Subgroups — #36471289

Order 22784.

554 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667952	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊#13083549	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018330	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667952	—	2	4	no	—
7	C₂×#15667952	—	2	2	no	—
8	C₂²×#13083549	C₂²	1	2	yes	—
9	C₂²×#13083549	C₂²	1	2	yes	—
10	C₂²⋊#13083549	—	2	4	no	—
11	C₂²⋊#13083549	—	2	2	no	—
12	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
15	C₈×D₄×C₈₉	—	2	2	no	—
16	C₈×D₄×C₈₉	—	2	2	no	—
17	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₈₉	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₈₉	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#15667952	—	2	8	no	—
22	C₂×#13083549	—	2	2	no	—
23	C₂×#13083549	—	2	2	no	—
24	C₂×#13083549	—	2	2	no	—
25	C₂×#13083549	—	2	2	no	—
26	C₂×#13083549	—	2	2	no	—
27	C₂×#13083549	D₄	1	4	yes	complemented
28	C₂²×D₄×C₈₉	C₈	1	1	yes	characteristic
29	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	—	2	2	no	—
32	C₂²×C₈×C₈₉	—	2	2	no	—
33	C₂²×C₈×C₈₉	—	2	2	no	—
34	C₂²×C₈×C₈₉	—	2	1	no	—
35	C₂²×C₈×C₈₉	—	2	1	no	—
36	C₂²×C₈×C₈₉	—	2	2	no	—
37	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	D₄	1	2	yes	—
41	C₂²×C₈×C₈₉	D₄	1	2	yes	—
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₄×D₄×C₈₉	—	2	2	no	—
44	C₄×D₄×C₈₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₈₉	C₈	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₈₉	C₈	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₈₉	—	2	2	no	—
50	C₂²⋊C₈×C₈₉	—	2	2	no	—
51	C₂²⋊C₈×C₈₉	—	2	2	no	—
52	C₄×C₈×C₈₉	—	2	1	no	—
53	C₄×C₈×C₈₉	—	2	1	no	—
54	#13083549	—	2	4	no	—
55	#13083549	—	4	2	no	—
56	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
57	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
58	C₂×D₄×C₈₉	—	2	2	no	—
59	C₂×D₄×C₈₉	—	2	2	no	—
60	C₂×D₄×C₈₉	—	2	2	no	—
61	C₂²×C₄×C₈₉	—	2	2	no	—
62	C₂²×C₄×C₈₉	—	2	2	no	—
63	C₂²×C₄×C₈₉	—	2	1	no	—
64	C₂²×C₄×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	—	2	2	no	—
75	C₂×C₈×C₈₉	—	2	2	no	—
76	C₂×C₈×C₈₉	—	2	4	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	2	1	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₂×C₈×C₈₉	—	4	1	no	—
83	C₂×C₈×C₈₉	—	4	2	no	—
84	C₂×C₈×C₈₉	—	4	1	no	—
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₈₉	#41	1	2	yes	—
90	C₄⋊C₄×C₈₉	—	2	2	no	—
91	C₂²⋊C₄×C₈₉	—	2	2	no	—
92	C₂²⋊C₄×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	—	2	1	no	—
94	C₄²×C₈₉	—	2	1	no	—
95	C₂³×C₈₉	—	2	1	no	—
96	C₂³×C₈₉	—	2	2	no	—
97	C₂³×C₈₉	—	2	2	no	—
98	C₂³×C₈₉	—	2	1	no	—
99	C₂³×C₈₉	—	2	1	no	—
100	C₂³×C₈₉	—	2	1	no	—
101	C₂³×C₈₉	—	2	2	no	—
102	C₂³×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	C₂²⋊C₈	1	2	yes	—
104	C₂³×C₈₉	M₃₂	1	2	yes	complemented
105	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
106	D₄×C₈₉	—	2	8	no	—
107	C₂×C₄×C₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	—	2	4	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₈₉	M₃₂	1	2	yes	complemented
122	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#131	1	2	yes	—
126	C₈×C₈₉	—	2	2	no	—
127	C₈×C₈₉	—	4	1	no	—
128	C₈×C₈₉	—	4	1	no	—
129	C₈×C₈₉	#120	1	1	yes	characteristic
130	C₈×C₈₉	#121	1	2	yes	—
131	C₈×C₈₉	#123	1	1	yes	characteristic
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	2	2	no	—
136	C₂²×C₈₉	—	2	4	no	—
137	C₂²×C₈₉	—	4	1	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	—	4	2	no	—
140	C₂²×C₈₉	—	4	2	no	—
141	C₂²×C₈₉	—	4	2	no	—
142	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₂²×C₈₉	C₈×D₄	1	2	yes	—
144	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
146	C₂²×C₈₉	#504	1	2	yes	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	4	1	no	—
150	C₄×C₈₉	—	4	1	no	—
151	C₄×C₈₉	#386	1	1	yes	characteristic
152	C₄×C₈₉	#408	1	1	yes	characteristic
153	C₄×C₈₉	#435	1	2	yes	—
154	#12977	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	4	2	no	—
156	C₂×C₈₉	—	4	2	no	—
157	C₂×C₈₉	#1869	1	1	yes	characteristic
158	C₂×C₈₉	#2112	1	1	yes	characteristic
159	C₂×C₈₉	#2113	1	1	yes	characteristic
160	C₂×C₈₉	#2168	1	2	yes	—
161	C₂×C₈₉	#2169	1	2	yes	—
162	C₂²×M₃₂	—	89	1	no	—
163	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
164	C₂×C₂²⋊C₁₆	—	89	2	no	—
165	C₂×C₂²⋊C₁₆	—	89	1	no	—
166	#1374	—	89	2	no	—
167	C₈₉	#12977	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
170	C₂×C₄×D₄	Dic₈₉	1	1	yes	characteristic
171	C₂×M₃₂	—	178	4	no	—
172	C₂×M₃₂	—	178	2	no	—
173	C₂²×C₁₆	—	89	2	no	—
174	C₂²×C₁₆	—	89	2	no	—
175	C₈×D₄	—	2	2	no	—
176	C₈×D₄	—	2	2	no	—
177	C₂×C₄⋊C₈	Dic₈₉	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	Dic₈₉	1	1	yes	characteristic
179	C₂×C₂²⋊C₈	Dic₈₉	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
181	C₂²⋊C₁₆	—	178	4	no	—
182	C₂²⋊C₁₆	—	178	2	no	—
183	C₂²×D₄	#10623494	1	1	yes	characteristic
184	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
193	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
194	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
196	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₂×C₄⋊C₄	#10623494	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	#10623494	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	#10623494	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
204	M₃₂	—	178	8	no	—
205	C₂×C₁₆	—	89	4	no	—
206	C₂×C₁₆	—	178	2	no	—
207	C₂×C₁₆	—	178	2	no	—
208	C₂×C₁₆	—	178	2	no	—
209	C₂×C₁₆	—	178	2	no	—
210	C₂×C₁₆	—	178	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₂⁴	C₂×#10623494	1	1	yes	characteristic
217	C₂⁴	C₂×#10623494	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²⋊Dic₈₉	1	2	yes	—
228	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
229	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
230	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
231	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
232	C₂²×C₄	#13083558	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
246	C₂×C₈	D₄×D₈₉	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂×C₈	#13083573	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₁₆	—	356	2	no	—
256	C₁₆	—	178	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	C₂²⋊#10623494	1	2	yes	—
266	C₂³	#15667952	1	2	yes	complemented
267	C₂³	C₂²×#10623494	1	1	yes	characteristic
268	D₄	—	2	8	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	C₂²⋊#10623494	1	2	yes	—
283	C₂×C₄	#15667952	1	2	yes	complemented
284	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
285	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
286	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
287	C₂×C₄	#15668019	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	4	1	no	—
291	C₈	#15667980	1	1	yes	characteristic
292	C₈	#15667984	1	2	yes	—
293	C₈	#15667993	1	1	yes	characteristic
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	1	no	—
304	C₂²	C₂×C₂²⋊#10623494	1	1	yes	characteristic
305	C₂²	D₄×#10623494	1	2	yes	—
306	C₂²	C₂×#15667952	1	1	yes	characteristic
307	C₂²	C₂×#15667952	1	1	yes	characteristic
308	C₂²	#19152015	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	#19151478	1	1	yes	characteristic
314	C₄	#19151565	1	1	yes	characteristic
315	C₄	#19151670	1	2	yes	—
316	C₂	—	4	2	no	—
317	C₂	—	4	2	no	—
318	C₂	#26020545	1	1	yes	characteristic
319	C₂	#26021863	1	1	yes	characteristic
320	C₂	#26021866	1	1	yes	characteristic
321	C₂	#26022078	1	2	yes	—
322	C₂	#26022084	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₈₉
commutator, socle	C₂³×C₈₉	C₂²×C₈
Fitting	C₂×C₈×D₄×C₈₉	C₂
prime radical	C₂²×D₄×C₈₉	C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#12977
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667952	C₂	semidirect
2. C₂×C₂²⋊#13083549	C₂	semidirect
3. C₂×C₂²⋊#13083549	C₂	semidirect
4. #26018330	C₂	semidirect
27. C₂×#13083549	D₄	semidirect
104. C₂³×C₈₉	M₃₂	semidirect
121. C₂×C₄×C₈₉	M₃₂	semidirect
167. C₈₉	#12977	semidirect
266. C₂³	#15667952	semidirect
283. C₂×C₄	#15667952	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667952	2
C₂×C₂²⋊#13083549	2
#26018330	2
C₂×C₈×D₄×C₈₉	2
#12977	89