Subgroups — #36559114

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26021188	C₂	1	2	yes	maximal
3	#26020970	C₂	1	2	yes	maximal
4	#26020809	C₂	1	4	yes	maximal
5	C₄×#15667941	C₂	1	2	yes	maximal
6	C₄×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
7	#26019997	C₂	1	2	yes	maximal
8	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₇₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
13	C₂×#15667941	C₂²	1	4	yes	—
14	C₂×C₄×#10623494	C₂²	1	2	yes	—
15	C₄×#13083553	C₄	1	2	yes	—
16	C₄×#13083553	C₄	1	2	yes	—
17	C₄×#13083553	C₂²	1	2	yes	—
18	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
19	C₄×C₄⋊Dic₈₉	C₄	1	1	yes	characteristic
20	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
21	C₄²×Dic₈₉	C₂²	1	2	yes	—
22	#19151353	C₂²	1	2	yes	—
23	#19151352	C₂²	1	4	yes	—
24	#19151352	C₂²	1	4	yes	—
25	#19151352	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₄³×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10623494	C₂³	1	4	yes	—
33	C₄×Dic₁₇₈	—	2	4	no	—
34	C₄×Dic₁₇₈	—	2	4	no	—
35	C₄×Dic₁₇₈	C₈	1	8	yes	complemented
36	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×#13083553	C₂×C₄	1	2	yes	—
40	C₂×#13083553	C₂×C₄	1	2	yes	—
41	C₂×#13083553	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
47	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
48	C₄⋊#10623494	—	2	4	no	—
49	#15667941	—	2	8	no	—
50	C₄×#10623494	—	2	4	no	—
51	#15667919	C₂×C₄	1	2	yes	—
52	#15667919	C₂×C₄	1	2	yes	—
53	#15667919	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	—	2	4	no	—
64	C₄×C₈×C₈₉	—	2	4	no	—
65	C₄×C₈×C₈₉	—	2	2	no	—
66	C₄×C₈×C₈₉	Q₈	1	4	yes	complemented
67	C₂×Dic₁₇₈	—	2	2	no	—
68	C₂×Dic₁₇₈	—	2	2	no	—
69	C₂×Dic₁₇₈	C₂×C₈	1	4	yes	—
70	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
71	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
72	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×Dic₈₉	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	—	2	4	no	—
75	C₂×#10623494	—	2	4	no	—
76	C₂×#10623494	—	2	4	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	C₂×C₈	1	4	yes	—
80	#13083553	—	2	4	no	—
81	#13083553	—	2	4	no	—
82	#13083553	C₂×C₈	1	8	yes	—
83	C₄×Dic₈₉	—	2	4	no	—
84	C₄×Dic₈₉	—	2	4	no	—
85	C₄×Dic₈₉	—	2	4	no	—
86	C₄×Dic₈₉	—	2	4	no	—
87	C₄×Dic₈₉	—	2	4	no	—
88	C₄×Dic₈₉	C₂×C₈	1	8	yes	—
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₈₉	#41	1	4	yes	—
102	C₄²×C₈₉	—	2	2	no	—
103	C₄²×C₈₉	—	2	2	no	—
104	C₄²×C₈₉	—	2	2	no	—
105	C₄²×C₈₉	C₂×C₈	1	4	yes	—
106	C₄²×C₈₉	M₁₆	1	4	yes	—
107	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
108	C₄²×C₈₉	#41	1	4	yes	—
109	C₄²×C₈₉	#41	1	2	yes	—
110	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	#117	1	2	yes	—
118	C₂×C₄×C₈₉	#117	1	2	yes	—
119	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₈₉	#131	1	2	yes	—
128	C₂×C₄×C₈₉	#131	1	2	yes	—
129	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
133	C₂×Dic₈₉	—	2	2	no	—
134	C₂×Dic₈₉	—	2	2	no	—
135	C₂×Dic₈₉	—	2	2	no	—
136	C₂×Dic₈₉	—	2	2	no	—
137	C₂×Dic₈₉	—	2	2	no	—
138	C₂×Dic₈₉	—	2	2	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	2	2	no	—
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
144	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
145	Dic₁₇₈	—	2	8	no	—
146	C₈×C₈₉	—	2	4	no	—
147	C₈×C₈₉	—	2	4	no	—
148	#10623494	—	4	4	no	—
149	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
150	C₂²×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²×C₈₉	C₄×#41	1	1	yes	characteristic
152	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
153	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
155	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	#431	1	4	yes	—
160	C₄×C₈₉	C₈×Q₈	1	4	yes	—
161	C₄×C₈₉	#446	1	4	yes	—
162	C₄×C₈₉	#529	1	2	yes	—
163	C₄×C₈₉	#533	1	2	yes	—
164	Dic₈₉	—	2	4	no	—
165	Dic₈₉	—	2	4	no	—
166	Dic₈₉	—	4	2	no	—
167	Dic₈₉	—	4	2	no	—
168	#21339	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	#2295	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
171	C₂×C₈₉	#2922	1	1	yes	characteristic
172	C₂×C₈₉	C₂×C₈×Q₈	1	1	yes	characteristic
173	C₂×C₈₉	C₂×#446	1	1	yes	characteristic
174	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
175	C₂×C₈₉	#2965	1	1	yes	characteristic
176	C₄²×Q₈	—	89	1	no	—
177	#1984	—	89	2	no	—
178	#1939	—	89	2	no	—
179	#1916	—	89	4	no	—
180	C₄×C₄⋊C₈	—	89	2	no	—
181	C₄×C₄⋊C₈	—	89	1	no	—
182	#1744	—	89	2	no	—
183	C₄²×C₈	D₈₉	1	1	yes	characteristic
184	C₈₉	#21339	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄×Q₈	—	89	1	no	—
186	C₂×C₄×Q₈	—	89	1	no	—
187	C₂×C₄×Q₈	—	89	1	no	—
188	C₂×C₄⋊C₈	—	89	4	no	—
189	C₂×C₄⋊C₈	—	89	2	no	—
190	C₂×C₄×C₈	—	89	2	no	—
191	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
193	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
195	C₄×C₄⋊C₄	—	89	1	no	—
196	C₄×C₄⋊C₄	—	89	1	no	—
197	C₄×C₄⋊C₄	—	89	2	no	—
198	C₄×C₄⋊C₄	—	89	2	no	—
199	C₄×C₄⋊C₄	—	89	1	no	—
200	C₄×C₄⋊C₄	—	89	2	no	—
201	C₄³	—	89	2	no	—
202	C₄³	C₂×D₈₉	1	1	yes	characteristic
203	#336	—	89	4	no	—
204	#336	—	89	4	no	—
205	#336	—	89	4	no	—
206	#336	—	89	2	no	—
207	C₂²×Q₈	—	89	1	no	—
208	C₂²×C₈	—	89	4	no	—
209	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
210	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
211	C₄×Q₈	—	89	8	no	—
212	C₄×Q₈	—	178	4	no	—
213	C₄×Q₈	—	178	4	no	—
214	C₂×C₄⋊C₄	—	89	2	no	—
215	C₂×C₄⋊C₄	—	89	1	no	—
216	C₂×C₄⋊C₄	—	89	2	no	—
217	C₂×C₄⋊C₄	—	89	2	no	—
218	C₂×C₄⋊C₄	—	89	1	no	—
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₄²	—	89	2	no	—
221	C₂×C₄²	—	89	2	no	—
222	C₂×C₄²	—	89	2	no	—
223	C₂×C₄²	—	89	2	no	—
224	C₂×C₄²	—	89	2	no	—
225	C₂×C₄²	—	89	2	no	—
226	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
227	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
228	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
229	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
233	C₄⋊C₈	—	178	8	no	—
234	C₄⋊C₈	—	178	4	no	—
235	C₄×C₈	—	2	4	no	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	—	178	4	no	—
239	C₄×C₈	Dic₁₇₈	1	4	yes	complemented
240	#95	—	89	2	no	—
241	#95	—	89	2	no	—
242	#95	—	89	2	no	—
243	C₂×Q₈	—	89	4	no	—
244	C₂×Q₈	—	178	2	no	—
245	C₂×Q₈	—	178	2	no	—
246	C₂²×C₄	—	89	2	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	—	89	2	no	—
249	C₂²×C₄	—	89	2	no	—
250	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	178	4	no	—
262	C₂×C₈	—	178	4	no	—
263	C₂×C₈	—	178	4	no	—
264	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
265	C₂×C₈	#13083574	1	4	yes	—
266	C₄⋊C₄	—	89	8	no	—
267	C₄⋊C₄	—	89	4	no	—
268	C₄⋊C₄	—	178	4	no	—
269	C₄⋊C₄	—	178	2	no	—
270	C₄⋊C₄	—	178	4	no	—
271	C₄⋊C₄	—	178	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	89	8	no	—
276	C₄²	—	178	4	no	—
277	C₄²	—	178	4	no	—
278	C₄²	—	178	4	no	—
279	C₄²	—	178	4	no	—
280	C₄²	—	178	4	no	—
281	C₄²	C₈×D₈₉	1	4	yes	—
282	C₄²	#13083557	1	4	yes	—
283	C₄²	C₂×Dic₁₇₈	1	2	yes	—
284	C₄²	#13083574	1	4	yes	—
285	C₄²	#13083574	1	2	yes	—
286	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
287	Q₈	—	178	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	89	4	no	—
295	C₂×C₄	—	89	4	no	—
296	C₂×C₄	—	178	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	2	no	—
299	C₂×C₄	—	178	2	no	—
300	C₂×C₄	—	178	2	no	—
301	C₂×C₄	—	178	2	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	—	178	2	no	—
305	C₂×C₄	—	178	2	no	—
306	C₂×C₄	#15667969	1	2	yes	—
307	C₂×C₄	#15667969	1	2	yes	—
308	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
309	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
311	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
312	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
313	C₂×C₄	C₂×#13083557	1	2	yes	—
314	C₂×C₄	C₂×#13083557	1	2	yes	—
315	C₂×C₄	C₂×#13083557	1	2	yes	—
316	C₂×C₄	#15668018	1	2	yes	—
317	C₂×C₄	#15668018	1	2	yes	—
318	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
319	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
320	C₂×C₄	C₂×#13083574	1	2	yes	—
321	C₂×C₄	C₂×#13083574	1	2	yes	—
322	C₈	—	2	4	no	—
323	C₈	—	2	4	no	—
324	C₈	—	356	4	no	—
325	C₂²	C₂×#15667969	1	1	yes	characteristic
326	C₂²	C₂×C₄×Dic₁₇₈	1	1	yes	characteristic
327	C₂²	C₄×#13083574	1	1	yes	characteristic
328	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
329	C₂²	C₂²×#13083557	1	1	yes	characteristic
330	C₂²	C₂×#15668018	1	1	yes	characteristic
331	C₂²	C₂×#15668018	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	356	2	no	—
336	C₄	—	356	2	no	—
337	C₄	—	178	4	no	—
338	C₄	—	178	4	no	—
339	C₄	#19151649	1	4	yes	—
340	C₄	C₈×Dic₁₇₈	1	4	yes	—
341	C₄	#19151728	1	4	yes	—
342	C₄	#19152132	1	2	yes	—
343	C₄	#19152155	1	2	yes	—
344	C₂	#26022729	1	1	yes	characteristic
345	C₂	C₂×#19151649	1	1	yes	characteristic
346	C₂	#26028894	1	1	yes	characteristic
347	C₂	C₂×C₈×Dic₁₇₈	1	1	yes	characteristic
348	C₂	C₂×#19151728	1	1	yes	characteristic
349	C₂	C₈×#13083574	1	1	yes	characteristic
350	C₂	#26029218	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂³×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#21339
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×Dic₁₇₈	C₈	semidirect
66. C₄×C₈×C₈₉	Q₈	semidirect
184. C₈₉	#21339	semidirect
239. C₄×C₈	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₁₇₈	2
#26021188	2
#26020970	2
#26020809	2
C₄×#15667941	2
C₄×C₄⋊#10623494	2
#26019997	2
C₄²×C₈×C₈₉	2
#21339	89