Subgroups — #36708148

Order 22784.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022078	C₂	1	2	yes	maximal, complemented
5	D₄×#13083549	C₂	1	2	yes	maximal, complemented
6	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022059	C₂	1	2	yes	maximal, complemented
8	#26022051	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022011	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2966	C₂	1	1	yes	maximal, characteristic
12	C₂×#15667952	C₂²	1	2	yes	—
13	C₂²×#13083549	C₂²	1	2	yes	—
14	C₂²×#13083549	C₂²	1	2	yes	—
15	C₄⋊#13083549	C₂²	1	1	yes	characteristic
16	C₄⋊#13083549	C₂²	1	2	yes	—
17	C₄⋊#13083549	C₂²	1	1	yes	characteristic
18	C₄⋊#13083549	C₂²	1	2	yes	—
19	C₄⋊#13083549	C₂²	1	2	yes	—
20	C₄⋊#13083549	C₂²	1	1	yes	characteristic
21	C₄⋊#13083549	C₂²	1	1	yes	characteristic
22	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
23	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
24	C₂²⋊#13083549	C₂²	1	2	yes	—
25	C₂²⋊#13083549	C₂²	1	2	yes	—
26	#19151374	C₂²	1	2	yes	—
27	C₄×#13083549	C₂²	1	1	yes	characteristic
28	C₄×#13083549	C₂²	1	2	yes	—
29	C₄×#13083549	C₂²	1	1	yes	characteristic
30	#529×C₈₉	C₄	1	1	yes	characteristic
31	#446×C₈₉	C₄	1	1	yes	characteristic
32	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#436×C₈₉	C₄	1	1	yes	characteristic
34	#435×C₈₉	C₄	1	2	yes	—
35	C₈×D₄×C₈₉	C₂²	1	2	yes	—
36	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
37	#433×C₈₉	C₄	1	2	yes	—
38	#432×C₈₉	C₄	1	1	yes	characteristic
39	#405×C₈₉	C₂²	1	2	yes	—
40	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
41	#15667952	—	4	2	no	—
42	C₂×#13083549	—	4	2	no	—
43	C₂×#13083549	—	4	2	no	—
44	C₂×#13083549	C₂³	1	1	yes	characteristic
45	C₂×#13083549	C₂³	1	1	yes	characteristic
46	C₂×#13083549	C₂³	1	2	yes	—
47	C₂×#13083549	C₂³	1	1	yes	characteristic
48	C₂×#13083549	C₂³	1	1	yes	characteristic
49	C₂×#13083549	C₂³	1	2	yes	—
50	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
54	#126×C₈₉	C₈	1	2	yes	—
55	#125×C₈₉	C₈	1	1	yes	characteristic
56	#124×C₈₉	C₈	1	1	yes	characteristic
57	#123×C₈₉	C₈	1	2	yes	—
58	C₂²⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
59	#121×C₈₉	C₈	1	1	yes	characteristic
60	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
63	#117×C₈₉	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
68	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
72	#97×C₈₉	C₂×C₄	1	2	yes	—
73	#97×C₈₉	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083549	—	2	2	no	—
80	#13083549	—	2	1	no	—
81	#13083549	—	2	2	no	—
82	#13083549	—	2	1	no	—
83	#13083549	—	2	1	no	—
84	#13083549	—	2	1	no	—
85	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
86	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
87	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
92	M₁₆×C₈₉	—	4	2	no	—
93	C₂×C₈×C₈₉	—	2	1	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
105	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
106	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
108	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
109	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
110	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
111	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
112	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
113	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
114	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
115	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
116	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
120	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
121	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	Q₈×C₈₉	—	2	2	no	—
123	D₄×C₈₉	—	2	4	no	—
124	D₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	2	no	—
145	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	#504	1	2	yes	—
153	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	#504	1	2	yes	—
162	C₄×C₈₉	#520	1	1	yes	characteristic
163	C₄×C₈₉	#568	1	1	yes	characteristic
164	#33508	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	—	4	2	no	—
167	C₂×C₈₉	#2988	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
170	#2966	D₈₉	1	1	yes	characteristic
171	#2183	—	89	1	no	—
172	Q₈×C₁₆	—	89	1	no	—
173	#2169	—	89	1	no	—
174	#2168	—	89	2	no	—
175	D₄×C₁₆	—	89	1	no	—
176	D₄×C₁₆	—	89	2	no	—
177	#2164	—	89	2	no	—
178	#2162	—	89	1	no	—
179	#2151	—	89	2	no	—
180	C₂×C₄⋊C₁₆	—	89	1	no	—
181	C₈₉	#33508	1	1	yes	characteristic, complemented, Sylow
182	#529	Dic₈₉	1	1	yes	characteristic
183	C₂×M₃₂	—	89	2	no	—
184	C₂²×C₁₆	—	89	2	no	—
185	C₂²×C₁₆	—	89	2	no	—
186	#446	Dic₈₉	1	1	yes	characteristic
187	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
188	#436	Dic₈₉	1	1	yes	characteristic
189	#435	Dic₈₉	1	2	yes	—
190	C₈×D₄	C₂×D₈₉	1	2	yes	—
191	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
192	#433	Dic₈₉	1	2	yes	—
193	#432	Dic₈₉	1	1	yes	characteristic
194	#405	C₂×D₈₉	1	2	yes	—
195	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
196	C₄⋊C₁₆	—	89	2	no	—
197	C₄⋊C₁₆	—	89	1	no	—
198	C₄⋊C₁₆	—	89	1	no	—
199	C₄⋊C₁₆	—	89	1	no	—
200	C₄⋊C₁₆	—	89	2	no	—
201	C₄⋊C₁₆	—	89	2	no	—
202	C₄⋊C₁₆	—	89	1	no	—
203	C₂²⋊C₁₆	—	89	1	no	—
204	C₂²⋊C₁₆	—	89	1	no	—
205	C₂²⋊C₁₆	—	89	2	no	—
206	C₂²⋊C₁₆	—	89	2	no	—
207	#346	—	89	2	no	—
208	C₄×C₁₆	—	89	1	no	—
209	C₄×C₁₆	—	89	1	no	—
210	C₄×C₁₆	—	89	2	no	—
211	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
212	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
215	#126	#10623494	1	2	yes	—
216	#125	#10623494	1	1	yes	characteristic
217	#124	#10623494	1	1	yes	characteristic
218	#123	#10623494	1	2	yes	—
219	C₂²⋊Q₈	#10623494	1	1	yes	characteristic
220	#121	#10623494	1	1	yes	characteristic
221	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
222	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
223	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
224	#117	C₂×Dic₈₉	1	2	yes	—
225	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
226	M₃₂	—	356	2	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	—	89	2	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	—	89	2	no	—
233	C₂×C₁₆	—	356	2	no	—
234	C₂×C₁₆	—	356	2	no	—
235	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
236	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
237	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
238	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
239	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
240	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
241	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
242	#97	C₂×Dic₈₉	1	2	yes	—
243	#97	C₂×Dic₈₉	1	2	yes	—
244	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₈₉	1	2	yes	—
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
250	C₂×D₄	C₂×#10623494	1	2	yes	—
251	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
252	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
253	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
256	M₁₆	—	4	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
266	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
267	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#10623494	1	2	yes	—
269	C₄⋊C₄	C₂×#10623494	1	2	yes	—
270	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
271	C₄⋊C₄	C₂×#10623494	1	2	yes	—
272	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#10623494	1	2	yes	—
274	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
275	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
276	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
277	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
278	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
279	C₄²	C₂×#10623494	1	1	yes	characteristic
280	C₄²	C₂×#10623494	1	1	yes	characteristic
281	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
282	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
283	C₄²	C₂²×Dic₈₉	1	2	yes	—
284	C₁₆	—	178	1	no	—
285	C₁₆	—	178	2	no	—
286	C₁₆	—	178	2	no	—
287	C₁₆	—	178	1	no	—
288	C₁₆	—	178	1	no	—
289	C₁₆	—	178	1	no	—
290	C₂³	C₂²×#10623494	1	2	yes	—
291	C₂³	C₂²×#10623494	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
302	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
303	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
304	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10623494	1	2	yes	—
306	C₂×C₄	C₂²×#10623494	1	2	yes	—
307	C₂×C₄	C₂²×#10623494	1	2	yes	—
308	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
309	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	#15668058	1	1	yes	characteristic
316	C₈	#15668059	1	1	yes	characteristic
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#19152015	1	2	yes	—
323	C₂²	C₂³×#10623494	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#19152015	1	2	yes	—
332	C₄	#19152081	1	1	yes	characteristic
333	C₄	#19152365	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	#26029363	1	1	yes	characteristic
337	C₂	C₂×#19152015	1	1	yes	characteristic
338	C₂	C₂×#19152015	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2966	C₂
prime radical	#121×C₈₉	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2966	D₈₉
hypocenter	C₈₉	#33508
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #26022161	C₂	semidirect
2. Q₈×#13083549	C₂	semidirect
3. #26022084	C₂	semidirect
4. #26022078	C₂	semidirect
5. D₄×#13083549	C₂	semidirect
6. D₄×#13083549	C₂	semidirect
7. #26022059	C₂	semidirect
8. #26022051	C₂	semidirect
9. #26022011	C₂	semidirect
10. C₂×C₄⋊#13083549	C₂	semidirect
181. C₈₉	#33508	semidirect

Maximal subgroups

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

subgroup	index
#26022161	2
Q₈×#13083549	2
#26022084	2
#26022078	2
D₄×#13083549	2
#26022059	2
#26022051	2
#26022011	2
C₂×C₄⋊#13083549	2
C₈₉×#2966	2
#33508	89