145	C₄	1	1	yes	characteristic
14	#19152014	—	2	8	no	—
15	C₂×#15667952	C₂²	1	2	yes	—
16	C₂×#15667951	C₂²	1	2	yes	—
17	C₂×#15667951	C₂²	1	1	yes	characteristic
18	C₂×#15667951	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
20	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
21	#19151676	C₄	1	1	yes	characteristic
22	#19151675	C₄	1	1	yes	characteristic
23	#19151673	C₄	1	2	yes	—
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
26	#19151620	C₄	1	1	yes	characteristic
27	#19151563	C₄	1	2	yes	—
28	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	#19151377	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	#19151373	C₂²	1	1	yes	characteristic
35	#19151373	C₂²	1	1	yes	characteristic
36	C₄×#13083549	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
38	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#13083574	C₂×C₄	1	2	yes	—
43	#15668018	—	2	4	no	—
44	#15668018	—	2	4	no	—
45	C₂×#13083558	C₂×C₄	1	2	yes	—
46	C₂×#13083557	C₂×C₄	1	2	yes	—
47	C₂×#13083557	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₈₉	C₂³	1	2	yes	—
50	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
51	#15668011	C₈	1	1	yes	characteristic
52	#15668008	C₈	1	1	yes	characteristic
53	#15667998	C₈	1	2	yes	—
54	#15667987	C₈	1	1	yes	characteristic
55	#15667985	C₈	1	2	yes	—
56	#15667983	C₈	1	1	yes	characteristic
57	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
58	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
59	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
60	#15667952	—	2	4	no	—
61	#15667951	—	2	2	no	—
62	#15667951	—	2	4	no	—
63	#15667951	D₄	1	4	yes	complemented
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	C₂×#13083549	C₂³	1	2	yes	—
66	C₂×#13083549	C₂³	1	1	yes	characteristic
67	C₁₆×D₈₉	—	2	4	no	—
68	C₁₆×D₈₉	—	2	4	no	—
69	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
70	#15667941	C₂×C₄	1	2	yes	—
71	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
72	#15667924	C₂×C₄	1	2	yes	—
73	#15667924	C₂×C₄	1	2	yes	—
74	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×C₁₆×C₈₉	—	2	4	no	—
80	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
82	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	#13083574	—	2	4	no	—
87	#13083574	—	2	4	no	—
88	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
89	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
90	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
91	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
92	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
93	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
94	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
98	#13083558	—	2	4	no	—
99	#13083557	—	2	4	no	—
100	#13083557	—	2	4	no	—
101	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
102	C₂×#10623494	C₂²×C₄	1	2	yes	—
103	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₈₉	—	2	2	no	—
105	C₈×D₈₉	—	2	2	no	—
106	C₈×D₈₉	—	2	2	no	—
107	C₈×D₈₉	C₂×D₄	1	2	yes	—
108	C₈×D₈₉	#41	1	2	yes	—
109	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
110	#13083553	C₂×C₈	1	2	yes	—
111	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
112	#13083551	C₂×C₈	1	2	yes	—
113	#13083551	C₂×C₈	1	2	yes	—
114	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
117	#13083549	—	2	2	no	—
118	#13083549	—	2	1	no	—
119	#13083549	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	—	2	2	no	—
124	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
128	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
129	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₁₆×C₈₉	—	2	2	no	—
131	C₁₆×C₈₉	—	2	1	no	—
132	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
133	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
134	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
135	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
136	D₄×C₈₉	—	2	2	no	—
137	C₂×C₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₈₉	—	2	4	no	—
144	C₂²⋊D₈₉	—	2	4	no	—
145	C₂²⋊D₈₉	—	4	2	no	—
146	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
148	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
149	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
150	D₃₅₆	—	2	4	no	—
151	D₃₅₆	—	2	2	no	—
152	C₄×D₈₉	—	2	2	no	—
153	C₄×D₈₉	—	2	2	no	—
154	C₄×D₈₉	—	2	2	no	—
155	C₄×D₈₉	—	4	1	no	—
156	C₄×D₈₉	C₄×D₄	1	2	yes	—
157	C₄×D₈₉	#131	1	2	yes	—
158	Dic₁₇₈	—	2	4	no	—
159	C₈×C₈₉	—	2	2	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
163	#10623494	—	2	2	no	—
164	#10623494	—	2	1	no	—
165	#10623494	C₄×D₄	1	2	yes	—
166	C₂²×C₈₉	—	2	2	no	—
167	C₂²×C₈₉	—	2	2	no	—
168	C₂²×C₈₉	—	2	2	no	—
169	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
170	C₂×D₈₉	—	2	1	no	—
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	2	no	—
173	C₂×D₈₉	—	2	2	no	—
174	C₂×D₈₉	—	2	2	no	—
175	C₂×D₈₉	C₈×D₄	1	2	yes	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	2	no	—
178	C₄×C₈₉	#504	1	2	yes	complemented
179	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
181	Dic₈₉	—	2	1	no	—
182	Dic₈₉	—	2	1	no	—
183	Dic₈₉	—	2	2	no	—
184	Dic₈₉	C₈×D₄	1	2	yes	—
185	#33438	—	89	1	no	maximal, Sylow
186	C₂×C₈₉	—	4	2	no	—
187	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
188	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
189	C₂×C₈₉	#3407	1	1	yes	characteristic
190	D₈₉	—	2	2	no	—
191	D₈₉	—	4	2	no	—
192	C₂×#504	—	89	2	no	—
193	#2931	—	89	1	no	—
194	#2169	—	89	1	no	—
195	#2169	—	89	1	no	—
196	#2168	—	89	2	no	—
197	#2168	—	89	2	no	—
198	D₄×C₁₆	—	89	1	no	—
199	D₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
200	C₂×C₄⋊C₁₆	—	89	1	no	—
201	#2113	—	89	2	no	—
202	C₄×M₃₂	—	89	1	no	—
203	C₈₉	#33438	1	1	yes	characteristic, complemented, Sylow
204	C₂×#131	—	89	2	no	—
205	#532	—	89	1	no	—
206	#504	—	178	8	no	—
207	C₂×M₃₂	—	89	2	no	—
208	C₂×M₃₂	—	89	1	no	—
209	C₂×M₃₂	—	89	1	no	—
210	C₂×M₃₂	—	89	2	no	—
211	C₂²×C₁₆	—	89	2	no	—
212	C₂²×C₁₆	—	89	2	no	—
213	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
214	#436	—	89	1	no	—
215	#436	—	89	1	no	—
216	#436	—	89	2	no	—
217	C₈×D₄	—	89	2	no	—
218	C₈×D₄	—	89	1	no	—
219	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
220	#423	—	89	1	no	—
221	#408	—	89	2	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	C₄⋊C₁₆	—	89	2	no	—
224	C₄⋊C₁₆	—	89	1	no	—
225	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
226	C₂²⋊C₁₆	—	89	2	no	—
227	C₂²⋊C₁₆	—	89	2	no	—
228	C₂²⋊C₁₆	—	89	2	no	—
229	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
230	#346	—	89	1	no	—
231	#346	—	89	1	no	—
232	C₄×C₁₆	—	89	1	no	—
233	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
234	C₂×#41	—	89	2	no	—
235	#131	—	178	4	no	—
236	#131	—	178	4	no	—
237	C₂×M₁₆	—	89	2	no	—
238	C₂×M₁₆	—	89	2	no	—
239	C₂×M₁₆	—	89	2	no	—
240	C₂²×C₈	—	89	1	no	—
241	C₂²×C₈	—	89	1	no	—
242	C₂²×C₈	—	89	2	no	—
243	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
244	C₄⋊Q₈	—	89	1	no	—
245	#127	—	89	1	no	—
246	#124	—	89	2	no	—
247	#121	—	89	1	no	—
248	#121	—	89	1	no	—
249	#121	—	89	2	no	—
250	C₄×D₄	—	89	2	no	—
251	C₄×D₄	—	89	1	no	—
252	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
253	C₂×C₄²	—	89	1	no	—
254	M₃₂	—	89	4	no	—
255	M₃₂	—	178	2	no	—
256	M₃₂	—	178	4	no	—
257	M₃₂	—	178	4	no	—
258	C₂×C₁₆	—	2	4	no	—
259	C₂×C₁₆	—	89	1	no	—
260	C₂×C₁₆	—	89	1	no	—
261	C₂×C₁₆	—	89	2	no	—
262	C₂×C₁₆	—	178	4	no	—
263	C₂×C₁₆	—	178	4	no	—
264	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
266	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
267	C₄⋊C₈	—	89	2	no	—
268	C₄⋊C₈	—	89	1	no	—
269	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
270	C₂²⋊C₈	—	89	2	no	—
271	C₂²⋊C₈	—	89	2	no	—
272	C₂²⋊C₈	—	89	2	no	—
273	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	1	no	—
276	C₄×C₈	—	89	1	no	—
277	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
278	#41	—	178	4	no	—
279	#41	—	178	4	no	—
280	C₂×Q₈	—	89	2	no	—
281	C₂×D₄	—	89	2	no	—
282	C₂×D₄	—	89	2	no	—
283	C₂×D₄	—	89	2	no	—
284	C₂×D₄	—	89	1	no	—
285	C₂×D₄	—	89	2	no	—
286	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	—	89	1	no	—
288	C₂²×C₄	—	89	1	no	—
289	C₂²×C₄	—	89	1	no	—
290	C₂²×C₄	—	89	2	no	—
291	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
292	M₁₆	—	178	4	no	—
293	M₁₆	—	178	4	no	—
294	M₁₆	—	178	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	—	89	2	no	—
299	C₂×C₈	—	89	2	no	—
300	C₂×C₈	—	89	1	no	—
301	C₂×C₈	—	89	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
307	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
308	C₄⋊C₄	—	89	2	no	—
309	C₄⋊C₄	—	89	1	no	—
310	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
311	C₂²⋊C₄	—	89	2	no	—
312	C₂²⋊C₄	—	89	2	no	—
313	C₂²⋊C₄	—	89	2	no	—
314	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
315	C₄²	—	89	1	no	—
316	C₄²	—	89	1	no	—
317	C₄²	—	89	1	no	—
318	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	—	89	2	no	—
322	C₁₆	—	178	2	no	—
323	C₁₆	—	178	1	no	—
324	C₁₆	D₄×D₈₉	1	2	yes	—
325	C₂³	—	89	1	no	—
326	C₂³	—	89	2	no	—
327	C₂³	C₂×C₈×D₈₉	1	2	yes	—
328	Q₈	—	178	4	no	—
329	D₄	—	2	2	no	—
330	D₄	—	356	2	no	—
331	D₄	—	178	4	no	—
332	D₄	—	178	4	no	—
333	D₄	—	178	4	no	—
334	D₄	—	178	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	89	1	no	—
338	C₂×C₄	—	89	2	no	—
339	C₂×C₄	—	89	2	no	—
340	C₂×C₄	—	89	1	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	2	no	—
343	C₂×C₄	—	356	1	no	—
344	C₂×C₄	—	178	2	no	—
345	C₂×C₄	—	178	2	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
349	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	89	2	no	—
354	C₈	—	178	2	no	—
355	C₈	—	178	1	no	—
356	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
357	C₈	#41×D₈₉	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	89	2	no	—
362	C₂²	—	178	2	no	—
363	C₂²	—	178	2	no	—
364	C₂²	—	178	2	no	—
365	C₂²	—	178	1	no	—
366	C₂²	—	178	1	no	—
367	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	89	2	no	—
371	C₄	—	178	2	no	—
372	C₄	—	178	1	no	—
373	C₄	—	178	1	no	—
374	C₄	#19152014	1	2	yes	complemented
375	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
376	C₄	#131×D₈₉	1	1	yes	characteristic
377	C₂	—	4	2	no	—
378	C₂	—	356	2	no	—
379	C₂	—	178	2	no	—
380	C₂	C₈×D₄×D₈₉	1	1	yes	characteristic
381	C₂	C₂×#19152014	1	1	yes	characteristic
382	C₂	#26033353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	D₄×C₁₆×C₈₉	C₂
prime radical	#19152145	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33438
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152014	C₂	semidirect
3. #26022085	C₂	semidirect
4. #26022084	C₂	semidirect
5. #26022080	C₂	semidirect
6. #26022077	C₂	semidirect
7. #26022071	C₂	semidirect
8. C₄⋊C₁₆×D₈₉	C₂	semidirect
9. #26021864	C₂	semidirect
10. C₄×#15667951	C₂	semidirect
11. D₄×C₁₆×C₈₉	C₂	semidirect
63. #15667951	D₄	semidirect
178. C₄×C₈₉	#504	semidirect
199. D₄×C₁₆	D₈₉	semidirect
203. C₈₉	#33438	semidirect
374. C₄	#19152014	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152014	2
#26028959	2
#26022085	2
#26022084	2
#26022080	2
#26022077	2
#26022071	2
C₄⋊C₁₆×D₈₉	2
#26021864	2
C₄×#15667951	2
D₄×C₁₆×C₈₉	2
#33438	89