Subgroups — #36708116

Order 22784.

532 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083573	C₂	1	1	yes	maximal, characteristic
2	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022084	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022082	C₂	1	2	yes	maximal, complemented
6	#26022081	C₂	1	2	yes	maximal, complemented
7	#26022049	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022048	C₂	1	2	yes	maximal, complemented
9	#26022007	C₂	1	2	yes	maximal, complemented
10	#26022006	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083573	C₄	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	2	yes	—
14	C₂×#15667951	C₂²	1	2	yes	—
15	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
16	#19151722	C₄	1	1	yes	characteristic
17	D₄×#10623494	C₄	1	1	yes	characteristic
18	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
19	#19151661	C₄	1	2	yes	—
20	#19151650	C₄	1	1	yes	characteristic
21	#19151648	C₄	1	2	yes	—
22	C₂×C₈×Dic₈₉	C₂²	1	2	yes	—
23	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151416	C₂²	1	2	yes	—
26	#19151416	C₂²	1	2	yes	—
27	#19151416	C₂²	1	2	yes	—
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151416	C₂²	1	1	yes	characteristic
30	C₂²⋊#13083549	C₂²	1	2	yes	—
31	#19151377	C₂²	1	1	yes	characteristic
32	#19151377	C₂²	1	1	yes	characteristic
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₈₉	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	C₈	1	1	yes	characteristic
42	C₂²×#10623494	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
46	#15667977	C₈	1	1	yes	characteristic
47	D₄×Dic₈₉	C₈	1	1	yes	characteristic
48	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
49	#15667972	C₈	1	2	yes	—
50	#15667970	C₈	1	1	yes	characteristic
51	#15667968	C₈	1	2	yes	—
52	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
54	#15667952	—	4	2	no	—
55	#15667951	—	4	2	no	—
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	2	yes	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
60	#15667941	C₂×C₄	1	2	yes	—
61	#15667941	C₂×C₄	1	2	yes	—
62	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
63	#15667924	C₂×C₄	1	2	yes	—
64	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
67	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
72	M₃₂×C₈₉	—	4	2	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083573	—	2	4	no	—
80	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
82	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
83	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×#10623494	—	4	2	no	—
88	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
89	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²×C₄	1	2	yes	—
91	C₈×D₈₉	—	2	1	no	—
92	C₈×D₈₉	—	2	1	no	—
93	C₈×D₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
95	#13083553	C₂×C₈	1	2	yes	—
96	#13083553	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
101	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
102	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
104	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
105	#13083549	—	2	2	no	—
106	#13083549	—	2	1	no	—
107	#13083549	—	2	1	no	—
108	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
117	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	2	no	—
120	C₁₆×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	C₂²×C₈	1	2	yes	—
122	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
123	D₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	2	4	no	—
131	C₂×Dic₈₉	—	2	4	no	—
132	C₂×Dic₈₉	—	2	4	no	—
133	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
136	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
137	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	2	2	no	—
142	Dic₁₇₈	M₃₂	1	4	yes	complemented
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
147	#10623494	—	2	1	no	—
148	#10623494	—	2	1	no	—
149	#10623494	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	C₂×M₃₂	1	2	yes	—
160	C₄×C₈₉	#520	1	1	yes	characteristic
161	C₄×C₈₉	#568	1	1	yes	characteristic
162	Dic₈₉	—	2	1	no	—
163	Dic₈₉	—	2	2	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	C₂×M₃₂	1	4	yes	—
166	Dic₈₉	#504	1	2	yes	—
167	#33505	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	4	2	no	—
169	C₂×C₈₉	#2988	1	1	yes	characteristic
170	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
171	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
172	D₈₉	—	4	1	no	—
173	C₈×#41	—	89	1	no	—
174	Q₈×C₁₆	—	89	1	no	—
175	Q₈×C₁₆	—	89	1	no	—
176	#2169	—	89	2	no	—
177	#2169	—	89	1	no	—
178	#2169	—	89	2	no	—
179	#2169	D₈₉	1	1	yes	characteristic, complemented
180	#2162	—	89	2	no	—
181	#2162	—	89	1	no	—
182	#2150	—	89	2	no	—
183	#2150	—	89	1	no	—
184	C₈₉	#33505	1	1	yes	characteristic, complemented, Sylow
185	C₄×#41	—	89	1	no	—
186	C₂×M₃₂	—	89	2	no	—
187	C₂×M₃₂	—	89	2	no	—
188	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
189	C₈×Q₈	—	89	1	no	—
190	C₈×Q₈	—	89	1	no	—
191	C₈×D₄	—	89	2	no	—
192	C₈×D₄	—	89	1	no	—
193	C₈×D₄	—	89	2	no	—
194	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
195	#431	—	89	1	no	—
196	#431	—	89	2	no	—
197	C₂×C₄×C₈	—	89	2	no	—
198	C₂×C₄×C₈	—	89	1	no	—
199	C₄⋊C₁₆	—	89	2	no	—
200	C₄⋊C₁₆	—	89	1	no	—
201	C₄⋊C₁₆	—	89	2	no	—
202	C₄⋊C₁₆	—	89	2	no	—
203	C₄⋊C₁₆	—	89	1	no	—
204	C₄⋊C₁₆	—	89	1	no	—
205	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
206	C₂²⋊C₁₆	—	89	2	no	—
207	C₂²⋊C₁₆	—	89	1	no	—
208	C₂²⋊C₁₆	—	89	1	no	—
209	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
210	C₄×C₁₆	—	89	2	no	—
211	C₄×C₁₆	—	89	2	no	—
212	C₄×C₁₆	—	89	1	no	—
213	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
214	C₂×#41	—	89	1	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	—	89	1	no	—
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
219	C₄×Q₈	—	89	1	no	—
220	C₄×Q₈	—	89	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	89	2	no	—
223	C₄×D₄	—	89	2	no	—
224	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
225	#117	—	89	1	no	—
226	#117	—	89	2	no	—
227	C₂×C₄²	—	89	2	no	—
228	C₂×C₄²	—	89	1	no	—
229	M₃₂	—	4	2	no	—
230	M₃₂	—	356	2	no	—
231	M₃₂	—	356	2	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	89	2	no	—
234	C₂×C₁₆	—	89	1	no	—
235	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	—	89	2	no	—
240	C₄⋊C₈	—	89	1	no	—
241	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
242	C₂²⋊C₈	—	89	2	no	—
243	C₂²⋊C₈	—	89	2	no	—
244	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
245	C₄×C₈	—	89	2	no	—
246	C₄×C₈	—	89	1	no	—
247	C₄×C₈	—	89	2	no	—
248	C₄×C₈	—	89	2	no	—
249	C₄×C₈	—	89	1	no	—
250	C₄×C₈	—	89	1	no	—
251	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	#41	—	178	4	no	—
253	C₂×Q₈	—	89	1	no	—
254	C₂×D₄	—	89	2	no	—
255	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	1	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	—	89	2	no	—
267	C₂×C₈	—	356	1	no	—
268	C₂×C₈	—	356	2	no	—
269	C₂×C₈	—	178	1	no	—
270	C₂×C₈	—	178	1	no	—
271	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	89	2	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	89	2	no	—
279	C₂²⋊C₄	—	89	2	no	—
280	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
281	C₄²	—	89	2	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	2	no	—
284	C₄²	—	89	1	no	—
285	C₄²	—	89	1	no	—
286	C₄²	—	89	2	no	—
287	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
288	C₁₆	—	2	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	178	2	no	—
292	C₁₆	—	178	1	no	—
293	C₁₆	—	178	1	no	—
294	C₂³	—	89	1	no	—
295	C₂³	C₂×C₈×D₈₉	1	2	yes	—
296	Q₈	—	89	4	no	—
297	D₄	—	2	2	no	—
298	D₄	—	178	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	178	2	no	—
307	C₂×C₄	—	178	4	no	—
308	C₂×C₄	—	178	4	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	1	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	2	no	—
320	C₈	—	178	1	no	—
321	C₈	#15668057	1	1	yes	characteristic
322	C₈	#15668059	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	178	1	no	—
327	C₂²	—	178	1	no	—
328	C₂²	—	178	1	no	—
329	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	89	4	no	—
333	C₄	—	89	2	no	—
334	C₄	—	178	1	no	—
335	C₄	—	178	1	no	—
336	C₄	—	178	2	no	—
337	C₄	M₃₂×D₈₉	1	2	yes	—
338	C₄	#19152077	1	1	yes	characteristic
339	C₄	#19152363	1	1	yes	characteristic
340	C₂	—	4	2	no	—
341	C₂	—	356	1	no	—
342	C₂	#26029359	1	1	yes	characteristic
343	C₂	C₂×M₃₂×D₈₉	1	1	yes	characteristic
344	C₂	#504×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2169	C₂
prime radical	C₂×#13083573	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33505
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
2. C₁₆×Dic₁₇₈	C₂	semidirect
3. #26022154	C₂	semidirect
4. #26022084	C₂	semidirect
5. #26022082	C₂	semidirect
6. #26022081	C₂	semidirect
7. #26022049	C₂	semidirect
8. #26022048	C₂	semidirect
9. #26022007	C₂	semidirect
10. #26022006	C₂	semidirect
11. C₈₉×#2169	C₂	semidirect
142. Dic₁₇₈	M₃₂	semidirect
179. #2169	D₈₉	semidirect
184. C₈₉	#33505	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083573	2
C₁₆×Dic₁₇₈	2
#26022154	2
#26022084	2
#26022082	2
#26022081	2
#26022049	2
#26022048	2
#26022007	2
#26022006	2
C₈₉×#2169	2
#33505	89