Subgroups — #36707993

Order 22784.

516 conjugacy classes of subgroups.

The subgroup lattice has 342 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	#26022084	C₂	1	2	yes	maximal, complemented
5	#26022078	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	#26021849	C₂	1	2	yes	maximal, complemented
10	C₄×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2966	C₂	1	1	yes	maximal, characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂×#15667952	C₂²	1	2	yes	—
15	C₂²×#13083549	C₂²	1	2	yes	—
16	C₄⋊#13083549	C₂²	1	1	yes	characteristic
17	C₄⋊#13083549	C₂²	1	2	yes	—
18	C₄⋊#13083549	C₂²	1	1	yes	characteristic
19	C₄⋊#13083549	C₂²	1	2	yes	—
20	C₂²⋊#13083549	C₂²	1	2	yes	—
21	C₂²⋊#13083549	C₂²	1	2	yes	—
22	C₂²⋊#13083549	C₂²	1	2	yes	—
23	#19151374	C₂²	1	2	yes	—
24	#19151374	C₂²	1	1	yes	characteristic
25	#19151374	C₂²	1	1	yes	characteristic
26	C₄×#13083549	C₂²	1	1	yes	characteristic
27	C₄×#13083549	C₂²	1	2	yes	—
28	C₄×#13083549	C₂²	1	1	yes	characteristic
29	C₄×#13083549	C₂²	1	2	yes	—
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	1	yes	characteristic
36	C₈×D₄×C₈₉	C₂²	1	2	yes	—
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	—	2	2	no	—
42	#15667952	—	2	2	no	—
43	#15667952	—	2	4	no	—
44	C₂×#13083549	—	2	4	no	—
45	C₂×#13083549	C₂³	1	2	yes	—
46	C₂×#13083549	C₂³	1	1	yes	characteristic
47	C₂×#13083549	C₂³	1	1	yes	characteristic
48	C₂×#13083549	C₂³	1	2	yes	—
49	C₂×#13083549	C₂³	1	1	yes	characteristic
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
55	#126×C₈₉	C₈	1	2	yes	—
56	#125×C₈₉	C₈	1	1	yes	characteristic
57	#124×C₈₉	C₈	1	1	yes	characteristic
58	#123×C₈₉	C₈	1	2	yes	—
59	C₂²⋊Q₈×C₈₉	C₈	1	1	yes	characteristic
60	#121×C₈₉	C₈	1	1	yes	characteristic
61	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
63	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
64	#117×C₈₉	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
69	C₄⋊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	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
73	#97×C₈₉	C₂×C₄	1	2	yes	—
74	#97×C₈₉	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₈₉	C₂³	1	2	yes	—
80	#13083549	—	2	2	no	—
81	#13083549	—	2	1	no	—
82	#13083549	—	2	1	no	—
83	#13083549	—	2	2	no	—
84	#13083549	#41	1	2	yes	—
85	#13083549	#41	1	2	yes	—
86	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
87	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
88	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
89	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
93	M₁₆×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
106	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
107	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
109	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
110	C₄⋊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	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
116	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
118	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
121	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₈₉	—	2	2	no	—
124	D₄×C₈₉	—	2	2	no	—
125	D₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	1	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	—	4	1	no	—
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	1	yes	characteristic
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	2	yes	—
139	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
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	#131	1	2	yes	—
146	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
147	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	2	no	—
154	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	2	no	—
162	C₄×C₈₉	#504	1	2	yes	—
163	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
164	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
165	#33483	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	4	1	no	—
167	C₂×C₈₉	—	4	2	no	—
168	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
170	C₂×C₈₉	#3407	1	1	yes	characteristic
171	#2966	D₈₉	1	1	yes	characteristic
172	#2183	—	89	1	no	—
173	Q₈×C₁₆	—	89	1	no	—
174	#2169	—	89	1	no	—
175	#2169	—	89	2	no	—
176	#2168	—	89	2	no	—
177	D₄×C₁₆	—	89	1	no	—
178	#2164	—	89	2	no	—
179	#2162	—	89	1	no	—
180	#2108	—	89	2	no	—
181	C₄×M₃₂	—	89	1	no	—
182	C₈₉	#33483	1	1	yes	characteristic, complemented, Sylow
183	#529	Dic₈₉	1	1	yes	characteristic
184	C₂×M₃₂	—	89	1	no	—
185	C₂×M₃₂	—	89	1	no	—
186	C₂×M₃₂	—	89	2	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	#446	Dic₈₉	1	1	yes	characteristic
189	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
190	#436	Dic₈₉	1	1	yes	characteristic
191	#435	Dic₈₉	1	2	yes	—
192	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
193	C₈×D₄	C₂×D₈₉	1	2	yes	—
194	#433	Dic₈₉	1	2	yes	—
195	#432	Dic₈₉	1	1	yes	characteristic
196	#405	C₂×D₈₉	1	2	yes	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
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	C₂²⋊C₁₆	—	89	2	no	—
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	—	89	2	no	—
205	#346	—	89	1	no	—
206	#346	—	89	1	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	C₄×C₁₆	—	89	2	no	—
212	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
213	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
216	#126	#10623494	1	2	yes	—
217	#125	#10623494	1	1	yes	characteristic
218	#124	#10623494	1	1	yes	characteristic
219	#123	#10623494	1	2	yes	—
220	C₂²⋊Q₈	#10623494	1	1	yes	characteristic
221	#121	#10623494	1	1	yes	characteristic
222	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
223	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
224	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
225	#117	C₂×Dic₈₉	1	2	yes	—
226	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
227	M₃₂	—	178	2	no	—
228	M₃₂	—	178	2	no	—
229	M₃₂	—	178	4	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	—	89	2	no	—
233	C₂×C₁₆	—	89	1	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	—	89	2	no	—
236	C₂×C₁₆	—	178	4	no	—
237	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
238	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
239	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
240	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
241	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
242	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
243	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
244	#97	C₂×Dic₈₉	1	2	yes	—
245	#97	C₂×Dic₈₉	1	2	yes	—
246	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
247	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₈₉	1	2	yes	—
251	C₂×Q₈	C₂×#10623494	1	1	yes	characteristic
252	C₂×D₄	C₂×#10623494	1	1	yes	characteristic
253	C₂×D₄	C₂×#10623494	1	2	yes	—
254	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
255	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
258	M₁₆	—	2	4	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
267	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
268	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂×#10623494	1	2	yes	—
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	C₄⋊C₄	C₂×#10623494	1	1	yes	characteristic
275	C₄⋊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	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
280	C₂²⋊C₄	C₂×#10623494	1	2	yes	—
281	C₄²	C₂×#10623494	1	1	yes	characteristic
282	C₄²	C₂×#10623494	1	1	yes	characteristic
283	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
284	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
285	C₄²	C₂²×Dic₈₉	1	2	yes	—
286	C₁₆	—	89	2	no	—
287	C₁₆	—	89	2	no	—
288	C₁₆	—	178	2	no	—
289	C₁₆	—	178	1	no	—
290	C₁₆	—	178	1	no	—
291	C₁₆	—	178	2	no	—
292	C₂³	C₂²×#10623494	1	2	yes	—
293	C₂³	C₂²×#10623494	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
304	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
306	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
307	C₂×C₄	C₂²×#10623494	1	2	yes	—
308	C₂×C₄	C₂²×#10623494	1	2	yes	—
309	C₂×C₄	C₂²×#10623494	1	2	yes	—
310	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
311	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	2	no	—
316	C₈	#15668019	1	2	yes	—
317	C₈	#41×D₈₉	1	1	yes	characteristic
318	C₈	#41×D₈₉	1	1	yes	characteristic
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	2	no	—
325	C₂²	C₂³×#10623494	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#19152015	1	2	yes	—
334	C₄	#41×Dic₈₉	1	1	yes	characteristic
335	C₄	C₂×#15668019	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#10623494	1	1	yes	characteristic
339	C₂	C₂×#19152015	1	1	yes	characteristic
340	C₂	#26033355	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	#33483
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. #26022084	C₂	semidirect
5. #26022078	C₂	semidirect
6. D₄×#13083549	C₂	semidirect
7. #26022059	C₂	semidirect
8. #26022051	C₂	semidirect
9. #26021849	C₂	semidirect
10. C₄×#15667952	C₂	semidirect
182. C₈₉	#33483	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
#26021849	2
C₄×#15667952	2
C₈₉×#2966	2
#33483	89